THE TRAJECTORY OPTIMIZATION &

SPACE LOGISTICS OF MINING

MISSIONS

Scott Dorrington

A thesis in fulfilment of the requirements for the degree of

Doctor of Philosophy

School of Mechanical and Manufacturing Engineering

Faculty of Engineering

University of New South Wales, Sydney

June 2019

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ORIGINALITY STATEMENT ‘I hereby declare that this submission is my own work and to the best of my knowledge it contains no materials previously published or written by another person, or substantial proportions of material which have been accepted for the award of any other degree or diploma at UNSW or any other educational institution, except where due acknowledgement is made in the thesis. Any contribution made to the research by others, with whom I have worked at UNSW or elsewhere, is explicitly acknowledged in the thesis. I also declare that the intellectual content of this thesis is the product of my own work, except to the extent that assistance from others in the project's design and conception or in style, presentation and linguistic expression is acknowledged.’

COPYRIGHT STATEMENT ‘I hereby grant the University of New South Wales or its agents a non-exclusive licence to archive and to make available (including to members of the public) my thesis or dissertation in whole or part in the University libraries in all forms of media, now or here after known. I acknowledge that I retain all intellectual property rights which subsist in my thesis or dissertation, such as copyright and patent rights, subject to applicable law. I also retain the right to use all or part of my thesis or dissertation in future works (such as articles or books).’

‘For any substantial portions of copyright material used in this thesis, written permission for use has been obtained, or the copyright material is removed from the final public version of the thesis.’

AUTHENTICITY STATEMENT ‘I certify that the Library deposit digital copy is a direct equivalent of the final officially approved version of my thesis.’

iv The Trajectory Optimization & Space Logistics of Missions Scott Dorrington – June 2019

ABSTRACT

Near- are expected to be rich in mineral resources which, if extracted, could have tremendous benefits in lowering the costs of conducting space exploration missions, and could potentially generate significant monetary returns. Despite this potential, there remain many unanswered questions surrounding asteroid mining: What is the best strategy? Which are the best targets? How much material can be retrieved? How much profit can be generated? This thesis aims to answer these questions, determining the optimal trajectory design, mission architecture, and asteroid targets to maximize the total net present value (NPV) of an asteroid mining operation.

A parametric economic model is formulated to assess the feasibility of numerous mission alternatives over a range of system, mission, and cost parameters. This analysis shows that the optimal strategy uses multiple return-trip missions, with propellant processed from asteroid resources.

New methods are then developed for the combinatorial trajectory optimization of multiple return-trip missions, identifying optimal flight itineraries that maximize NPV, rather than minimizing delta-V. These trajectories account for the distribution of asteroid-derived resources to a propellant depot and customer between consecutive trips. A location-routing problem is developed to identify the optimal orbital location of the propellant depot, and the routing of spacecraft throughout the supply chain network to maximize total sellable delivered to customers.

These methods are then applied to assess the economic values of over 100 candidate asteroids, with realistic trajectories over multiple return-trip missions. These candidates were filtered from the entire list of 19,880 near-Earth asteroids known at the time of writing, as having either known/assumed C-complex taxonomic classes, low delta-Vs, or close to Earth.

The results produced a list of 35 asteroids with positive NPVs. These missions could be achieved with capital costs of around $400M, generating large profits ranging from $198.27M to $1547.7M over 20 , with positive returns after the first or second trips. These results confirm the economic viability of asteroid mining missions, however it is noted that additional data and understanding of asteroid composition is needed to reduce the large uncertainty in the presence of resources before asteroid mining should commence.

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ACKNOWLEDGEMENTS

I would like to thank my supervisors, Dr John Olsen and Dr Nathan Kinkaid, for their support and guidance throughout this degree, and their valuable insight into the PhD process. To Dr Olsen in particular, for taking over main supervisory duties following Dr Kinkaid’s departure from the university. Thanks are also given to Dr Naomi Tsafnat for her teaching of the course that first inspired me to peruse this field of research, and for allowing me to tutor the course over the past few years.

I would also like to thank Prof. Andrew Dempster and other staff at the Australian Centre for Space Engineering Research for their interest and support in my research; and the other PhD students at the Centre and other faculties perusing other fields of Off- Earth mining and space-related research.

Special thanks are given to friends and colleagues in my own faculty. To Ali Ahmed for his friendship and encouragement over the years, and to William Crowe for countless discussions over our mutual research interests in all things asteroids. Thanks are also given to other space friends I have met around the world, and to my friends back home for providing much needed distractions from the worlds of academia and space.

Finally, I would like to give the thanks to my family for their endless support and encouragement throughout this PhD, my undergrad degree, and my entire life.

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CONTENTS

1 INTRODUCTION ...... 1

1.1 MOTIVATION FOR ASTEROID MINING ...... 1 Feasibility ...... 1

1.2 AIMS OF THESIS ...... 2

1.3 OVERVIEW OF CHAPTERS ...... 3

1.4 METHODOLOGY ...... 4 Trajectory Design Methods ...... 4 Economic Models ...... 5 Graph Theory Methods ...... 5

1.5 PUBLICATIONS PRODUCED FROM THIS WORK ...... 6 Journal Paper: ...... 6 Refereed Conference Paper: ...... 6 Conference Papers (abstract refereed): ...... 6 2 ASTEROID PROPERTIES & RESOURCES ...... 8

2.1 SOURCES OF ASTEROID OBSERVATIONAL DATA ...... 8 2.1.1 Photometric Observations ...... 9 2.1.2 Spectroscopic Observations ...... 11 2.1.3 Spacecraft Missions ...... 14

2.2 ORBITAL PROPERTIES ...... 17 2.2.1 Classical Orbital Elements ...... 17 2.2.2 Designation and Numbering Convention ...... 19 2.2.3 Other Orbital Properties ...... 19 2.2.4 Asteroid Groups ...... 22

2.3 DELTA-V ESTIMATES ...... 28 2.3.1 Hohmann Transfers ...... 28 2.3.2 Shoemaker-Helin Equations ...... 31

2.4 PHYSICAL PROPERTIES ...... 33 2.4.1 Viewing Geometry ...... 33 2.4.2 Phase Curve ...... 34 2.4.3 ...... 35 2.4.4 Mean Diameter ...... 36 2.4.5 Internal Structure ...... 38

2.5 CHEMICAL COMPOSITION ...... 41

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2.5.1 Minerals ...... 42 2.5.2 Meteorite Taxonomy ...... 43

2.6 ASTEROID PARENT BODIES ...... 47 2.6.1 Water in C-Type Asteroids ...... 48

2.7 USEFUL RESOURCES AND THEIR ABUNDANCES ...... 50 2.7.1 Precious Metals ...... 51 2.7.2 Water-Based Propellant ...... 51 2.7.3 Abundance of Water in Near-Earth Asteroids ...... 52

2.8 ASTEROID MINING STUDIES ...... 53

2.9 CONCLUSION ...... 57 3 PARAMETRIC ECONOMIC ANALYSIS ...... 58

3.1 INTRODUCTION ...... 58

3.2 COMPONENTS OF AN ASTEROID MINING INDUSTRY ...... 59 3.2.1 Prospecting ...... 59 3.2.2 Exploration Missions ...... 60 3.2.3 Mining Missions ...... 62 3.2.4 Distribution ...... 62 3.2.5 Return to Asteroid ...... 62

3.3 MISSION ALTERNATIVES ...... 63 3.3.1 Distribution Network Design ...... 63 3.3.2 Trajectory Design (Flight Itinerary) ...... 64 3.3.3 Extraction/Mining Alternatives ...... 64 3.3.4 Propellant Supply Alternatives ...... 66 3.3.5 Summary of Mission Alternatives ...... 68 3.3.6 Exploration Approach (Mining Campaign) ...... 70

3.4 FIGURES OF MERIT FOR EVALUATING MISSION ALTERNATIVES ...... 71 3.4.1 System Parameters ...... 72 3.4.2 Mission Parameters ...... 73 3.4.3 Mass Parameters ...... 75 3.4.4 Economic Parameters ...... 78

3.5 FIGURE OF MERIT EXPRESSIONS FOR MISSION ALTERNATIVES ...... 80 3.5.1 In Situ Propellant Production (Option 1A) ...... 82 3.5.2 In Situ Propellant Production with Reserve Propellant (Option 1B) ...... 88 3.5.3 Earth-Based Propellant Supply (Options 2A & 2B) ...... 91 3.5.4. Reference Mission ...... 96 viii

3.6 EVALUATION OF MISSION ALTERNATIVES ...... 97 3.6.1 Values Used in Trade Study ...... 97 3.6.2 Feasibility of Single-Trip Missions ...... 101 3.6.3 Single-Trip Break-Even Mass ...... 105 3.6.4 Feasibility of Multi-Trip Missions ...... 109 3.6.5 Expectation Value of Multi-Trip Missions ...... 116

3.7 CONCLUSION ...... 122 4 PARAMETRIC MINING RATE MODEL ...... 124

4.1 ASTEROID GEOLOGICAL ANALYTICAL MODEL ...... 124 4.1.1 Bulk Properties ...... 125 4.1.2 Particle Size Distribution ...... 125 4.1.3 Mineral Concentration ...... 126

4.2 MINING EQUIPMENT AND OPERATIONS ...... 127

4.3 ENERGY AND TIME REQUIREMENTS OF OPERATIONS ...... 130 4.3.1 Block Definitions ...... 130 4.3.2 Regolith and Rock Fractions ...... 131 4.3.3 Transportation ...... 133 4.3.4 Fragmentation ...... 134 4.3.5 Excavation ...... 135 4.3.6 Chemical Processing ...... 135 4.3.7 Unloading ...... 137

4.4 MINE DESIGN ...... 137 4.4.1 Mine Volume ...... 139 4.4.2 Total Energy and Time Requirements ...... 140

4.5 NUMERICAL EXAMPLE ...... 141

4.6 DISCUSSION ...... 143

4.7 CONCLUSION ...... 145 5 SUPPLY CHAIN NETWORK ...... 146

5.1 INTRODUCTION ...... 147 5.1.1 Asteroid Mining ...... 147 5.1.2 Transportation Problems ...... 148 5.1.3 Space Transportation Networks ...... 149

5.2 ASTEROID MINING SUPPLY CHAIN NETWORK ...... 150 5.2.1 Basic Network ...... 151 5.2.2 Expanded Network ...... 153

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5.2.3 Problem Formulation ...... 159

5.3 ORBITAL TRANSFERS ...... 162 5.3.1 Asteroid to Arrival ∆푉푡푖 ...... 162 5.3.2 Asteroid to Arrival to Depot ∆푉푡푖푗 (Route 1) ...... 163 5.3.3 Arrival to Departure to Asteroid ∆푉푖푚푛푡 (Route 2) ...... 166 5.3.4 Depot to Customer Orbit ∆푉푗푘 ...... 169 5.3.5 Arrival to Depot ∆푉푖푗푚 (Route 2) ...... 169

5.4 ASTEROID MINING LOCATION-ROUTING PROBLEM ...... 169 5.4.1 Optimal Orientation Angles ...... 170 5.4.2 Facility Location Problem ...... 174 5.4.3 Problem Scenarios ...... 183

5.5 NUMERICAL EXAMPLE ...... 184 5.5.1 Results ...... 186 5.5.2 Sensitivity Analysis ...... 191

5.6 DISCUSSION ...... 197 5.6.1 Need for Stockpiling ...... 197 5.6.2 Determining Specific Sale Price ...... 198 5.6.3 Wait-Time Constraints ...... 199 5.6.4 Scheduled Maintenance ...... 201 5.6.5 Computational Complexity ...... 202 5.6.6 Improvements to the Model ...... 203

5.7 CONCLUSION ...... 204 6 COMBINATORIAL TRAJECTORY OPTIMIZATION ...... 205

6.1 INTRODUCTION ...... 205

6.2 LAMBERT’S PROBLEM ...... 206 6.2.1 Transfer Geometry ...... 206 6.2.2 Lambert Solving Algorithms ...... 208

6.3 PORKCHOP PLOTS ...... 211 6.3.1 Launch Opportunities ...... 212 6.3.2 Type I and Type II Transfers ...... 213 6.3.3 Multi-revolution Transfers ...... 215

6.4 DELTA-VS OF HYPERBOLIC DEPARTURE/ARRIVAL TRAJECTORIES ...... 216 6.4.1 Earth-to-Asteroid (launch from Earth) ...... 217 6.4.2 Earth-to-Asteroid rendezvous (launch from Earth) ...... 218

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6.4.3 Asteroid-to-Earth rendezvous (to Earth orbit) ...... 219 6.4.4 Earth-to-Asteroid rendezvous (from Earth orbit) ...... 222

6.5 PROCEDURE TO GENERATE PORKCHOP PLOTS ...... 223 Step 1: Generate Search Space ...... 223 Step 2: Compute Transfer Geometry ...... 225 Step 3: Solve Lambert’s Problem ...... 226 Step 4: Repeat for Asteroid-to-Earth transfer ...... 226 Step 5: Locate Minima ...... 226 6.5.1 Programming Implementation ...... 227 6.5.2 Example Porkchop Plots ...... 229 6.5.3 Computation Time ...... 231 6.5.4 Search Space Pruning ...... 232 6.5.5 Primer Vector Analysis ...... 234

6.6 FLIGHT ITINERARY OPTIMIZATION PROBLEM ...... 237

6.7 TRAJECTORY COMBINATIONS USING NETWORK GRAPHS ...... 239 6.7.1 Objective Functions ...... 241 6.7.2 Two-Step Graph ...... 242

6.8 DEFINING THE GRAPHS ...... 243 6.8.1 Adjacency Matrices and Edge Costs ...... 243 6.8.2 Constructing Graphs with Arbitrary Trip Numbers ...... 250 6.8.3 Edge Cost Matrices ...... 251 6.8.4 Shortest Path Problem ...... 252 6.8.5 Longest Path Problem ...... 254 6.8.6 Computational Complexity ...... 254

6.9 CONCLUSION ...... 255 7 ASTEROID CANDIDATES ...... 257

7.1 CANDIDATE SELECTION STUDIES ...... 257

7.2 DATA SOURCES & PROCESSING ...... 259 7.2.1 MPC Orbital Elements Data ...... 260 7.2.2 Asteroid Lightcurve Data ...... 260 7.2.3 Trajectory Browser ...... 262 7.2.4 Combining Data ...... 263

7.3 ASTEROID CANDIDATE GROUPS ...... 264 7.3.1 Known & Assumed C-Complex Asteroids ...... 264 7.3.2 Potential C-Complex Asteroids ...... 266

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7.3.3 Distribution of Candidate Groups ...... 270

7.4 LAUNCH OPPORTUNITIES ...... 271 7.4.1 Known and Assumed C-complex ...... 272 7.4.2 Arjunas ...... 275 7.4.3 Co-orbitals ...... 277 7.4.4 Low Delta-Vs ...... 279

7.5 NET PRESENT VALUE RESULTS ...... 281 7.5.1 All-pairs Longest Path Solutions ...... 281 7.5.2 Global Optimal NPV Solutions ...... 284 7.5.3 Trends in Average Trip Parameters ...... 285 7.5.4 NPV Trends with Orbital Parameters ...... 287 7.5.5 Tisserand Parameter ...... 289

7.6 NPV ADJUSTED FOR FINITE WATER CONTENT IN SMALL ASTEROIDS ...... 295 Constraints Due to Limited Launch Data Range ...... 298

7.7 IDEAL CANDIDATES FOR ASTEROID MINING MISSIONS ...... 299 7.7.1 Asteroids with Maximum NPV and Profit ...... 301 7.7.2 Asteroid with Maximum Expected NPV and Profit ...... 303

7.8 CONCLUSION ...... 306 8 CONCLUSION...... 308

8.1 SUMMARY OF CHAPTERS ...... 308 Chapter 1: Introduction ...... 308 Chapter 2: Asteroid Properties and Resources ...... 308 Chapter 3: Parametric Economic Analysis ...... 309 Chapter 4: Parametric Mining Rate Model ...... 310 Chapter 5: Supply Chain Network ...... 310 Chapter 6: Combinatorial Trajectory Optimization ...... 311 Chapter 7: Asteroid Candidates ...... 311 Answers to Questions ...... 312

8.2 FEASIBILITY OF ASTEROID MINING ...... 314

8.3 DISCUSSION ...... 314 Alternative Objectives ...... 314 Asteroids – A Limited Resource ...... 315

8.4 FUTURE WORK ...... 315 Selection of Exploration Strategy ...... 315 Improved Parametric Modelling ...... 316 xii

Extended Launch Date Range ...... 316 Computing Expected NPVs from Time-Expanded Networks ...... 317 Generalized Interplanetary Supply Chain Network ...... 317 9 REFERENCES ...... 318 10 APPENDICES ...... 333

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LIST OF TABLES

TABLE 2.1 SUMMARY OF THE MAJOR ASTEROID TAXONOMIC CLASSES [33-36]...... 13

TABLE 2.2 SPACECRAFT MISSIONS TO ASTEROIDS & [42]...... 15

TABLE 2.3 DEFINITIONS OF THE CLASSICAL ORBITAL ELEMENTS...... 17

TABLE 2.4 MINERAL GROUPS FOUND IN [116]...... 42

TABLE 2.5 BULK CHEMICAL COMPOSITION OF REPRESENTATIVE CHONDRITIC METEORITES [118]...... 43

TABLE 2.6 METEORITE TAXONOMY OF MASON [118, 128]...... 45

TABLE 2.7 ASTEROID RESOURCES AND THEIR USES [3]...... 50

TABLE 3.1 SUMMARY OF MISSION ALTERNATIVE CHOICES...... 69

TABLE 3.2 SUMMARY OF MISSION ALTERNATIVES...... 69

TABLE 3.3 QUALITATIVE COST-RISK ANALYSIS...... 71

TABLE 3.4 SYSTEM PARAMETERS...... 72

TABLE 3.5 TRAJECTORY PARAMETERS...... 73

TABLE 3.6 MASS PARAMETERS...... 75

TABLE 3.7 SPECIFIC COST PARAMETERS...... 78

TABLE 3.8 PROBABILITIES AND VALUES OF EACH OUTCOME OF THE DECISION TREE FOR

THE ISPP APPROACH...... 87

TABLE 3.9 PROBABILITIES AND VALUES OF EACH OUTCOME OF THE DECISION TREE FOR

THE ISPP APPROACH WITH RESERVE PROPELLANT...... 90

TABLE 3.10 PROBABILITIES AND VALUES OF EACH OUTCOME OF THE DECISION TREE FOR

THE ISPP APPROACH WITH RESERVE PROPELLANT...... 95

TABLE 3.11 SUMMARY OF EQUATIONS FOR TOTAL SINGLE-TRIP NPV...... 101

TABLE 3.12 BREAK-EVEN MASS EXPRESSIONS FOR THE DIFFERENT MISSION

ALTERNATIVES...... 106

TABLE 3.13 SUMMARY OF NPV EXPRESSIONS OVER MULTIPLE TRIPS...... 110

TABLE 3.14 SUMMARY OF NPV EXPRESSIONS OVER MULTIPLE TRIPS...... 116

TABLE 4.1 ASTEROID GEOLOGICAL PARAMETERS...... 124 xiv

TABLE 4.2 MAJOR SYSTEM COMPONENTS...... 127

TABLE 4.3 SUBSYSTEM PARAMETERS...... 128

TABLE 4.4 SEQUENCE OF ROVER ACTIONS IN A SINGLE MINING TRIP...... 129

TABLE 4.5 ASTEROID BULK PROPERTIES...... 141

TABLE 4.6 SYSTEM PARAMETERS...... 141

TABLE 4.7 MINE SHAPE PARAMETERS...... 142

TABLE 5.1 CLASSIFICATION OF PROBLEM SCENARIOS...... 183

TABLE 5.2 OPTIMIZATION PROBLEMS FOR DIFFERENT SCENARIOS...... 184

TABLE 5.3 ASTEROID-TO-EARTH TRANSFER (ARRIVAL AT EARTH)...... 185

TABLE 5.4 EARTH-TO-ASTEROID TRANSFER (DEPARTURE FROM EARTH)...... 185

TABLE 5.5 PARAMETERS USED IN EXAMPLE...... 186

TABLE 5.6 TRAJECTORY DETAILS AND SELLABLE CALCULATED FOR THREE

ADDITIONAL ASTEROIDS...... 197

TABLE 6.1 PROPERTIES OF TRANSFER TRAJECTORIES...... 208

TABLE 6.2 EDGE COST FUNCTIONS FOR (T = 1) STAY EDGES...... 244

TABLE 6.3 EDGE COST FUNCTIONS FOR (T > 1) STAY EDGES...... 245

TABLE 6.4 EDGE COST FUNCTIONS FOR WAIT EDGES...... 245

TABLE 6.5 EDGE COST FUNCTIONS FOR TRIP EDGES...... 247

TABLE 6.6 NODE COST FUNCTIONS DEFINED OVER SINGLE TRANSFER NODE...... 248

TABLE 6.7 EDGE COST FUNCTIONS OVER TRIP EDGES...... 248

TABLE 7.1 CHANGES MADE TO THE TAXONOMIC CLASS...... 262

TABLE 7.2 KNOWN C-COMPLEX ASTEROIDS...... 265

TABLE 7.3 ASSUMED C-COMPLEX ASTEROIDS...... 266

TABLE 7.4 CO-ORBITAL ASTEROIDS...... 268

TABLE 7.5 ARJUNA ASTEROIDS...... 268

TABLE 7.6 LOW DELTA-V ASTEROIDS...... 269

TABLE 7.7 FLIGHT ITINERARY RESULTS FOR CANDIDATES WITH POSITIVE NPVS...... 284

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TABLE 7.8 SUMMARY OF ORBITAL PARAMETERS SPACE FOR POTENTIALLY POSITIVE NPVS...... 294

TABLE 7.9 NPV RESULTS ADJUSTED FOR FINITE WATER CONTENT IN SMALLER ASTEROIDS...... 300

TABLE 7.10 OPTIMAL NPV FLIGHT ITINERARY FOR MISSION TO ASTEROID (141424) 2002 CD...... 305

TABLE 8.1 ANSWERS TO QUESTIONS OF ASTEROID MINING...... 313

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LIST OF FIGURES

FIGURE 2.1 MOTION OF A SUSPECTED ASTEROID OVER SUCCESSIVE OBSERVATIONS [18]. 10

FIGURE 2.2 DISCOVERY STATISTICS OF NEAR-EARTH ASTEROIDS [24]...... 11

FIGURE 2.3 REFLECTANCE SPECTRA OF SEVERAL ASTEROIDS FROM SMASS (SOLID LINE)

AND ECAS (OPEN SQUARES) DATA [25]...... 14

FIGURE 2.4 CLASSICAL ORBITAL ELEMENTS (COES) [59]...... 18

FIGURE 2.5 ORBITAL PARAMETER SPACE OF NEA GROUPS...... 22

FIGURE 2.6 LOCATION OF ASTEROID BELTS IN THE [68]...... 23

FIGURE 2.7 DISTRIBUTION OF SEMI-MAJOR AXIS AND INCLINATION [71]...... 23

FIGURE 2.8 DEFINITIONS OF NEAR-EARTH ASTEROID GROUPS [72]...... 24

FIGURE 2.9 CLASSES OF CO-ORBITAL MOTION SHOWN IN AN EARTH- ROTATING

REFERENCE FRAME [81]...... 26

FIGURE 2.10 TRAJECTORY OF ASTEROID 2006 RH120 SHOWING ITS TEMPORARY CAPTURE

INTO EARTH ORBIT BETWEEN 2006 AND 2007...... 27

FIGURE 2.11 LEFT: PERIAPSIS TRANSFER. RIGHT: APOAPSIS TRANSFER...... 29

FIGURE 2.12 DELTA-V OF HOHMANN-LIKE TRANSFER TO ASTEROIDS...... 30

FIGURE 2.13 HISTOGRAM OF HOHMANN DELTA-VS FOR VARIOUS ASTEROID GROUPS. .... 30

FIGURE 2.14 SHOEMAKER-HELIN DELTA-VS FOR ALL ASTEROIDS...... 32

FIGURE 2.15 HISTOGRAM OF SHOEMAKER-HELIN DELTA-VS FOR VARIOUS ASTEROID

GROUPS...... 32

FIGURE 2.16 VIEWING GEOMETRY OF ILLUMINATED BODY...... 33

FIGURE 2.17 PHASE CURVE OF ASTEROID [91, 92]...... 34

FIGURE 2.18 LIGHT CURVE OF ASTEROID 44 NYSA [91]...... 35

FIGURE 2.19 DIAMETER RANGES FOR ASSUMED RANGES AS A FUNCTION OF

ABSOLUTE ...... 37

FIGURE 2.20 SEQUENCE OF IMAGES DURING APPROACH TO ASTEROID 162173

RYUGU ...... 38

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FIGURE 2.21 LEFT: GLOBAL VIEW OF ASTEROID . RIGHT: GLOBAL VIEW OF

ASTEROID ...... 38

FIGURE 2.22 IMAGES OF ASTEROID 162173 RYUGU TAKEN FROM THE HAYABUSA2

SPACECRAFT SHOWING BOULDER SIZE DISTRIBUTIONS ON TWO SCALES ...... 39

FIGURE 2.23 DISTRIBUTION OF DIAMETER AND PERIOD FOR MBS AND NEAS. 41

FIGURE 2.24 IMAGES OF METEORITES. TOP LEFT: A POLISHED AND ETCHED SLICE OF THE

HENBURY IRON METEORITE (IRON, IIIAB) [125]. TOP RIGHT: A SLICE OF THE

GLORIETA MOUNTAIN STONY-IRON METEORITE (PALLASITE, PMG) [126]. BOTTOM

LEFT: A SLICE OF THE ALLENDE STONY METEORITE (CV3 CARBONACEOUS

CHONDRITE) [127]. BOTTOM RIGHT: A MAGNIFIED VIEW OF THE ALLENDE

METEORITE SHOWING GLASSY CHONDRULES [126]...... 44

FIGURE 2.25 METEORITE TAXONOMY OF WEISBERG ET AL. [123]...... 46

FIGURE 2.26 NORMALIZED REFLECTANCE SPECTRA FOR TWO ASTEROIDS AND THEIR

ANALOGOUS METEORITES [131]. BOTH SETS ARE NORMALIZED AT 0.55 ΜM. SECOND

SET IS OFFSET BY 0.5...... 48

FIGURE 2.27 WATER DISTRIBUTION IN CARBONACEOUS CHONDRITE METEORITES [1, 149]...... 53

FIGURE 3.1 REFERENCE SPACE MISSION ARCHITECTURE OF AN ASTEROID MINING MISSION...... 63

FIGURE 3.2 DECISION TREE SHOWING THE VARIOUS MISSION ALTERNATIVES...... 70

FIGURE 3.3 COMBINED MASSES OF THE MINING SPACECRAFT AND ASTEROID MATERIAL AT

DIFFERENT STAGES OF THE MISSION...... 76

FIGURE 3.4 DECISION TREE REPRESENTATION OF EXPECTED VALUE...... 80

FIGURE 3.5 DECISION TREE SHOWING OUTCOMES FOR MULTIPLE MINING TRIPS...... 86

FIGURE 3.6 TIME OF FLIGHT VS DELTA-V FOR HOHMANN TRANSFERS...... 100

FIGURE 3.7 TOTAL PROFIT OVER A RANGE OF DELTA-VS AND SPECIFIC IMPULSES...... 102

FIGURE 3.8 NPV AND MISSION DURATION FOR 푭푻 = ퟐ N...... 104

FIGURE 3.9 NPV AND MISSION DURATION FOR 푭푻 = ퟏퟎ N...... 105

FIGURE 3.10 ZERO-PROFIT BREAK-EVEN EXPRESSED OVER A RANGE OF

DELTA-VS AND SPECIFIC IMPULSES...... 107 xviii

FIGURE 3.11 BEMR FOR ZERO NPV...... 109

FIGURE 3.12 CUMULATIVE PROFIT OVER A 10-TRIP MINING CAMPAIGN...... 111

FIGURE 3.13 CUMULATIVE DURATION OVER 10 MINING TRIPS...... 111

FIGURE 3.14 CUMULATIVE NPV OVER 10 MINING TRIPS...... 112

FIGURE 3.15 CASH-FLOW DIAGRAM FOR ASTEROID WITH DELTA-V 5KM/S, WITH RETURN

MASSES OF ퟏퟎퟎ 풎풅풓풚...... 113

FIGURE 3.16 CASH-FLOW DIAGRAM FOR ASTEROID WITH DELTA-V 5KM/S, WITH RETURN

MASSES OF ퟓퟎ 풎풅풓풚...... 115

FIGURE 3.17 PROBABILITY OF CONSECUTIVE FAILURES AS A FUNCTION OF PROBABILITY OF

SUCCESS...... 118

FIGURE 3.18 EXPECTED NPV FOR AN ASTEROID OF UNKNOWN TAXONOMY CLASS...... 118

FIGURE 3.19 EXPECTED NPV FOR A KNOWN C-TYPE ASTEROID...... 119

FIGURE 3.20 CASH-FLOW DIAGRAM FOR KNOWN C-TYPE ASTEROID WITH DELTA-V

5KM/S, WITH 풑풔 = 25%...... 120

FIGURE 3.21 CASH-FLOW DIAGRAM FOR KNOWN C-TYPE ASTEROID WITH DELTA-V

5KM/S, WITH 풑풔 = 50%...... 121

FIGURE 4.1 TIMELINE OF OPERATIONS OVER MULTIPLE TRIPS...... 129

FIGURE 4.2 UNIT BLOCK DEFINITIONS...... 130

FIGURE 4.3 PACKING OF REGOLITH AND ROCK IN HOLDING TANK...... 133

FIGURE 4.4 GEOMETRY OF A SPHERICAL CAP...... 138

FIGURE 4.5 GEOMETRY OF OPEN PIT MINE WITH CONSTANT WALL SLOPE...... 139

FIGURE 4.6 TOTAL OPERATION TIMES AS A FUNCTION OF AVAILABLE ROVER POWER. ... 142

FIGURE 4.7 TOTAL MINING RATE AS A FUNCTION OF AVAILABLE ROVER POWER...... 143

FIGURE 5.1 LAYOUT OF THE SUPPLY CHAIN NETWORK...... 151

FIGURE 5.2 SUPPLY CHAIN NETWORK WITH CANDIDATE DEPOT AND

LOCATIONS...... 153

FIGURE 5.3 CANDIDATE VEHICLE ROUTES THROUGH THE NETWORK...... 156

FIGURE 5.4 GRID OF CANDIDATE FACILITY LOCATION COMBINATIONS...... 158

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FIGURE 5.5 GEOMETRY OF THE ARRIVAL B-PLANE...... 162

FIGURE 5.6 GEOMETRY OF THE ELLIPTICAL PARKING ORBIT CASE FOR ROUTE 2 SHOWING

THE ORBITAL MANOEUVRES OF THE MINING SPACECRAFT...... 168

FIGURE 5.7 PLANE CHANGE DELTA-V AS A FUNCTION OF ARRIVAL ORIENTATION ANGLE

FOR CIRCULAR AND ELLIPTICAL PARKING ...... 172

FIGURE 5.8 GEOMETRY OF OPTIMAL ARRIVAL PARKING ORBIT ORIENTATIONS FOR ROUTE

1. TOP: CIRCULAR PARKING ORBIT (MIN. PLANE CHANGE ANGLE). BOTTOM:

ELLIPTICAL PARKING ORBIT (PERIAPSIS COINCIDES WITH PLANE CROSSING)...... 173

FIGURE 5.9 CONTOUR PLOTS OF SELLABLE MASS OVER DEPOT AND PARKING ORBIT RADII

FOR AN EQUATORIAL DEPOT ORBIT, SHOWN FOR THE VARIOUS ROUTES AND PARKING

ORBIT SHAPES...... 188

FIGURE 5.10 MAXIMUM SELLABLE MASS AT EACH DEPOT ORBIT RADIUS, SHOWN FOR THE

DIFFERENT ROUTES AND PARKING ORBIT CASES...... 189

FIGURE 5.11 MAXIMUM TOTAL SELLABLE MASS AS A FRACTION OF THE INITIAL LOAD

EXPRESSED FOR VARYING VALUES OF SPECIFIC IMPULSE...... 191

FIGURE 5.12 MAXIMUM TOTAL SELLABLE MASS FRACTION EXPRESSED AS A FUNCTION OF 퐞퐱퐩 (ퟏ/흊풆)...... 192

FIGURE 5.13 MAXIMUM TOTAL SELLABLE MASS FRACTION FOR VARYING VALUES OF

MINING SPACECRAFT DRY MASS...... 193

FIGURE 5.14 MAXIMUM TOTAL SELLABLE MASS FRACTION FOR VARYING VALUES OF

TRANSPORT SPACECRAFT DRY MASS...... 194

FIGURE 5.15 MAXIMUM SELLABLE MASS VARIATION WITH CUSTOMER ORBIT...... 194

FIGURE 5.16 CONTOUR PLOT OF SELLABLE MASS FOR THE ELLIPTICAL CASE OF ROUTE 1,

WITH A LOWERED PERIAPSIS...... 196

FIGURE 6.1 LEFT: GEOMETRY OF THE TRANSFER. RIGHT: UNIQUE PROPERTIES OF AN

EXAMPLE ...... 207

FIGURE 6.2 TIME OF FLIGHT AS A FUNCTION OF 풛 FOR ONE PARTICULAR TRANSFER

GEOMETRY [58]...... 211

FIGURE 6.3 PORKCHOP PLOT SHOWING 2018 OPTIMAL EARTH-TO- TRANSFERS [234]...... 213

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FIGURE 6.4 THE 180O TRANSFER FOR EARTH-TO-MARS TRAJECTORIES SHOWING THE

REQUIRING 90O INCLINATION [233]...... 214

FIGURE 6.5 COMPARISON OF SINGLE- AND MULTI-REVOLUTION TRANSFERS WITH THE

SAME LAUNCH AND ARRIVAL DATES...... 216

FIGURE 6.6 GEOMETRY OF THE HYPERBOLIC DEPARTURE FROM A ...... 218

FIGURE 6.7 GEOMETRY OF THE LAUNCH TRAJECTORY [236]...... 219

FIGURE 6.8 GEOMETRY OF CAPTURE MANOEUVRE OF THE ASTEROID-TO-EARTH (TO

EARTH ORBIT) TRANSFER...... 222

FIGURE 6.9 LD/AD GRID SHOWING TIME OF FLIGHT CONTOURS AND CONSTRAINTS...... 224

FIGURE 6.10 LD/TOF GRID SHOWING CONTOURS OF TRANSFER ANGLE SEPARATING THE

TYPE I AND TYPE II TRANSFERS...... 225

FIGURE 6.11 EXAMPLE PORKCHOP PLOT FOR THE EARTH-TO-ASTEROID (LAUNCH FROM

EARTH) TRANSFER TO ASTEROID 101955 BENNU...... 229

FIGURE 6.12 DETAILED VIEW OF THE PORKCHOP PLOT SHOWING SEVERAL BASINS OF

ATTRACTION OF TYPE I, II AND III TRANSFERS...... 230

FIGURE 6.13 DISTANCES IN LD AND AD FROM THE INITIAL GUESS TO THE CONVERGED

SOLUTION...... 231

FIGURE 6.14 DELTA-V VS INCLINATION...... 232

FIGURE 6.15 THE FUNDAMENTAL [229]...... 233

FIGURE 6.16 DELTA-V COMPUTED WITH THE FUNDAMENTAL ELLIPSE VS ACTUAL DELTA- V...... 234

FIGURE 6.17 EXAMPLE PRIMER VECTOR OF A LOCALLY OPTIMAL TRAJECTORY...... 236

FIGURE 6.18 EXAMPLE PRIMER VECTOR OF A NON-OPTIMAL TRAJECTORY...... 236

FIGURE 6.19 TIME-EXPANDED NETWORK OF POSSIBLE COMBINATIONS OF SELECTED

TRAJECTORIES...... 240

FIGURE 6.20 A TWO-STEP GRAPH OF THE NETWORK...... 242

FIGURE 6.21 TWO-STEP GRAPH FOR A 10-TRIP MINING CAMPAIGN TO ASTEROID (459872)

2014 EK24, WITH TRIP EDGES SHOWN IN BLUE, AND ALL SHORTEST PATH ITINERARIES

FROM THE FIRST STARTING 푬푨푳 NODE HIGHLIGHTED IN RED...... 253

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FIGURE 7.1 SEGMENTATION OF ASTEROIDS BASED ON TAXONOMIC CLASS...... 264

FIGURE 7.2 ORBITAL ELEMENT DISTRIBUTION OF THE ASTEROID CANDIDATE GROUPS. .. 270

FIGURE 7.3 DISTRIBUTION OF SHOEMAKER-HELIN DELTA-V AND DIAMETER FOR THE

ASTEROID CANDIDATE GROUPS...... 271

FIGURE 7.4 DELTA-V DISTRIBUTION OF 푬푨푳 LAUNCH OPPORTUNITIES FOR KNOWN AND

ASSUMED C-COMPLEX ASTEROIDS...... 272

FIGURE 7.5 LAUNCH DATE AND DELTA-V DISTRIBUTION OF 푬푨푳 LAUNCH OPPORTUNITIES

FOR KNOWN AND ASSUMED C-COMPLEX ASTEROIDS...... 273

FIGURE 7.6 VARIATION OF RANGE, RELATIVE , AND DELTA-V OF 푬푨푳

OPPORTUNITIES TO ASTEROID (341843) 2008 EV5...... 275

FIGURE 7.7 LAUNCH DATE AND DELTA-V DISTRIBUTION OF 푬푨푳 LAUNCH OPPORTUNITIES

FOR THE ARJUNA ASTEROIDS...... 276

FIGURE 7.8 DELTA-V DISTRIBUTION OF 푬푨푳 LAUNCH OPPORTUNITIES FOR THE ARJUNA

ASTEROIDS...... 276

FIGURE 7.9 DELTA-V DISTRIBUTION OF 푬푨푳 LAUNCH OPPORTUNITIES FOR THE CO-

ORBITAL ASTEROIDS...... 277

FIGURE 7.10 LAUNCH DATE AND DELTA-V DISTRIBUTION OF 푬푨푳 LAUNCH

OPPORTUNITIES FOR THE CO-ORBITAL ASTEROIDS...... 278

FIGURE 7.11 RANGE, RELATIVE MEAN LONGITUDE, AND 푬푨푳 DELTA-VS TO THE QUASI-

SATELLITE (469219) 2016 HO3...... 279

FIGURE 7.12 LAUNCH DATE AND DELTA-V DISTRIBUTION OF 푬푨푳 LAUNCH

OPPORTUNITIES FOR THE LOW DELTA-V ASTEROIDS...... 280

FIGURE 7.13 DISTRIBUTION OF AVERAGE ONE-WAY DELTA-V, AVERAGE TRIP DURATION,

AND TOTAL NPV FOR THE ALL-PAIRS MAXIMUM NPV FLIGHT ITINERARIES FOR FOUR

REPRESENTATIVE ASTEROIDS...... 283

FIGURE 7.14 GLOBAL OPTIMAL NPV VS AVERAGE ONE-WAY DELTA-V...... 285

FIGURE 7.15 AVERAGE TRIP DURATION AND AVERAGE ONE-WAY DELTA-V...... 286

FIGURE 7.16 NPV TRENDS WITH SEMI-MAJOR AXIS, ECCENTRICITY, AND INCLINATION. 288

FIGURE 7.17 NPV TRENDS IN EARTH TISSERAND PARAMETER (EXPRESSED IN TERMS OF

NORMALIZED ENCOUNTER VELOCITY 푼)...... 289 xxii

FIGURE 7.18 NPV TRENDS WITH SHOEMAKER-HELIN DELTA-V...... 290

FIGURE 7.19 COMPARISON OF SHOEMAKER-HELIN DELTA-VS TO AVERAGE ONE-WAY

DELTA-VS...... 290

FIGURE 7.20 NPV TRENDS WITH SYNODIC PERIOD...... 291

FIGURE 7.21 NPV TRENDS WITH AVERAGE ABSOLUTE SEPARATION FROM EARTH...... 292

FIGURE 7.22 TRAJECTORIES OF TWO ARJUNA ASTEROIDS HAVING THE HIGHEST (LEFT)

AND LOWEST (RIGHT) NPV RESULTS, OVER THE PERIOD 2020 TO 2050, SHOWN IN AN

EARTH-SUN CO-ROTATING REFERENCE FRAME...... 293

FIGURE 7.23 TRAJECTORIES OF TWO CO-ORBITAL ASTEROIDS HAVING THE POSITIVE (LEFT)

AND NEGATIVE (RIGHT) NPV RESULTS, OVER THE PERIOD 2020 TO 2050, SHOWN IN

AN EARTH-SUN CO-ROTATING REFERENCE FRAME...... 294

FIGURE 7.24 TOTAL WATER CONTENT IN ASTEROID CANDIDATES...... 296

FIGURE 7.25 NPV TRENDS WITH ASTEROID DIAMETER. TOP: NPV RESULTS FOR A FULL 10

TRIPS. BOTTOM: MAXIMUM TRIP NUMBERS FOR SMALL ASTEROIDS CONSTRAINED BY

TOTAL WATER CONTENT...... 297

FIGURE 7.26 TIME-EXPANDED NETWORK FOR A 10-TRIP MISSION TO THE ARJUNA-TYPE

ASTEROID (459872) 2014 EK24. TWO LONGEST PATH TREES BEGINNING EARLY AND

LATE IN THE LAUNCH DATE RANGE ARE HIGHLIGHTED IN YELLOW AND GREEN. THE

GLOBAL OPTIMAL NPV FLIGHT ITINERARY IS HIGHLIGHTED IN RED...... 299

FIGURE 7.27 TOTAL NPVS AND PROFITS OF POSITIVE NPV ASTEROIDS...... 301

FIGURE 7.28 CASH-FLOW DIAGRAM FOR 4-TRIP MISSION THE MAXIMUM NPV CANDIDATE,

ASTEROID 2012 TF79...... 302

FIGURE 7.29 CASH-FLOW DIAGRAM FOR 10-TRIP MISSION TO THE MAXIMUM PROFIT

CANDIDATE, ASTEROID (469219) 2016 HO3...... 303

FIGURE 7.30 TOTAL EXPECTED NPVS AND EXPECTED PROFITS OF THE POSITIVE NPV

ASTEROIDS...... 304

FIGURE 7.31 CASH-FLOW DIAGRAM FOR A 6-TRIP MISSION TO THE KNOWN B-TYPE

ASTEROID (141424) 2002 CD (MAXIMUM EXPECTED NPV AND PROFIT CANDIDATE)...... 305

xxiii

LIST OF ABBREVIATIONS AND ACRONYMS

AD Arrival Date

AE Asteroid-to-Earth

ARM Asteroid Redirect Mission

BEMR Brea-Even Mass Ratio

CCD Charge Coupled Device

COEs Classical Orbital Elements

CR3BP Circular Restricted Three Body Problem

DAG Directed Acyclic Graph

EA Earth-to-Asteroid

EAL Earth-to-Asteroid (Launch from Earth)

ECAS Eight-Color Asteroid Survey

EMV Expected Monetary Value

ET Time

FOM Figure of Merit

FTE Full Time Equivalent

GEO

GTO Geostationary Transfer Orbit

IRAS Infrared Astronomical

IRR Internal Rate of Return

ISPP In Situ Propellant Production

ISRU In Situ Resource Utilization

ISU International Space University

JED Julian Ephemeris Date

JPL Jet Propulsion Laboratory

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LCDB Asteroid Lightcurve Database

LD Launch Date

LEO

LOX/LH2 Liquid Oxygen / Liquid Hydrogen

LRP Location-Routing Problem

LSST Large Synoptic Survey Telescope

MB Main Belt

MPBR Mass Payback Ratio

MPC Minor Center

MPCORB Center Orbital Elements Database

MR Mining Rate

NAIF Navigation and Ancillary Information Facility

NASA National Aeronautics and Space Administration

NEA Near-Earth Asteroid

NEO Near-Earth Object

NHATS Near-Earth Object Human Space Flight Accessible Targets Study

NPV Net Present Value

SDSS Sloan Digital Sky Survey

SE Stay Edge

SH Shoemaker-Helin

SM Sellable Mass

ST Stay Time

SMASS Small Main-Belt Asteroid Spectroscopic Survey

SOI Sphere of Influence

SPK Spacecraft Planet Kernels

TB Trajectory Browser

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TE Trip Edge

TOF Time of Flight

TSP Travelling Salesman Problem

USCM8 Unmanned Space Vehicle Cost Model v.8

VRP Vehicle-Routing Problem

WE Wait Edge

WT Wait Time

WISE Wide-field Infrared Survey Explorer

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LIST OF APPENDICES

APPENDIX A ANALYTICAL DELTA-V ESTIMATES ...... 334

APPENDIX B TWO PARAMETER H-G PHASE CURVE MODEL ...... 340

APPENDIX C COST ESTIMATE FOR MINING SPACECRAFT ...... 343

APPENDIX D LOCATION-ROUTING PROBLEM ALGORITHMS ...... 345

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Chapter 1: Introduction

1 INTRODUCTION

1.1 Motivation for Asteroid Mining Space exploration activities have traditionally been highly expensive, due to the large energy and cost requirements of escaping the Earth’s well. Asteroid mining has been proposed as a means to significantly reduce the cost of conducting space missions by sourcing resources from in space, thereby avoiding the need to deliver them by costly rocket launches from Earth.

There are a myriad of near-Earth asteroids accessible with low delta-Vs, some of which are considered more accessible than the [1, 2]. These asteroids are expected to be rich in mineral resources such as water (in the form of hydrated minerals), nickel-iron metals, silicates, precious metals, and other volatiles [3]. These resources can be processed into construction materials and propellants which, if delivered to Earth orbit, could create a space-based infrastructure facilitating the construction and refuelling of spacecraft in orbit [4]. Having these capabilities available in Earth orbit would have tremendous benefits for space exploration, as it is generally regarded that:

"Get to low-Earth orbit and you're halfway to anywhere in the solar system."

– Robert A. Heinlein.

Feasibility The concept of asteroid mining has, until recently, been constrained to the realm of science fiction – often dismissed as impossible. However, recent technical and economic feasibility studies have begun to indicate that asteroid mining has the potential to become a highly profitable endeavour. In one such early study on space colonization, O’Neill [5] noted:

"As sometimes happens in the hard sciences, what began as a joke had to be taken more seriously when the numbers began to come out right" – Gerard O'Neill, 1974.

Scott Dorrington – June 2019 1 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

Asteroid mining has received renewed interest in recent years, with the founding of multiple commercial asteroid mining companies, and large-scale investments from venture capital and governments such as Luxembourg. Investment bank Goldman Sachs recently published a note to its clients, favouring investment into asteroid mining [6]:

“Space mining could be more realistic than perceived. Water and platinum group metals that are abundant on asteroids are highly disruptive from a technological and economic standpoint. Water is easily converted into rocket fuel, and can even be used unaltered as a propellant. Ultimately being able to stockpile the fuel in LEO [low earth orbit] would be a game changer for how we access space. And platinum is platinum. According to a 2012 Reuters interview with Planetary Resources, a single asteroid the size of a football field could contain $US25bn- $US50bn worth of platinum.”

Despite the positive outlook on asteroid mining, claims of the potential monetary value of individual asteroids are highly simplified, often citing the price of platinum or other resources in the current world market. There have been several studies conducted on the technical and economic feasibility of asteroid mining missions (discussed in chapter 2). These studies often have a strong on one of the multiple disciplines involved in asteroid mining, while making simplifications or assumptions in the other disciplines. For instance, economic studies often use simplified trajectory estimates and make assumptions about the frequency of launch opportunities for return trip missions; while detailed trajectory design studies try to minimize the total mission delta-V at the expense of long mission durations that would lead to longer payback periods when considering the time-value of money.

1.2 Aims of Thesis The goal of this thesis is to develop new methods of evaluating the feasibility of asteroid mining from a multi-disciplinary approach, considering technical factors such as trajectory design, mission architecture, and mining operations; along with economic factors such as the capital cost to build, launch and operate the mining spacecraft, and the expected revenues generated from the sale of asteroid-derived material in a future space-based market. Emphasis will be placed on optimizing mission and trajectory design to maximize the total net present value.

2 Scott Dorrington – June 2019 Chapter 1: Introduction

The thesis aims to: • consider a range of possible mission alternatives to determine the optimal layout and operating procedure of an asteroid mining industry; • develop new methods of optimizing multiple return trip trajectories that maximizes profit or net present value (rather than minimizing total delta-V as is the current standard); • apply these methods to candidate asteroid to assess their potential economic value; • identify the best prospective targets for future asteroid mining missions; and • demonstrate that asteroid mining can, in-fact, be a profitable endeavour (a fact that is still debated).

The thesis will aim to answer questions such as: • How much asteroid material can be extracted during a given stay-time at an asteroid? • How much of this mass can be delivered to customer spacecraft in Earth orbit? • What is the best location for a propellant depot in Earth orbit to stockpile returned resources for distribution to customer spacecraft? • Which asteroids are the best targets? • How much profit can be made from mining a given asteroid target? • How many return trips are required to recover the capital investment?

1.3 Overview of Chapters Chapter 2 will present a literature review on the current state of knowledge of asteroid orbital, physical, and compositional properties. This is intended to introduce the reader to a number of concepts and data sources used throughout the thesis. A review will also be presented on technical and economic studies that have been conducted on the feasibility of asteroid mining missions.

Chapter 3 will outline a generalized space mission architecture for an asteroid mining industry. Several mission alternatives will be identified that could be selected for different elements of the design. A parametric economic model will be generated to produce economic figures of merit that can be used to assess the feasibility of these alternatives. These will be used to draw conclusions about the optimal strategies that should be employed, and the conditions under which they can be feasible.

Scott Dorrington – June 2019 3 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

Chapter 4 will outline a parametric model to compute the achievable mining rate for extracting and processing asteroid resources as a function of spacecraft subsystem parameters, asteroid geological parameters, and specific energy of operations.

Chapter 5 will develop a mathematical model of an asteroid mining industry as a supply chain network consisting of various orbital nodes, and transfer edges. Graph theory methods used in studying terrestrial supply chain networks will be adapted to the orbital network in order to determine the optimal layout of orbital infrastructure and routing of spacecraft throughout the network.

Chapter 6 will present methods to determine optimal launch dates of transfer trajectories to/from individual asteroids. A time-expanded network graph will be used to enumerate all possible combinations of successive transfers, with paths through the network describing potential flight itineraries that could be selected for multiple return trip missions. Edge cost functions will be generated to compute the total profit or net present value of each path, from which a longest path problem can be solved to compute the optimal flight itinerary.

In chapter 7, lists of asteroid candidates will be generated from two databases containing orbital elements and photometric properties of near-Earth asteroids. Methods described in the previous chapters will be applied to each candidate asteroid to calculate their total net present value. These will be used to assess the economic values of over 100 asteroids, using realistic trajectories over multiple return trip missions.

Chapter 8 will summarize the major conclusions drawn from work produced in this study, and suggest future work that can improve upon these methods.

1.4 Methodology

Trajectory Design Methods Much of the analysis in this thesis relies on the computation of trajectories to and from asteroids. These will be computed using a combination of analytical and numerical methods. Initial delta-V estimates will be produced using the Hohmann transfer and Shoemaker-Helin equations [7], that are widely used for preliminary trajectory design studies. While lists of Shoemaker-Helin delta-Vs are available online (such as Benner [8]), they are only updated periodically. In this work, Shoemaker-Helin delta-Vs will be computed directly from asteroid orbital elements obtained from the , so as to include the most up to date data on all known near-Earth asteroids.

4 Scott Dorrington – June 2019 Chapter 1: Introduction

Shoemaker-Helin delta-Vs can be computed with little computation time, making these methods applicable to the processing of the tens of thousands of known asteroids.

These delta-Vs, along with photometric data available for a limited number of asteroids, will be used to produce smaller lists of candidate asteroids for which numerical trajectory design methods will be applied. Impulsive transfers will be calculated using a Lambert solver [9], with asteroid ephemeris data obtained from the JPL Horizons system [10]. A “porkchop plot” program will be developed in MATLAB to display the variation of objective functions of trajectories with launch and arrival dates ranging from 2020 to 2050. Separate sets of optimal launch opportunities will be identified over this time frame for four different types of heliocentric transfers: 1. Earth-to-Asteroid flyby (launch from Earth); 2. Earth-to-Asteroid rendezvous (launch from Earth); 3. Asteroid-to-Earth rendezvous (to Earth orbit); and 4. Earth-to-Asteroid rendezvous (from Earth orbit).

Each transfer will use a separate objective function defining the total delta-V of the transfer, including plane change manoeuvres for those departing from or arriving at parking orbits.

Economic Models The economic feasibility of asteroid mining missions will be evaluated using two separate approaches. First, a parametric economic model will be produced in order to calculate operational and economic figures of merit as a function of critical system, mission, and cost parameters. This will be used to perform a trade study to consider the feasibility of different mission alternatives over a wide range of parameter values (primarily delta-V).

Secondly, the same methods will be applied to trajectories computed using the numerical trajectory optimization methods described above. This will account for more realistic mission parameters such as the frequency of launch opportunities and mission durations for multiple return trip missions. These methods will be applied to lists of asteroid candidates to evaluate their potential economic values.

Graph Theory Methods The thesis will also make use of several graph theory methods and algorithms to visualize and analyse combinations of decision alternatives. Chapter 3 will make use of

Scott Dorrington – June 2019 5 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions decision trees to enumerate all possible outcomes from a set of sequential decisions to be made during an asteroid mining campaign. Chapter 5 will use a network graph to model the supply chain logistics of the distribution of material from an asteroid source to customer spacecraft. Chapter 6 will use a time-expanded network to consider combinations of successive transfers.

1.5 Publications Produced from this Work Work from this thesis has resulted in the following publications:

Journal Paper: • S. Dorrington, J. Olsen, A location-routing problem for the design of an asteroid mining supply chain network, Acta Astronautica, 157 (2019) 350-373.

Refereed Conference Paper: • S. Dorrington, N. Kinkaid, J. Olsen, Trajectory Design and Economic Analysis of Asteroid Mining Missions to Asteroid 2014 EK24, The Third International Future Mining Conference, Sydney, Australia, November 2015.

Conference Papers (abstract refereed): • S. Dorrington, N. Kinkaid, J. Olsen, Trajectory Opportunities to Arjuna-type Asteroids for Asteroid Mining Missions, IAC-15-D4.3.9, 66th International Astronautical Congress, Jerusalem, Israel, 2015, 12 – 16 October. • S. Dorrington, J. Olsen, N. Kinkaid, Architecture for an Asteroid Mining Industry, IAC-16-D4.5.1, 67th International Astronautical Congress, Guadalajara, Mexico, 2016, 26 – 30 September.

• S. Dorrington, J. Olsen, Mining Requirements for Asteroid Ore Extraction, IAC-17- D4.5.2, 68th International Astronautical Congress, Adelaide, Australia, 2017, 25 – 29 September.

• S. Dorrington, J. Olsen, Logistics Problems in the Design of an Asteroid Mining Industry, IAC-18-D4.5.15, 69th International Astronautical Congress, Bremen, Germany, 2018, 1 – 5 October.

6 Scott Dorrington – June 2019 Chapter 1: Introduction

The work from the 2015, 2016, and 2017 conference papers form the basis for the parametric analyses presented in chapters 3 and 4. The 2018 conference paper was intended as a preliminary executive summary of the various aspects considered in this thesis. The 2015 Future Mining Conference paper focused on explaining aspects of trajectory and mission design to the mining community. The work of chapter 5 has been published in its entirety as a journal article. Some of the introductory material from the journal paper have been incorporated into this chapter. Chapters 3, 6, and 7 are expected to form the basis of future journal publications.

Scott Dorrington – June 2019 7 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

2 ASTEROID PROPERTIES & RESOURCES

This chapter presents a literature review covering the current state of knowledge of the orbital, physical, and compositional properties of asteroids. The first section describes the sources of observational data that are available from ground and space-based observations. The second section discusses the orbital properties of the major asteroid groups, including analytical methods that can be used to estimate their accessibility for spacecraft missions. The third section describes physical properties of asteroids that can be inferred from observations. The fourth section discusses the chemical composition of meteorites, and their connections to asteroid parent bodies. The final sections discuss the abundance and uses of potential resources present in asteroids, and a review of studies on the technical and economic feasibility of asteroid mining missions.

This chapter is intended to introduce the reader to a number of concepts and data sources that will be used throughout the thesis. Asteroid mining is a developing, multi- disciplinary field covering areas of astronomy, geology, mining engineering, economics, and astronautics. As such, it is not possible to cover all of these areas in this short chapter. Other sources, such as Burbine [11], discuss many of these topics in greater detail. The book series Asteroids I – IV [12-15] also contains collections of numerous papers written by authorities in asteroid sciences, presenting a periodic update of the state of the art of data sources, analysis methods, and interpretations of asteroids and their connection to meteorites (many of these papers are referenced in this chapter).

2.1 Sources of Asteroid Observational Data Information on the orbital and physical properties of asteroids comes mostly from remote sensing sources: photometric and spectroscopic observations from ground and

8 Scott Dorrington – June 2019 Chapter 2: Asteroid Properties & Resources space-based telescopes at visual and infrared wavelengths. From these observations, inferences can be made as to the size, shape, and composition of asteroids.

2.1.1 Photometric Observations Photometric astronomy focuses on the measurement of the brightness or magnitude of light over various bandwidths. Optical telescopes capture light from celestial sources to form an of the night sky over a restricted field of view. This image is captured and recorded by a charge-coupled device (CCD) that quantifies the magnitude of light illuminating various pixels across the image. From this image, two pieces of information can be observed about asteroids: • their or brightness; and • their celestial coordinates (declination and right ascension).

Due to the small size and large relative distances of asteroids, their images are only recorded by a single pixel, from which it is not possible to resolve anything about the shape or size. However, from several observations over time, it is possible to track the motion of an asteroid with respect to the background catalogue of , to identify its orbital properties. Several other physical properties can also be inferred from the variation of the magnitude over time.

Asteroid Discovery and Tracking from Photometric Data Ground-based photometric observations of asteroids are primarily used for the discovery of new asteroids, and the continued tracking of catalogued asteroids. The methods used in discovering new asteroids have not changed substantially over time, all relying on identifying objects that move relative to the catalogued stars with fixed celestial coordinates. However, improvements have been made in the data collection and processing methods.

Early astronomical surveys were performed by direct visual observations, with the coordinates of each tabulated by hand. This lead to the discovery of the first asteroid 1 by Giuseppe Piazzi in 1801, when its coordinates were found to have moved over successive nights [16].

The detection of asteroids has been made easier through the introduction of photographic methods, where several images of the same segment of sky can be visually compared. Modern search campaigns use computer software such as Astrometrica [17], that allow image sets to be compared by “blinking” images in rapid

Scott Dorrington – June 2019 9 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions succession. This software uses image search algorithms to highlight previously known asteroids, and autonomously identify suspected asteroids. Human input is still required to either confirm or reject the suspected object, and to search for objects not detected by the software. Several citizen scientist campaigns such as “Find an Asteroid” and "Asteroid Zoo" are organised each to help in the processing of these image sets. Figure 2.1 shows three successive observations showing the motion of a suspected asteroid (image colour has been inverted).

Figure 2.1 Motion of a suspected asteroid over successive observations [18].

Asteroid Search Campaigns There are currently a number of asteroid search campaigns dedicated to the discovery and tracking of near-Earth asteroids. These search campaigns consist of collections of ground-based telescopes, making systematic observations to produce image sets over different areas of the sky.

The motivation for these search campaigns is primarily to identify near-Earth asteroids that pose a potential collision risk with Earth. In 1998, NASA was issued a mandate by the US Congress to discover 90% of all potentially hazardous asteroids with diameters greater that 1 km. This target was successfully met, and current efforts are directed at discovering 90% of asteroids with diameters greater than 140 m [19]. Since this goal was issued, the number of asteroid discoveries has significantly increased over time. At the time of writing, a total of 19,819 near-Earth asteroids have been discovered, including 895 with diameters greater than 1 km, and 8,541 with diameters greater than 140 m (including the > 1km asteroids). Figure 2.2 shows a plot of the cumulative number of discovered near-Earth asteroids.

While there are a larger number of asteroids with diameters smaller than 140 m, they become increasingly difficult to observe due to the decreasing detection efficiency of ground-based telescopes at dimmer magnitudes [20]. The limiting magnitude of these telescopes is determined by the diameter of the lens, and atmospheric conditions at the observation site [21]. The telescopes used in asteroid search campaigns

10 Scott Dorrington – June 2019 Chapter 2: Asteroid Properties & Resources have diameters ranging from 0.5 to 1.8 m, with limiting magnitudes of 19 and 22.6, respectively [16]. These correspond to asteroid with estimated diameters of 420 m and 85 m (from [22] using an assumed albedo of 0.25 – discussed in section 2.3.4). The addition of larger telescopes such as the upcoming Large Synoptic Survey Telescope (LSST) are expected to improve the observation efficiency of smaller asteroids [23].

Figure 2.2 Discovery statistics of near-Earth asteroids [24].

2.1.2 Spectroscopic Observations Reflectance spectroscopy involves studying the fraction of incident solar radiation reflected off celestial objects as a function of wavelength [25]. Spectroscopic observations of asteroids can be made in one of two ways.

Detailed spectral data can be obtained using long-slit spectrographs, that observe a narrow one-dimensional section of the telescope’s field of view. This light is refracted – through the use of a prism or diffraction grating – to produce a two-dimensional image of the spectrum that can be captured on a CCD [25]. This method has the benefit of sampling the spectra of the background sky along the length of the slit, making it easier to remove signal noise introduced by the Earth’s atmosphere. This method was used in the Small Main-Belt Asteroid Spectroscopic Survey (SMASS) and its follow up survey (SMASSII) to obtain reflectance spectra of over 1000 small (D < 20 km) main belt asteroids [26].

Scott Dorrington – June 2019 11 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

Low resolution spectral data can be produced from photometric observations over restricted wavelengths using a series of band pass filters. From these observations, a set of “color” indices are produced from differences of the observed magnitudes of the different bandwidths. For example, 푈 − 퐵, 푈 − 푉, and 퐵 − 푉, for the Ultraviolet (푈), Blue (퐵), and Visual (푉) filters of the standard UBV photometric system [27]. These color indices can be used to plot a low-resolution asteroid reflectance spectrum normalized at the visual V magnitude. This method was used in the Eight-Color Asteroid Survey (ECAS) to produce spectral data of 582 asteroids over eight bands (including the UBV filters) in the visual and infrared spectrum from 0.34 to 1.04 μm [28].

While this method produces lower resolution spectra, data can be obtained from wide field of view images and hence does not require dedicated observations of individual asteroids. This allows spectral data to be obtained from other photometric surveys not specifically designed for asteroid observations. Several studies have used asteroid color indices processed from the Sloan Digital Sky Survey (SDSS) (for example [29]). This survey observes the whole sky in five band filters from 0.3 to 1.123 μm [30].

In both methods, relative reflectance spectra are obtained by dividing the spectral data by that of a solar analogue star (with emissions spectra similar to the Sun). The spectra are then typically normalized to unity at a wavelength of 0.55 μm.

Asteroid Taxonomy Several taxonomic systems have been developed to categorize asteroids based on their photometric and spectroscopic properties. These systems have been generated from statistical cluster analyses on different data sets. The most widely used systems are the Tholen taxonomy [31], and the SMASSII/Bus taxonomy [32]. (See Bus [32] for a detailed review of the history of asteroid taxonomic systems.)

The Tholen taxonomy used principle component analysis of color indices from the Eight-Color Asteroid Survey to identify 14 distinct spectral classes (A, B, F, C, G, D, E, M, P, Q, R, S, T, and V). The SMASSII/Bus taxonomy further sub-divided these classes into 26 distinct spectral classes, based on the presence of absorption features and slope trends in the detailed spectral data from the SMASSII survey.

The Bus-DeMeo taxonomy [33] extends the SMASSII taxonomy to introduce broader taxonomic complexes of classes sharing similar features. The major complexes include the C-complex (containing the Tholen B, C, F, and G classes), the S-complex

12 Scott Dorrington – June 2019 Chapter 2: Asteroid Properties & Resources

(containing the Tholen S-class), and the X-complex (containing the Tholen E, M, and P classes). Table 2.1 summarizes the spectral features of the Tholen and SMASSII classes included in these complexes (reproduced from Cellino et al. [34]). Representative spectra in the infrared wavelengths (0.45 to 2.45 μm) are taken from [35] (modified from DeMeo et al. [33]). Approximate number distributions of the complexes amongst the near-Earth asteroid population are taken from Stuart & Binzel [36], corrected for the observational bias of high albedo objects (low albedo C-type asteroids are harder to detect, and hence under-represented in observation numbers).

Figure 2.3 shows normalized reflectance spectra from SMASS (solid line) and color indices from the Eight-Color Asteroid Survey (open squares) for several asteroids of different Tholen taxonomic classes. This figure is adapted from Bus et al. [25].

Table 2.1 Summary of the major asteroid taxonomic classes [33-36].

SMASSII/ Tholen Rep. % Complex Bus Spectral Features Classes Spectra NEAs Classes Low albedo. Linear, generally featureless spectra. Differences B, C, B, C, C , C b 10% in UV absorption features and F, G C , C , C h g hg presence/absence of narrow absorption feature near 0.7 μm. Moderate albedo. Moderately steep reddish slope downward of 0.7 μm; moderate to steep S, S , S , absorption longward of 0.75 S S a k 22% Sl, Sq, Sr μm; peak of reflectance at 0.73 μm. Bus subgroups intermediate

between S and A, K, L, Q, R classes. From low albedo (P) to very high albedo (E). Generally featureless spectrum with X, X , X , X E, M, P c e 34% reddish slope; differences in Xk subtle absorption features and/or spectral curvature and/or peak relative reflectance.

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Figure 2.3 Reflectance spectra of several asteroids from SMASS (solid line) and ECAS (open squares) data [25].

Space-based observations Infrared observations from ground-based telescopes are limited to a number of narrow bandwidths due to the opacity of the Earth’s atmosphere in infrared. Space-based telescopes such as the Wide-field Infrared Survey Explorer (WISE) spacecraft also contribute to the observations, allowing imaging of the sky in a wider range of infrared wavelengths [37]. The Infrared Astronomical Satellite (IRAS) surveyed the sky in 1983 at four wavelength bands centred at 12, 25, 60, and 100 μm [38]. This data has been used to derive albedo and diameter estimates for 1811 asteroids [39].

While current asteroid search campaigns are conducted for government or academic research, commercial opportunities may exist in future to provide prospecting data for asteroid mining companies. Several commercial companies have business plans in asteroid prospecting. Planetary Resources has announced plans to deploy a fleet of space-based telescopes in Earth orbit [40]. The also has plans for a positioned in the inner Solar System that would provide a better vantage point for viewing asteroids in the Earth's orbit [41].

2.1.3 Spacecraft Missions Interplanetary spacecraft missions have been deployed to provide up-close observations of several asteroids and comets. Details of these missions and their targets are listed in table 2.2 (adapted from [42]).

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Table 2.2 Spacecraft Missions to Asteroids & Comets [42].

Spacecraft Asteroid/ Mission Type Date (DD/MM/YY) Sample return 24/09/23 OSIRIS-REx 101955 Bennu Surface 2020 (expected) Orbiter 03/12/18 – Present Sample return Dec 2020 Hayabusa2/ 162173 Ryugu Surface 21/02/19 Mascot Orbiter 27/06/18 – Present 67P/Churyumov- Surface 12/11/15 – 09/07/15 / Gerasimenko Orbiter 06/08/14 – 30/09/16 Flyby 10/07/10 2867 Šteins Flyby 05/09/08 1 Ceres Orbiter 06/03/15 - Present 4 Orbiter 16/07/11 – 05/09/12 9P/Temple 1 Flyby 14/02/11 Sample return 15/01/06 81P/Wild 2 Flyby 02/01/04 Flyby 02/11/02 Sample return 13/06/10 Orbiter 12/09/05 – 25/04/06 / 103P/Hartley Flyby 04/11/10 EPOXI 9P/Temple 1 Flyby/Impact 04/07/05 19P/Borrelly Flyby 22/09/01 Flyby 28/06/99 NEAR Orbiter/ 14/02/00 – 01/03/01 Shoemaker Flyby 27/06/97 Flyby 28/08/93 Galileo Flyby 29/10/91 1P/Halley Flyby 14/03/86 1P/Halley Flyby 11/03/86 2 1P/Halley Flyby 09/03/86 Suisei 1P/Halley Flyby 08/03/86 1P/Halley Flyby 06/03/86 ISEE 3/ICE 21P/Giacobini-Zinner Flyby 11/09/85 The majority of these missions have been flyby missions, designed as secondary objectives that can be carried out on the way to their primary target. Targets for asteroid flyby missions are typically selected based on their proximity to the nominal trajectory of the primary mission. For example, the Rosetta spacecraft conducted three Earth manoeuvres (as well as one Mars gravity assist), making two crossings of the main on its way to rendezvous with comet 67P/Churyumov-

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Gerasimenko [43]. During these crossings, the spacecraft conducted flybys of two main belt asteroids: • 2867 Šteins at a distance of 800 km (relative speed 8.6 km/s); and • 21 Lutetia at a distance of 3,170 km (relative speed 15 km/s) [44].

These two asteroids were selected from a list of candidate asteroids after the launch of the spacecraft, requiring the lowest delta-V correction to the nominal trajectory [45]. During these flybys, Rosetta’s scientific instruments collected high resolution imaging of the asteroids over multiple wavelengths. A range of in situ measurements were also made of the dust, plasma, magnetic, and radiation environment [46]. Details of the data and results can be found in two special issues of the Planetary and Space Science journal [47, 48].

There have also been several asteroids selected as primary targets for dedicated missions. These asteroids are selected based on their scientific interest and relative accessibility (low delta-V trajectories). The Dawn spacecraft used a low-thrust trajectory to rendezvous with two of the largest minor in the main asteroid belt: 1 Ceres and . This was the first planetary spacecraft mission to rendezvous with two separate planetary bodies [49]. The spacecraft spent several orbiting each of the asteroids, producing high resolution imaging of the entire surface at multiple wavelengths. A ray and neutron spectrometer was also used to map the surface elemental abundances of major rock-forming elements (O, Mg, Al, Si, Ca, Ti, and Fe), light elements found in ices (H, C, and N), long-lived radioactive elements (K, Th, and U), and elements (Gd and Sm) [50, 51]. Surface maps of this data can be explored through the interactive web-based tools Ceres Trek [52] and Vesta Trek [53].

Spacecraft missions have also been conducted to return samples of asteroid material. The Hayabusa mission returned a small sample (less than 1 milligram) of regolith from the S-type near-Earth asteroid 25143 Itokawa [54]. Two further sample return missions are currently in progress. The Hayabusa2 mission is currently in orbit around the C-type near-Earth asteroid 162173 Ryugu. It recently deployed three smaller probes to land on the asteroids and take in situ measurements of its surface. In February 2019, the spacecraft descended to the surface and collected a sample of regolith. A second sub- surface sample is planned to be obtained from a crater recently created by a carry-on impactor, before returning the samples to Earth in December 2020 [55]. Watanabe et al. [56] provide an overview of the Hayabusa2 mission. The OSIRIS-REx mission is also

16 Scott Dorrington – June 2019 Chapter 2: Asteroid Properties & Resources currently in orbit around the B-type near-Earth asteroid 101955 Bennu. A sample collection is planned for mid 2020, before returning to Earth in September 2023 [57].

2.2 Orbital Properties

2.2.1 Classical Orbital Elements The orbits of asteroids can be described by the six Classical Orbital Elements (COEs): semi-major axis 푎, eccentricity 푒, inclination 푖, 휔, longitude of the ascending node 훺, and 휈 (or alternatively, 푀) at a reference . These elements are described in table 2.3 and visualised in figure 2.4. Asteroid orbital elements can be computed from several photometric observations over time, using Gauss’ method of orbit determination that uses a least squares method to find a best fit orbit [58]1.

Table 2.3 Definitions of the Classical Orbital Elements.

Symbol Element Description 푎 Semi-major axis Size of the orbit. 푒 Eccentricity Shape of the orbit. 푖 Inclination Inclination to the plane. Orientation of periapsis from 휔 Argument of periapsis ascending node. Orientation of the orbit from a 훺 Longitude of the ascending node reference direction. 휈 True anomaly Location in orbit at a given time.

1 The method of fitting orbits from observations, was first developed by Newton, and used by Halley to conclude that the comet he observed in 1682 was in fact the same as those previously observed in 1380, 1456, 1531 and 1607, from which he predicted its return in 1758 (thereafter named Halley’s comet, or 1P/Halley). Orbit determination methods were further developed by Euler, Lambert, Lagrange, Laplace, and Gauss. Gauss’ method is considered the most modern and detailed method. A full description of the details and history of these methods is described in Bate et al. [58] R.R. Bate, D.D. Mueller, J.E. White, Fundamentals of astrodynamics, Courier Corporation, 1971.

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Figure 2.4 Classical Orbital Elements (COEs) [59].

The Minor Planet Center (MPC) maintains a database of orbital elements for all known asteroids [60]. This data also includes the , perihelion and aphelion.

Orbital elements can be used together with numerical N-body integration methods to produce ephemeris data of the asteroids position and velocity vectors over time. Asteroid ephemerides used in this thesis are generated using the JPL Horizons system [10].

Orbit Uncertainty As with any curve fitting method, the accuracy of the orbital elements is determined by the number and spacing of observations used to find the best fit orbit. For each asteroid in its catalogue, the MPC issues an orbit uncertainty parameter – an integer value specifying the level of accuracy of the fitted orbital elements. The integer ranges from 0 to 9, with 0 indicating very small uncertainty and 9 an extremely large uncertainty [61]. This value is calculated from the fitted orbital elements and the uncertainties in the and time of periapsis passage.

The orbital elements of asteroids tend to vary over time due to gravitational interactions with the major planets, and other perturbing forces – particularly solar radiation pressure. If these asteroids are not continuously tracked, their orbits will become increasingly uncertain, and can even become lost.

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2.2.2 Designation and Numbering Convention Asteroids are catalogued by the MPC using a system of provisional designations, numbers, and names. Upon their first discovery, unless they can be identified immediately with an existing object, new asteroids with at least two nights of observations are provided with a provisional designation [62]. The standard format for this designation consists of a 4-digit number indicating the year of discovery, followed by two letters specifying the half- of discovery and the order of discovery in that half-month period. Additional numbers are appended (using subscripts where possible) if there are more than 25 asteroids discovered in each period.

Once an asteroid has been observed for over four oppositions, it is given a permanent designation, or number, according to the order in which it is confirmed [63]. If the asteroid is of particular significance, it can be issued with a name to replace its provisional designation. Asteroid naming is administrated by the International Astronomical Union. Naming conventions exist for several groups of asteroids, for example, asteroids are named after heroes from the Trojan war.

Asteroids are conventionally referenced first by their number, then by their name. For example, the first asteroid discovered is referred to as 1 Ceres. Numbered asteroids without a name, are referred to by their number (specified in brackets), followed by their provisional designation. For example, (459872) 2014 EK24.

2.2.3 Other Orbital Properties There are several other characteristic orbital properties that can be derived from an asteroid’s orbital elements.

Periapsis and Apoapsis The periapsis 푞 and apoapsis 푄 define the minimum and maximum distances of an elliptical orbit:

푞 = 푎(1 − 푒) , (2.1)

푄 = 푎(1 + 푒) . (2.2)

These parameters can alternatively be referred to as perihelion and aphelion for heliocentric orbits; or perigee and apogee for geocentric orbits. is a generalized term referring to the of an ellipse and can be applied to any orbit.

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Orbital Period and The orbital period 푇 can be computed from the semi-major axis using Kepler’s 3rd law:

푎3 푇 = 2휋√ , (2.3) 휇 where 휇 is the gravitational parameter of the Sun.

This may also be expressed as the mean motion 푛, defining the average angular velocity of the asteroid:

2휋 휇 푛 = = √ . (2.4) 푇 푎3

Synodic Period

The Synodic period 푇푠푦푛 gives the period of the resonant motion with respect to Earth.

It is calculated from the orbital period of the asteroid 푇 and orbital period of Earth 푇1:

1 푇 = . 푠푦푛 1 1 (2.5) | − | 푇1 푇

The synodic period determines the wait-time between Hohmann transfer opportunities to each asteroid. For most of the planets, the synodic period ranges over a few years (: 1.59 years, Mars: 2.14 years, and Jupiter: 1.09 years). Main belt asteroids (푎 = 2.2 to 3.3 AU) have synodic periods ranging from 1.2002 to 1.4419 years.

Asteroids in Earth-similar orbits (푎 = 0.9 to 1.1 AU) have synodic periods of the order of decades (5.8408 and 7.5068 years at the edge, with synodic period approaching infinity at 푎 = 1 AU). For these asteroids, there is a long wait-time between Hohmann transfers, however other launch opportunities are available each year.

Tisserand Parameter The Tisserand parameter is defined from the orbital elements of the asteroid, with respect to the semi-major axis 푎푝 of a reference planet (most commonly Jupiter or the Earth):

푎 푎 푝 2 푇푝 = + 2√ (1 − 푒 )푐표푠(푖) . (2.6) 푎 푎푝

While the orbital elements of an asteroid may change considerably due to gravitational perturbations with planets, the Tisserand parameter remains relatively unchanged. The

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Jovian Tisserand parameter 푇퐽 (defined with respect to Jupiter’s orbit 푎퐽 = 5.2 AU) is commonly used to distinguish between asteroid groups having different gravitational interactions with Jupiter [64]. Objects with 푇퐽 > 3 show little gravitational influence with Jupiter, with orbits either totally interior or totally exterior to Jupiter’s orbit.

Objects with 2 < 푇퐽 < 3 show strong gravitational interactions with Jupiter. These are associated with short-period comets that have been gravitationally captured by Jupiter into the inner Solar System (the Jupiter Family Comets).

Relative Encounter Velocity The Tisserand parameter can also be used to estimate the relative encounter velocity 푣 (and its normalized value 푈) during a close approach to the reference planet:

푣 푈 = = √3 − 푇푝 , (2.7) 푣푝 where 푣푝 is the orbital velocity of the reference planet.

Using the Tisserand parameter with respect to Earth, this gives an estimate for the relative velocity of an asteroid during its close encounter with Earth. For example, asteroids with 2.995 < 푇퐸 < 3 have encounter velocities of approximately 푣 < 2.1 km/s [65]. This velocity is a good indicator of the delta-V required to capture the asteroid during its flyby (assuming a negligible delta-V phasing manoeuvre to move the asteroid into an intersecting path).

Minimum Orbital Intersection Distance (MOID) The Minimum Orbital Intersection Distance (MOID) defines the closest point between the orbits of the asteroid and Earth (or another planetary body). While this cannot be computed through analytical methods, there are several numerical methods that may be used (for example [66]).

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2.2.4 Asteroid Groups Asteroids have been categorized into several groups based on ranges of orbital parameters (particularly semi-major axis and perihelion distance). Figure 2.5 shows the distribution of semi-major axis and eccentricity of all 18,796 catalogued asteroids in the MPC orbital elements database (MPCORB). (These have been limited to asteroids in the inner Solar System with 푎 ≤ 6 AU.)

Contours of characteristic semi-major axis, perihelion, aphelion, and Jovian Tisserand parameter are displayed to separate the major orbital groups defined by Warner et al. [67] used in the Asteroid Lightcurve Database (LCDB). These groups are arranged in order of increasing semi-major axis.

Figure 2.5 Orbital parameter space of NEA groups.

Main Belt Asteroids The majority of asteroids in the Solar System are located in two asteroid belts: • the “Main belt” (MB), located between the orbits of Mars and Jupiter (with 2.1 < 푎 < 3.3); and • the “Kuiper belt”, extending out past the orbit of (푎 > 30 AU).

Figure 2.6 shows the locations of these asteroid belts with respect to the other planets in the Solar System.

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Figure 2.6 Location of asteroid belts in the Solar System [68].

Main belt asteroids are further segmented into numerous sub-groups of asteroid families. These families are identified by clusters of asteroids sharing similar orbital elements, and are named after prominent asteroids in each family. These families are thought to have been created through catastrophic collisions that broke up a larger primordial parent asteroid [69]. Divisions in the families occur at 2:1, 3:1, 5:2, and 13:6 orbital resonances with Jupiter, producing orbital regions that are devoid of asteroids (the “Kirkwood gaps”) [70]. Figure 2.7 shows the distribution of semi-major axis and inclination for all asteroids in the MPCORB database, from which these divisions and families can be more clearly seen.

Figure 2.7 Distribution of semi-major axis and inclination [71].

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Near-Earth Objects Near-Earth Objects (NEOs) describe both asteroids and comets with perihelion distances less than 1.3 AU and orbital periods less than 200 years [72]. The near-Earth asteroids (NEAs) are categorised into four main groups: Armors, Apollos, Atens, and Atiras. These divisions are based on their semi-major axis, perihelion, and aphelion (summarized in figure 2.8).

Figure 2.8 Definitions of near-Earth asteroid groups [72].

Both the Apollo and Aten class asteroids have orbits that cross that of the Earth (Earth- crossing asteroids), posing a potential collision risk. Asteroids in these groups with minimum orbit intersection distance 푀푂퐼퐷 < 0.05 AU and absolute magnitude 퐻 < 22 are designated as Potentially Hazardous Asteroids (PHAs). These asteroids are continuously tracked, with lists generated of their close approach dates [73]. Asteroids with larger MOIDs may also collide in the future as gravitational interaction with the Earth can cause gradual changes to the perihelion and longitude of the ascending node, placing them onto intersecting orbits [74].

Asteroids on intersecting orbits have a finite collision probability per orbit, leading to an expected lifetime of around 100 million years before collision with Earth [74]. As this lifetime is significantly shorter than the age of the Solar System, it has been suggested that the near-Earth asteroid population is being continuously re-supplied by main belt asteroids that enter orbital resonances with Jupiter (i.e. the missing asteroids in the

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Kirkwood gaps). Numerical studies have shown that main belt asteroids in Jupiter- resonant orbits can migrate to the near-Earth region through successive gravitational interactions with Jupiter, following paths in the 푎-푒 space that approximately conserve the Jovian Tisserand parameter [75]. These asteroids can then be “extracted” from the orbital resonances through gravitational interactions with the Earth.

Trojan Asteroids Trojan asteroids are a large group of asteroids that share the same semi-major axis as Jupiter, while leading or trailing its position by 60o [76]. This motion is caused by a 1:1 mean motion resonance with Jupiter, where the asteroids follow “Tadpole” orbits that librate about the stable L4 and L5 Lagrange points (stable orbits in the Sun-Jupiter- Asteroid three-body system).

Jupiter Trojans are conventionally named for Trojan and Greek soldiers in Homer’s

Iliad, with the “Trojan camp” orbiting the L4 point and the “Greek camp” orbiting the

L5 point [77]. The only exceptions are those asteroids that were named before the introduction of this convention: (in the Trojan camp); and 642 Hektor and 617 (in the Greek camp).

Trojan asteroids have also been found orbiting with other planets including Mars [78] and Neptune [79]. A single , asteroid 2010 TK7, has been discovered orbiting Earth’s L4 point [80].

Earth Co-Orbital Asteroids Trojan asteroids follow one of several classes of co-orbital motion possible in a three- body system, including Horseshoe, Tadpole, Quasi-satellite, and Passing orbits. These classes are distinguished by the long-term motion of their relative mean longitude (angular separation from a line connecting the two primary bodies). Figure 2.9 shows the motion of these orbits in the Earth-Sun rotating reference frame (adapted from Dobrovolskis et al. [81]).

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Figure 2.9 Classes of co-orbital motion shown in an Earth-Sun rotating reference frame [81].

Over the duration of multiple orbits, the mean longitude of these orbits librate about the stable Lagrange points, periodically approaching Earth. The relative mean longitude of o Tadpole orbits librate about ±60 (i.e. the L4 or L5 points). These are the orbits followed by the Trojan asteroids described above. The relative mean longitude of Horseshoe o orbits librate about 180 (i.e. the L3 point). The relative mean longitude of Quasi- o satellite orbits librate about 0 (around both the L1 and L2 points). This relative motion appears to be orbing the second body, when viewed in a rotating reference frame – hence the name Quasi-satellite. Passing orbits are those that do not experience any of these resonant motions, with the relative mean longitude gradually completing a full revolution over a long synodic period.

There are several near-Earth asteroids that have been observed to follow one or more of these co-orbital motions [82]. These asteroids have been considered as a separate sub- group of NEAs (sometimes referred to as “Arjunas”) with very Earth-like orbital parameters [83]. Asteroids can experience transitions between these co-orbital regimes due to gravitational interactions with the Earth. For example, the asteroid 2002 AA29 was first observed in a , and has now transitioned into a Quasi-satellite orbit, always remaining very close to Earth (within 0.2 AU) [84].

Asteroids that pass close enough to Earth can also be temporarily captured through interactions of the invariant manifolds of the Earth-Sun L1 or L2 Lagrange points. The near-Earth asteroid 2006 RH120 was captured into an orbit around the Earth between

2006 and 2007. Figure 2.10 shows the trajectory of asteroid 2006 RH120 in an Earth-Sun

26 Scott Dorrington – June 2019 Chapter 2: Asteroid Properties & Resources rotating frame over the period from 1993 to 2027 (generated using ephemeris data from the JPL Horizons system [10]). A single invariant manifold of the L2 point is shown in green for reference. This trajectory was propagated from the eigenvector of the L2 point using the Circular Restricted Three Body Problem (CR3BP). In the left of the figure, the Horseshoe motion of the asteroid is seen to closely follow the invariant manifold on its approach to Earth. The right of the figure shows a detailed view of the period between

2006 to 2007, when it was temporarily captured through the Earth-Sun L2 point.

Due to their close proximity to Earth, and their potential to be temporarily captured, co- orbital asteroids have been considered potential candidates for asteroid mining missions. Studies have shown that near-Earth asteroids can be artificially captured by providing a small delta-V that changes their trajectories to intercept one of these invariant manifolds [85]. Once temporarily captured, these asteroids can be permanently captured into stable Earth orbits with small delta-Vs on the order of 100’s of m/s [86]. The down side to these asteroids is that they tend to be much smaller than other near-Earth asteroids (typically of the order of 5-10 m) [82].

Figure 2.10 Trajectory of asteroid 2006 RH120 showing its temporary capture into Earth orbit between 2006 and 2007.

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2.3 Delta-V Estimates The energy required to reach an asteroid is measured in the delta-V (ΔV) – the incremental velocity changes required to change an object’s orbit. The total delta-V of impulsive transfer trajectories will depend on the relative positions of the Earth and asteroid at the time of launch and arrival. These can be found through solving a two- point boundary value problem known as Lambert’s problem (discussed in chapter 6).

While these methods require numerical methods, analytical estimates of the delta-V can be made from the orbital elements of the asteroid and Earth (assumed to be in a circular orbit of radius 1 AU). The two most common methods include the Hohmann transfer and the Shoemaker-Helin equations. These equations are described below and detailed in Appendix A.

2.3.1 Hohmann Transfers The most common type of orbital transfer problem studied in orbital mechanics is the transfer between two circular orbits in the same orbital plane (coplanar). The optimal (minimum delta-V) transfer between coplanar circular orbits has been shown to be a two-impulse 180o transfer with the apses tangential to the initial and final orbit [87] 2. This transfer is known as a Hohmann transfer.

The Hohmann transfer is often used as a first approximation for estimating the delta-V of transfers between planets (having approximately coplanar and circular orbits). As the orbits of asteroids are highly elliptical, Hohman-like circular-to-elliptical transfers can also be considered from the Earth’s radius to either the asteroid’s periapsis or apoapsis (shown in figure 2.11).

The optimal transfer strategy will depend on the relative apsidal distances of the Earth and asteroid. In general, for non-intersecting orbits (Amors and Atiras), the optimal transfer is from the higher apoapsis to the lower periapsis, and for intersecting orbits (Apollos and Atens), the optimal transfer connects the circular orbit to the apoapsis of the ellipse [88, 89].

2 For transfers in which the ratio of the two orbital radii are large (푟2/푟1 > 11.95 or 푟2/푟1 < 0.084), a three-impulse “bi-elliptic” transfer is more efficient than the Hohmann transfer. As we are interested in asteroids within the main asteroid belt (푎 < 3 AU), we will not consider the bi-elliptic transfer.

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Figure 2.11 Left: Periapsis transfer. Right: Apoapsis transfer.

The equations for periapsis and apoapsis Hohman-like transfers are summarized in Appendix A.2. Using these equations, the approximate delta-V require to reach the main belt asteroids can be calculated as ranging from 9.34 km/s at the inner limit (2.2 AU) to 22.04 km/s at the outer limit (3.3 AU). From this, we can conclude that the main belt asteroids would require large delta-V requirements to reach, and are thus not good candidates for mining missions. As such, asteroid candidates considered in this thesis will be limited to the near-Earth asteroids.

Figure 2.12 shows the delta-V estimates of Hohmann-like circular-to-elliptical transfers calculated for all asteroids in the MPCORB database. For each asteroid, delta-Vs for both the periapsis and apoapsis transfer are calculated, and the minimum result is displayed. Characteristic contours of semi-major axis, periapsis and apoapsis are shown to divide the asteroids into their separate classes. Contours of the Jovian Tisserand parameter are shown to indicate the region 2 ≤ 푇퐽 ≤ 3 that is thought to be associated with Jupiter Family Comets. Figure 2.13 shows a histogram of the distribution of Hohmann delta-Vs amongst the near-Earth asteroids, Mars-crossers, and main belt asteroids.

From these plots, it can be seen that all NEAs can be reached with delta-V less than 16 km/s, with the mean delta-V of around 8 km/s. From this, we can conclude that a significant number of NEAs are considered more accessible than the main belt asteroids.

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Figure 2.12 Delta-V of Hohmann-like transfer to asteroids.

Figure 2.13 Histogram of Hohmann delta-Vs for various asteroid groups.

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2.3.2 Shoemaker-Helin Equations The Hohmann transfer is an idealized approximation that assumes the initial and final orbits are coplanar and (due to the approximate circular orbit of the Earth) coaxial. In most cases, asteroids will have a slight inclination relative to the ecliptic, and the initial and final orbits will not be coplanar. As shown in figure 2.7, the inclinations of the NEAs are distributed from zero to 90o (there are even a few retrograde asteroids with higher inclinations). Transfers between non-coplanar orbits require a plane change manoeuvre to remove the relative inclination between the orbits.

The Shoemaker-Helin equations [7] are commonly used to estimate the delta-V of transfers to asteroid based on their orbital elements. The transfer trajectories used in the Shoemaker-Helin equations are assumed to rendezvous at the asteroid’s apoapsis or periapsis (similar to those described above), with a transfer inclination half that of the asteroid (i.e. the trajectory that is halfway between the asteroid and Earth’s orbital planes). Velocities of the transfer orbit are calculated from the relative encounter velocities at the minimum orbit intersection distance. Full details of the Shoemaker- Helin equations are described in Appendix A.3.

While the Shoemaker-Helin equations are more accurate than the Hohmann transfer, they also make a number of assumptions such as the argument of periapsis being zero. In this case, the line of nodes is parallel to the apse line, such that the node crossings occur at periapsis and apoapsis. As the orbital velocity is minimum at apoapsis, it is the best place to carry out a plane change manoeuvre. This is an idealized case, giving the minimum delta-V for this type of transfer. Astronomical observations of asteroid orbital elements reveal that the right ascension of the ascending node 훺, and the argument of periapsis 휔 are both uniformly distributed across all angles [90]. As a result, the node crossing will generally not occur at periapsis or apoapsis, resulting in a higher transfer delta-V.

Despite these limitations, the Shoemaker-Helin equations are often used as a filter to identify accessible near-Earth asteroids from the large number of asteroids. Several asteroid mining studies have used a list of Shoemaker-Helin delta-Vs calculated by Benner [8].

Figure 2.14 shows a plot of the Shoemaker-Helin delta-V calculated for all asteroids in the MPCORB database. For clarity, this plot only shows asteroids with delta-Vs within the same limits as the Hohmann transfer plot (< 16 km/s). There are a large number of

Scott Dorrington – June 2019 31 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions high inclination asteroids dispersed through the region that have higher delta-Vs. Figure 2.15 shows a histogram of the distribution of Shoemaker-Helin delta-Vs for the near- Earth, Mars-crossers, and main belt asteroids.

Figure 2.14 Shoemaker-Helin delta-Vs for all asteroids.

Figure 2.15 Histogram of Shoemaker-Helin delta-Vs for various asteroid groups.

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2.4 Physical Properties Asteroids and other non-emitting celestial objects are observed through reflected off their surfaces. The apparent magnitude or brightness of an object is primarily determined by the size and shape of the object, the optical properties of the surface, and the viewing geometry relative to the observer and illumination source. Photometric observations of asteroids can therefore provide clues about the asteroid’s size and shape, physical properties, and surface composition.

2.4.1 Viewing Geometry The viewing geometry is characterized by the heliocentric distance 푟 between the Sun and asteroid, the geocentric distance ∆ between the Earth and asteroid, and the phase angle 훼 – the included angle formed from between the Sun, asteroid, and Earth. This geometry is shown in figure 2.16. The phase angle determines the proportion of the surface that is illuminated, as viewed by the observer. A phase angle of zero will show maximum illumination (such as the ), while a phase angle of 1800 will show minimum illumination (such as the ). These are observed at opposition and superior conjunction, when the asteroid is aligned with the Earth-Sun vector.

Figure 2.16 Viewing geometry of illuminated body.

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2.4.2 Phase Curve Observations of an asteroid over long time frames allow for the generation of a phase curve – the observed magnitude potted over a range of phase angles. Figure 2.17 shows an example phase curve generated from observational data of asteroid 44 Nysia (this figure is taken from Buchheim [91] using data reproduced from Harris et al. [92]). Note that the y-axis in this figure has been inverted to show lower magnitudes (brighter observations) at the top – this is the convention for plots of magnitude.

Figure 2.17 Phase curve of asteroid 44 Nysa [91, 92].

A best fit curve can be fitted from observations using an empirical formula. The standard model endorsed by the International Astronomical Union (IAU) is the two- parameter (퐻-퐺) model [93]:

퐻(훼) = 퐻 − 2.5 log10[(1 − 퐺)훷1(훼) + 퐺훷2(훼)] , (2.8) where 퐺 is the slope parameter, and 퐻 is the magnitude at zero phase angle. The functions 훷1 and 훷2 are defined in Appendix B. An alternative three-parameter models also exist including two slope parameters at low and high phase angles [94].

The phase curve can be used to compute the apparent visual magnitude 푉 of an asteroid as viewed from Earth as a function of its position:

푉 = 퐻(훼) + 5 log10(푟∆) , (2.9) where:

34 Scott Dorrington – June 2019 Chapter 2: Asteroid Properties & Resources

푟 is the heliocentric (Sun-to-Asteroid) distance (AU); and

∆ is the geocentric (Earth-to-Asteroid) distance (AU).

This can be used to predict the visibility of asteroids over their orbits, allowing the planning of observations. For asteroids without observed phase curves, a slope parameter of 퐺 = 0.15 is often assumed for these calculations.

2.4.3 Light Curve Detailed observations over short time frames can produce a time-varying plot of the magnitude known as a light curve. Light curves show small scale fluctuations in the apparent magnitude due to the changing reflecting cross-section as an irregular shaped asteroid rotates.

Measurements of the peak-to-peak distance of the light curve can provide estimates of the asteroid’s period of rotation. The relative magnitudes of the peaks and troughs can also provide some information as to the shape and spin axis orientation of the asteroid. Image inversion methods can also be used to compute upper limits of the shapes of the asteroid (a convex hull) that could produce the same light curve [95]. Figure 2.18 shows an example light curve of asteroid 44 Nysia showing observed visual magnitudes fluctuating about a mean value (taken from Buchheim [91]).

Figure 2.18 Light curve of asteroid 44 Nysa [91].

Scott Dorrington – June 2019 35 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

2.4.4 Mean Diameter The mean diameter of an asteroid can be estimated from the observed magnitude 퐻 at zero phase angle, and the (V-band) 푝푉 [96]:

1329 퐷 = 10−0.2퐻 [푘푚] . (2.10) √푝푉

The albedo of an object is a measure of the brightness of its surface: low albedo objects have dark surfaces, while high albedo objects have bright surfaces. There are two definitions for albedo, depending on the viewing geometry.

The bond albedo 퐴 is defined as the fraction of incident solar energy that is reflected off its surface, ranging from 0 (total absorption) to 1 (total reflection). This measurement of albedo will vary with the observed phase angle. The geometric albedo 푝 is the albedo that would be seen at zero phase angle. This is calculated from bond albedo using a phase 푞, computed from the slope parameter of the phase curve:

퐴 ≈ 퐴푉 = 푞푝푉 , (2.11) where:

푞 = 0.290 + 0.684퐺 . (2.12)

Harris [96] assume that the bolometric bond albedo (measured over all wavelengths) is approximately equal to the visual bond albedo.

Geometric albedo and diameter data for over 2000 asteroids is available in the Supplemental IRAS Minor Planet Survey (SIMPS) database – processed from observations from the IRAS satellite [97]. This data, along with observational data from multiple data sets, has been compiled into a single Asteroid Lightcurve Database (LCDB) [67].

For asteroids without observed albedo measurements an assumed albedo is used for diameter calculations, based on the average albedo of asteroids of different taxonomic classes. Most estimates use the assumed geometric albedo of 푝푉 = 0.18 ± 0.09 for S- type asteroids, which are the most populous asteroids in the NEA and inner main belt regions [98]. The diameters of low albedo C-type asteroids are calculated using an assumed albedo value of 푝푉 = 0.058 ± 0.024. These asteroids are most populous in the mid to outer main belt, and those in the outer Solar System.

36 Scott Dorrington – June 2019 Chapter 2: Asteroid Properties & Resources

Figure 2.19 shows a plot of the observed magnitude and estimated mean diameter for NEAs with observed light curve data, obtained from the asteroid LCDB [99]. While a small number of these NEAs have accurate diameter and albedo measurements, it can be seen that a large number use an assumed albedo of 0.2, based on the assumed S-type taxonomic class. Only a small fraction of the known NEAs are represented in the LCDB.

Figure 2.19 Diameter ranges for assumed albedo ranges as a function of absolute magnitude.

Due to the large uncertainty in albedo, remote observations can only provide an approximate diameter range. More detailed diameter measurements and shape models can be obtained through observations during an asteroid’s close approach to Earth, or during a spacecraft flyby or rendezvous. Figure 2.20 shows a sequence of images of increasing resolution taken by the Hayabusa2 spacecraft on its approach to asteroid 162173 Ryugu. A detailed global view of the asteroid is shown in the left of figure 2.21. The right of figure 2.21 shows a global view of the asteroid 101955 Bennu composed of images taken by the OSIRIS-REx spacecraft at a distance of 24 km. While both images are shown at the same size, 162173 Ryugu (diameter ~900 m) is almost twice the size of 101955 Bennu (diameter ~500 m).

Scott Dorrington – June 2019 37 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

Figure 2.20 Sequence of images during Hayabusa2 approach to asteroid 162173 Ryugu 3.

Figure 2.21 Left: Global view of asteroid 162173 Ryugu 4. Right: Global view of asteroid 101955 Bennu 5.

2.4.5 Internal Structure The internal structure of asteroids is difficult to discern from observations alone; however, can be inferred from the bulk density of the asteroid, calculated from the total mass and volume. Spacecraft flybys afford the opportunity to measure the mass of an asteroid from the trajectory perturbations before and after the flyby [100]. Asteroid mass data is sparse as only a small number of spacecraft missions to asteroids have been carried out. Mass determinations may also be calculated from observed perturbations of asteroid encounters with other bodies such as Earth, Mars, and other asteroids [101].

3 Image Credit: Phil Stooke / JAXA, University of Tokyo, Kochi University, Rikkyo University, Nagoya University, Chiba Institute of Technology, Meiji University, University of Aizu and AIST. 4 Image Credit: JAXA, University of Tokyo, Kochi University, Rikkyo University, Nagoya University, Chiba Institute of Technology, Meiji University, University of Aizu, AIST. 5 Image Credit: NASA / Goddard / University of Arizona.

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Knowledge of the mass and size of an asteroid allow for the calculation of the bulk density, and porosity (the volume fraction of voids). Asteroids with high bulk density are thought to be monolithic asteroids, with an interior composed of solid rock (low porosity). Asteroids with low bulk density are thought to be “” asteroids – aggregates of boulders, rocks, gravel, and regolith held together by weak gravitational and surface forces. These asteroids have large portions of voids between their constituent particles (high porosity).

Rubble pile asteroids have a range of particle sizes across their surface and interior. Analysis of images and surface samples of asteroid 25143 Itokawa, taken by the Hayabusa spacecraft, revealed particle sizes ranging from large boulders of the order of 10’s of metres to fine-grained regolith of the order of microns [102]. Similar size distributions have been observed on asteroid 162173 Ryugu by the Hayabusa2 spacecraft. Figure 2.22 shows boulders on asteroid 162173 Ryugu of the scale of 102 and 100 m.

Figure 2.22 Images of asteroid 162173 Ryugu taken from the Hayabusa2 spacecraft showing boulder size distributions on two scales 6.

Statistical measurements of boulder sizes across the surface of 25143 Itokawa show a power-law size distribution of the form г−푛, where г is the particle diameter and 푛 is a

6 Image Credit: JAXA, University of Tokyo, Kochi University, Rikkyo University, Nagoya University, Chiba Institute of Technology, Meiji University, University of Aizu, AIST

Scott Dorrington – June 2019 39 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions power index. A number of fits have been generated over various particle size ranges, show power indices of -3.6 to -2.8 [103].

Sánchez & Scheeres [104] outline an analytical formulation of particle size distribution in rubble pile asteroids, wherein particle sizes are distributed from a minimum particle size г0 to a maximum particle size г1, with a cumulative number density distribution following a г−3 power law (taking the average of the range of power indices). The number density function (giving the number of particles of a given size г) can be expressed as:

1 푛(г) = 3푁 г3 , (2.13) 1 1 г4 where 푁1 is a fitting constant defining the number of particles with the maximum size

г1.

Inferences from Spin Rate The internal structure can also be inferred from knowledge of an asteroid’s size and rotation rate (from observed light curve data). Rubble pile asteroids that are held together by weak gravitational forces are expected to only be stable at low rotation rates. A critical 푃푐 can be determined by equating the gravitational and centripetal accelerations at the asteroid’s equator (at the mean radius) [105]:

퐺푚 2 3휋 3.3 ℎ푟푠 = 휔푐 푟 ⇒ 푃푐 = √ ≈ , (2.14) 푟2 퐺휌 √휌 where 퐺 is the gravitational constant, 푚 is the mass of the asteroid, 푟 is the mean radius, and 휌 is the bulk density (g/cm3).

Asteroids are expected to have bulk densities ranging from 2 – 3 g/cm3 [106], equating to critical rotation periods of 2.33 – 1.9 hrs. This critical period is a “spin-barrier” that separates rubble pile asteroids from monolithic asteroids. It is thought that asteroids rotating at faster rates must be monolithic, as a rubble pile asteroid would have its constituent particles thrown apart by centripetal forces.

Figure 2.23 shows the distribution of diameter and rotation period for main belt and near-Earth asteroids in the asteroid LCDB. Only asteroids with a quality code greater than 2 are displayed, as lower values indicate incomplete light curves with large uncertainties (up to 30%) in rotational period. This data set and condition is typically used in most statistical studies of asteroid rotation rates (for example, [107-109]).

40 Scott Dorrington – June 2019 Chapter 2: Asteroid Properties & Resources

Asteroids with diameters greater than 40 km show a Maxwellian distribution of spin rates, indicating that their rotation is the result of collisions with other asteroids [110]. The majority of large asteroids with diameters greater than 150 m rotate slowly with periods greater than around 2.2 . This value is consistent with the expected range of bulk densities, providing evidence that the majority of large asteroids are rubble piles [105].

Smaller asteroids show non-Maxwellian distributions of spin rates, with excesses of slow and fast rotating asteroids. These differences to the distribution are thought to be caused by the Yarkovsky–O'Keefe–Radzievskii–Paddack (YORP) effect – a torque generated by scattering and thermal re-emission of solar radiation off the asteroid’s surface [111]. Very small asteroids with diameters less than 150 m are seen to have fast rotation rates, with rotational periods tending to decrease with diameter. The YORP effect has an increasing effect on smaller asteroids with smaller moments of inertia, leading to faster rates of change in rotation [112].

Figure 2.23 Distribution of diameter and rotation period for MBs and NEAs.

2.5 Meteorite Chemical Composition Meteorites are the remnants of asteroid material that has survived entry through the Earth’s atmosphere to impact on the ground. Meteorites therefore provide an important source of data for the likely chemical composition of their asteroid parent bodies.

Scott Dorrington – June 2019 41 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

2.5.1 Minerals Over 25,000 meteorites have been found and studied, revealing a wide range of chemical compositions, with approximately 275 mineral species identified (see Rubin [113, 114] for a complete list). These minerals are predominantly composed of the elements Fe, Mg, Si, and O, making up over 90% of the meteorite’s mass [115]. These mostly exist in the form of native metals and their oxides, and as silicates – most commonly olivine (Mg, Fe)2SiO3 and pyroxene (Fe2, Mg2, CaMg, CaFe)Si2O6. Table 2.4 provides definitions of common mineral groups found in meteorites [116].

Table 2.4 Mineral groups found in meteorites [116].

Group Description Native Elements Elements not combined with any other elements Compounds containing various cations combined with SiO , SiO or Silicates 2 3 SiO4 Oxides and Compounds of cations (most commonly Fe) with O2- or OH- hydroxides Sulfides and Compounds of cations (usually Fe) with - or P6-, respectively. phosphides 2- Carbonates Compounds of cations (usually Ca, Mg, or Fe) with CO3 2- Sulfates Compounds of cations (usually Ca or Mg) with SO4 3- Phosphates Compounds of cations (usually Ca) with PO4 Halides Compounds of cations (usually Na or K) with Cl-, Br-, or F-

Carbonaceous chondrites have a significant fraction of carbon (up to 3 wt%), giving them a dark appearance (lower albedo). Carbonaceous chondrites have also been found to contain large amounts of volatiles including water in the form of hydrated minerals – minerals containing H2O or OH compounds. These hydrated minerals are found chemically bonded between sheets of phyllosilicates (clay minerals) such as Serpentine:

(Mg, Fe)6Si4O10(OH)8 and other non-silicate minerals such as Epsomite: MgSO4.7H2O.

These hydrated minerals were formed by aqueous alteration of anhydrous silicates in the presence of liquid water – thought to be produced from heating of water-ice in the asteroid (see Rivkin et al. [117] and the references therein). For this reason, carbonaceous chondrites and their suspected asteroid parent bodies (C-type asteroids) are of primary interest to asteroid mining as sources of water.

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Table 2.5 shows bulk chemical compositions of several representative chondritic meteorites of different types (discussed in the next section). This table is reproduced from Mueller & Saxena [118], with modern taxonomic classes of each meteorite (specified in brackets) obtained from the Meteoritical Bulletin Database [119]. The

compositions of silicates SiO2 and water H2O have been highlighted.

Table 2.5 Bulk chemical composition of representative chondritic meteorites [118].

Ordinary Ordinary Type-III Type-II Type-I Enstatite Type: H L carbonaceous carbonaceous carbonaceous (EL6) (H6) (L6) (CO3) (CM2) (CI1) Daniel’s Meteorite: Oakley Kyushu Warrenton Mighei Orgueil Kuil Fe 23.70 15.15 6.27 4.02 0.00 0.00 Ni 1.78 1.88 1.34 1.43 0.00 0.00 Co 0.12 0.13 0.05 0.09 0.00 0.00 FeS 8.09 6.11 5.89 5.12 3.66(5)a 5.65a SiO2 38.47 36.55 39.93 34.82 27.81 21.74 TiO2 0.12 0.14 0.14 0.15 0.08 0.07 Al2O3 1.78 1.91 1.86 2.18 2.15 1.59 Composition (wt%) MnO 0.02 0.32 0.33 0.20 0.21 0.18 FeO 0.23 10.21 15.44 24.34 27.34 22.86 MgO 21.63 23.47 24.71 23.57 19.46 15.24 CaO 1.03 2.41 1.70 2.17 1.66 1.18 Na2O 0.64 0.78 0.74 0.69 0.63 0.71 K2O 0.16 0.20 0.13 0.23 0.05 0.07 P2O5 trace 0.30 0.31 0.20 0.30 0.27

H2O 0.34 0.21 0.27 0.10 12.86 19.17 Cr2O3 0.23 0.52 0.54 0.58 0.36 0.35 NiO 0.11 – – 0.00 1.53 1.19 CoO – – – 0.00 0.07 0.06 C 0.32 – 0.03 0.19 2.48 2.99 Organic – – – – – 6.71 Matter Total 99.89 100.29 99.67 100.08 101.00 100.03 Mason & Prior Wiik Reference Wiik Wiik [121] Wiik [121] Wiik [121] [120] [121] [122] a Wiik reported all S as FeS but it is given here as S, and the corresponding Fe is reported as FeO.

2.5.2 Meteorite Taxonomy There have been several taxonomic systems developed over the years for cataloguing meteorites into classes of similar physical and mineralogical properties. Early taxonomies divided meteorites into three broad classes (stones, stony-irons, and irons) that can be discerned from visual inspection of the meteorite based on their gross properties. These taxonomic systems were initially developed for cataloguing meteorite

Scott Dorrington – June 2019 43 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions collections in various museums (see Weisberg et al. [123] for a history on meteorite taxonomic systems).

Irons are pure (~99 wt%) metallic alloys of nickel and iron [124]. Once cut and polished, they are easily identified from their silvery colour, and distinguishable cross- hatched "Windmanstätten" markings [11]. Stones have a dark, granular structure composed dominantly of silicate minerals, with small particles of metals and sulfides. Stoney meteorites often contain small (millimetre-scale) glassy spheres known as “chondrules”. Stony-irons are a mixture of the two – silicate granules in a metallic matrix. Figure 2.24 shows images of these different meteorite types.

Figure 2.24 Images of meteorites. Top Left: A polished and etched slice of the Henbury iron meteorite (Iron, IIIAB) [125]. Top Right: A slice of the Glorieta Mountain stony-iron meteorite (Pallasite, PMG) [126]. Bottom Left: A slice of the Allende stony meteorite (CV3 carbonaceous chondrite) [127]. Bottom Right: A magnified view of the Allende meteorite showing glassy chondrules [126].

The three meteorite groups can be further divided into several sub-groups sharing similar mineralogic and petrologic properties. Table 2.6 shows the taxonomic classes defined by Mason [128] (reproduced from Mueller & Saxena [118]). In this taxonomy, iron meteorites are sub-divided based on their relative nickel and iron content, and

44 Scott Dorrington – June 2019 Chapter 2: Asteroid Properties & Resources crystal structure. Stoney-iron meteorites are similarly sub-divided based on the relative proportions of olivine and pyroxene in their silicate component.

Table 2.6 Meteorite Taxonomy of Mason [118, 128].

Stoney meteorites are broadly divided into chondrites (further divided into Carbonaceous, Ordinary, and Enstatite chondrites) and achondrites. These groups were originally defined based on the presence or absence of chondrules (hence the name); however modern taxonomic systems use the terms to distinguish between a meteorite’s relative elemental abundances. Chondrites show elemental abundances closely matching those observed in the Sun. They are thought to be formed from undifferentiated primitive material from which the Solar System formed [129]. For this reason, chondritic meteorites are important in the study of Solar System formation. Achondrites show elemental abundances that differ from that the Sun. These are thought to be formed from fragments of asteroid parent bodies that were differentiated into compositionally distinct layers [130].

Modern taxonomic systems such as that of Weisberg et al. [123], use a hierarchical structure consisting of classes, clans, groups, and subgroups to divide the meteorites based on varying degrees of similarities in their chemical compositions, and oxygen isotopic properties. (See Weisberg et al. [123] for details of these divisions, and a review of the history and evolution of meteorite taxonomic systems.)

Scott Dorrington – June 2019 45 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

Figure 2.25 shows the taxonomic system of Weisberg et al. [123], with the following abbreviations for meteorite groups:

• Primitive Achondrites: URE – ureilite, BRA – brachinite, ACA – acapulcoites, LOD – lodranite, WIN – winonaite; • Achondrites: ANG – angrite, AUB – , EUC – eucrite, DIO – diogenite, HOW – howardite, MES – mesosiderite, MG PAL – main-group pallasite, ES PAL – Eagle Station pallasite, PP PAL – pyroxene pallasite; • Mars: SHE – shergottite, NAK – nakhlite, CHA – chassignite, OPX – orthopyroxenite.

Figure 2.25 Meteorite taxonomy of Weisberg et al. [123].

Carbonaceous Chondrites In the early taxonomic systems, carbonaceous chondrites were divided into sub-groups of petrological types I – III defining the degree to which their chemical composition has been altered by secondary processes such as heating and aqueous alteration (chemical changes due to the presence of water). Modern taxonomic systems divide carbonaceous chondrites into eight groups (CI, CM, CO, CV, CK, CR, CH, and CB). These are named

46 Scott Dorrington – June 2019 Chapter 2: Asteroid Properties & Resources after similarities to prominent meteorites [123]: CI (Ivuna), CM (Mighei), CO (Ornans), CV (Vigarano), CK (Karoonda), CR (Renazzo), CB (Bencubbin), and CH (high Fe; ALH 85085).

Of these, the CI, CM and CR meteorites are the most interesting for asteroid mining purposes, showing the highest degrees of aqueous alteration and H2O content. The CI carbonaceous chondrites correspond to petrological type I, showing the highest degree of aqueous alterations, with H2O content as high as 19 wt% (shown in table 2.5). CM and CR carbonaceous chondrites show lower degrees of aqueous alterations (petrologic types I and II). The petrological group is often used to subdivide the various meteorite groups according to their hydrated mineral content (e.g. CM1 and CM2 chondrites).

2.6 Asteroid Parent Bodies The similar chemical composition of meteorite groups suggest that they may originate from common asteroid parent bodies. Current estimates suggest that the 22,500 meteorite samples found on Earth could originate from as little as 100 – 150 asteroid parent bodies [131]. (These figures refer to the original primordial asteroids, of which numerous present- asteroids are expected to be fragments – such as the asteroid families discussed above.)

Based on similarities in the reflectance spectra of meteorites and asteroids, a number of linkages have been proposed between meteorites groups, and large main belt asteroids. Burbine et al. [131] discuss evidence for four of the main speculative asteroid-meteorite linkages: • S-type asteroids as the parent bodies of ordinary chondrites; • C-type asteroids as the parent bodies of CM carbonaceous chondrites; • V-type asteroids (in particular 4 Vesta) as the parent bodies of basaltic achondrites; and • M-type asteroids as the parent bodies of irons and Enstatite chondrites.

These linkages are based on similarities in the overall shape of the reflectance spectra in the visible and infrared, and the strength of characteristic absorption features. Figure 2.26 (taken from Burbine et al. [131]) shows comparisons between the reflectance spectra of two C-complex asteroids (shown in black) and their analogous meteorites (shown in grey):

Scott Dorrington – June 2019 47 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

• (Tholen G-type or Bus Ch-type) and CM2 meteorite LEW 90500; and • (Tholen C-type) and a heated sample of (ungrouped) C1 meteorite Y- 82162.

Figure 2.26 Normalized reflectance spectra for two asteroids and their analogous meteorites [131]. Both sets are normalized at 0.55 μm. Second set is offset by 0.5.

2.6.1 Water in C-Type Asteroids Hydrogen and oxygen are the first and third most abundant elements in the universe, and stable compounds of these two elements (OH and H2O) are expected to be common throughout the Solar System [117].

Water Ice It is hypothesised that asteroids were formed in the early Solar System as mixtures of water-ice and anhydrous silicates [117]. At distances beyond about 2.7 AU the and pressure of the solar nebula are low enough to allow water to exist as solid ice [132]. This so-called “snow line” falls in the middle of the main asteroid belt, separating icy bodies in the outer Solar System, from dry bodies in the inner Solar System.

Evidence of water-ice has been observed in a number of planetary and objects in the outer Solar System (e.g. [133-135]). For asteroids in the inner Solar System, water-ice located near the surface of asteroids is gradually lost to space due to sublimation. However, main belt asteroids may still have substantial deposits of ice beneath their surfaces.

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Theoretical models have been developed to predict the loss rate of sub-surface ice in main belt asteroids as a function of heliocentric distance, and a number of dynamical, physical, and thermal properties. Schorghofer [136] used one such model to show that water-ice can persist in the top few meters of main belt asteroids over billions of years, provided that the mean surface temperature is less than about 145 K.

In a further developed model, Schorghofer [137] predicted that water-ice can be expected in the top half-meter of the surface of asteroid 1 Ceres at latitudes greater than 600. Fanale & Salvail [138] similarly predicted that water-ice should be present in the top 1 – 10 m at latitudes greater than 40o. These predictions were confirmed by the Dawn spacecraft that showed direct observations of exposed surface ice in a 10 km wide crater on the C-type main belt asteroid 1 Ceres [139].

Many early studies of asteroid mining made the assumption that water would be present in the form of water-ice dispersed in the regolith of C-type near-Earth asteroids. This misconception was identified as one of the findings of the 2016 ASIME conference (Asteroid Science Intersections with In-Space Mine Engineering), that brought together engineers and planetary scientists. In a white paper produced from the same conference, it was noted that due to their proximity to the Sun, near-Earth asteroids have higher average surface than main belt asteroids (280 K at 1 AU with 7% albedo), with an ice loss rate of around 330 m per million years [140]. It is therefore unlikely that any near-Earth asteroids possess substantial sub-surface ice deposits due to their small sizes, unless they have only recently migrated from the main belt within the last million years.

Hydrated Minerals The expected source of water in near-Earth asteroids is in the form of hydrated minerals in C-type asteroids. C-type asteroids have been linked to carbonaceous chondrites due to their low albedo and similar spectral slopes, weak absorption features at 0.7 μm, and strong absorption features at 3 μm [131]. Both of these absorption features have been attributed to the presence of phyllosilicate minerals.

As hydrated minerals (and some ices that are not expected to be stable at the surface) are the only minerals that give rise to the 3 μm absorption feature, the observation of this feature in asteroid spectra is thought to be diagnostic of the presence of water in the form of hydrated minerals [117]. The 0.7 μm feature may also be diagnostic of the presence of phyllosilicates, however there are other minerals that give rise to absorption

Scott Dorrington – June 2019 49 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions features at this wavelength (see Rivkin et al. [117] for a discussion on the presence of hydrated minerals in asteroids).

Due to the limited data available on asteroid material, the precise composition of asteroid surfaces is still highly uncertain. DeMeo et al. [141] discuss a number of open questions on the composition of asteroids such as the robustness of the meteorite- asteroid connections, and compositional differences that may occur in the different asteroid groups (NEAs and main belts).

At the time of writing, initial results from the OSIRIS-REx mission have shown spectral evidence for the presence of aqueously altered hydrated minerals on the B-type near- Earth asteroid 101955 Bennu [142]. The Hayabusa2 spacecraft has also observed similar spectral evidence for hydrated minerals (in lower concentrations) on the surface of C-type near-Earth asteroid 162173 Ryugu [143]. More conclusive evidence is expected to be found once samples of these asteroids are returned to Earth.

2.7 Useful Resources and their Abundances The minerals found in meteorites could potentially have many practical uses in the support of space exploration activities, such as sources of propellant, construction material, and life support consumables. Table 2.7 lists a number of resources expected to be present in asteroids, and their uses (reproduced from Ross [3]).

Table 2.7 Asteroid resources and their uses [3].

Volatiles

Primary Use Molecules

Life Support H2O, N2, O2 Propellant H2, O2, CH4, CH3OH Agriculture Co2, NH4OH, NH3 Oxidizer H2O2 Refrigerant SO2 Metallurgy CO, H2S, Ni(CO)4, Fe(CO)5, H2SO4, SO3

Metals and Semiconductors

Primary Use Molecules Construction Fe, Ni Precious Metals Au, Pt, Pd, Os, Ir, Rh, Ru, Re, Ge Semiconductors Si, Al, P, Ga, Ge, Cd, Cu, As, Se, In, Sb, Te

50 Scott Dorrington – June 2019 Chapter 2: Asteroid Properties & Resources

2.7.1 Precious Metals Based on the elemental abundances of chondritic meteorites, asteroids are thought to possess abundant supplied of precious metals in higher concentrations than are found on Earth. Kargel [144] estimated that LL ordinary chondrites could contain 50 – 200 ppm of precious metals, and a single 1 km sized S-type asteroid (the suspected parent bodies of ordinary chondrites) could contain over 400,000 tonnes of gold and platinum group metals. If this material was to be returned to Earth, it could potentially be worth over $5 trillion at current market prices. Elvis [1] calculated that a smaller 100 m sized asteroid could contain around 23.6 tonnes of precious metals (worth an estimated $1.18 billion), based on a more conservative concentration estimates of 10 ppm. The Asterank database [145] maintains a list of the estimated value of asteroids based on their size, expected mineral content (from taxonomic class), and current market prices.

Despite these large potential supplies, the economic viability of returning asteroid- derived platinum to Earth is uncertain, due to the large expected costs and complexity of carrying out missions to retrieve it, compared to well-tested traditional terrestrial mining methods. It has also been suggested that an increase in the global supply of platinum would place downward pressure on the market price, making it harder to recover the capital investment of an asteroid mining mission [144].

2.7.2 Water-Based Propellant Water has been considered to be the first mineral to be mined from asteroids, due to its potential to be used as a propellant for spacecraft. Water may be used as a propellant in several ways. The first method is to use thermal energy to vaporize the water, producing an exhaust plume of steam. These steam-powered propulsion systems could provide high thrust levels, with specific impulses around 200 s [146].

Water can further be electrolyzed – by means of solar energy – to produce gaseous hydrogen and oxygen (GH2/GO2) propellant. Studies have suggested that electrolysis- based propulsion systems could be developed to produce high thrust levels, with specific impulses of over 350 s [147].

The H2 and O2 gasses could be further liquified to produce liquid hydrogen and oxygen

(LOX/LH2) propellant, with specific impulses up to 450 s. LOX/LH2 is a common propellant used by many commercial spacecraft. The United Launch Alliance (ULA) has also issued a preliminary pricing estimate that it would be willing to pay for space- derived LOX/LH2 propellant delivered to Earth orbit ($3M/tonne in Low Earth Orbit or

Scott Dorrington – June 2019 51 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

$1M/tonne in Geostationary Orbit) [148]. This offer represents the first steps in the development of an orbital propellant market, providing additional incentives for asteroid mining companies.

While LOX/LH2 propellant provides higher specific impulses and improved storage density than steam or electrolysis-based propulsion, studies have shown that there would be a high relative hardware mass required for the processing and storage of cryogenic propellants [146].

High specific impulse electric propulsion systems typically use noble gasses such as Argon and Neon as their propellants. These elements are only present in trace amounts in chondritic meteorites, and would require the processing of large amounts of asteroid ore to extract in usable quantities.

2.7.3 Abundance of Water in Near-Earth Asteroids As will be discussed in chapter 7, only 219 of the 19,880 near-Earth asteroids in the MPCORB database have sufficient observational data to assign a taxonomic class. Given this limited observational capability, the chemical composition of any particular asteroid candidate is highly uncertain. It is therefore more appropriate to use probabilistic methods to estimate the likely quantity of resources in asteroids.

Elvis [1] outlined a simple probabilistic method of estimating the total fraction of ore- bearing asteroids 푃표푟푒 in the near-Earth asteroid population as the product of the probability 푃푡푦푝푒 that an asteroid is of the resource-bearing type, and the probability

푃푟푖푐ℎ that this type of asteroid is sufficiently rich in the desired resource:

푃표푟푒 = 푃푡푦푝푒 × 푃푟푖푐ℎ . (2.15)

As noted in section 2.1.2, current de-biased estimates of the distribution of taxonomic classes suggest that C-type asteroids make up only around 10% of the total near-Earth asteroid population (hence, 푃푡푦푝푒 = 0.1) [36]. The fraction of C-type asteroids with high water content can be estimated from the water distribution in carbonaceous chondrite meteorites, shown in figure 2.27 (taken from Elvis [1], using data obtained from Jarosewich [149]).

52 Scott Dorrington – June 2019 Chapter 2: Asteroid Properties & Resources

Figure 2.27 Water distribution in carbonaceous chondrite meteorites [1, 149].

From this distribution, Elvis [1] suggested a value of 푃푟푖푐ℎ = 0.25 as the probability of finding high concentrations of hydrated minerals (calculated from the 31% of samples with H2O > 6 wt%, and adjusting for higher concentrations that would occur due to water-ice). From these values, it is estimated that only 2.5% of near-Earth asteroids are expected to have rich concentrations of water. With the current number of known asteroids, this suggests only around 500 asteroids that could be potential targets for asteroid mining missions. While this number is expected to increase with the discovery of additional NEAs, there will still only be a relatively small, finite number of water- rich NEAs that could be considered as targets for asteroid mining missions.

2.8 Asteroid Mining Studies Numerous studies have been conducted on asteroid mining, focusing on various aspects of the technical and economic feasibility. These studies can be broadly divided into those considering detailed trajectory and mission designs to identified asteroid targets, and those providing more generic parametric modelling.

Detailed trajectory studies employ both analytical and numerical trajectory optimization methods to find optimal trajectories for flyby, rendezvous, or sample return missions to near-Earth asteroids (e.g. [150-153]). The most comprehensive list of optimal trajectories is maintained by the Trajectory Browser tool [154], containing multiple launch opportunities for a large number of NEAs from 2010 to 2040. These studies are commonly conducted to provide realistic delta-V estimates in the final stages of the target selection process for scientific asteroid missions, including sample return

Scott Dorrington – June 2019 53 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions missions (discussed in chapter 7). These studies most commonly aim to minimize the total delta-V of the trajectory, often making use of long duration missions utilizing low- thrust trajectories, and multiple gravity assist manoeuvres. Solar sailing methods that do not require any propellant have also been considered for the design of asteroid sample return missions (e.g. [155]).

Trajectory design studies have also been conducted to assess the technical feasibility of capturing and returning small asteroids into orbit around Earth. As previously noted, small asteroids can be directed into temporarily captured orbits through the use of invent manifolds in the Earth-Sun L1 and L2 Lagrange points [85], from which they can be permanently captured into stable Earth orbits with small delta-Vs on the order of 100’s of m/s [86]. (See Sánchez et al. [156] for a review on asteroid retrieval missions.)

The most advanced of these studies were produced for the NASA Asteroid Redirect Mission (ARM), that proposed retrieving a whole small asteroid (or a boulder from a larger asteroid) into orbit around Earth [157-159]. This mission progressed to Phase B studies prior to its cancellation, producing detailed systems designs, and prototypes of robotic capture systems that were tested on a full-scale mock-up of an asteroid boulder [160]. Landau et al. [161] conducted a study on low-thrust trajectories to several asteroid candidates making use of low-thrust trajectories powered by solar electric propulsion, demonstrating that it would be feasible to retrieve small asteroids up to 10,000 kg to Earth orbit.

These whole-asteroid capture missions have been proposed as one such method in which asteroid mining can be achieved, where the processing of material occurs in Earth orbit. The alternative method involves the in situ processing of material in the asteroids original orbit [162]. These alternative approaches are discussed further in chapter 3.

While trajectory design studies for scientific missions aim to minimize individual parameters, such as total mission delta-V, for commercial ventures, such as asteroid mining missions, more appropriate metrics should be used that consider not only the operational factors, but also the economic factors in the mission design. Ramohalli et al. [163] introduced the concept of “figures of merit” (FOMs) – quantitative metrics computed from combinations of critical mission and system parameters – that can be used as an objective function in the evaluation and optimization of space mission architectures. FOMs can be developed to include factors such as specific impulse, delta- Vs, mass ratios, mission costs, reliability, repairability, and risk. An example scenario

54 Scott Dorrington – June 2019 Chapter 2: Asteroid Properties & Resources was presented to evaluate alternative architectures for a Mars sample return mission using alternative propellants (cryogenic or storable), and varying degrees of in situ propellant production (ISPP) – return-trip oxidizer and/or propellant processed at Mars, or completely supplied from Earth. The FOM used to evaluate these alternatives was the mass ratio of the returned sample to the total launch mass of the rocket at Earth, modified to include scaling factors for reliability, repairability, and inverse risk. Preliminary results indicated that the alternative using in situ processed oxidizer and

Earth-sourced storable propellant (CH4) was more efficient than the traditional, baseline alternative using cryogenic propellant and oxidizer supplied from Earth. The case study demonstrated the benefit in using FOMs in identifying better mission designs than if system parameters were considered individually (for example, the baseline Earth- sourced propellant case can provide higher specific impulses, but is not the optimal solution). While this case study focused on Mars sample return missions, it can be equally applicable to asteroid mining missions.

In his book Mining the Sky, [164] provides a broad overview of the potential resources present in asteroids and other planetary bodies, and how they could be exploited in the near future. Lewis [164] considers the feasibility of asteroid mining missions in terms the mass payback ratio – the mass of material that can be returned from asteroids compared to the mass of the spacecraft required to retrieve it (comparable to the FOM used by Ramohalli et al. [163], without the additional factors). Lewis [164] speculates that utilizing ISPP, mass payback ratios of 3:1 could be achievable with each round-trip mission to a near-Earth asteroid, and payback ratios of up to 1000:1 could be achievable over multiple round-trip missions (based on order of magnitude delta-V and propellant mass estimates). In a follow-up book Asteroid Mining 101, Lewis [124] discusses further details of the resources present in asteroids, and a number of operational concepts that would be required, such as capturing and de- spinning a tumbling asteroid, and mineral beneficiation.

The International Space University (ISU) has produced several white paper studies on asteroid mining from a multi-disciplinary approach; considering the scientific and engineering challenges, business cases, as well as the legal, ethical, and social aspects of asteroid mining [165-167]. The 1990 study [165] focused on the feasibility of a multiple return trip mission to extract water (assumed to be in the form of water-ice) from the near-Earth asteroid . Oxnevad [168] developed a parametric economic model to evaluate the feasibility of this mission design, computing the costs, revenues,

Scott Dorrington – June 2019 55 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions and net present value (NPV) as a function of the delta-V, and other mission and system parameters. This model was used to perform a trade study on the economic feasibility of the mission under different parameters. The study concluded that the project could only be feasible (produce positive NPVs) if masses greater than 30 to 50 tonnes were returned in each return trip.

NPV was also used by Sonter [169] as the primary economic FOM for evaluating asteroid mining missions. This study used analytic trajectory design methods to develop general mission scenarios for the different classes of near-Earth asteroids. Stay-times at the asteroid between these transfers were also discussed, but not calculated. It was noted that trajectories arriving close to an asteroid’s aphelion (furthest distance from the Sun) allows a much longer “aphelion mining ” than those arriving at an asteroid perihelion (closest distance to the Sun). Several case studies were considered in which the NPV was calculated for returning 1,000 tonnes of asteroid ore using a 5 tonne spacecraft (a mass payback ratio of 200), with the use of ISPP for the return propellant. Sonter’s findings indicated that these missions would be technically feasible, however a significant amount of the material collected at the asteroid would have to be used as propellant to return it (as much as 60% of the mined material).

Craig et al. [170] conducted a study of using multiple spacecraft to extract nickel-iron metal from the M-type asteroid (6178) 1986 DA. The study assumed each spacecraft return one-third of its mass in asteroid resources (20 tonnes) in each return trip. The NPV method was used to calculate the cash flows generated over a 100-year mining operation, finding that the payback period before a positive NPV can be achieved was around 80 years. A second case study to a hypothetical, more easily accessible asteroid returned a payback period of 8.5 years. While accurate delta-V estimates were obtained from Benner [8] (using the Shoemaker-Helin equations), assumptions were made about the mission duration and frequency of launch opportunities (one return trip per year for asteroid (6178) 1986 DA, and two return trips per year for the hypothetical asteroid). Only a few small (5–20 m) asteroids have been discovered so far (using the Trajectory Browser tool [154]) that have total return-trip mission durations of six months required for this second case study, each having only one or two launch opportunities during their decades long synodic periods.

Many of these mission designs assume the use of a single large, monolithic spacecraft; however, studies have suggested that asteroid mining could be achieved with the use of multiple smaller spacecraft. Calla et al. [171] conducted a trade study on using swarms

56 Scott Dorrington – June 2019 Chapter 2: Asteroid Properties & Resources of small spacecraft (< 500 kg), concluding that a total of 250 spacecraft would be needed to reach a break-even point within 10 years of operation, with a total capital cost of around $4 billion.

2.9 Conclusion This chapter has provided a review of some of the important orbital, physical, and compositional properties of asteroids that can be inferred from remote sensing observations, and studies of meteorites and asteroid samples. The chapter also introduced two data sets that will be used in the following chapters: • The Minor Planet Center orbital elements database (MPCORB) [60]; and • The Asteroid Lightcurve Database (LCDB).

Delta-V calculations from asteroid orbital elements obtained from the MPCORB database show that the near-Earth asteroids have significantly lower delta-Vs than Mars-crossers, or the main belts. From this, we can conclude that the main candidates for asteroid mining missions will be near-Earth asteroids, defined as those having perihelion distances less than 1.3 AU. Due to their high magnitudes (low expected diameters), limited observational data is available for these asteroids. Probabilistic estimates based on the water content of carbonaceous chondrites, and the expected distribution of C-type asteroids suggest that only around 2.5% of the near-Earth asteroids are expected to have rich sources of water. This water is expected to be in the form of hydrated minerals, rather than solid water-ice.

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3 PARAMETRIC ECONOMIC ANALYSIS

3.1 Introduction The viability of asteroid mining missions will depend both on their operational and economic performance. Operational factors include the complexity and duration of the mission, and the mass of asteroid resources that can be delivered to customers. Economic factors include the capital and operational costs of performing the mission, and the revenue generated from the sale of the returned material. Each of these factors will be dependent on the design and sizing of the spacecraft vehicles, mission and trajectory design, and the overall logistics of the asteroid mining process.

In this chapter, a space mission architecture will be presented, describing the various elements of an asteroid mining operation. Alternative approaches are identified for several aspects of the mission design, such as the mining/extraction method, propellant supply method, and exploration strategy. These alternatives are evaluated using a number of operational and economic figures of merit, defined as parametric functions of critical system and mission parameters. A trade study is then performed to assess the feasibility of the mission alternatives over a range of parameter values.

The aim of this is to present a generalized framework that can be used to represent, analyse, and select between numerous design decisions. This may be used as a tool for evaluating different system designs, or for deciding the range over which mission alternatives will be economically viable. This architecture will be used throughout this thesis, with various chapters focusing on the design alternatives of different components.

58 Scott Dorrington – June 2019 Chapter 3: Parametric Economic Analysis

3.2 Components of an Asteroid Mining Industry The basic aim of asteroid mining is to extract mineral resources from asteroids and deliver them to customers spacecraft in Earth orbit. As in the terrestrial mining industry, asteroid mining will require a number of operational phases, including: 1. Prospecting – locating asteroids with potential ore deposits; 2. Exploration – investigating asteroids up-close with spacecraft (flyby or orbiter/lander); 3. Mining – extracting ore from the asteroid and returning it to Earth orbit; 4. Distribution – delivering products derived from the extracted ore to customers; and 5. Return to asteroid – repeating the mining and distribution steps for additional mining trips.

These operations will consist of multiple interplanetary missions, involving the use of several spacecraft vehicles and other infrastructure both in space and on the ground. The operations may be performed by a single asteroid mining company, or by several companies, leading to the development of an asteroid mining industry that has the potential to become a significant component of a larger, developing cislunar economy. In this thesis, the focus is on the operational logistics of the asteroid mining operation, rather than the business organisation.

Mining Campaign For the purposes of this thesis, the collection of missions involved in the asteroid mining process is referred to as a “mining campaign”. The mining campaign specifies the number and sequencing of missions, defining the overall strategy employed by the asteroid mining companies. For example, a mining campaign may begin directly with mining missions to an identified asteroid candidate, or may include preliminary exploration missions (flyby and/or sampling). Multiple mining missions could be carried out either by launching new mining spacecraft, or by multiple mining trips performed by a single mining spacecraft. These subsequent trips could return to the same asteroid, or target a new asteroid candidate.

3.2.1 Prospecting The first phase in the mining campaign will be to identify asteroids with the presence of an ore resource – a mineral concentration high enough to be considered commercially profitable [172, 173]. The prospecting phase will require two processes. The first

Scott Dorrington – June 2019 59 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions process will be the discovery and tracking of new asteroids. This will be performed by asteroid search campaigns (such as those discussed in chapter 2) using ground-based photometric observations. The second process will involve detailed follow-up photometric and spectroscopic observations of individual asteroids. This will produce detailed light curves and reflectance spectra, from which an asteroid taxonomy class can be assigned. This information can be used to select candidate asteroids meeting a number of conditions such as correct spectral type, taxonomy class, size, and expected presence of ore resources.

Observations obtained during the prospecting phase can give indications as to an asteroid’s taxonomic class and chemical composition. For example, the presence of the 1.9 and 3 μm absorption features in asteroid reflectance spectra are considered to be diagnostic of the presence of hydrated minerals (sources of water) [117]. Studies have shown that these features are seen in a significant portion of low-albedo C-type (carbonaceous) asteroids that are of particular interest to asteroid mining.

However, similar absorption features associated with other minerals are also seen throughout the infra-red and visible spectra, leading to the chance of mis-diagnosing an asteroid with the presence of hydrated minerals. Similarly, asteroids that do have hydrated minerals may not show these spectral indicators.

3.2.2 Exploration Missions The uncertainty in an asteroid’s chemical composition could be reduced by sending an exploration mission to investigate the asteroid up-close. These may take the form of either a flyby, an orbital, or surface sampling mission, or even a sample return mission. Exploration missions would decrease the uncertainty of the presence of asteroid resources, at the cost of additional capital investment. If the resource is determined to not be present at any stage of the mining campaign, the asteroid target may be rejected, saving the cost of deploying an unsuccessful mining mission.

Flyby Missions Flyby missions would allow a short period of up-close study using remote sensing methods. Optical observations during approach could be used to develop a detailed light curve, determining the shape and rotation of the asteroid. Surface mapping of the asteroid’s surface could also be achieved using a number of instruments, such as Neutron Spectrometers that may estimate the concentration and distribution of hydrated

60 Scott Dorrington – June 2019 Chapter 3: Parametric Economic Analysis minerals in the asteroid’s surface [51]. The coverage and resolution of the mapping would depend on the spatial and spectral resolution of the instruments, and the geometry of the flyby trajectory.

Flyby missions may be accomplished at low delta-Vs as they are not required to match the asteroid’s velocity on arrival. These missions could therefore be conducted by smaller spacecraft at a much lower cost than sampling missions. Flyby missions could also be designed to visit multiple asteroids. This has been considered for several mission proposals such as the KaNaRiA mission, proposed to visit numerous asteroids in the main belt [162]. The design of multiple asteroid flyby missions presents a challenge in trajectory design, determined by the sequencing of multiple trajectory segments. This problem has been the focus of previous Global Trajectory Optimization Competitions [174], from which several papers were produced.

Asteroids that pass within the Earth’s sphere of influence also present opportunities of achieving rapid, low-cost flyby missions without the need for long duration interplanetary transfers [175]. These missions would also give valuable information of the potentially hazardous asteroids that may collide with the Earth on future close encounters.

While flyby missions would give more detailed photometric and spectroscopic data than ground-based observations, the presence of mineral resources must still be inferred from the observational data. This may still leave a level of uncertainty in the presence and concentration of mineral resources in the asteroid.

Sampling Missions Sampling missions would require larger delta-Vs than flyby missions, and would therefore be considerably more expensive. However, they would allow a much longer window of study at the asteroid. Prior to landing, sampling missions could perform extensive surface mapping from orbit, from which a landing sites can be selected. The sampling spacecraft could then analyse drill samples of the asteroid material taken at multiple locations across the asteroid’s surface. This could provide ground proofing of the presence and concentration of mineral resources in the asteroid.

Sampling missions could also be designed to visit numerous asteroids. However, due to the larger delta-Vs, fewer targets could be visited by a single spacecraft. The WINE mission has been proposed to use propellant extracted from asteroids to move between numerous asteroids [176]. (This concept is known as in situ propellant production –

Scott Dorrington – June 2019 61 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions defined later as a potential propellant supply alternativity.) This could allow for the exploration of large numbers of asteroid candidates with a single spacecraft.

3.2.3 Mining Missions Following the confirmation of the presence of a desired ore resource, a mining mission could be sent to extract and return these resources to Earth orbit. A mining mission will consist of an Earth-to-Asteroid heliocentric transfer trajectory; a stay-time at the asteroid during which mining operations are undertaken; and a return Asteroid-to-Earth heliocentric transfer.

The delta-Vs and times of flight of each of the heliocentric transfers will depend on the asteroid’s orbital elements, and will vary over successive launch opportunities. During the stay-time at the asteroid, the mining equipment will extract a mass of asteroid material to be returned to Earth. This mass will be dependent on the stay-time at the asteroid, and the mining rate achievable by the mining equipment.

3.2.4 Distribution The mining spacecraft will be responsible for the transportation of a shipment of asteroid ore back to an arrival parking orbit around the Earth. From here, the resources are required to be distributed to customer spacecraft. As there may be many customer spacecraft in various orbits and locations, a warehouse, or depot can be used as an intermediate destination of the asteroid ore. Asteroid ore will be delivered to the depot, where it will be stockpiled, and possibly further processed into usable products. From here, the resources could be distributed to the customers by the mining spacecraft, or by smaller transport spacecraft. Depending on the distribution of customer spacecraft, this may require additional planning of transport operations; assigning and scheduling transport spacecraft to service selected customers.

3.2.5 Return to Asteroid Following a successful mining mission, the mining spacecraft may be refuelled and sent back to the asteroid for subsequent missions. If water-based propellant is the resource being delivered to the depot, the mining spacecraft may retain enough propellant to return to the asteroid, with the excess resources delivered to customers for sale. This would remove the need for refuelling missions to be launched from Earth prior to the next mining trip. Figure 3.1 displays a generalised space mission architecture

62 Scott Dorrington – June 2019 Chapter 3: Parametric Economic Analysis representing the mission elements of an asteroid mining campaign over multiple mining trips.

Figure 3.1 Reference space mission architecture of an asteroid mining mission.

3.3 Mission Alternatives For each of the mission elements in the space mission architecture, there may be a number of alternative approaches to select from. These may include system design choices such as the sizing of the different spacecraft; operational choices such as the mining methods employed; and logistics decisions such as the design of the distribution system and selection of exploration strategy.

3.3.1 Distribution Network Design There are two key design decisions to be made in the layout and operations of the distribution network: 1. The routing of the mining and transport spacecraft between asteroid, depot, and customers; and 2. The orbital locations (shape and orientation) of the depot and arrival/departure parking orbits.

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The optimal design of this distribution network is the major focus of chapter 5, where the distribution network is modelled as a network graph, with nodes representing the orbital locations of the depot, asteroid, and customers, and edges representing transfer trajectories between these orbits. A location-routing problem is formulated to find the optimal location of network nodes and spacecraft routing strategy that maximizes the total sellable mass delivered to the customers over multiple mining trips.

3.3.2 Trajectory Design (Flight Itinerary) A major aspect of the mission performance will be determined by the design of the heliocentric trajectories between the Earth and target asteroid. The launch and arrival dates of successive transfers will determine factors such as the delta-V and times of flight, the stay-time at the asteroid, and the wait-time at Earth before the next mining trip can commence.

The performance of an asteroid mining campaign will therefore be highly dependent on the selection of the launch and arrival dates of the successive transfers of multiple mining trips (i.e. the “flight itinerary”). The selection of an optimal flight itinerary is the major focus of chapter 6, where multiple local optimal launch opportunities are identified for target asteroids by plotting the delta-V of transfers over a wide range of launch and arrival dates. Graph theory methods are then applied to consider combinations of numerous transfers, from which optimal flight itineraries can be selected that maximize the total profit or net present value of the mining campaign.

3.3.3 Extraction/Mining Alternatives This chapter focuses on the logistics of the exploration and mining phases. In the following sections, three key design decisions are considered for the extraction/mining method; the propellant supply method; and the sequencing of exploration and mining missions.

There are two general strategies that can be considered for the mining operations, concerning how the resources are to be extracted or collected: 1. Whole asteroid capture; or 2. In situ processing.

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Whole Asteroid Capture The first approach is to collect asteroid material in its raw form by either grabbing a large boulder from the asteroid’s surface, or capturing a small asteroid in its entirety by use of a bag or tether. This method is generally considered for most asteroid mining studies, and was also proposed for the now-cancelled NASA Asteroid Redirect Mission (ARM) [159].

This approach is perhaps the simplest and lowest cost approach, as it requires minimal mining operations at the asteroid rendezvous location. It also may reduce the initial dry mass of the spacecraft as the mass of the bag and tether would be considerably less than that of heavy mining equipment.

Other than the initial selection of the asteroid target, this approach will have no control over the size of the material being returned to Earth. As there are large uncertainties in an asteroid’s size and density, the asteroid may be larger or smaller than expected, or there may not be the presence of appropriately sized boulders on the surface. This would introduce the risk of decreased revenues for smaller than expected asteroids, or a mission failure if the asteroid is too large to be retrieved with the available propellant supply.

This approach also presents additional challenges in de-spinning the asteroid prior to capture, and the attitude control of the asteroid during the return transfer to Earth. Several approaches have been considered for this including impulsive pushing, de- spinning, and asteroid tugs. The effectiveness of these methods will depend on the rotation state of the asteroid, that may be rotating about a principle axis, or tumbling in all directions [177]. Scheeres & Schweickart [178] provide a review of some of the technical challenges in controlling the translational and rotational motion of whole asteroids. Alternative solutions involve using a gravity tractor method, where the spacecraft uses the gravitational attraction between the asteroid and spacecraft to pull the asteroid along [179].

In Situ Processing The second approach to asteroid mining is to use surface mining methods such as drilling or excavating to extract and process material from the asteroid orebody. In this approach, the raw asteroid material could be processed into either a refined ore, or a readily usable material in its pure form. This approach would add value to the material

Scott Dorrington – June 2019 65 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions being retrieved, potentially increasing the sale price, and avoiding the transportation of non-useful by-products.

However, this would require carrying both mining and processing equipment, and necessary power sources for both processes. The amount of material that can be extracted would also be dependent on the stay-time at the asteroid, the mining rate achievable by the mining method and equipment, and the maximum capacity that the mining spacecraft can contain.

Processing at the asteroid affords the option of carrying out multiple trips to and from the asteroid, bringing back small amounts in each trip. The mining equipment could be left at the asteroid to continue extraction, further increasing the duration of the mining operations.

3.3.4 Propellant Supply Alternatives Two approaches could also be considered for the supply of propellant for the mission: 1. Earth-based propellant supply (EBPS); or 2. In situ propellant production (ISPP).

Earth-Based Propellant Supply (EBPS) For the first approach, all propellant for the return trip to the asteroid is launched from Earth with the mining spacecraft. This method is the standard approach that has been implemented for all space missions that have been conducted to date, and is commonly considered in most asteroid mining studies (including the ARM mission).

The standard EBPS approach could use either chemical or electrical propulsion. Chemical propulsion would allow for short duration impulsive transfers; however, the low specific impulse would require a large amount of propellant to be supplied at launch. Electric propulsion systems allow for a much higher fuel efficiency, reducing the propellant mass at launch, however the low-thrust trajectories would require longer times of flight, particularly for the return transfer.

All sample return missions that have been conducted to date have brought back only small quantities of resources that are negligible compared to the mass of the spacecraft. For example, the Hayabusa mission was designed to retrieve a sample of a few grams of asteroid material, using a 510 kg spacecraft (consisting of 380 kg dry mass and 130 kg of propellant) [180]. (In fact, the mission was only successful in returning less than one milligrams of material.) This equates to a ratio of return mass to launch mass of

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1.96 × 10−6. To retrieve this sample, the propellant mass made up 25% of the total launch mass (typical of planetary missions).

In order to become profitable, asteroid mining missions would need to return large shipments of asteroid material that could be one or two orders of magnitude greater than the mass of the spacecraft retrieving it. The propellant required to return masses of these scales would make up a significant proportion of the initial launch mass.

For example, initial feasibility studies of the Asteroid Redirect Mission considered targeting a 500 tonne asteroid using an 17 tonne spacecraft (consisting of a 4.1 tonne dry mass, 12 tonnes of Xenon propellant, and 900 kg of Hydrazine propellant) [159]. The total mission duration was expected to be around 6 to 10 years. For this configuration, the asteroid mass would be 29.4 times the total launch mass of the spacecraft, and the mass of propellant would make up 70% of the total launch mass.

In situ Propellant Production (ISPP) The second approach is one in which the propellant required for the return Asteroid-to- Earth transfer is processed from material at the asteroid (in situ propellant production) [181]. The propellant choices will be limited to materials that are present in the asteroid material. As discussed in chapter 2, C-type asteroids are expected to contain large amounts of volatiles such as water in the form of hydrated minerals that can be used in steam, electrolysis, or LOX/LH2 propulsion systems.

Using this approach, a mining spacecraft could be launched with only enough propellant to reach the asteroid, lowering the cost to that of a one-way mission. However, a significant portion of the material extracted from the asteroid would need to be consumed as propellant over the return Asteroid-to-Earth transfer, reducing the amount of material that is delivered to customers. For example, a near-Earth asteroid with return delta-V of 4 km/s would require 60% of the material to be consumed to return the load to Earth orbit (using chemical propulsion with 4.4 km/s exhaust velocity). This fraction would likely become even greater once the delta-Vs of distributing the material to customers is taken into consideration.

Using the ISPP approach, the mining spacecraft could use a further portion of the returned material as propellant to return to the asteroid (or to a different target asteroid). This would allow for subsequent mining trips to be accomplished with no additional costs (other than a reduced revenue from the preceding trip). Any number of mining trips could be made, so long as the mining spacecraft is still operational, and the

Scott Dorrington – June 2019 67 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions asteroid is still within reach (launch opportunities exist with low delta-V trajectories). Using the standard Earth-based propellant supply approach, additional mining trips would require a refuelling mission to be launched from Earth at the cost of additional capital.

This ISPP approach has been proposed for the WINE mission, where a small CubeSat- size spacecraft could conduct sampling missions on numerous asteroids (potentially continuing indefinitely) [176]. The method has been successfully demonstrated in initial tests on asteroid simulant.

ISPP with Reserve Propellant The ISPP approach is considerably more risky than the standard propellant supply case in that it relies on the presence of propellant at the asteroid. In the event that this propellant is not found (such as if the taxonomic class was mis-identified), or not present in sufficient quantities, the mining spacecraft would be lost at the asteroid without the means of returning to Earth.

For the EBPS approach, the asteroid material could still be returned if the wrong taxonomic class was found. However, a mission failure could occur, if the asteroid is found to be too large to be retrieved with the available propellant. In this case, the spacecraft would have sufficient propellant to return to Earth without any asteroid material. To mitigate the risks of an unsuccessful ISPP mission, an alternative ISPP approach can be considered where the spacecraft is launched with enough reserve propellant to return to Earth with no asteroid material in the case of a mission failure.

3.3.5 Summary of Mission Alternatives The design of an asteroid mining mission can be made from two sets of decisions: the extraction/mining method, and the propellant supply method. Table 3.1 summarizes the mission alternatives of each.

From these choices, four mission alternatives cases can be identified for further study in this chapter. These are listed in table 3.2, and displayed using a decision tree in figure 3.2. An additional alternative is also added representing a reference mission, where the same mass of material is delivered directly to the customers from Earth, without the need to visit an asteroid. This approach could only be considered for resources that are available on Earth.

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Table 3.1 Summary of mission alternative choices.

Extraction/Mining Description Alternatives Option 1: Whole asteroid The mining spacecraft will capture an asteroid in its entirety by use of a net, tether or bag, and return the return whole asteroid to Earth. The mining spacecraft will process a small portion of Option 2: In situ processing asteroid ore from the main body, and only return that.

Propellant Supply Description Alternatives All propellant required for the transfers is supplied from Option 1: Earth-Based Earth at the beginning of the mission. Refuelling Propellant Supply (SFE) missions are conducted prior to each successive return trip. Propellant for the initial Earth-to-Asteroid transfer is supplied from Earth for the first mission. Propellant for Option 2a: In situ Propellant the Asteroid-to-Earth transfer is processed at the Production (ISPP) asteroid. Propellant for the Earth-to-Asteroid transfer for subsequent mining missions is taken from the propellant returned in the previous mining mission. Option 2b: In situ Propellant Similar to ISPP, however, enough propellant is provided Production with Reserve for the mining spacecraft to return to Earth in the event (ISPP) that no propellant is found at the asteroid.

Table 3.2 Summary of Mission Alternatives.

Extraction/Mining Propellant Supply Alternative Method Method ISPP 1A ISPP with reserve In situ Processing 1B propellant EBPS 2A Whole Asteroid Return EBPS 2B Earth-sourced EBPS Ref

Scott Dorrington – June 2019 69 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

Figure 3.2 Decision tree showing the various mission alternatives.

3.3.6 Exploration Approach (Mining Campaign) Given the high level of uncertainty in the presence of resources within an asteroid, beginning a mining campaign directly with a mining mission is expected to be highly risky. This risk may be mitigated by gaining further information on the asteroid, by sending lower-cost exploration missions (either flyby or sampling).

If an exploration mission detects signs indicating the presence of a desired resource, the certainty of the presence of the ore would increase. This would increase the probability that a subsequent mining mission would be successful in finding and extracting the desired resource.

If an exploration mission does not show signs of the presence of the resource, the resource may still be present, and additional exploration missions could be sent to investigate further. Alternatively, the mining campaign could be terminated, or re- directed to another asteroid candidate. In this case, the mining campaign would lose the capital investment of the exploration missions, however it would save a potential loss of the higher capital of an unsuccessful mining mission.

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In either outcome, sending exploration missions reduces the total risk at the cost of additional capital. Determining the number and type of exploration missions needed before mining can commence should be determined by a cost-risk analysis. Table 3.3 shows the qualitative cost and risk for a number of possible approaches to the exploration process of an asteroid mining campaign. A more detailed analysis can be conducted by enumerating the potential outcomes (success and failure) of each successive exploration mission using a decision tree, and computing the total expectation value of each approach. This approach is commonly applied in the design of optimal exploration strategies in the mineral exploration for terrestrial mines [182].

Table 3.3 Qualitative cost-risk analysis.

Approach Cost Risk Flyby – Sample – Mine High Low Sample – Mine Medium Medium Flyby – Mine Medium Medium Mine Low High

3.4 Figures of Merit for Evaluating Mission Alternatives The mission alternatives identified in table 3.2 may be evaluated and compared by creating a parametric model of the space mission architecture. System parameters describe the key system drivers such as the specific impulse and dry masses of the various spacecraft. Mission parameters describe the delta-Vs and durations of the trajectories involved in the mission. Cost parameters describe the specific costs and revenues as a function of the system and mission parameters. These parameters may be used to develop a number of “figures of merit” – quantitative metrics describing the cost and performance of the mission design [163].

Parametric-based models have been used in several previous asteroid mining feasibility studies (discussed in chapter 2) [168-170]. These models vary in their formulation and complexity, depending on the assumed mining approach (single trip or multiple trips), and the cost estimating relationships of the system and mission parameters. Common figures of merit used in these studies include the asteroid return mass, mass payback ratio, payback period, internal rate of return (IRR), net present value (NPV), and expectation value.

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This section defines the operational and economic figures of merit considered in this thesis, and their relationship to identified critical system, mission, and cost parameters. Differences in the formulation that occur for the various mission alternatives are introduced in the following section.

3.4.1 System Parameters Table 3.4 describes the system parameters considered in the model.

Table 3.4 System Parameters.

Component Parameter Units Description

푚푑푟푦,푀푆 kg Dry mass of mining spacecraft. Specific impulse of mining spacecraft 퐼 s 푠푝 propulsion system. Mining Thrust of mining spacecraft propulsion Spacecraft 퐹 N 푇 system.

푀푚푎푥 kg Maximum capacity of resource tank.

푃푀푆 W Available power of the mining spacecraft. Mining 푚푀퐸 kg Mass of mining equipment Equipment 푀푅 kg/day Mining rate of the mining equipment/rover. (Rover) 푃푅 W Available power of the mining equipment.

In this table, the mining spacecraft and mining equipment are treated as two separate vehicles. As previously mentioned, the mining equipment will be carried to the asteroid as an additional payload of the mining spacecraft on its first mining trip. Depending on the mining approach, the mining equipment may be returned to Earth or left at the asteroid to continue mining operations for subsequent mining trips.

The mining rate 푀푅 will be dependent on the design of the subsystems of the mining equipment, and the operations involved in the mining process. A separate parametric model for the mining rate is presented in chapter 4.

The maximum mass capacity defines a limit on the mass of resources that can be carried by the spacecraft. This may be limited by the performance of the spacecraft (such as the available thrust of the propulsion system), or by the physical dimensions of the spacecraft (such as the maximum volume of the bag or storage tank that the resources are held in during the return transfer).

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3.4.2 Mission Parameters Table 3.5 lists a set of mission parameters describing properties of the Earth-to-Asteroid and Asteroid-to-Earth transfers.

Table 3.5 Trajectory Parameters.

Parameter Units Description

∆푉퐸퐴 km/s Delta-V of Earth-to-Asteroid transfer.

∆푉퐴퐸 km/s Delta-V of Asteroid-to-Earth transfer.

∆푉푡표푡 km/s Total delta-V of return transfer.

푇푂퐹퐸퐴 Years Time of flight of Earth-to-Asteroid transfer.

푇푂퐹퐴퐸 Years Time of flight of Asteroid-to-Earth transfer.

푇푠푡푎푦 Years Stay-time at the asteroid.

푇푐푎푝 Years Time required to capture the asteroid.

푇푤푎푖푡 Years Wait time at Earth. 푇 years Total duration of the mission.

Impulsive and Low-thrust Trajectories The transfers to and from the asteroid could be performed using either high-thrust impulsive transfers making use of chemical propulsion systems, or by continuous low- thrust trajectories making use of electric propulsion systems.

For impulsive transfers, the delta-Vs and times of flight will be determined by the launch and arrival dates of the transfers, and the relative positions of the Earth and target asteroid. As discussed in chapter 2, analytical estimates for the delta-Vs can be made as a function of the asteroid’s orbital parameters using either the Hohmann transfer or Shoemaker-Helin equations (detailed in Appendix A). Using these approximations, the time of flight can be estimated as half the period of the transfer orbit.

Low-thrust transfers are achieved by applying continuous thrust over the duration of the transfer, with the magnitude and direction of the thrust acceleration 푎푇 determined by solving an optimal control problem. Analytical estimates for the total delta-V can be made from an asteroid’s orbital parameters using the Edelbaun equation [183, 184]. The total duration of the transfer can be found by equating the integral of the acceleration profile over the duration of the transfer to the total delta-V:

푡푓 푡푓 퐹푇 ∆푉 = ∫ 푎푇(푡). 푑푡 = ∫ 푑푡 . (3.1) 0 0 푚(푡).

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This time of flight will be highly dependent on the specific design of the transfer. Order of magnitude estimates for the time of flight can be made by assuming that the total delta-V will be applied by a constant acceleration 푎̅푇 over the duration of the transfer [184]:

∆푉 푇푂퐹 = . (3.2) 푎̅푇

The acceleration level will vary over the duration of the transfer as propellant mass is consumed. However, lower limits of the acceleration can be computed from the thrust magnitude and the total mass at the beginning of each transfer (the launch mass 푚0 for the Earth-to-Asteroid transfer, and the combined wet mass and return mass 푀푅 for the Asteroid-to-Earth transfer). Using these assumptions, the total times of flight for the two transfers can be expressed as:

∆푉퐸퐴 ∆푉퐸퐴 푇푂퐹퐸퐴 = = 푚0 , (3.3) 푎̅푇 퐹푇 and:

∆푉퐴퐸 ∆푉퐴퐸 푇푂퐹퐴퐸 = = (푚푑푟푦 + 푀푅 + 푚푝,퐴퐸) , (3.4) 푎̅푇 퐹푇 where 푚푝,퐴퐸 is the propellant mass required for the Asteroid-to-Earth transfer.

Mission Duration For the ISPP alternatives, the total mission duration will be determined by the times of flight of the two transfers, and the stay-time at the asteroid. As the ISPP method is limited to chemical propellants available in the asteroid material, the transfers will be impulsive, having relatively short durations. However, long stay-times will be required to extract and process the asteroid material. The total stay-time will be determined by the total mass of material to extract 푀퐸푥 and the mining rate 푀푅 of the mining equipment. Hence, the total mission duration can be expressed as:

푀 푇 = 푇푂퐹 + 푇 + 푇푂퐹 = 푇푂퐹 + 퐸푥 + 푇푂퐹 . (3.5) 퐸퐴 푠푡푎푦 퐴퐸 퐸퐴 푀푅 퐴퐸

For the EBPS alternative, the total mission duration will similarly be found by the times of flight of the two transfers, and the time required to capture the asteroid:

푇 = 푇푂퐹퐸퐴 + 푇푐푎푝 + 푇푂퐹퐴퐸 . (3.6)

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The capture time 푇푐푎푝 is expected to be much shorter than the required time to process asteroid material for the ISPP alternatives. As the EBPS approach can use either chemical or electric propulsion, two methods are used to compute the times of flight of the transfers. For chemical propulsion, the times of flight will be equal to the impulsive transfers (half the period of the transfer orbit), while for electric propulsion, the times of flight will be determined by Eqs. 3.3 and 3.4.

3.4.3 Mass Parameters Table 3.6 describes a set of characteristic masses involved in the mission. These include the extraction mass, the return mass, the propellant mass, and the masses 푚0 to 푚3 defining the combined masses of the mining spacecraft and asteroid material at different stages of the mission. These masses are illustrated in figure 3.3, showing example return trajectories to asteroid .

Table 3.6 Mass Parameters.

Parameter Description

푚푑푟푦 Total dry mass of the mining spacecraft (including mining equipment). Extraction Mass. Mass of asteroid material extracted/collected during 푀 퐸푥 mining operations.

푀푅 Return mass. Mass of asteroid material returned to Earth orbit.

푚푝 Propellant mass required to carry out the heliocentric transfers. Launch mass of the mining spacecraft. Includes the dry mass, mass of mining 푚 0 equipment, and propellant mass.

푚1 Mass of mining spacecraft at asteroid arrival. Mass of mining spacecraft at the asteroid departure. Includes mass of 푚 2 asteroid resources to be returned.

푚3 Mass of mining spacecraft and return mass at Earth arrival.

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Figure 3.3 Combined masses of the mining spacecraft and asteroid material at different stages of the mission.

Total Dry Mass As will be seen in the following sections, the propellant mass and launch masses can be expressed as linear functions of the spacecraft dry mass. It is convenient to introduce a term describing the total dry mass 푚푑푟푦 of the mining spacecraft including the mining equipment:

푚푑푟푦 = 푚푑푟푦,푀푆 + 푚푀퐸 . (3.7)

This allows the same equations to be applied, with the dry mass specified as either 푚푑푟푦 or 푚푑푟푦,푀푆 , in cases when the mining equipment is being carried by the mining spacecraft or left at the asteroid.

Extraction Mass

The extraction mass 푀퐸푥 is defined here as the total mass of asteroid material that is extracted during the mining mission. For the in situ processing approach, the extraction mass will be determined by the product of the stay-time and the mining rate 푀푅 achievable by the mining equipment, and further limited by the maximum mass capacity

푀푚푎푥 of the mining spacecraft:

푀퐸푥 = 푚푖푛{푇푠푡푎푦푀푅, 푀푚푎푥} . (3.8)

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For the whole asteroid return case, the extraction mass will be equal to the total mass of the asteroid (or boulder). This can be considered as a special case of the in situ processing option in which the mining rate is effectively instantaneous.

Return Mass and Sellable Mass

The return mass 푀푅 will be the mass of material that is returned to Earth orbit. The sellable mass 푆푀 is defined as the mass that can be sold to customers (after the extraction of propellant required to return to the asteroid). For the Earth-based propellant supply option, the return mass and sellable mass will be equal to the extraction mass. For the ISPP options, the return mass will depend on the fraction of extracted mass that is consumed as propellant in the return Asteroid-to-Earth transfer. These masses are computed in section 3.5 for the different mission alternatives.

Mass Ratios The various mass parameters may be normalized by a characteristic mass, producing a non-dimensional mass ratio. Lewis [164] introduced the concept of a Mass Payback

Ratio (MPBR) defining the return mass as a fraction of the launch mass 푚0:

푀 푀푃퐵푅 = 푅 . (3.9) 푚0

This ratio is comparable to the for launch vehicles. The MPBR may be useful in expressing the masses that are achievable from different asteroid targets and mission designs, including over multiple return trips. For the two examples described previously, the Hayabusa mission would have an 푀푃퐵푅 of 1.99 × 10−6, and the ARM mission would have a 푀푃퐵푅 of 29.4. Lewis discussed mass payback ratios of up to 1000, which are achievable with multiple trips to near-Earth asteroids.

The use of launch mass as the reference mass is problematic, as the launch mass will change with the delta-Vs. The initial launch mass consists of the dry mass and the propellant mass. This propellant mass is dependent on the delta-V of the transfer, and the mass of material being returned.

As will be shown in the following sections, many of the mass parameters and figures of merit may be expressed as linear functions of the spacecraft dry mass. This makes the spacecraft dry mass a more appropriate reference mass, as it will remain fixed for a particular system design while the propellant mass will vary with the asteroid target.

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3.4.4 Economic Parameters The evaluation of a successful mission design should not only consider mission performance, but also economic factors such as the capital costs required to conduct the mission, the revenues received from the sale of the returned material, and the duration and risk of the mission.

Capital and Revenue

The capital cost 퐶0 of the mining mission will include the cost of constructing and launching the spacecraft on its first mission to the asteroid. This cost can be decomposed into a series of cost components, each dependent on one or more of the system or mission parameters. These costs include: • the production cost (dependent on the dry mass of the spacecraft); • the propellant cost (dependent on the required propellant mass); • the launch cost (dependent on the launch mass); and • the operations cost (dependent on the duration of the mission).

These costs can be expressed using cost estimation relations, where the cost is expressed as the product of system parameters and specific cost functions (detailed in table 3.7):

퐶0 = 푚푑푟푦푐푝푟표푑 + 푚0푐푙 + 푚푝푐푝 + 푐표푝푠푇 (3.10) = (푐푝푟표푑 + 푐푙)푚푑푟푦 + (푐푙 + 푐푝)푚푝 + 푐표푝푠푇 .

The revenue 푅 generated from the mission can similarly be expressed as the product of the return mass 푀푅 and the specific sale price of the asteroid material:

푅 = 푀푅푐푠푎푙푒 . (3.11)

Table 3.7 Specific cost parameters.

Parameter Description

푐푝푟표푑 Specific production cost ($/kg) of the mining spacecraft.

푐푙 Specific launch cost ($/kg).

푐푝 Specific propellant cost ($/kg).

푐표푝푠 Specific operations cost ($/yr).

푐푠푎푙푒 Specific sale price ($/kg) of asteroid material in orbit.

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Profit and Net Present Value The profit 푃 of the mining mission can be calculated as the difference of the capital cost and the revenue:

푃 = 푅 − 퐶0 . (3.12)

The net present value (NPV) is a metric similar to profit that accounts for the time value of money. The NPV is calculated by discounting the receipts received from the future sale of returned asteroid material to the present value, taking into account the initial capital investment and discount rate:

푅 푁푃푉 = −퐶 + , (3.13) 0 (1 + 푟)푇 where 푟 is the discount rate, and 푇 is the duration of the mission in years.

The NPV can be found by substitute the expressions for 퐶0 and 푅:

푀 푐 푁푃푉 = −(푐 + 푐 )푚 − (푐 + 푐 )푚 − 푐 푇 + 푅 푠푎푙푒 . (3.14) 푝푟표푑 푙 푑푟푦 푙 푝 푝 표푝푠 (1 + 푟)푇

In order to be profitable, the NPV of the proposed project should be positive.

There are several other metrics that can be computed from the NPV. The internal rate of return (IRR) is the discount rate that is required to give an NPV of zero. Similarly, the payback period is the time required to recover the initial capital investment. NPV has been regarded as one of the best figures of merit for the economic study of asteroid mining missions [3].

Expected Profit and Expected Net Present Value The Profit and NPV described above are computed assuming that the mission is successful in returning a shipment of asteroid material. However, there is also a chance that the mission would be unsuccessful. For instance, if the asteroid is discovered to not contain the desired material, or if it is too large to be retrieved in its entirety. In such cases, the resulting outcome will not include the revenues, and may result in the complete loss of the mining spacecraft.

For events that include multiple outcomes (such as success and failure), an expectation value can be computed as the sum of the outcomes multiplied by their associated probabilities. This method can be applied to both the profit and net present value, giving the expected profit 〈푃〉:

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〈푃〉 = 푝푠(푅 − 퐶0) + (1 − 푝푠)(−퐶0) (3.15) = −퐶0 + 푝푠푅 , and the expected net present value 〈푁푃푉〉:

푅 〈푁푃푉〉 = 푝 (−퐶 + ) + (1 − 푝 )(−퐶 ) 푠 0 (1 + 푖)푇 푠 0 (3.16) 푅 = −퐶 + 푝 , 0 푠 (1 + 푖)푇 where 푝푠 is the probability of success, and (1 − 푝푠) is the probability of failure. In both cases, the outcome of a failed mission will be the negative of the capital cost.

The expected value method is frequently used in the decision-making processes of the terrestrial mining industry, where they are referred to as the Expected Monetary Value (EMV) [182].

Decision Trees The process of evaluating the expected value can be visually displayed using a decision tree, where decision alternatives are represented as branches originating from square nodes, and alternative outcomes are represented as branches originating from circular nodes. Figure 3.4 shows an example of the expected value calculated for a single decision alternative.

Figure 3.4 Decision tree representation of expected value.

3.5 Figure of Merit Expressions for Mission Alternatives The Figures of Merit derived in the previous section are highly dependent on the mass parameters such as the propellant mass, return mass, and extraction mass. These masses will vary for the different mining approaches, and hence must be calculated separately.

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The launch, propellant, and return masses for the different mission alternatives can be computed using the Tsiolkovsky rocket equation, that relates the initial 푚푖 and final 푚푓 masses of a spacecraft over a transfer with delta-V ∆푉:

−∆푉 푚푓 = 푒 휈푒 , (3.17) 푚푖 where 휐푒 is the exhaust velocity of the propulsion system.

From this equation, the propellant mass consumed over the transfer can be expressed as a function of either the initial or final mass:

−∆푉 ∆푉 휈 휈 푚푝 = 푚푖 − 푚푓 = 푚푖 (1 − 푒 푒 ) = 푚푓 (푒 푒 − 1) . (3.18)

In traditional space missions, the only changes in mass occur due to the consumption of propellant, hence the total propellant mass can be calculated from the total mission delta-V. In the asteroid mining case, there is a change in mass mid-way through the mission introduced by the extraction of asteroid material. This requires solving the mass expressions over multiple segments of the mission.

In table 3.6, the masses 푚0 to 푚3 were described, defining the combined mass of the mining spacecraft and asteroid material at different stages of the mission. Applying the rocket equation over the various transfer segments of the mission will produce a set of simultaneous equations for the masses 푚0 to 푚3, which can be solved to find 푚0 and

푚푝.

The mass 푚1 at the arrival at the asteroid will be dependent on the initial launch mass and the delta-V of the Earth-to-Asteroid transfer:

−∆푉퐸퐴 휈 (3.19) 푚1 = 푚0푒 푒 .

The mass 푚2 at the departure of the asteroid will be the sum of this mass and the mass of asteroid material 푀퐸푥 extracted during the stay-time:

−∆푉퐸퐴 휈 (3.20) 푚2 = 푚1 + 푀퐸푥 = 푚0푒 푒 + 푀퐸푥 .

The mass 푚3 at arrival at Earth can similarly be found from the initial mass 푚2 and the delta-V of the Asteroid-to-Earth transfer:

−∆푉퐴퐸 −∆푉퐸퐴 −∆푉퐴퐸 휈 휈 휈 푚3 = 푚2푒 푒 = (푚0푒 푒 + 푚퐸푥) 푒 푒 . (3.21)

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Rearranging this equation, produces an expression for the initial launch mass 푚0:

∆푉푡표푡 ∆푉퐸퐴 휈 휈 (3.22) 푚0 = 푚3푒 푒 − 푀퐸푥푒 푒 .

The propellant mass can also be found as:

∆푉푡표푡 ∆푉퐸퐴 휈 휈 (3.23) 푚푝 = 푚0 − 푚푑푟푦 = 푚3푒 푒 − 푚퐸푥푒 푒 − 푚푑푟푦 .

Eqs. 3.19 to 3.23 are valid for all of the mission alternatives; however, they will differ when expressed as functions of the return mass and spacecraft dry mass, so must be computed separately.

3.5.1 In Situ Propellant Production (Option 1A) For the ISPP case, the propellant for the return trip is produced at the asteroid. As such, only enough propellant is required to be supplied at launch to reach the asteroid. The minimum launch mass required to reach the asteroid will be found if there is no remaining propellant at the asteroid arrival (i.e. 푚1 = 푚푑푟푦). Substituting this into Eq. 3.19 results in an expression for the launch mass as a function of the dry mass:

∆푉퐸퐴 휈 (3.24) 푚0 = 푚푑푟푦푒 푒 .

Substituting this as the initial mass in Eq. 3.18, the total propellant mass can be expressed as:

∆푉퐸퐴 휈 푚푝 = 푚푝,퐸퐴 = 푚푑푟푦 (푒 푒 − 1) . (3.25)

The total capital cost of the mission can be found by substituting these expressions into Eq. 3.10:

∆푉퐸퐴 휈 퐶0 = 푚푑푟푦 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] + 푐표푝푠푇 . (3.26)

The propellant mass required for the return Asteroid-to-Earth transfer will be taken from the mass extracted at the asteroid. As a result, the mass returned to Earth orbit will be less than that extracted at the asteroid.

The mass 푚2 at the asteroid departure will consist of the dry mass and the extracted asteroid mass (i.e. 푚2 = 푚푑푟푦 + 푀퐸푥). The mass 푚3 at Earth arrival will consist of the dry mass and the return mass (i.e. 푚3 = 푚푑푟푦 + 푀푅). Substituting these into Eq 3.21,

82 Scott Dorrington – June 2019 Chapter 3: Parametric Economic Analysis produces an expression for the return mass in terms of the dry mass and extracted asteroid mass:

−∆푉퐴퐸 −∆푉퐴퐸 휈 휈 푀푅 = 푚푑푟푦 (푒 푒 − 1) + 푀퐸푥푒 푒 . (3.27)

Alternatively, this equation may be rearranged to express the required extraction mass to achieve a specified return mass:

∆푉퐴퐸 ∆푉퐴퐸 휈 휈 푀퐸푥 = 푚푑푟푦 (푒 푒 − 1) + 푀푅푒 푒 . (3.28)

The total mission duration can be found from the times of flight of the two transfers (both assumed to be impulsive transfers) and the time required to extract this mass:

1 ∆푉퐴퐸 ∆푉퐴퐸 푇 = 푇푂퐹 + 푇푂퐹 + [푚 (푒 휈푒 − 1) + 푀 푒 휈푒 ] . (3.29) 퐸퐴 퐴퐸 푀푅 푑푟푦 푅

The propellant mass consumed over the Asteroid-to-Earth transfer can be found from Eq. 3.18 as:

−∆푉퐴퐸 휈 푚푝,퐴퐸 = (푚푑푟푦 + 푚퐸푥) (1 − 푒 푒 ) . (3.30)

Dividing Eqs. 3.30 and 3.27 by 푀퐸푥, we can find the fractions of the extracted mass used as propellant and returned to Earth orbit:

푚 푚 −∆푉퐴퐸 푝,퐴퐸 푑푟푦 휈 푓푝 = = (1 + ) (1 − 푒 푒 ) , (3.31) 푀퐸푥 푀퐸푥 and:

푀 푚 −∆푉퐴퐸 −∆푉퐴퐸 푅 푑푟푦 휈 휈 푓푅 = = (푒 푒 − 1) + 푒 푒 . (3.32) 푀퐸푥 푀퐸푥

As expected, these fractions both add to unity.

The revenue generated from the sale of the return mass can be found by:

−∆푉퐴퐸 −∆푉퐴퐸 휈 휈 푅 = 푐푠푎푙푒푀푅 = 푐푠푎푙푒 [푚푑푟푦 (푒 푒 − 1) + 푀퐸푥푒 푒 ] . (3.33)

The expressions for capital and revenue may be used to express the total profit as:

∆푉퐸퐴 휈 푃 = 푐푠푎푙푒푀푅 − 푚푑푟푦 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] − 푐표푝푠푇 , (3.34)

Scott Dorrington – June 2019 83 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions and total NPV as:

∆푉 푐푠푎푙푒푀푅 퐸퐴 푁푃푉 = − 푚 [(푐 + 푐 ) + (푐 + 푐 ) (푒 휈푒 − 1)] − 푐 푇 . (3.35) (1 + 푟)푇 푑푟푦 푝푟표푑 푙 푙 푝 표푝푠

Multiple Trips As previously mentioned, using the ISPP method, the mining spacecraft can retain a portion of the returned material to refuel for the next mining trip to the asteroid. In doing so, the total revenue of a mining trip (for all but the final mission) would be reduced by the propellant mass of the next Earth-to-Asteroid transfer (from Eq. 3.25):

∆푉퐸퐴 휈 푅 = 푐푠푎푙푒(푀푅 − 푚푝) = 푐푠푎푙푒 [푀푅 − 푚푑푟푦,푀푆 (푒 푒 − 1)] . (3.36)

In this expression, the dry mass is replaced with the dry mass of the mining spacecraft

푚푑푟푦,푀푆. As previously mentioned, if multiple trips are conducted to the same asteroid, the mining equipment may be left at the asteroid to continue extraction for the next mining trip. This would reduce the amount of dry mass needed to be carried back and forth between the asteroid and Earth on subsequent mining missions, resulting in a slightly higher return mass.

Using these adjustments, the total profit for a mining campaign consisting of 푁 mining trips can be found by summing the total capitals and revenues over all mining trips. In the final trip, the full amount of returned mass may be sold to produce revenue:

푁−1 ∆푉퐸퐴 휈 푃 = ∑ 푐푠푎푙푒 [푀푅 − 푚푑푟푦,푀푆 (푒 푒 − 1)] + 푐푠푎푙푒푀푅 − 퐶0 푡=1

∆푉퐸퐴 휈 (3.37) = 푐푠푎푙푒 [푁푀푅 − (푁 − 1)푚푑푟푦,푀푆 (푒 푒 − 1)]

∆푉퐸퐴 휈 − 푚푑푟푦 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] − 푐표푝푠푇푁 ,

where 푇푁 is the total cumulative duration of all 푁 mining trips.

Similarly, the total NPV of 푁 mining trips can be expressed as:

84 Scott Dorrington – June 2019 Chapter 3: Parametric Economic Analysis

푁−1 ∆푉퐸퐴 휈 −푇푡 푁푃푉 = ∑ (푐푠푎푙푒 [푀푅 − 푚푑푟푦,푀푆 (푒 푒 − 1)] (1 + 푟) ) 푡=1

−푇푁 + 푐푠푎푙푒푀푅(1 + 푟) (3.38)

∆푉퐸퐴 휈 − 푚푑푟푦 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] − 푐표푝푠푇푁 ,

where 푇푡 is the cumulative duration of each mining trip 푡, measured in years from the start of the mining campaign:

푇푡 = (푡 − 1)푇푤푎푖푡 1 ∆푉퐴퐸 ∆푉퐴퐸 (3.39) + 푡 (푇푂퐹 + 푇푂퐹 + [푚 (푒 휈푒 − 1) + 푀 푒 휈푒 ]) . 퐸퐴 퐴퐸 푀푅 푑푟푦 푅

In the case that 푁 = 1, these expressions reduce to the same as those for a single trip mission.

Expectation Value As mentioned previously, the expected value of the profit and NPV can be computed from the sum of the outcomes multiplied by their associated probabilities. For the case of a single trip mission, these may be found by substituting the expressions for capital and revenue into Eqs. 3.15 and 3.16.

For the case of multiple mining trips, there will be additional decisions to be made following the outcome of each mining trip. If a mining trip is successful, with a probability of success 푝푠, it would confirm that there is a presence of resources on that asteroid. It can then be assumed that each subsequent mining trip would also be successful (with a 100% probability of success)7.

If a mining trip is unsuccessful, the mining spacecraft will be lost at the asteroid. A decision should then be made whether to launch a new mining spacecraft to continue the mining campaign. This new mission would require an additional capital investment, would target a different asteroid candidate, and would have the same outcomes and

7 There may be other reasons that subsequent mining missions would fail, for instance due to the reliability of the spacecraft and mining equipment. This could be accounted for in future with additional outcome branches in the decision tree.

Scott Dorrington – June 2019 85 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions probabilities as the initial mission. This process could continue until the end of the mining campaign (푁 mining trips).

In this analysis, it is assumed that the decision is made to continue sending mining missions in the event of failure until a total of 푁 mining trips are completed (whether successful or not). Figure 3.5 shows a decision tree enumerating these decisions and outcomes.

Figure 3.5 Decision tree showing outcomes for multiple mining trips.

The first outcome of this tree is the event in which the mining spacecraft was successful on its first mining trip (from which it is assumed all 푁 mining trips are also successful). The value of this outcome will be the total profit from 푁 successful mining trips (calculated from Eq. 3.37), and the associated probability will be equal to the probability of success 푝푠. The total mission duration will be 푁 times the duration of a successful mission 푇푠, calculated from the wait-time following the previous trip, the total time of flight, and the stay-time:

1 ∆푉퐴퐸 ∆푉퐴퐸 푇 = 푇 + 푇푂퐹 + 푇푂퐹 + [푚 (푒 휈푒 − 1) + 푀 푒 휈푒 ] . (3.40) 푠 푤푎푖푡 퐸퐴 퐴퐸 푀푅 푑푟푦 푅

As each new mission could commence immediately after the end of a failed mission, there would be no wait-time for the first successful mission, hence a single 푇푤푎푖푡 should be subtracted from the total mission duration over 푁 mining trips.

The second outcome of the tree is the event in which the first mining trip failed, while the second mining trip (and all following trips) was successful. The value of this outcome will be calculated from the capital for two mining spacecraft, and the revenues

86 Scott Dorrington – June 2019 Chapter 3: Parametric Economic Analysis

from (푁 − 1) successful mining trips. The associated probability will be (1 − 푝푠)푝푠 (the product of the probability of failure of the first trip and success of the second). The total duration will be determined by that of (푁 − 1) successful trips, and a single

unsuccessful trip 푇푓, calculated simply as the time of flight of the Earth-to-Asteroid transfer:

푇푓 = 푇푂퐹퐸퐴 . (3.41)

Each subsequent outcome, 푚, will be the event in which 푚 mining trips failed, followed by (푁 − 푚) successful trips. The value of each outcome will include 푚 capital investments, and the revenue from (푁 − 푚) successful mining trips, and the associated 푚 probability will be (1 − 푝푠) 푝푠. The total probabilities, durations, capital, and revenues of each outcome are displayed in table 3.8.

Table 3.8 Probabilities and values of each outcome of the decision tree for the ISPP approach.

# f* # s** Prob. Duration Capital Revenue

0 푁 푝푠 푁푇푠 − 푇푤푎푖푡 −퐶0 − 푐표푝푠푇 (푁 − 1)푅 + 푅푁

1 푁 − 1 (1 − 푝푠)푝푠 푇푓 + (푁 − 1)푇푠 − 푇푤푎푖푡 −2퐶0 − 푐표푝푠푇 (푁 − 2)푅 + 푅푁

2 2푇푓 + (푁 − 2)푇푠 2 푁 − 2 (1 − 푝푠) 푝푠 −3퐶0 − 푐표푝푠푇 (푁 − 3)푅 + 푅푁 − 푇푤푎푖푡 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

푚 푚푇푓 + (푁 − 푚)푇푠 −(푚 + 1)퐶0 푚 푁 − 푚 (1 − 푝푠) 푝푠 (푁 − 푚 − 1)푅 + 푅푁 − 푇푤푎푖푡 − 푐표푝푠푇 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 푁 푁 0 (1 − 푝푠) 푁푇푓 −푁퐶0 − 푐표푝푠푇 0 * Number of failed missions. ** Number of successful missions. The total expected profit for an 푁-trip mining campaign can then be found by the sum over all outcomes (푚 from zero to 푁 failures):

푁−1 푚 〈푃〉 = ∑ (1 − 푝푠) 푝푠[−(푚 + 1)퐶0 − 푐표푝푠(푚푇푓 + (푁 − 푚)푇푠 − 푇푤푎푖푡) 푚=0 (3.42) 푁 + (푁 − 푚 − 1)푅 + 푅푁] + (1 − 푝푠) (−푁퐶0 − 푐표푝푠푁푇푓) .

In this equation, 푅푁 is the revenue from the final trip including all return mass (from Eq. 3.33) and 푅 is the revenue from other trips, where the propellant mass is extracted from

Scott Dorrington – June 2019 87 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions the return mass (from Eq. 3.36). The capital cost 퐶0 for each new mission is calculated from Eq. 3.26, without the operations costs, as they have been included in the 〈푃〉 expression.

For the case of the expected net present value 〈푁푃푉〉, each of the revenues from the

(푁 − 푚) successful trips would be discounted at different times (푚푇푓 + 푇푡), and hence require an additional summation over the successful mining trips:

푁−1 푚 〈푁푃푉〉 = ∑ (1 − 푝푠) 푝푠 [−(푚 + 1)퐶0 − 푐표푝푠(푚푇푓 + (푁 − 푚)푇푠 − 푇푤푎푖푡) 푚=0 푁−푚−1 푅 푅푁 (3.43) + ∑ ( ) + ] (1 + 푟)푇푡,푚 (1 + 푟)푇푁,푚 푡=1 푁 + (1 − 푝푠) (−푁퐶0 − 푐표푝푠푁푇푓) ,

where 푇푡,푚 = 푚푇푓 + 푇푡 is the cumulative duration of each successful trip 푡 following 푚 failed trips (푇푡 is found from Eq. 3.39); and 푇푁,푚 = 푚푇푓 + (푁 − 푚)푇푠 − 푇푤푎푖푡.

Similarly, the expected total duration 〈푇〉 can be found from the product of the duration and probability, summed over each of these outcomes:

푁−1 푚 푁 〈푇〉 = ∑ (1 − 푝푠) 푝푠(푚푇푓 + (푁 − 푚)푇푠 − 푇푤푎푖푡) + (1 − 푝푠) (푁푇푓) . (3.44) 푚=0

3.5.2 In Situ Propellant Production with Reserve Propellant (Option 1B) For the second ISPP case (Option 1B), the mining spacecraft is launched with enough propellant to return to Earth in the event that there is no propellant found at the asteroid.

For this case, the initial launch mass 푚0 and propellant mass 푚푝 can be found using Eqs. 3.24 and 3.25, with the delta-V replaced with the total delta-V of the Earth-to- Asteroid and Asteroid-to-Earth transfers:

∆푉푡표푡 휈 (3.45) 푚0 = 푚푑푟푦푒 푒 , and:

∆푉푡표푡 휈 푚푝 = 푚푝,퐸퐴 = 푚푑푟푦 (푒 푒 − 1) . (3.46)

From this, the capital will be expressed as:

88 Scott Dorrington – June 2019 Chapter 3: Parametric Economic Analysis

∆푉푡표푡 휈 퐶0 = 푚푑푟푦 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] + 푐표푝푠푇 . (3.47)

In the event of a successful mission, the return mass, revenue, duration, and the propellant mass for the Asteroid-to-Earth transfer will be the same as those described in Eqs. 3.27 to 3.30. From this, the total profit can be expressed as:

∆푉푡표푡 휈 푃 = 푐푠푎푙푒푀푅 − 푚푑푟푦 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] − 푐표푝푠푇 , (3.48) and the total NPV as:

∆푉 푐푠푎푙푒푀푅 푡표푡 푁푃푉 = − 푚 [(푐 + 푐 ) + (푐 + 푐 ) (푒 휈푒 − 1)] − 푐 푇 . (3.49) (1 + 푟)푇 푑푟푦 푝푟표푑 푙 푙 푝 표푝푠

Multiple Trips For the deterministic case with multiple mining trips (not accounting for failed trips), it is assumed that the additional propellant is only needed for the first trip to the asteroid. In all subsequent mining trips, the propellant mass required for the next Earth-to- Asteroid transfer is extracted from the returned mass, reducing the revenues for all but the final mining trip. The total profit and NPV for this alternative can be computed by

Eqs. 3.37 and 3.38, with the ∆푉퐸퐴 in the second term (the capital component) replaced with ∆푉푡표푡.

Expectation Value The decisions and outcomes for this mission alternative will also be the same as those represented in the decision tree shown in figure 3.5. For the first mining trip, the mining spacecraft will be launched from Earth with enough propellant to reach the asteroid, and reserve propellant to return to Earth orbit. The capital of this first trip will be that expressed in Eq. 3.47. The event of a successful mining trip will confirm the presence of the propellant resource on the target asteroid, and all subsequent mining trips will be assumed to also be successful. As such, subsequent missions only need to be refuelled with the propellant needed to return to the asteroid.

In the event of a failed mission, the mining spacecraft can use its reserve propellant to return to Earth orbit. This would increase the duration of a failed mission to include the wait-time at Earth prior to commencement of the mission, and the total time of flight of the two transfers. It may also require an additional wait-time at the asteroid 푇푤,퐴푠푡 prior to the return trip:

Scott Dorrington – June 2019 89 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

푇푓 = 푇푤푎푖푡 + 푇푂퐹퐸퐴 + 푇푂퐹퐴퐸 + 푇푤,퐴푠푡 . (3.50)

Once returned to Earth orbit, additional new missions could be deployed using the same mining spacecraft, without the need to construct and launch a new one from the Earth. The new mission would need to be refuelled with enough propellant for a return trip to a new target asteroid.

For successful missions, this mass is taken out of the return mass from the previous mission, reducing the revenue from the previous mission. However, following an unsuccessful mission, this would need to be borrowed from a stockpile of propellant held at the propellant depot. This can be accounted for with an additional capital

investment or refuelling cost 퐶푅, calculated from the propellant mass and the specific sale price:

∆푉푡표푡 휈 퐶푅 = 푐푠푎푙푒푚푑푟푦 (푒 푒 − 1) . (3.51)

This refuelling cost is equal to the reduction in revenue from the successful missions, hence the revenue 푅 for successful missions could alternatively be represented as the

full revenue 푅푁 minus this refuelling cost.

Table 3.9 summarizes the probabilities, durations, capital, and revenue for each outcome in the decision tree.

Table 3.9 Probabilities and values of each outcome of the decision tree for the ISPP approach with reserve propellant.

# f* # s** Prob. Duration Capital Revenue

0 푁 푝푠 푁푇푠 − 푇푤푎푖푡 −퐶0 − 푐표푝푠푇 (푁 − 1)푅 + 푅푁

1 푁 − 1 (1 − 푝푠)푝푠 푇푓 + (푁 − 1)푇푠 − 푇푤푎푖푡 −퐶0 − 퐶푅 − 푐표푝푠푇 (푁 − 2)푅 + 푅푁 2 2 푁 − 2 (1 − 푝푠) 푝푠 2푇푓 + (푁 − 2)푇푠 − 푇푤푎푖푡 −퐶0 − 2퐶푅 − 푐표푝푠푇 (푁 − 3)푅 + 푅푁 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 푚 푚 푁 − 푚 (1 − 푝푠) 푝푠 푚푇푓 + (푁 − 푚)푇푠 − 푇푤푎푖푡 −퐶0 − 푚퐶푅 − 푐표푝푠푇 (푁 − 푚 − 1)푅 + 푅푁 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

푁 −퐶0 − (푁 − 1)퐶푅 푁 0 (1 − 푝푠) 푁푇푓 − 푇푤푎푖푡 0 − 푐표푝푠푇 * Number of failed missions. ** Number of successful missions.

90 Scott Dorrington – June 2019 Chapter 3: Parametric Economic Analysis

The expected profit for this approach can then be found using a similar method as that described above:

푁−1 푚 〈푃〉 = ∑ (1 − 푝푠) 푝푠[−퐶0 − 푚퐶푅 − 푐표푝푠(푚푇푓 + (푁 − 푚)푇푠 − 푇푤푎푖푡) 푚=0 (3.52) + (푁 − 푚 − 1)푅 + 푅푁] 푁 + (1 − 푝푠) (−퐶0 − (푁 − 1)퐶푅 − 푐표푝푠푁푇푓) , and the total expected NPV as:

푁−1 푚 〈푁푃푉〉 = ∑ (1 − 푝푠) 푝푠 [−퐶0 − 푚퐶푅 − 푐표푝푠(푚푇푓 + (푁 − 푚)푇푠 − 푇푤푎푖푡) 푚=0 푁−푚 푅 푅 + ∑ + 푁 ] (3.53) (1 + 푟)푇푡,푚 (1 + 푟)푇푁,푚 푡=1

푁 + (1 − 푝푠) (−퐶0 − (푁 − 1)퐶푅 − 푐표푝푠(푁푇푓 − 푇푤푎푖푡)) ,

where 푇푡,푚 = 푚푇푓 + 푇푡 is the cumulative duration of each successful trip 푡 following 푚 failed trips (푇푡 is found from Eq. 3.39); and 푇푁,푚 = 푚푇푓 + (푁 − 푚)푇푠 − 푇푤푎푖푡.

The expected total duration is further expressed as:

푁−1 푚 〈푇〉 = ∑ (1 − 푝푠) 푝푠(푚푇푓 + (푁 − 푚)푇푠 − 푇푤푎푖푡) 푚=0 (3.54) 푁 + (1 − 푝푠) (푁푇푓 − 푇푤푎푖푡) .

Using this approach, the mining campaign can continue in the event of a mission failure, without the need to construct and launch a new mining spacecraft from Earth. While the initial capital investment for this approach is higher than the standard ISPP approach, the additional capital to launch new missions is lower. It is expected that this method will be beneficial in the case of high uncertainties of the presence of asteroid resources.

3.5.3 Earth-Based Propellant Supply (Options 2A & 2B) For the Earth-based propellant supply case, all propellant for the mission is required to be supplied at the beginning of the mission. This applies to both the in situ processing and the whole asteroid capture approaches.

Scott Dorrington – June 2019 91 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

The minimum launch mass required to complete the mission will be achieved if there is no remaining propellant at the arrival at Earth (i.e. 푚3 = 푚푑푟푦 + 푀푅). Substituting this into Eq. 3.22 results in an expression for the launch mass in terms of the dry mass and the return mass:

∆푉푡표푡 ∆푉푡표푡 ∆푉퐸퐴 휈 휈 휈 푚0 = 푚푑푟푦푒 푒 + 푀푅 (푒 푒 − 푒 푒 ) . (3.55)

The propellant mass can also be expressed as:

∆푉푡표푡 ∆푉푡표푡 ∆푉퐸퐴 휈 휈 휈 푚푝 = 푚0 − 푚푑푟푦 = 푚푑푟푦 (푒 푒 − 1) + 푀푅 (푒 푒 − 푒 푒 ) . (3.56)

From this, the total capital can be expressed as:

∆푉푡표푡 휈 퐶0 = 푚푑푟푦 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] (3.57) ∆푉푡표푡 ∆푉퐸퐴 휈 휈 + 푀푅(푐푙 + 푐푝) (푒 푒 − 푒 푒 ) + 푐표푝푠푇 .

Eq. 3.56 gives the total mass of propellant that is required to be supplied at the beginning of the mission. The propellant masses consumed over the individual Earth-to-

Asteroid and Asteroid-to-Earth transfers (푚푝,퐸퐴 and 푚푝,퐴퐸) can be found by substituting Eq. 3.23 into Eqs. 3.19 and 3.21:

∆푉푡표푡 ∆푉푡표푡 ∆푉퐸퐴 −∆푉퐸퐴 휈 휈 휈 휈 푚푝,퐸퐴 = [푚푑푟푦푒 푒 + 푀푅 (푒 푒 − 푒 푒 )] (1 − 푒 푒 ) , (3.58) and:

∆푉퐴퐸 휈 푚푝,퐴퐸 = (푚푑푟푦 + 푀푅) (푒 푒 − 1) . (3.59)

For the EBPS case, all of the material extracted from the asteroid will be returned to Earth orbit:

푀퐸푥 = 푀푅 . (3.60)

The revenue will then be expressed as:

푅 = 푐푠푎푙푒푀푅 . (3.61)

Using Eqs. 3.57 and 3.61, the total profit can be expressed as:

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∆푉푡표푡 ∆푉퐸퐴 휈 휈 푃 = 푀푅 [푐푠푎푙푒 − (푐푙 + 푐푝) (푒 푒 − 푒 푒 )] (3.62) ∆푉푡표푡 휈 − 푚푑푟푦 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] − 푐표푝푠푇 , and the total NPV as:

∆푉 ∆푉 푐푠푎푙푒 푡표푡 퐸퐴 푁푃푉 = 푀 [ − (푐 + 푐 ) (푒 휈푒 − 푒 휈푒 )] 푅 (1 + 푟)푇 푙 푝 (3.63) ∆푉푡표푡 휈 − 푚푑푟푦 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] − 푐표푝푠푇 .

Multiple Trips For the Earth-based propellant supply option, following a successful mining trip, the mining spacecraft would have used all its propellant in returning a mass of asteroid material. Additional mining trips could be sent by refuelling the spacecraft with a mass of propellant 푚푝 launched from Earth. This propellant would need to be delivered to by an additional spacecraft capable of manoeuvring from the deployed launch parking orbit to rendezvous with the mining spacecraft (using the approach of the reference mission discussed in the next section). As a result, the cost of refuelling 퐶푅 would be comparable to the total capital cost of launching a new mining spacecraft.

As the EBPS approach is expected to take a long duration in returning a large shipment of asteroid material, it is expected that following a single mission, the mining spacecraft would be at the end of its design life, making it beneficial to replace it with a new spacecraft for each subsequent mining trip.

Using these assumptions, the total profit over 푁 mining can be computed as:

∆푉푡표푡 ∆푉퐸퐴 휈 휈 푃 = 푁푀푅 [푐푠푎푙푒 − (푐푙 + 푐푝) (푒 푒 − 푒 푒 )] (3.64) ∆푉푡표푡 휈 − 푁푚푑푟푦 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] − 푐표푝푠푇푁 , and the total NPV as:

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푁 푀 푐 ∆푉푡표푡 ∆푉퐸퐴 푅 푠푎푙푒 휈 휈 푁푃푉 = [∑ ] − 푁푀푅(푐푙 + 푐푝) (푒 푒 − 푒 푒 ) (1 + 푟)푇푡 푡=0 (3.65) ∆푉푡표푡 휈 − 푁푚푑푟푦 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] − 푐표푝푠푇푁 .

Expectation Value As with the second ISPP case, the mining spacecraft would have enough propellant to return to Earth following a failed mission. As the spacecraft would not be carrying any additional mass, the return trip could be carried out in a shorter duration. The total duration of a failed mission could then be found by:

∆푉퐸퐴 ∆푉퐴퐸 푇푓 = 푇푤푎푖푡 + 푚0 + (푚푑푟푦 + 푚푝,퐴퐸) . (3.66) 퐹푇 퐹푇

Given this shorter duration, it could be beneficial to send the same mining spacecraft on the next mission following a resupply of propellant launched from Earth.

Eq. 3.59 described the propellant mass required for the return Asteroid-to-Earth transfer. This equation can be used to find the total propellant mass saved in returning to Earth with no additional return mass (i.e. with 푀푅 = 0):

∆푉퐴퐸 ∆푉퐴퐸 휈 휈 ∆푚푝,퐴퐸 = (푚푑푟푦 + 푀푅) (푒 푒 − 1) − (푚푑푟푦 + 0) (푒 푒 − 1) (3.67) ∆푉퐴퐸 휈 = 푀푅 (푒 푒 − 1) .

This difference will define the remaining propellant mass of the mining spacecraft following an unsuccessful mining trip. The additional propellant mass 푚푝,푅푒 required to resupply this spacecraft for the next trip can be found by the propellant required for a new mission (found using Eq. 3.56), less this remaining propellant mass:

푚푝,푅푒 = 푚푝 − ∆푚푝,퐴퐸

∆푉푡표푡 ∆푉푡표푡 ∆푉퐸퐴 ∆푉퐴퐸 = 푚 (푒 휈푒 − 1) + 푀 (푒 휈푒 − 푒 휈푒 ) − 푀 (푒 휈푒 − 1) 푑푟푦 푅 푅 (3.68)

∆푉푡표푡 ∆푉푡표푡 ∆푉퐸퐴 ∆푉퐴퐸 휈 휈 휈 휈 = 푚푑푟푦 (푒 푒 − 1) + 푀푅 (푒 푒 − 푒 푒 − 푒 푒 + 1) .

The refuelling cost 퐶푅 can be computed as the capital cost of the reference mission (Eq.

3.76, described in the next section), where the return mass is replaced with 푚푝,푅푒.

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The expected profit can be computed using a similar method to that used for the ISPP case with reserve propellant. In computing these values, the following rules are assumed: • Following each successful mission, a new mining spacecraft is launched at a cost

of 퐶0; and • Following an unsuccessful mission, the mining spacecraft is refuelled with a

resupply mission launched from Earth at a cost of 퐶푅.

Using these rules, the probabilities, durations, capital, and revenues of the outcomes are listed in table 3.10:

Table 3.10 Probabilities and values of each outcome of the decision tree for the ISPP approach with reserve propellant.

# f* # s** Prob. Duration Capital Revenue

0 푁 푝푠 푁푇푠 − 푇푤푎푖푡 −푁퐶0 − 푐표푝푠푇 푁푅

1 푁 − 1 (1 − 푝푠)푝푠 푇푓 + (푁 − 1)푇푠 − 푇푤푎푖푡 −퐶푅 − (푁 − 1)퐶푅 − 푐표푝푠푇 (푁 − 1)푅

2 2푇푓 + (푁 − 2)푇푠 2 푁 − 2 (1 − 푝푠) 푝푠 −2퐶푅 − (푁 − 2)퐶푅 − 푐표푝푠푇 (푁 − 2)푅 − 푇푤푎푖푡 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮

푁 (1 푚푇푓 + (푁 − 푚)푇푠 −푚퐶푅 − (푁 − 푚)퐶0 푚 푚 (푁 − 푚)푅 − 푚 − 푝푠) 푝푠 − 푇푤푎푖푡 − 푐표푝푠푇 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 푁 푁 0 (1 − 푝푠) 푁푇푓 − 푇푤푎푖푡 −퐶0 − (푁 − 1)퐶푅 − 푐표푝푠푇 0 * Number of failed missions. ** Number of successful missions.

The expected profit can then be expressed as:

푁−1 푚 〈푃〉 = ∑ (1 − 푝푠) 푝푠[−(푁 − 푚)퐶0 − 푚퐶푅 푚=0 (3.69) − 푐표푝푠(푚푇푓 + (푁 − 푚)푇푠 − 푇푤푎푖푡) + (푁 − 푚)푅] 푁 + (1 − 푝푠) [−퐶0 − (푁 − 1)퐶푅 − 푐표푝푠(푁푇푓 − 푇푤푎푖푡)] ,

where 푅 is the total revenue of each successful trip (from Eq. 3.61); and 퐶0 is the capital cost of each new mining spacecraft (from Eq. 3.57), excluding the operations costs.

The expected NPV can similarly be expressed as:

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푁−1 푚 〈푁푃푉〉 = ∑ (1 − 푝푠) 푝푠 [−(푁 − 푚)퐶0 − 푚퐶푅 푚=0 푁−푚 푅 (3.70) − 푐표푝푠(푚푇푓 + (푁 − 푚)푇푠 − 푇푤푎푖푡) + ∑ ] (1 + 푟)푇푡,푚 푡=1 푁 + (1 − 푝푠) [−퐶0 − (푁 − 1)퐶푅 − 푐표푝푠(푁푇푓 − 푇푤푎푖푡)] .

The expected duration will have the same expression as that for the ISPP case with reserve propellant (Eq. 3.54), using the new values for 푇푓.

3.5.4. Reference Mission

The reference mission approach is defined as one in which the total return mass 푀푅 is delivered directly to orbit by a launch vehicle from Earth. To form a comparison, it is assumed that the same mining spacecraft is used, with no mining equipment attached.

The spacecraft will be launched into the same parking orbit for which the specific launch cost 푐푙 is defined – assumed to be a Geostationary Transfer Orbit (GTO). From here, the spacecraft would perform a combined plane change manoeuvre at apoapsis to transfer into a Geostationary Orbit (GEO). The delta-V of this manoeuvre can be found by:

2 2 ∆푉퐴푝표 = √푣1 + 푣1 − 2푣1푣2 cos(∆푖) , (3.71) where ∆푖 is the inclination change between GTO and GEO. The velocities 푣1 and 푣2 are the velocities of GTO and GEO orbits at apoapsis 푟퐺퐸푂:

휇 2휇 푣1 = √2 ( − ) , (3.72) 푟퐺퐸푂 (푟퐺퐸푂 + 푟푝) and:

휇 푣2 = √ , (3.73) 푟퐺퐸푂 where 푟푝 is the periapsis of the GTO parking orbit.

For the SpaceX Falcon 9 rocket, the nominal periapsis altitude is 185 km, and inclinations range up to 28.5 degrees [185]. For the maximum inclination, this equates to a delta-V of 1.8375 km/s.

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The minimum launch mass required to complete the mission will be achieved if there is no remaining propellant at the customer orbit (i.e. 푚1 = 푚푑푟푦,푀푆 + 푚푅). Substituting this into Eq. 3.19 results in an expression for the launch mass:

∆푉퐴푝표 휈 (3.74) 푚0 = (푚푑푟푦,푀푆 + 푀푅)푒 푒 .

The propellant mass required for the manoeuvre can be found by:

∆푉퐴푝표 휈 (3.75) 푚푝 = 푚0 − 푚푑푟푦,푀푆 = (푚푑푟푦,푀푆 + 푀푅)푒 푒 − 푚푑푟푦,푀푆 .

The total capital of the mission can be found by:

∆푉퐴푝표 휈 퐶0 = 푚푑푟푦,푀푆 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] (3.76) ∆푉퐴푝표 휈 + 푀푅(푐푙 + 푐푝) (푒 푒 − 1) + 푐표푝푠푇 .

This equation can be used to compute the total capital of a refuelling mission for the

EBPS option, where 푀푅 is replaced by 푚푝. The return mass and revenue will be the same as those for the other mission alternatives.

3.6 Evaluation of Mission Alternatives In this section, the figures of merit are calculated over a range of parameter values to assess the feasibility of the different mission alternatives.

3.6.1 Values Used in Trade Study

System Parameters The system and cost parameters will be fixed by the design of the spacecraft. For this study, a single fixed value for dry mass of 1000 kg is used. This is a similar size to the Hayabusa and OSIRIS-REx missions. The dry mass of the mining spacecraft is further approximated at 250 kg, giving a total combined spacecraft dry mass of 1250 kg. As will be shown, using a ratio of the extraction mass to total spacecraft dry mass can make the NPV and further derived metrics invariant to the spacecraft dry mass.

For the ISPP approaches, the maximum capacity of mining spacecraft will be determined by the dimensions of the resource tank, storing the extracted asteroid material during its transportation back to Earth. If the resource tank is constructed as a rigid structure, this volume will be restricted by the dimensions of the fairing of the

Scott Dorrington – June 2019 97 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions launch vehicle. (Additional capacity could conceivable be achieved through the in-space assembly or manufacturing of larger resource tanks, or with a deployable structure such as a bag that can be folded into the fairing.) The SpaceX Falcon 9 has a fairing made up of a central cylindrical section of inner diameter 4.6 m and height 6.7 m, and an upper conic section of diameter 1.45 m, and height 4.3 m [185].

This gives a total volume of 173.72 m3 for the upper limit of the resource tank capacity. This volume will be further reduced by the volume of the mining spacecraft and mining equipment. Assuming the spacecraft is of comparable dimensions to the Hayabusa spacecraft (a 1 m x 1.1m x 1.6 m hexagon [180]), the total volume would be around a few cubic meters. From this, an upper limit of 160 m3 can be assumed for the capacity of the resources tank. Assuming the resource is liquid water, having a density of 1000 kg/m3, this amounts to a maximum capacity of 160 tonnes.

The study will consider a range of specific impulses typically representative of chemical propulsion (300 – 450 s) and electric propulsion (1000 – 3000 s). As the ISPP case is restricted to propellants available at the asteroid, a single value of 450 s is used.

All low-thrust missions that have been launched to date have had very small thrust levels, typically of the order of 100’s of micro Newtons. In order to achieve reasonable mission durations with the large return masses, larger thrust levels of the order of 1 or 10’s of Newtons will be required. For example, the ARM mission considered using four 10 kW Hall effect thrusters (plus one spare) with an assumed thrust efficiency of 60% [158]. While the total thrust assumed in this study is not explicitly stated, it can be calculated from the thrust to power ratios of 푇/푃 = 46 to 68 mN/kW of the NASA- 300M Hall effect thruster [186]. This amounts to a total thrust of 1.88 to 2.72 N. For this trade study, two thrust levels of 2 and 10 N are considered for the EBPS cases using electric propulsion.

Cost functions.

The specific launch cost 푐푙 can be calculated from the total launch cost, and the payload performance of the launch vehicle. For this study, it is assumed that the mining spacecraft is initially deployed into a Geostationary Transfer Orbit (GTO). This is a common parking orbit used for launch vehicles. The values are calculated from performance of the SpaceX Falcon 9 launch vehicle, with a total launch cost of $62M, and payload performance of 8,300 kg to GTO [187]. This gives a specific launch cost of $7,469.88/kg.

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To remain competitive, the sale price of asteroid material would need to be lower than the cost of launching the same material from Earth (calculated using the reference mission alternative), thereby undercutting the costs offered by the commercial launch industry. In the following analysis it is assumed that specific sale price is 90% of the specific launch cost. At this sale price, all comparable masses delivered using the reference mission launched from Earth will produce negative profits and NPVs. (As will be discussed in chapter 5, the sale price will likely be determined by a number of economic and market factors. A more significant reduction in sale price may be required to attract customers to offset the higher risk of asteroid mining compared to the well-established terrestrial launch industry.)

The specific production cost is calculated from the estimated cost of the mining spacecraft using established parametric cost estimation models. Two cost estimation models were used for a 1000 kg dry mass spacecraft with 5 kW power.

An initial cost estimate of $213M was generated using the Unmanned Space Vehicle Cost Model v.8 (USCM8). An alternative cost estimate of $509.9M was found using the NASA QuickCost model. These cost calculations are shown in Appendix C, using algorithms described in Wertz et al. [188]. The second cost estimate is higher as it includes the development cost. This is a non-recurring cost that would be incurred in the research and development stages of the asteroid mining campaign. Once an established asteroid mining industry enters regular operations, the production cost of each additional unit is expected to decrease, following a learning curve.

For this study, an intermediate estimate of $300M is used to calculate a specific production cost of $300,000/kg. This value is also assumed for the specific production cost of the mining equipment.

The specific operations cost can be calculated based on the total salary of the operational staff required to monitor and control the spacecraft throughout its mission. Due to the remote distances of the asteroid, it is expected that much of the mining operations would require a high level of automation, and hence a minimal operational staff. The specific operations cost can be estimated from the full time equivalent (FTE) salary of the operational staff (i.e. the salary of a full-time worker employed for one year). Wertz et al. [188] provides an estimated FTE salary for spacecraft operations of $200,000 in FY2010 (or $243,580 in FY2020). Assuming staff take shifts such that at least two spacecraft operators are constantly monitoring the spacecraft, the specific

Scott Dorrington – June 2019 99 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions operations cost is calculated at $487,160/year. As the duration of each mission is expected to be on the order of years to decades, the full operations cost will be 2 to 3 orders of magnitude less than both the construction and launch costs.

Delta-Vs Rather than relying on delta-Vs calculated for particular asteroid targets, the figures of merit are calculated over a range of delta-Vs that might be expected for target asteroids. As noted in chapter 2, the delta-Vs of transfers to asteroids typically range from approximately 2 km/s for asteroids in Earth-like orbits, up to 20 km/s for main belt asteroids. An upper limiting delta-V of 10 km/s is applied. To simplify calculations, it is assumed that this delta-V is equally representative of both the Earth-to-Asteroid and

Asteroid-to-Earth. Hence, ∆푉퐸퐴 = ∆푉퐴퐸 and ∆푉푡표푡 = 2∆푉퐴퐸. While these delta-Vs may differ for actual trajectories, they are expected to be approximately equal to each other.

Duration For impulsive transfer trajectories, the times of flight will be half the period of the transfer orbit. Analytical estimates for both the delta-V and time of flight may be found using a Hohmann transfer from a circular orbit of 1 AU to a circular orbit of an arbitrary heliocentric radius 푟2 . The calculation of the delta-V and time of flight for such a transfer are described in Appendix A.1. Figure 3.6 shows a plot of the delta-V and time of flight calculated for a final orbital radius ranging from 0.5 to 1 AU (inner transfer) and from 1 to 2.5 AU (outer transfer). As can be seen, for a given delta-V value, two values for the time of flight exist, depending on the direction of the transfer (to an inner planet, or to an outer planet).

Figure 3.6 Time of flight vs delta-V for Hohmann transfers.

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Over the range of 0.9 to 1.1 AU (the range of near-Earth asteroids close to Earth), the time of flight varies from 0.46 to 0.58 years. From this, an average transfer time of 0.5 years can be assigned for impulsive transfers.

3.6.2 Feasibility of Single-Trip Missions Table 3.11 summarizes the equations describing the total NPV of the three mission alternatives for a single mining trip (ignoring the probability of success and failure). The expressions for the total profit can be derived from these equations by setting the discount rate 푟 to zero.

Table 3.11 Summary of equations for total single-trip NPV.

Alternative Expression

∆푉 푐푠푎푙푒푀푅 퐸퐴 푁푃푉 = − 푚 [(푐 + 푐 ) + (푐 + 푐 ) (푒 휈푒 − 1)] − 푐 푇 (1 + 푟)푇 푑푟푦 푝푟표푑 푙 푙 푝 표푝푠

ISPP where:

1 ∆푉퐴퐸 ∆푉퐴퐸 푇 = 2푇푂퐹 + [푚 (푒 휈푒 − 1) + 푀 푒 휈푒 ] 퐼푚푝 푀푅 푑푟푦 푅

∆푉 푐푠푎푙푒푀푅 푡표푡 푁푃푉 = − 푚 [(푐 + 푐 ) + (푐 + 푐 ) (푒 휈푒 − 1)] − 푐 푇 (1 + 푟)푇 푑푟푦 푝푟표푑 푙 푙 푝 표푝푠

ISPP w/res where:

1 ∆푉퐴퐸 ∆푉퐴퐸 푇 = 2푇푂퐹 + [푚 (푒 휈푒 − 1) + 푀 푒 휈푒 ] 퐼푚푝 푀푅 푑푟푦 푅

∆푉 ∆푉 푐푠푎푙푒 푡표푡 퐸퐴 푁푃푉 = 푀 [ − (푐 + 푐 ) (푒 휈푒 − 푒 휈푒 )] 푅 (1 + 푟)푇 푙 푝

∆푉푡표푡 휈 − 푚푑푟푦 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] − 푐표푝푠푇

EBPS where:

2푇푂퐹퐼푚푝 + 푇푐푎푝 , 푓표푟 푐ℎ푒푚푖푐푎푙 푝푟표푝. 푇 = { ∆푉퐸퐴 ∆푉퐴퐸 푚0 + (푚푑푟푦 + 푀푅 + 푚푝,퐴퐸) + 푇푐푎푝 , 푓표푟 푒푙푒푐푡푟푖푐 푝푟표푝. 퐹푇 퐹푇

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From these equations, it can be seen that for a set of defined parameter values, the profit and NPV can be controlled by varying the return mass. For the ISPP options, the return mass can be controlled by the stay-time at the asteroid. For the Earth-based propellant supply case (whole-asteroid return), the return mass can be controlled by selecting asteroid targets of an expected size and density. This presents a challenge of deciding how much mass should be targeted during a mining mission.

Total Profit An initial analysis can be applied by choosing an arbitrary value for the return mass. Figure 3.7 shows a plot of the total profit over a range of delta-Vs for a return mass of 100 times the combined dry mass of the mining spacecraft and mining equipment. The two ISPP cases are shown with black lines, while coloured contours are shown for the EBPS case with varying specific impulses.

Figure 3.7 Total profit over a range of delta-Vs and specific impulses.

As the revenue is only dependent on the return mass, the total revenue will be the same in all cases over the range of delta-Vs (a value of $840M). The differences in the plot can then be attributed to the varying capital costs. From the figure, it can be seen that in all cases, the profit decreases with an increasing delta-V. This is due to an increasing propellant mass required to be launched from Earth.

For the EBPS cases, the total propellant mass, and hence the profit, will also decrease with higher specific impulses. For cases using electric propulsion (specific impulses

102 Scott Dorrington – June 2019 Chapter 3: Parametric Economic Analysis greater than 1000 s), the figure shows a positive profit is achievable over a wide range of delta-Vs. For the cases using chemical propulsion, only low delta-Vs less than around 2 km/s would be able to return a profit.

While the ISPP cases also have low specific impulses, there is a significant saving in the launch cost achieved by sourcing the propellant for the return trip from asteroid resources. This leads to larger profits over a wide range of delta-Vs. However, the required extraction mass for the ISPP case would be significantly larger than the return mass (around 312 times the dry mass at 5 km/s, or 972 times the dry mass at 10 km/s). In order to return this mass, the maximum capacity of the mining spacecraft would need to be significantly increased for larger delta-V trajectories.

From this analysis, it can be concluded that for a mission with a targeted return mass of 100 times the dry mass of the spacecraft, the ISPP case and the EBPS cases using electric propulsion would be feasible in returning profits in a single-trip mining mission, over a wide range of delta-Vs. Earth-based propellant supply using chemical propulsion would only be feasible for asteroids with very low delta-Vs less than 2 km/s. However, as noted in chapter 2, there are not expected to be any asteroids accessible with such low delta-Vs.

Total NPV Using total profit as the figure of merit simplifies the analysis in removing the dependence on additional parameters such as thrust level that determine the mission duration. Figure 3.8 shows a plot of the NPV calculated for the same return mass, with a thrust of 2 N, and a discount rate of 20%. In all cases, the total capital is the same as those calculated in the previous plot, however the revenues are discounted to account for the duration of the mission (shown at the bottom of figure 3.8).

In both the ISPP cases, and the EBPS cases using chemical propulsion, the mission durations are short (on the order of a few years). As a result, the revenues are discounted at only a small rate, and the NPV shows similar trends to the profit (with slightly lower values).

For the cases using electrical propulsion, the large return mass requires a long time of flight for the return Asteroid-to-Earth transfer (on the order of decades at this thrust level). This has the effect of greatly reducing the time-value of the revenue, such that the NPV is negative over a larger range of delta-Vs.

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Figure 3.8 NPV and mission duration for 푭푻 = ퟐ N.

The mission duration can be reduced by increasing the thrust level of the spacecraft. Figure 3.9 shows the NPV and mission durations for the case of 10 N thrust.

In this case, the mission durations for the electric propulsion cases are reduced to the order of 3 decades, resulting in higher NPVs over a wider range of delta-Vs. If the thrust level were increased further, the mission duration could be reduced enough such that the EBPS electric propulsion case generates higher NPVs than the ISPP case. However, increasing thrust would require a larger propulsion and power supply systems, increasing the total dry mass of the spacecraft. These feedback dependences could be accounted for in future with a more detailed parametric model of the spacecraft subsystems.

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Figure 3.9 NPV and mission duration for 푭푻 = ퟏퟎ N.

3.6.3 Single-Trip Break-Even Mass The previous analysis was performed for an arbitrary asteroid return mass of 100 times the spacecraft dry mass. If the same analysis is performed for higher or lower return masses, similar trends are observed, with the entire profit and NPV curves shifted up or down.

While the target asteroid return mass will be a design choice for the mission and trajectory design, characteristic values can be found from the upper and lower limits of the return mass. For the ISPP case, the upper limit will be determined by the maximum capacity of the spacecraft.

The lower limit of the extraction mass can be determined by the minimum mass required to return a profit with only a single mining trip. This break-even point can be

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∗ found by the critical return mass 푀푅 required to produce a NPV of zero This produces a new figure of merit that is comparable to the internal rate of return, that specifies the critical discount rate required to achieve zero NPV. Table 3.12 summarizes the expressions for the break-even mass for the different mission alternatives (found by solving the NPV equations in table 3.11).

Table 3.12 Break-even mass expressions for the different mission alternatives.

Alternative Break-Even Mass (푵푷푽 = ퟎ)

∆푉퐸퐴 푐 휈 표푝푠 (푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1) + 푇 푚푑푟푦 ISPP 푀∗ = 푚 푅 푑푟푦 푐 (1 + 푟)−푇 푠푎푙푒 [ ]

∆푉푡표푡 푐 휈 표푝푠 (푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1) + 푇 푚푑푟푦 ISPP w/res 푀∗ = 푚 푅 푑푟푦 푐 (1 + 푟)−푇 푠푎푙푒 [ ]

∆푉푡표푡 푐 휈 표푝푠 (푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1) + 푇 푚푑푟푦 ∗ EBPS 푀 = 푚푑푟푦 푅 ∆푉푡표푡 ∆푉퐸퐴 푐 (1 + 푟)−푇 − (푐 + 푐 ) (푒 휈푒 − 푒 휈푒 ) [ 푠푎푙푒 푙 푝 ]

Due to the use of specific cost functions, these equations can be expressed as a linear function of the spacecraft dry mass. Dividing these equations by the dry mass can produce a non-dimensional mass ratio, defined here as the break-even mass ratio (BEMR):

∗ 푀푅 퐵퐸푀푅 = . (3.77) 푚푑푟푦

While the previously defined mass payback ratio (Eq. 3.9) is normalized by the launch mass, which implicitly varies with the return mass, this new ratio removes the dependence of the spacecraft dry mass, such that it can be applied to a range of spacecraft classes from microsatellites to explorer class missions.

It should be noted that while the operations cost is expressed as a specific function of the mission duration, it is not explicitly a function of the spacecraft dry mass, and hence is not invariant to the spacecraft dry mass. However, the increase in operations cost is

106 Scott Dorrington – June 2019 Chapter 3: Parametric Economic Analysis expected to be small in comparison to the other cost factors as it scales linearly with −푇 mission duration, while the revenues scale exponentially (i.e. the 푐푠푎푙푒(1 + 푟) term).

Zero-Profit BEMR In the above expressions, the total duration is itself a function of the return mass. As a result, these expressions are implicit functions of the return mass. The approach may be simplified by removing the (1 + 푟)−푇 term in the denominator, giving the break-even mass required to give a profit of zero. Figure 3.10 shows a plot of the zero-profit break- even mass ratio as a function of delta-V, for the same specific impulse contours as those used in the previous figures.

Figure 3.10 Zero-profit break-even mass ratio expressed over a range of delta-Vs and specific impulses.

From this new figure, it can be seen that each of the specific impulse contours for the EBPS case demonstrate an asymptotic relation. For small delta-Vs, the BEMR is positive, indicating that return masses above this value will return positive profits. For larger delta-Vs, the BEMR is seen to be negative – meaning that the incremental increase in capital cost required to return a larger asteroid mass outweighs the incremental increase in revenue. As such, it is not possible to generate a positive profit (and therefore NPV), regardless of the mass of returned material.

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The asymptotic behaviour in the EBPS case is caused by a singularity in the BEMR equation, when the denominator equals zero. For the zero-profit case, the (1 + 푟)−푇 term in the denominator can be ignored, and an analytical expression can be produced describing the limiting maximum delta-V reachable for which it is possible to break- even (i.e. generate positive profits) in a single trip:

1 4푐푠푎푙푒 ∆푉푙푖푚 = 휐푒퐿푛 [ (1 + √1 + )] . (3.78) 2 (푐푙푎푢푛푐ℎ + 푐푝)

This expression could alternatively be used to express the limiting specific impulse required to reach an asteroid with a specified delta-V.

The asymptotic behaviour is not seen in the ISPP curves, where the BEMR monotonically increases with the delta-V. From this, we can conclude that: • the ISPP approach is always capable of returning a positive profit, so long as the extraction mass required to achieve this BEMR is less than the maximum capacity of the mining spacecraft.

Given these analytical expressions, the zero-profit break-even mass ratio is a useful metric that can be used to assess the feasibility of the mission alternatives. For example, for an asteroid with a one-way transfer delta-V of 5 km/s, the standard propellant supply case is not feasible for specific impulses below 1100 s. From this we can infer that it is not feasible to generate a positive profit in a single-trip mission using chemical propulsion, where all the propellant is supplied from Earth at the start of the mission. Using electric propulsion, a mining mission to this asteroid could be feasible, so long as it returns more than around 80 times the spacecraft dry mass for a specific impulse of 2000 s, or 61 times the spacecraft dry mass for a specific impulse of 3000 s.

Zero-NPV BEMR As previously mentioned, the BEMR expressions defined in table 3.12 are implicit functions of the return mass. These equations can be solved using numerical root finding methods. Figure 3.11 shows the zero-NPV BEMR calculated for the same thrust and discount rate settings used in figure 3.8 (퐹푇 = 10 푁 and 푟 = 20%). In this figure, only the positive BEMR values are shown.

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Figure 3.11 BEMR for zero NPV.

These results show similar trends to those seen in the previous case, with the limiting delta-Vs shifted to significantly lower values. Limiting delta-Vs are also observed for both the ISPP cases. For delta-Vs above these values, the NPV is negative – meaning that over the additional time required to extract and process additional resources, the incremental increase in revenue is outweighed by the incremental increase in the interest incurred on the capital. While analytical expressions for these limiting delta-Vs are difficult to express, the values are found to be highly dependent on the mining rate.

For the same asteroid with delta-V of 5 km/s, the standard propellant supply case would require specific impulses greater than 3400 s in order to return a positive NPV. For specific impulses of around 450 s, typical of chemical propulsion, the maximum reachable delta-V is around 1.8 km/s. As there generally are not any asteroids with delta-Vs this low, we can conclude that: • it is not feasible to break-even or produce a positive NPV with a single-trip mining mission using chemical propellant supplied from Earth.

3.6.4 Feasibility of Multi-Trip Missions Following a successful mining trip, a mining spacecraft using the standard EBPS option would need to be completely resupplied with propellant launched from Earth. As this refuelling cost would be comparable to that of the initial capital investment, it would be

Scott Dorrington – June 2019 109 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions beneficial to send a new mining spacecraft on subsequent mining missions. In this case, each new mining trip in a mining campaign would need to return masses greater than the break-even mass to recover their capital investment.

For the ISPP cases, the mining spacecraft can refuel with propellant returned from the previous trip. Each mining trip would return additional shipments of asteroid material, generating a stream of multiple revenues, further increasing the total profit and NPV. Table 3.13 summarizes the NPV expressions for the two ISPP cases over 푁 mining trips. The expressions for profit can be derived by setting the discount rate to zero.

Table 3.13 Summary of NPV expressions over multiple trips.

Alternative Multi-trip NPV

푁−1 ∆푉퐸퐴 휈 −푇푡 푁푃푉 = ∑ (푐푠푎푙푒 [푀푅 − 푚푑푟푦,푀푆 (푒 푒 − 1)] (1 + 푟) ) 푡=1

−푇푁 ISPP + 푐푠푎푙푒푀푅(1 + 푟)

∆푉퐸퐴 휈 − 푚푑푟푦 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] − 푐표푝푠푇푁

푁−1 ∆푉퐸퐴 휈 −푇푡 푁푃푉 = ∑ (푐푠푎푙푒 [푀푅 − 푚푑푟푦,푀푆 (푒 푒 − 1)] (1 + 푟) ) 푡=1

−푇푁 ISPP w/res + 푐푠푎푙푒푀푅(1 + 푟)

∆푉푡표푡 휈 − 푚푑푟푦 [(푐푝푟표푑 + 푐푙) + (푐푙 + 푐푝) (푒 푒 − 1)] − 푐표푝푠푇푁

where 푇푡 is the cumulative duration of each mining trip 푡:

1 ∆푉퐴퐸 ∆푉퐴퐸 푇 = (푡 − 1)푇 + 푡 (푇푂퐹 + 푇푂퐹 + [푚 (푒 휈푒 − 1) + 푀 푒 휈푒 ]) 푡 푤푎푖푡 퐸퐴 퐴퐸 푀푅 푑푟푦 푅

Cumulative Profit Figure 3.12 shows the cumulative profits generated over 10 mining trips, with each trip returning 100 times the dry mass of the spacecraft. In this figure, solid lines indicate the standard ISPP case, and dashed lines show the ISPP case with reserve propellant. In this plot, the total profit is seen to increase by the same increment with each additional trip.

Each of these mining trips would also increase the cumulative duration 푇푡 of the mission (shown in figure 3.13).

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Figure 3.12 Cumulative profit over a 10-trip mining campaign.

Figure 3.13 Cumulative duration over 10 mining trips.

Cumulative NPV In terms of the NPV, the revenues from each additional trip will be discounted at an ever-increasing rate, as the cumulative mission duration increases. This has the effect of diminishing the present value of the revenue for each additional trip. This effect is commonly seen in perpetuity bonds, where the present value of a constant stream of identical cash-flows diminishes over time [189].

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Figure 3.14 shows the cumulative NPV calculated for each of these mining trips. In this figure, it can be seen that the contours are approaching a limiting NPV.

Figure 3.14 Cumulative NPV over 10 mining trips.

This limit is caused by the sum of an infinite series having a finite value. If each trip has an equal revenue of 푅 and duration of 푇, the limit of the NPV can be expressed as:

∞ 1 푅 푁푃푉 = −퐶 + 푅 ∑ = −퐶 + . (3.79) 푙푖푚 0 (1 + 푟)푛푇 0 (1 + 푟)푇 − 1 푛=1

This effect can be seen more clearly by plotting the revenues of successive mining trips for a single delta-V. Figure 3.15 shows a cash-flow diagram showing the values and times of the revenues over the 10 mining trips to an asteroid with a one-way delta-V of 5 km/s. In this figure, cash-flows are shown with red and green arrows, indicating the capital (negative cash-flow) and revenues (positive cash-flows). The cumulative profits and NPVs are shown in black and blue lines, with the cases for ISPP and ISPP with reserve propellant shown with dashed and dotted lines. Both ISPP cases show similar trends in profit and NPV, with the later showing slightly weaker performance due to the excess capital cost of launching the reserve propellant during the first mining trip.

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Figure 3.15 Cash-flow diagram for asteroid with delta-V 5km/s, with return masses of

ퟏퟎퟎ 풎풅풓풚.

The total number of trips that each mining spacecraft could complete would be limited by the design life and reliability of the spacecraft. Assuming a design life of 20 years (comparable to communications satellites) and an asteroid with a one-way delta-V of 5 km/s, a single mining spacecraft could potentially complete seven return trips to the asteroid, returning 700 times the dry mass of the spacecraft, with a total profit of $5.4Bn or a positive total NPV of $895.5M. At this delta-V, the zero-NPV BEMRs for the two ISPP cases are 68.3 and 81.13; hence, both cases will return a positive NPV after the first trip.

For the EBPS option, to maximize the profits, each spacecraft should aim to return as much mass as possible, with a total mission duration within the design life of the spacecraft. For the case of an electric propulsion system with specific impulse 3000 s, and thrust 10 N, a maximum return mass of 716 times the spacecraft dry mass could be returned from this asteroid within a 20-year mission. This mission would have a total capital cost of $1.859Bn, producing a revenue of $6.017Bn, for a total profit of $4.158Bn. However, as the limiting delta-V for the zero-NPV BEMR is less than 5 km/s, this mission would return a negative NPV of –$1.702Bn, and hence would not be economically viable.

The diminishing present value of multiple cash-flows was identified by Sonter [169] as a reason for favouring single-trip mining missions over multiple trips. However, as seen

Scott Dorrington – June 2019 113 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions from this comparison, the multi-trip ISPP cases not only generate higher profits than the single-trip EBPS approach, but they can also generate positive NPVs. From this, we can conclude that: • both ISPP cases with multiple mining trips are more favourable than a single- trip EBPS mission.

Number of Mining Trips For the ISPP cases, the total number of mining trips that can be achieved within the design life of the mining spacecraft will be determined by the return mass extracted in each trip. Larger return masses would require extended stay-times at the asteroid, reducing the total number of mining trips. Smaller masses would require shorter stay- times at the asteroid, increasing the total number of mining trips.

In the previous case, the total extraction mass required to deliver this return mass was found to be 312 times the dry mass of the spacecraft. This would require a resource tank with a larger volume capacity than could fit within the fairing of a single launch vehicle.

Reducing the return mass to 50 times the spacecraft dry mass would reduce the extraction mass to a more feasible level of 157.47 times the spacecraft dry mass. Figure 3.16 shows a cash-flow diagram for this new case, for the same 5 km/s asteroid.

In this case, the mission durations are shorter, and a total of 9 missions could be completed within the 20-year design life. While this case can deliver two additional revenues, each revenue will be half of those in the previous case. As each of these revenues are returned at slightly shorter durations, they will be discounted at a lower rate, producing a total NPV that is slightly greater than half that of the previous case. While in the first case, the NPV becomes positive following the first mining trip, this second case would require two trips before breaking even (i.e. the payback period is increased).

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Figure 3.16 Cash-flow diagram for asteroid with delta-V 5km/s, with return masses of

ퟓퟎ 풎풅풓풚.

Reducing the return mass further shows that this trend continues, with more total trips, lower total NPVs, and larger payback periods. Even if the return mass was reduced to zero (i.e. zero stay-time), there would still be a finite number of trips that could be achieved within the given lifetime, determined by the total time of flight of the return transfer:

퐿푖푓푒 퐿푖푓푒 푁푚푎푥 = = (3.80) 푇푂퐹퐸퐴 + 푇푂퐹퐴퐸 2푇푂퐹퐼푚푝

From this, we can draw the following conclusions for the ISPP case over multiple mining trips: • it is more beneficial to spend a longer time at the asteroid to retrieve larger shipments than it is to complete more trips with smaller shipments (such that less time is wasted during transfers to and from the asteroid); • each mining spacecraft should aim to stay at the asteroid long enough to completely fill its resource tank to the maximum capacity; and • increasing the maximum capacity of the mining spacecraft will allow for larger NPVs.

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3.6.5 Expectation Value of Multi-Trip Missions In all the analysis performed so far, the ISPP case with reserve propellant has been seen to follow the same trends as the standard ISPP case, with a slightly reduced performance due to the excess capital.

The potential benefit of the second case is that it reduces the potential losses in the event of an unsuccessful mission. This can be examined by considering the expectation values of the profit and NPV over 푁 mining trips. These equations are summarized in table 3.14.

Table 3.14 Summary of NPV expressions over multiple trips.

Alternative Multi-trip NPV

푁−1 푚 〈푁푃푉〉 = ∑ (1 − 푝푠) 푝푠 [−(푚 + 1)퐶0 − 푐표푝푠(푚푇푓 + (푁 − 푚)푇푠 − 푇푤푎푖푡) 푚=0 푁−푚 ISPP 푅 푅푁 + ∑ ( ) + ] (1 + 푟)푇푡,푚 (1 + 푟)푇푁,푚 푡=1 푁 + (1 − 푝푠) (−푁퐶0 − 푐표푝푠푁푇푓)

푁−1 푚 〈푁푃푉〉 = ∑ (1 − 푝푠) 푝푠 [−퐶0 − 푚퐶푅 − 푐표푝푠(푚푇푓 + (푁 − 푚)푇푠 − 푇푤푎푖푡) 푚=0 푁−푚 푅 푅 ISPP w/res + ∑ + 푁 ] (1 + 푟)푇푡,푚 (1 + 푟)푇푁,푚 푡=1

푁 + (1 − 푝푠) (−퐶0 − (푁 − 1)퐶푅 − 푐표푝푠(푁푇푓 − 푇푤푎푖푡))

where 푇푡,푚 = 푚푇푓 + 푇푡 is the cumulative duration of each successful trip 푡 following 푚 failed trips ( 푇푡 is the same as that defined in table 3.13); and

푇푁,푚 = 푚푇푓 + (푁 − 푚)푇푠 − 푇푤푎푖푡.

Probability of Success The expected values will be highly dependent on the estimated probability of success for each mission. Assuming there are no failures in the spacecraft systems and no navigational problems in reaching the asteroid, the probability of success would be determined by the probability of finding the presence of the desired ore resource in the asteroid.

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As discussed in chapter 2, the fraction of water-bearing asteroids can be estimated from the expected 10% of near-Earth asteroids that are C-type asteroids, and the expected

25% of those asteroids that have high water content (> 6 wt% H2O). Hence, if no prior information is known about the target asteroid (i.e. no taxonomic class), the probability of a successful mission would be around 2.5%. If the asteroid is known to be a C-type asteroid (based on indicators from photometric and spectroscopic data), the probability of success would be around 25%.

In the equations presented in table 3.14, the first term describes the summation over outcomes in which 푚 mining trips failed and (푁 − 푚) mining trips were successful. This can be used to give the total probability that at least one mining trip would succeed in an 푁-trip mining campaign:

푁−1 푚 푃푟(푁 − 푚 ≥ 1|푁) = ∑ (1 − 푝푠) 푝푠 . (3.81) 푚=0

For an asteroid of unknown taxonomic class (푝푠 = 2.5%), over 10 mining trips, there would be a very low chance (22.27%) of a successful mission. For a known C-type asteroid (푝푠 = 25%), there would be a very high chance (94.36%) of a successful mission. This method can alternatively be used to specify how many mining trips would likely be required before a successful trip, with a specified level of confidence. For a known C-type asteroid, we could expect to see at least one success after 9 trips, with a confidence of 90%. For an unknown taxonomic class, 91 trips would be required for the same level of confidence.

These numbers agree with those predicted by Elvis & Esty [190] who used a similar probabilistic method based on the binomial distribution to find the expected number of exploration missions required to find a single ore-bearing asteroid.

Figure 3.17 shows a plot of the probability of at least one successful trip in a campaign with 푁 = 1 to 10 trips, over a range of 푝푠 values.

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Figure 3.17 Probability of consecutive failures as a function of probability of success.

Expected NPV for Asteroids of Unknown Taxonomic Class Figures 3.18 show a plot of the expected NPV of the two ISPP cases over a range of delta-Vs to asteroids with unknown taxonomic classes.

Figure 3.18 Expected NPV for an asteroid of unknown taxonomy class.

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From this figure, the expected NPV of both ISPP cases is found to be negative over the entire range of delta-Vs. Positive values for the expected NPV can be achieved by greatly increasing the return mass and mining rate. If the return mass is increased to 400 times the spacecraft dry mass, the second ISPP case can return positive values after 10 trips over a very small range of delta-Vs (less than 0.3 km/s). Increasing the return mass above 1000 times the spacecraft dry mass, the values become negative again.

Extending the range of delta-Vs to reasonable values of 3-4 km/s would require increasing the mining rate by an order of magnitude to 8,000 kg/day. However, large extraction masses of around 1000 times the spacecraft dry mass would be required in each trip. No values of the return mass or mining rate are capable of producing positive values for the standard ISPP case, where the expected NPV becomes increasingly negative with each additional trip.

While these return masses and mining rates could conceivably be achieved with well- designed mining equipment, it is safe to conclude that: • asteroids with unknown taxonomic class are not ideal candidates for asteroid mining, as they would produce negative expected NPVs.

Expected NPV for Known C-type Asteroids Figure 3.19 show a plot of the expected NPV of the two ISPP cases over a range of delta-Vs to asteroids of known C-type taxonomic class.

Figure 3.19 Expected NPV for a known C-type asteroid.

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From this figure, it can be seen that the standard ISPP case only provides positive expected NPVs over low delta-Vs (less than 1 km/s), while the ISPP case with reserve propellant provides a much larger expected NPV over a wider range of delta-Vs (up to 5.31 km/s).

While the results show that positive expected NPVs are achievable, the values are considerably less than those calculated from the deterministic case. Figure 3.20 shows a cash-flow diagram of the expected profit and NPV for an asteroid with one-way delta-V of 5 km/s. The cases for ISPP and ISPP with reserve propellant are shown with dashed and dotted lines, respectively. (The cash-flows of capital and revenues show their deterministic values.) This figure is comparable to the deterministic case, shown in figure 3.15.

Figure 3.20 Cash-flow diagram for known C-type asteroid

with delta-V 5km/s, with 풑풔 = 25%.

From this figure, it can be seen that the ISPP case with reserve propellant would expect to see positive NPVs after the 6th mining trip, with a payback period of around 11 years (compared to a single trip of 2.5 years for the deterministic case). A total of 9 mining trips could be expected within the 20 year design life of the mining spacecraft with a total expected profit of $4.46Bn and total expected NPV of $72.42M.

These values can be improved by increasing the return mass and mining rate in a similar manner to that discussed for the unknown asteroid. In doing so, both cases of the ISPP are seen to increase, with the reserve propellant case always having a higher value than the standard ISPP case.

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The best method of improving the results comes by increasing the probability of success. It is expected that as more asteroids are explored and mined, an improved understanding will be gained of the true concentration and distribution of mineral resources throughout the near-Earth asteroid, and the C-type asteroid populations. This larger data set could reveal statistically significant trends between water content and observable photometric or spectroscopic properties. These could be used to develop diagnostic methods (such as Bayesian inference) to distinguish between C-type asteroids with low and high water content from observational data with a higher certainty.

The probability of success could conceivably be increased to 50%, by limiting asteroid candidates to C-type asteroids that show strong 0.7 and 3 μm absorption features. This would likely filter out the C-type asteroids with expected low water content. Figure 3.21 shows the same cash-flow diagram with a probability of success of 50%. From this figure, the mining campaign would expect to break-even after its second trip, and would produce higher expected profit of $5.24Bn and expected NPV of $496M after 8 trips (around 20 years).

Figure 3.21 Cash-flow diagram for known C-type asteroid

with delta-V 5km/s, with 풑풔 = 50%.

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From the analysis of expected NPV, we can draw the following conclusions: • it is beneficial to provide reserve propellant for an ISPP mission to mitigate the risks of not finding the desired resource at the target asteroid; and • any improvement in the accuracy of predicting high water content amongst C- type asteroids will lead to higher expected NPVs.

3.7 Conclusion This chapter outlined a generalized space mission architecture that can be used to describe the elements of an asteroid mining operation, from which alternative approaches were identified for several aspects of the mission design.

A parametric economic model was formulated to compute operational and economic figures of merit from critical system, mission, and specific cost parameters. These were used to perform a trade study to assess the feasibility of the mission alternatives over a range of parameters for both single-trip and multi-trip mining missions.

A break-even analysis identified the minimum return masses required to recover all capital investment in a single-trip mission, with propellant either supplied from Earth or processed from asteroid material using ISPP. The results showed a limiting delta-V above which missions are not capable of generating positive NPVs, requiring infinite return masses to break-even. This occurred at around 1.8 km/s for missions using chemical propulsion, with propellant supplied from Earth. Electric propulsion was found to be feasible up to around 4.5 km/s, requiring return masses of around 100 times the spacecraft dry mass to break-even. Missions using return propellant processed from asteroid material were found to be feasible up to larger delta-Vs of 8.8 km/s; however, would require the extraction of larger masses of material to produce the return propellant.

Further analysis for the ISPP cases calculated the cumulative profit and NPV over multiple return trip mission, each bringing back a small shipment of resources (limited by a maximum capacity of the mining spacecraft). This showed that it would be possible to conduct up to 10 return trips within the 20-year design life of the spacecraft, generating large profits of the order of billions of dollars, and generating positive NPVs after the first or second trip. This concluded that multi-trip missions using ISPP are more favourable than single-trip missions returning larger shipments, where the propellant is completely supplied from Earth.

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This analysis was further extended to consider the expected NPV, accounting for the uncertainty in the presence of hydrated minerals in the target asteroid. This revealed that it would be infeasible to produce positive expected NPVs for asteroids of unknown taxonomic classes, and could only be feasibly for known C-complex asteroids with very low delta-Vs less than 1 km/s. Positive expected NPVs could be achieved using an alternative ISPP mission design in which reserve propellant is supplied allowing it to return to Earth orbit in the event of a failed mission. The results conclude that further data and understanding of asteroid chemical composition would be needed to reduce the uncertainty to levels capable of producing positive expected NPVs over a wider range of delta-Vs.

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4 PARAMETRIC MINING RATE MODEL

In the previous chapter, mission alternatives using in situ processing (such as the two ISPP alternatives) would require the mining equipment to extract and process large volumes of asteroid ore. In this chapter, a parametric model is presented in which the mining rate 푀푅 is calculated as a function of three components:

• Geological parameters of the asteroid; • Subsystem parameters of the mining equipment; and • Specific energy & time estimates for mining operations.

4.1 Asteroid Geological Analytical Model The geology of the asteroid is modelled as a spherical rubble pile asteroid, with properties defined in table 4.1. These properties were described in chapter 2.

Table 4.1 Asteroid geological parameters.

Parameter Units Description 푅 m Mean radius 푀 kg Mass of asteroid

г0 m Minimum particle size

г1 m Maximum particle size 푛 - Size distribution power index -3 휌푔 kg.m Average grain density

푐푚푟 wt% Concentration of mineral resource

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4.1.1 Bulk Properties From these parameters, a number of bulk properties of the asteroid can be derived, including the bulk volume 푉푏, bulk density 휌푏, porosity 푃, and particle volume fraction

푓. The bulk volume 푉푏 describes the volume containing all particles within the asteroid:

4휋 푉 = 푅3 . (4.1) 푏 3

The bulk density 휌푏 of the asteroid is the total mass divided by the bulk volume:

푀 휌푏 = . (4.2) 푉푏

The bulk volume consists of a volume 풱 of particles of various sizes, and a volume 푉푉 of voids or pores between the particles8. The volume fraction of particles in the asteroid

푓 can be calculated from the ratio of the bulk density to an average grain density 휌푔, estimated from the measured grain density of meteorite samples showing similar spectral properties to that of the asteroid:

휌푏 풱 푓 = = . (4.3) 휌푔 푉푏

The complement of the particle volume fraction gives the porosity 푃 of the asteroid (the volume fraction of voids):

휌푏 푃 = 1 − 푓 = 1 − . (4.4) 휌푔

Asteroids with low porosity are thought to be monolithic asteroids, with an interior composed of solid rock. Asteroids with high porosity are thought to be “rubble pile” asteroids – aggregates of boulders, rocks, gravel, and regolith held together by weak gravitational and surface forces.

4.1.2 Particle Size Distribution It is assumed that the sizes of particles throughout a rubble pile asteroid follows a number density distribution 푛(г). For the purposes of these calculation, we will assume a power-law distribution, as described by Sánchez & Scheeres [104]:

8 In this chapter, all bulk volumes are represented with the symbol 푉, and all particle volumes with the symbol 풱, to distinguish between the inclusion and exclusion of void volumes, respectively.

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1 푛(г) = 3푁 г3 , (4.5) 1 1 г4 where the constant 푁1 is a fitting constant defining the number of particles with the maximum size г1. This value may be fitted from observations of the asteroid surface to give the particle size distribution across the surface. While this distribution has been found for particles on the surface, we may assume that it is representative of the distribution of particle sizes throughout the interior of the asteroid (with a different fitting constant).

The total volume of particles in the asteroid (excluding the volume of voids) can be found from a weighted integral of the number density function:

г1 4휋 г1 풱 = ∫ г3푛(г). 푑г = 4휋푁 г3 ln ( ) . (4.6) 3 1 1 г г0 0

Assuming a constant grain density over all particle sizes, this value can be used to calculate the total mass of the asteroid. For the distribution throughout the interior of the asteroid, the constant 푁1 can be found by equating the mass found from the bulk volume and density, to that found from the total particle volume and density:

3 휌푏 푅 1 푁1 = . 3 г1 (4.7) 휌푔 3г1 ln ( ) г0

4.1.3 Mineral Concentration The mineralogical composition of the individual particles is assumed to be similar to that of the various meteorite types found on Earth. Meteorites show a conglomerate structure, with a mixture of various mineral crystals, and a rock or metallic matrix. The average grain density will be determined by the proportions and densities of the individual mineral grains, the density of the matrix, and the microporosity. Let 푐푚푟 represent the concentration (by weight) of a desired mineral resource within the asteroid. The total mass of the resource in the asteroid can be estimated from the bulk properties of the asteroid. The mass that can be extracted will be dependent on the efficiency 휀푚푟 of the equipment and chemical process used to extract these resources from the ore:

푀푚푟 = 휀푚푟푐푚푟휌푏푉푏 . (4.8)

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4.2 Mining Equipment and Operations The extraction of mineral resources will require a number of mining operations to be carried out on numerous blocks of asteroid ore. The operations are expected to be similar to those used in terrestrial mining, namely fragmentation, excavation, transportation, and chemical processing.

The design of the mining equipment is expected to differ from that of spacecraft and planetary rovers, with the need to be far more robust (and likely heavier) to withstand the forces involved in the mining operations. The equipment design and function are also expected to differ greatly from that used in terrestrial mining due to the unique environmental conditions of the asteroid, such as the lack of appreciable gravity and atmosphere.

Rather than designing individual systems, we can instead identify the major components of the mining spacecraft and mining equipment (or rover) (presented in table 4.2). The critical parameters of these subsystems used in the following parametric model are listed in table 4.3.

Table 4.2 Major system components.

Mining Spacecraft

Component Description Spacecraft that carries all components between Earth and the Spacecraft Bus asteroid. Composes all subsystems needed for the heliocentric transfer. Power supply for the main spacecraft. Provides power to run Spacecraft the main spacecraft subsystems, as well as the chemical Power Supply processing systems. Container to store the processed resources to be returned to Earth. May include segmented sections for storing various Resource Tank types of resources, including readily-usable propellant for use in the return heliocentric transfer. Reaction Chamber to contain chemical reaction for processing asteroid Chamber ore into desired products.

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Mining Equipment (Rover)

Component Description The mobile component containing the mining equipment Rover required for extraction and transportation of ore. Rover Power supply for the rover subsystems. Provides power to run Power Supply the various mining operations. Rover System for moving the rover over the asteroid. A number of Locomotion techniques may be used, such as wheels, legs or thrusters. System Equipment required for fragmentation and excavation of asteroid ore. Fragmentation equipment may require of a drill Mining or cutting tool to break large rocks into smaller sizes. Equipment Excavation equipment will consist of a bucket and arm to transfer the fragmented ore into the rover’s holding tank. A smaller resource tank to hold the extracted ore for Holding Tank transportation back to the reaction chamber.

Table 4.3 Subsystem parameters.

Component Parameter Units Description

푚푑푟푦,푀푆 kg Dry mass of mining spacecraft Mining 푃 W Available power of the spacecraft Spacecraft 푆 3 푉푅푇 m Volume of resource tank

푚푀퐸 kg Mass of mining equipment 3 Mining 푉퐻푇 m Volume of holding tank Equipment ℎ퐻푇 m Height of holding tank (Rover) 3 푉퐸푥 m Volume of excavator bucket

푃푅 W Available power of the rover

Asteroid ore is expected to be distributed across the asteroid’s surface, and interior. As such, the rover will need to carry out multiple trips to extract unit blocks of asteroid ore. During a single trip, the rover will travel to a block, and extract the ore into the rover's holding tank. It will then transport the ore back to the spacecraft, where it will be unloaded into the reaction chamber. The rover will then depart on its next trip, while the spacecraft begins processing the ore. Upon the second return to the spacecraft, the rover will need to dispose of any waste material left over from processing, before the next load can be processed. This may require an additional transportation operation to a

128 Scott Dorrington – June 2019 Chapter 4: Parametric Mining Rate Model designated refuse pile, or could simply be dumped to space. The sequence of operations is summarized in table 4.4. A timeline of these operations over two trips is displayed in figure 4.1.

Table 4.4 Sequence of rover actions in a single mining trip.

No. Operation Description Transport 1 Travel from spacecraft to block of ore. (out) Free ore from surface and break down large rocks into smaller 2 Fragmentation sizes. 3 Excavation Digging up ore and depositing it into the holding tank. Transport 4 Return to the spacecraft with the extracted load. (back) Transfer the ore from the rover’s holding tank to the reaction 5 Unloading chamber. Remove and dispose of waste from previous processing. The spacecraft will process the ore to extract the mineral 6 Processing resources. (Repeat for next trip)

Figure 4.1 Timeline of operations over multiple trips.

Scott Dorrington – June 2019 129 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

4.3 Energy and Time Requirements of Operations

4.3.1 Block Definitions To quantify the energy and time requirements to extract a volume of ore, a block body model is used to divide the bulk volume of the asteroid into unit blocks of equal volume (shown in figure 4.2). Each block 푖 may be characterized by the following parameters:

푉푖 bulk volume of block;

푓 particle volume fraction of block;

풱푖 = 푓푉푖 volume of ore in block;

푚푖 = 휌푏V푖 = 휌푔풱푖 mass of ore in block;

휆푖 relative longitude of block from spacecraft position;

푧푖 depth of centre of block below spacecraft position;

ℎ푖 height of block; and

푤푖 width of block (or spacing of blocks).

Figure 4.2 Unit block definitions.

Due to the small size of the blocks in comparison to the asteroid, the particle size distribution (and therefore the porosity) is not expected to be equal in each block. As demonstrated in figure 4.2, some blocks may contain only small particles, while others may contain mostly large boulders (or partial boulders). Spatial variations of particles

130 Scott Dorrington – June 2019 Chapter 4: Parametric Mining Rate Model may also contribute to variations in particle distribution throughout the asteroid. Surface observations of asteroid 25143 Itokawa revealed rough regions of numerous boulders, and smooth regions of fragmented debris, with little variation of the chemical composition across the surface [103].

To simplify calculations, it is assumed that the particles in the asteroid will be distributed randomly, such that on average, each unit volume has the same particle size distribution, grain density, porosity, and mineral compositions as that of the entire asteroid.

4.3.2 Regolith and Rock Fractions The volume of each block should be chosen such that it can be carried by the rover in a single trip. This volume will be dependent on the volume of the rover’s holding tank

푉퐻푇 and the packing density of ore in the tank.

Due to the volume and size constraints of the holding tank, large boulders are required to be broken down in order to fit into the holding tank. We define a maximum particle size 퐷, above which particles are to be broken down. This size divides the ore in each block into two regimes:

• Regolith - containing small particles (г0 ≤ г ≤ 퐷); and

• Rock - containing large particles (퐷 ≤ г ≤ г1).

The volume fractions of regolith and rock particles in each block can be calculated from integrals of the number density function:

퐷 풱푅푒푔 1 4휋 ln(퐷/г0) 푓 = = ∫ г3푛(г). 푑г = , (4.9) 푅푒푔 풱 풱 3 ln(г /г ) г0 1 0 and:

г1 ( ) 풱푅표푐푘 1 4휋 3 ln г1/퐷 푓푅표푐푘 = = ∫ г 푛(г). 푑г = , (4.10) 풱 풱 퐷 3 ln(г1/г0) where 풱 is the total volume of particles in the asteroid (from Eq. 4.6).

The bulk volume that the regolith and rock will occupy in the holding tank will be determined by the packing density of particles in a rectangular (or cylindrical) container. The packing density 휂 defines the ratio of the volume of particles to the volume of the container.

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After they are broken down, the rocks are assumed to form randomly packed uniform spheres of diameter 퐷, with a total volume unchanged during the process. The bulk volume of the rocks is expected to increase after fragmentation, due to the voids between the particles. The packing density for random loose packing of uniform spheres has been found experimentally to be 0.64 [191], hence: 휂푅표푐푘 = 0.64.

The regolith will be excavated directly into the holding tank, and will retain the original size distribution (truncated at the maximum size 퐷). Numerical simulations of random packing of spheres with power-law size distributions show a maximum packing density of 0.7 for a power index of -3.3 [192]. We can expect that the packing density of the regolith in the holding tank will be somewhere between this value and the volume fraction of particles in the entire asteroid. If the rock was placed on the bottom of the holding tank, denser packing could result from regolith filling the pores between the rocks. From the packing density and volume fractions of regolith and rock, we can calculate the bulk volume of the regolith and rock in the holding tank:

풱푅푒푔 푓푅푒푔V푖 V푅푒푔 = = , (4.11) 휂푅푒푔 휂푅푒푔 and:

풱푅표푐푘 푓푅표푐푘V푖 V푅표푐푘 = = . (4.12) 휂푅표푐푘 휂푅표푐푘

The size of the blocks can then be determined such that the sum of the bulk volume of regolith and rock are equal to the volume of the holding tank:

−1 푓푅푒푔 푓푅표푐푘 푉푖 = ( + ) V퐻푇 . (4.13) 휂푅푒푔 휂푅표푐푘

This volume might be smaller than the volume of the holding tank, as the fragmented rocks would occupy a larger volume in the holding tank due to the increased porosity. Figure 4.3 shows the packing of regolith and rock into the holding tank.

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Figure 4.3 Packing of regolith and rock in holding tank.

4.3.3 Transportation Various modes of locomotion can be considered for the rover. For driving or walking, the velocity of the rover 푣 will likely be slow, restricted by the of the asteroid 푣푒푠푐 (on the order of 10’s of cm/s for small asteroids). If propulsion is used, higher velocities may be attained, however this would require the consumption of propellant, and so is not favourable. (Extracted regolith may also be used as reaction mass for a , however large amounts of regolith would be required.)

For flat terrain, driving is the most energy efficient mode of transport, however overcoming obstacles in the terrain may be easier with a walking rover. The power requirements for a rover to drive over the surface can be estimated from balancing the torque provided by the drive motor to the rolling resistance force caused by the wheels sinking into the regolith:

휏 = 푑푤퐶푟푟푔퐴(푚푅 + 푚푖) , (4.14) where:

푚푅 is the mass of the rover;

푚푖 is the mass of ore load being transported;

푑푤 is the wheel diameter;

푔퐴 is the gravitational acceleration of the asteroid; and

퐶푟푟 = √푑푤/푧푤 is the rolling resistance coefficient.

The distance 푧푤 that the rover’s wheels sink into the regolith may vary depending on the geological and mechanical properties of the asteroid surface and regolith. From this torque, we can calculate the velocity of the rover as a function of the power 푃푅 available

Scott Dorrington – June 2019 133 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions from the rover's power supply. This velocity is limited by the escape velocity 푣푒푠푐 of the asteroid:

푃푅 푣푇 = 푚푖푛 { , 푣푒푠푐} . (4.15) 퐶푟푟푔퐴(푚푅 + 푚푖)

The rover’s power supply is likely to be sized according to the requirements of more energy demanding operations of fragmentation and excavation. As such, there is expected to be adequate power to achieve driving velocities equal to the escape velocity of the asteroid.

The time required to transport each block to and from the spacecraft will be determined by the velocity of the rover, and the distance of the block from the spacecraft. The distance can be broken down into two components: the relative depth of the block below the spacecraft 푧푖 , and the distance across the surface, defined by the relative longitude 휆푖, calculated from the great circle arc separating the two points:

cos 휆 = sin 휃1 ∙ sin 휃2 + cos 휃1 ∙ cos 휃2 ∙ cos(휑1 − 휑2) . (4.16)

The distance of the block will depend on the path taken by the rover, and may be dependent on the sequencing of blocks removed by the rover. The upper limit of the distance will be the sum of the vertical and horizontal components:

푑(푧푖, 휆푖) = |푧푖| + |푅휆푖| . (4.17)

4.3.4 Fragmentation Fragmentation is the process of freeing ore from surrounding rock and breaking it into fragments of manageable size. This may be achieved through various mechanical processes, or through blasting. Blasting of asteroid ore would create problems with containing the ejected material, and the need to supply explosives from Earth. The amount of explosives required to fragment a 100 m diameter asteroid is approximately 310 tonnes [193].

Mechanical fragmentation is achieved through drilling or cutting processes that apply a force that exceeds the compressive strength of the rock. The specific energy 푆퐸 required to fragment a unit volume of rock is dependent on the strength of the rock, the mean particle size of fragments, and the geometry of the drill bit or cutting head. Specific energies for mechanical drilling range from 2x108 to 5x108 J/m3 [194].

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For an average fragment size 퐷, the total work needed to fragment all particles within a block can be calculated from the volume fraction of rocks in the block (Eq. 4.10):

ln(г1/퐷) 휌푏 푊푓푟푎푔 = 푓푅표푐푘풱푖푆퐸(퐷) = 푉푖. 푆퐸(퐷) , (4.18) ln(г1/г0) 휌푔 where 푆퐸(퐷) is the specific energy (J/m3) of fragmentation for particle size 퐷.

The time required for fragmentation can be estimated from the energy requirements, and the power available from the rover:

푊푓푟푎푔 푇푓푟푎푔 = . (4.19) 푃푅

4.3.5 Excavation Once the ore has been fragmented, it must be scooped up by the excavator and deposited into the rover’s holding tank for transportation back to the spacecraft. The height lifted will be the sum of the height of the holding tank ℎ퐻푇 and half the height of the block ℎ푖. The energy required to carry out this operation can be estimated from the work required to lift a mass of ore 푚푖 to this height:

ℎ 푊 = 푚 푔 (ℎ + 푖) . (4.20) 퐸푥 푖 퐴 퐻푇 2

The total time required to excavate the block will be determined by the vertical velocity that the ore is lifted, and the total number of lifts required (limited by the volume of the excavator 푉퐸푥):

푉퐻푇 (2ℎ퐻푇 + ℎ푖) 푇퐸푥 = , (4.21) 푉퐸푥 푣푧 where 푣푧 is the vertical velocity of lifting. The vertical velocity is required to be limited by the escape velocity of the asteroid. If the ore is not enclosed by the excavator, higher velocities would result in particles drifting off. If the ore is enclosed, the lifting motion of the excavator’s arm could create a torque that tips the rover.

4.3.6 Chemical Processing The methods used to process the asteroid ore will depend on the types of mineral resources desired to be extracted. Volatiles such as water vapour and CO2 can be extracted through the process of pyrolysis, where chemical bonds are broken down at high temperatures.

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Volatiles of different types are released at different rates dependent on the temperature and duration of heating. Pyrolysis is often used to study the chemical composition of meteorites and asteroid simulants, where samples are heated at gradually increasing temperatures, and the volatile fractions produced are measured with a mass spectrometer [195]. Pyrolysis tests of carbonaceous chondrite meteorites show that water vapour begins to be released at temperatures around 200oC, with the cumulative yield of water vapour increasing to 100% at temperatures of 900oC [196].

The high temperatures required for pyrolysis can be achieved by concentrating sunlight into the reaction chamber using large mirrors. Concentrated sunlight can cause localized “spalling”, where small fragments are chipped off the surface, releasing volatiles in the process.

The commercial company TransAstra has demonstrated this method with large scale tests of samples of asteroid simulant [197]. The experiment used a primary mirror with a collection area of 140 m2 and secondary reflector of area 80 m2 to direct sunlight into a chamber with an average intensity of 150 W/cm2. During the test, a sample with an initial size of 1.32 kg produced 220 g of spall material and 130 g of H2O and CO2 volatiles (37% volatile yield). Further large scale tests (on 100 kg samples) are expected to produce processing rates of 10’s of tonnes of water per week.

The specific energy required for processing can be estimated by the enthalpy of dehydration ∆ℎ푝푟표푐 of hydrated silicates (found to be around 2800 kJ/kg) [197]. The specific energy of processing can then be found from this enthalpy and the bulk density of the block:

푊푝푟표푐 = 휌푏∆ℎ푝푟표푐 . (4.22)

Using typical values of bulk density of the order of 2000 kg/m3, the energy requirements for processing would be around an order of magnitude greater than that required for fragmentation. Therefore, the power available from the spacecraft for processing should also be an order of magnitude greater than that of the rover.

Calculation of the time required to carry out this processing would be dependent on the geometry of the concentrating mirrors, and other thermal and chemical properties of the reaction. For example, the rates of chemical reactions are dependent on the surface area of the reactant material. Additional mineral beneficiation steps such as comminution

136 Scott Dorrington – June 2019 Chapter 4: Parametric Mining Rate Model could be added to crush the ore into finer particles, increasing the speed and efficiency of the chemical reaction.

Ideally, the processing system should be designed such that the processing time is equal to the sum of the fragmentation, excavation, and transportation times, such that a load of asteroid ore can be processed before the arrival of the next load.

4.3.7 Unloading The processing of asteroid ore will produce waste in the form of spall particles. This must be disposed of before the next load of ore is processed.

The simplest method for disposal of this waste material is to eject it to space at velocities greater than the escape velocity of the asteroid. Some of the material may settle back on the asteroid, which may interfere with the mining process by creating an additional layer of waste material over the orebody, or by coating the solar panels and mirrors, thereby reducing the available power. The waste material may alternatively be consolidated in a refuse pile on the surface of the asteroid; however, this may also interfere with the extraction of ore beneath it.

An alternative strategy could be to contain the waste material in a bag or net. This would mitigate the interference of waste material on the mining process, and would allow the opportunity to retrieve the material in future missions. (As noted by Ross [3], waste material may be useful as radiation shielding and could therefore be sold to the human space exploration market.)

The energy and time required to load and unload the ore and waste is assumed to be negligible compared to that required for processing.

4.4 Mine Design

In order to produce a desired mass of mineral resources 푀푚푟, a large volume of asteroid ore must be mined. The total volume of the mine will be dependent on the concentration of mineral resources and the efficiency of the processing equipment:

푀푚푟 푉푚푖푛푒 = . (4.23) 푐푚푟휌푏휀푝푟표푐

This volume can be extracted using surface mining methods, where an area around a position on the surface is excavated down to a constant depth. On a flat surface, the extent of the area could be defined by a number of geometric shapes. Projecting these

Scott Dorrington – June 2019 137 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions areas onto the surface of the sphere will result in spherical polygons with non-Euclidean geometry. A circular area projected on the sphere produces a spherical cap, while a rectangular area produces a spherical rectangle with edges defined by great circle arcs.

Consider a surface mine defined by a spherical cap with an epicentre at the spherical coordinates (푅, 휃0, 휑0) on the surface of the asteroid. The extent of the cap can be defined by a cone subtended by an angle 2훼, with a constant depth 푍 across the region, forming a segment of a spherical shell (shown in figure 4.4).

Figure 4.4 Geometry of a spherical cap.

Extending the shape of the surface mine to lower depths may cause problems with the stability of the walls of the mine. The mine can be stabilized by extracting an addition volume to produce walls of constant slope. This method is often used in terrestrial open pit mining. In spherical coordinates, the shape of the wall can be defined by a logarithmic spiral with a constant slope 훾 (shown in figure 4.5):

푟 = 푎푒푏휃 , (4.24) where:

1 푏 = 휋 , (4.25) tan (2 − 훾)

푎 = (푅 − 푍)푒−푏훼 . (4.26)

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Figure 4.5 Geometry of open pit mine with constant wall slope.

4.4.1 Mine Volume The volume of the spherical cap and wall sections of the mine can be calculated by a volume of revolution in local spherical coordinates (푟′, 휃′, 휑′) centred at the epicentre:

. ′2 ′ ′ ′ ′ 2휋 3 3 푉푐푎푝 = ∭ 푟 sin 휃 . 푑푟 . 푑휃 . 푑휑 = (1 − cos 훼)[푅 − (푅 − 푍) ] , (4.27) 3 푉표푙 where:

푉표푙 = {(푟′, 휃′, 휑′) | (푅 − 푍) ≤ 푟′ ≤ 푅, 0 ≤ 휃′ ≤ 훼, 0 ≤ 휑′ ≤ 2휋} , (4.28) and:

. ′2 ′ ′ ′ ′ 푉푤푎푙푙 = ∭ 푟 sin 휃 . 푑푟 . 푑휃 . 푑휑 푊 2휋푎3 (4.29) = [푒3푏훽(3푏 sin 훽 − cos 훽) − 푒3푏훼(3푏 sin 훼 − cos 훼)] 3(9푏2 + 1) 2휋 + (cos 훽 − cos 훼)(푅 − 푍)3 , 3 where:

′ 푊 = {(푟′, 휃′, 휑′) | (푅 − 푍) ≤ 푟′ ≤ 푎푒푏휃 , 훼 ≤ 휃′ ≤ 훽, 0 ≤ 휑′ ≤ 2휋} , (4.30)

1 훽 = ln(푅/푎) . (4.31) 푏

The extent and depth of the mine can be chosen in any combination (훼, 푍), such that the total volume contains the desired mass of mineral resources (Eq. 4.23). Deeper depths would require multiple layers of ore blocks to be removed, making the terrain difficult

Scott Dorrington – June 2019 139 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions for the rover to traverse. Shallower depths will require the rover to cover larger areas of the surface, increasing the total travel distance. As the extent of the mine approaches 180o, the total volume and surface area will approach that of a spherical shell.

4.4.2 Total Energy and Time Requirements In section 4.3, equations were defined for the energy requirements to perform operations on unit blocks of asteroid ore. The total energy and time required to extract the entire volume of the mine can then be found from the sum of these values over all blocks. Assuming all blocks have the same properties, the total energy and time to extract the mine can be estimated from the total number of blocks in the mine (rounded up to the nearest integer):

푉 푁 = 푚푖푛푒 . (4.32) 푉푖

The total energy and time to fragment and excavate the ore in the mine would then be the found by multiplying the corresponding equations by the number of blocks.

The total travel time can be calculated from the average travel distance of the blocks, defined by the position of the centroid of the mine. Due to the symmetry of the mine, the centroid will be located below the epicentre of the mine, at a depth of:

. 1 푧̅ = 푅 − 푟̅ = 푅 − ∭ 푟′3 sin 휃′ . 푑푟′. 푑휃′. 푑휑′ 푉푐푎푝 푉표푙 (4.33) 3 [푅4 − (푅 − 푍)4] = 푅 − . 4 [푅3 − (푅 − 푍)3]

The total travel time can then be found from:

푧̅ + 푅휆0 푇푇 = 2 , (4.33) 푣푇 where 휆0 is the relative longitude of the epicentre from the spacecraft.

As discussed, we can assume that the processing equipment is sized such that the total processing time is equal to the sum of the fragmentation, excavation, and transportation times.

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4.5 Numerical Example To demonstrate the methods developed in this section, we can estimate the total time required to extract a mass of 30 tonnes of water from a carbonaceous asteroid. Consider an asteroid with bulk properties (taken from values of asteroid 25143 Itokawa) presented in table 4.5, system parameters in table 4.6, and mine shape parameters in table 4.7.

Table 4.5 Asteroid bulk properties.

Parameter Description Value Units Note 푀 Mass of asteroid 3.51x1010 kg * 7 3 푉푏 Bulk volume 1.84x10 m * 푅 Mean radius 163.77 m * 3 휌푏 Bulk density 1950 kg/m * 3 휌푔 Average grain density 2270 kg/m ** -6 г0 Minimum particle size 1x10 m ***

г1 Maximum particle size 20 m *** 푛 Size distribution power index 3 - ***

푐푚푟 Mineral concentration 10 wt% **** * Value for 25143 Itokawa [102] ** Average grain density of CI meteorite [198] *** Using power law from [104] **** Water concentration from [117]

Table 4.6 System Parameters.

Parameter Description Value Units

푚푅 Mass of rover 100 kg 3 푉퐻푇 Volume of holding tank 1 m

ℎ퐻푇 Height of holding tank 0.5 m 3 푉퐸푥 Volume of excavator bucket 0.004 m

퐶푟푟 Rolling resistance coefficient 0.1 -

휀푝푟표푐 Efficiency of processing equipment 0.9 - 퐷 Diameter of rocks 0.1 m 8 3 푆퐸푓푟푎푔 Specific energy of fragmentation 3x10 J/m

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Table 4.7 Mine shape parameters.

Parameter Description Value Units 푍 Depth 1 m 훾 Wall slope 45 deg

휆0 Relative longitude of epicentre from spacecraft 3 deg

Using the equations presented in previous sections, the following properties can be calculated: • Volume of mine: 174.74 m3; • Diameter of mine: 13.91 m; • Average travel distance: 9.1 m; • Total number of blocks: 229; and • Number of lifts of excavator per block: 250.

The total times required to complete the various operations are displayed in figure 4.6 as a function of the available rover power. Dividing the total mass of resources extracted (30 tonnes) by these times, we can estimate the daily mining rate of the rover as a function of available power (shown in figure 4.7).

Figure 4.6 Total operation times as a function of available rover power.

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Figure 4.7 Total mining rate as a function of available rover power.

4.6 Discussion The results of the numerical example show that fragmentation is the most energy and time intensive process. As such, the rover’s power supply should be sized to these requirements. Near-Earth asteroids have average semi-major axes of around 1 AU, making solar power an ideal power source for the rover. Large solar arrays may interfere with the transportation of the rover over the surface, and as a result, solar array areas (and therefore power levels) would be limited. Solar array areas of 5 – 10 m2 could be feasible, producing powers of 1.22 – 2.24 kW with mining rates of 217 – 409 kg/Day (assuming a solar panel efficiency of 18%).

The power available from solar arrays will be dependent on the position of the rover on the asteroid and the orientation of the asteroid’s spin axis. The best case would result from high latitude positions, with a spin axis that is aligned along the sun-asteroid vector, such that the solar arrays are in constant sunlight. The worst case would result from equatorial latitudes with a spin axis perpendicular to the sun-asteroid vector, where approximately half of the available stay-time would be in shadow, and the solar array area would need to be doubled to provide the same mining rate. For a 5 m2 solar panel in full sunlight, the desired mass of 30 tonnes of water could be extracted during a 6 month stay-time.

Having a separate mining vehicle also affords the option of leaving it at the asteroid to continue ore extraction while the spacecraft returns the load to Earth. This would extend the timeframe available to mine beyond the stay-time restricted by the launch and arrival dates of the heliocentric transfers. This option would require an additional

Scott Dorrington – June 2019 143 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions resource tank to be delivered to the asteroid during the first trip. Upon returning to the asteroid, the spacecraft could swap the empty resource tank for the full one, and depart immediately for Earth, greatly reducing the round trip time.

In calculating the volume and extent of the mine, it was assumed that the particle size distribution and mineral concentrations were constant in each block. Exploration of the asteroid may reveal concentrations of mineral resources throughout the asteroid. In this case, a spatial distribution function for the mineral concentration and other properties could be generated as a function of the position in the asteroid 푐푚푟(푟, 휃, 휑). Calculation of the total mineral mass would then require integration over the volume of the mine.

For large asteroids of the order of 100’s of meters, the mass of ore extracted in a single mining trip is likely to be negligible in comparison to the mass of the asteroid. Over multiple mining trips, or for smaller asteroids, extraction of large proportions of the asteroid’s mass will likely change the gravitational acceleration 푔퐴 . Removal of material would also change the internal mass distribution, affecting the mass moment of inertia and overall structural stability of the rubble pile asteroid. The incremental disposal of waste material to space would also act as a mass driver, generating small delta-Vs that may affect the rotational dynamics and (on a very small scale) the of the asteroid. (These effects would likely be negligible due to the low particle velocities in comparison to the exhaust plumes of chemical propellants, and may be controlled by the timing and direction of the ejections.)

In considering the environmental impacts, the waste material generated from mining an entire asteroid will be significantly large. If this mass is ejected to space, the material will likely disperse throughout the asteroid’s heliocentric orbit. If the asteroid is in an Earth-crossing orbit, this would lead to an increased probability that some of the waste material would impact with the Earth in the form of an artificially generated meteor shower. While the material would likely be too small to survive entry through the atmosphere, it could cause an increased hazard to Earth orbiting satellites. Over long time periods, interactions with the and other solar and gravitational forces could cause the material to migrate to other orbits, further dispersing the debris. This material may also eventually be accreted by other asteroids.

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4.7 Conclusion This chapter has outlined analytical methods that can be used to calculate the energy and time requirements to extract a given mass of mineral resources from asteroid ore. This was used to form a parametric model to estimate the mining rate that could be achievable as a function of spacecraft subsystem and asteroid geological parameters.

A numerical example showed that for an asteroid of the size of 25143 Itokawa (~300 m diameter), a spherical shaped surface mine with a depth of 1 m and diameter of 9.1 m would be required to extract 30 tonnes of water from the asteroid ore. Power supplied with a 5 m2 solar array, generating 1.22 kW could achieve this extraction during a 6 month stay-time at the asteroid (a mining rate of 217 kg/day). Increasing the rover power supply up to 5 kW can produce larger mining rates of 800 kg/day (this value will be used in analysis in the subsequent chapters).

The energy requirements for chemical processing of regolith were found to be an order of magnitude larger than those required for fragmentation and excavation, therefore the spacecraft power should also be an order of magnitude greater than that of the rover.

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5 SUPPLY CHAIN NETWORK

This chapter considers the design of a supply chain network for the delivery of asteroid- derived resources to customers in Earth orbit 9. The network features orbital nodes consisting of an asteroid source, an orbital propellant depot, arrival and departure parking orbits, and customer spacecraft. Edges between nodes represent transfer trajectories modelled as impulsive manoeuvres. Two classes of spacecraft vehicles are assigned to transport the product through various edges in the network. The optimal location of nodes, and vehicle routes through the network are determined, with the objective of maximizing either the total sellable mass delivered to customers, or the total net present value of the asteroid mining venture. The depot and parking orbit locations are to be selected from a discrete set of candidate locations with variable orbital radius and orientation. Vehicle routes are to be chosen from two candidate routes that ensure reusability over multiple mining trips. A mathematical formulation is presented to describe the objective function as a function of the candidate node locations and vehicle routes. A set of constrained optimization problems is developed to determine the optimal location and orientation of parking orbits, depot location, and vehicle routes though the network (a location-routing problem). Solution methods are developed using integer programming. A numerical example is presented to illustrate the method for the case of maximizing total sellable mass delivered from a near-Earth asteroid to a single customer in Geostationary Orbit.

9 This chapter has been reproduced in its entirety (with minor format changes) from work by the author published in the following journal paper: [199] S. Dorrington, J. Olsen, A location-routing problem for the design of an asteroid mining supply chain network, Acta Astronautica, 157 (2019) 350-373.

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5.1 Introduction Recent years have seen an increase in commercial activities taking place in Earth orbit. Reducing launch costs, and the miniaturization of satellites is driving the development of new space ventures that had previously not been considered technically or economically viable. One notable venture that has been receiving renewed attention is on-orbit servicing.

There are a number of services that satellites could provide to customers in Earth orbit, including: on-orbit refuelling of satellites to extend their operational life; on-orbit maintenance of satellites to prevent satellite loss due to deployment or component failures; on-orbit construction of satellites and space stations through additive manufacturing; and transportation of goods to various orbits around cislunar space.

To date, these services have not been widely implemented due to the large costs associated with the resupply of materials (particularly propellant) required to support these operations. While complex components such as electronics would be required to be supplied from Earth, raw materials such as water-based propellants and metals for construction materials could be sourced from near-Earth asteroids [3].

There are a myriad of near-Earth asteroids accessible with low delta-Vs, some of which are considered more accessible than the Moon [1, 2]. These asteroids are thought to contain high abundances of resources such as water (in the form of hydrated minerals), nickel-iron metals, platinum group metals, and other volatiles [3]. Resources can be extracted from these asteroids and delivered to customers in Earth orbit, avoiding the high energy and cost expenditure of escaping the Earth’s gravity well. With careful planning, it may become more efficient to source these resources from near-Earth asteroids, rather than launching them from Earth.

5.1.1 Asteroid Mining The principle aim of asteroid mining is to supply asteroid-derived materials to customers in Earth orbit. The goal is to create a self-sustaining space industry that is less dependent of supplies from Earth. An asteroid mining industry would require the establishment of an orbital infrastructure consisting of propellant depots, and a fleet of mining and servicing spacecraft. Asteroid resources would be delivered to orbital depots, where they would be stockpiled for distribution to customers. The design of this

Scott Dorrington – June 2019 147 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions supply chain will require consideration of the location of facilities, spacecraft routing, and manoeuvre operations.

Asteroid mining missions would differ significantly from traditional space missions in two main ways. The first is the need to return large masses that may be many times the mass of the spacecraft retrieving them. This would require large amounts of propellant for the return trip which, if supplied from Earth, would become prohibitively large. A solution to this is to produce the propellant required for the return trip from materials at the asteroid (in situ propellant production) [181]. In this case, a mining spacecraft could be launched with only enough propellant to reach the asteroid, lowering the cost to that of a one-way mission. A portion of the material extracted from the asteroid would need to be consumed as propellant for the return trip. The fraction of propellant required for the return trip would be dependent on the return delta-V (∆푉), and the efficiency of the propulsion system (measured by the specific impulse 퐼푠푝, or the exhaust velocity 휐푒). As a result, the mass of the load of asteroid material will decrease over the trip. For example, a near-Earth asteroid with return delta-V of 4 km/s would require 60% of the material to be consumed to return the load to Earth orbit (using chemical propulsion with 4.4 km/s exhaust velocity). This fraction would likely become even greater once plane change manoeuvres in Earth orbit are taken into consideration.

The second major difference in asteroid mining missions is the need for multiple return trips. This presents new challenges in preliminary mission design with the need to consider the cadence of multiple optimal launch opportunities, and the on-orbit operations between the return of the first mission, and the departure of the next.

5.1.2 Transportation Problems Transportation and supply chain problems have been well studied in Operations Research [200]. These problems deal with the transportation of resources or goods from one location to another at a minimal cost. Transportation networks are commonly described by a series of nodes defining locations of facilities, and connecting edges describing transportation routes between nodes. Common problems in supply chain network design include the facility location problem, the vehicle routing problem, and the location-routing problem.

The facility location problem aims to determine the optimal location of facilities (supply points) to minimize transportation costs to a set of customers (demand points). Facility and customer nodes are defined by geographical locations, and the transportation costs

148 Scott Dorrington – June 2019 Chapter 5: Supply Chain Network over edges are expressed as a function of the distance between nodes (either by Euclidean distance, or a number of other metrics [201]).

The vehicle-routing problem (VRP) is concerned with determining optimal routes of vehicles to transport material between various nodes, with known locations. The traveling salesman problem (TSP) is a classic example of a VRP, with the aim of determining the shortest route to visit each of a set of customers.

The location-routing problem (LRP) combines these two problems, requiring the simultaneous selection of facility locations, and vehicle routing between facilities. A number of variants of the location-routing problem have been classified based on features of the problem such as number of facilities, size of vehicle fleets, and the hierarchical structure of facilities in the network [202, 203].

Transportation problems are commonly formulated and solved using graph theory and integer programming. Numerous mathematical formulations have been developed for each type of problem, primarily differing with the choice of decision variables. Facility location problems are often formulated with decision variables representing the locations of the facilities selected from a set of candidate locations. Vehicle routing problems often use vehicle or commodity flow formulations with decision variables indicating the flow of vehicles or commodities along edges between nodes. This approach is also used in networks requiring the selection of multiple facilities, such as in multi-level distribution networks [204-207].

5.1.3 Space Transportation Networks The concepts and methods used to study terrestrial transportation networks may be applied to networks in space. Space transportation networks may be described with nodes representing orbital locations (as well as launch and landing sites on planetary bodies), and edges representing transfer trajectories between orbits. Orbital mechanics introduces a number of key differences that increase the complexity of transportation problems, including: 1. The positions of satellites are described by orbits in five dimensions (the classical orbital elements); 2. The positions of satellites are constantly moving in their orbit making it a time- varying problem; 3. The edge cost between orbits is a function of the delta-V of the orbital transfer; and 4. There are multiple paths (transfers) between two orbits with varying delta-V and transfer times.

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Several studies have considered applying vehicle routing problems to networks in space [208-212]. These studies use a multi-commodity flow formulation to determine routes that maximize the flow of commodities through a network of pre-defined orbital nodes. Numerous commodities can be considered such as crew, consumables, and payloads, as well as the vehicles transporting them over various paths through the network. Dynamic network models have also been developed, where each of the orbital nodes is duplicated at various time steps throughout the mission timeline. These time-expanded networks can be used to model numerous transfer edges between orbital nodes with different departure and arrival dates [213]. These studies have been used to explore potential architectures in the design of space missions, with applications to human and Mars.

Other studies of space transportation logistics have focused on adapting the traveling salesman problem for studies of multiple satellite rendezvous missions including on- orbit servicing and refuelling [214-216], and active removal [217-219]. A single-facility location problem has also been developed for determining the optimal radius of an orbital factory in heliocentric orbit [220].

The space transportation networks developed in [208-212] contain only a limited number of orbital nodes, defining generic orbits such as Low Earth Orbit, Geostationary Transfer Orbit, and Geostationary Orbit (as well as comparable orbits around the Moon and Mars), each with a defined orbital radius and shape. While these nodes allow for the determination of optimal routing strategies of spacecraft through the network, there is little ability in the optimization of the shape and orientation of the individual orbits.

In this chapter, we will develop a space transportation network with orbital nodes defining generic operations conducted in the space mission (such as parking orbits, and incoming/outgoing hyperbolas). Numerous candidate nodes are added of each orbit type, representing various parameters of the orbit such as orbital radius and orientation. This approach allows not only for the determination of optimal spacecraft routes, but also for the optimal design of the orbits and trajectories of the spacecraft along these routes.

5.2 Asteroid Mining Supply Chain Network We wish to design a supply chain network that delivers asteroid-derived material to customers in Earth orbit. In designing the supply chain, we desire to determine the optimal location of orbital facilities, manoeuvre operations, and vehicle routing through

150 Scott Dorrington – June 2019 Chapter 5: Supply Chain Network the network. The optimal design is achieved by solving a location-routing problem with the objective of maximizing either the total sellable mass of product delivered to the customers, or the total net present value of the operation.

In this chapter, we outline a mathematical formulation that describes the supply chain network as a location-routing problem, and introduce solution methods using integer programming.

5.2.1 Basic Network We model the flow of materials in an asteroid mining operation as a single-product, multi-stage supply chain network (shown in figure 5.1). The network consists of a set of orbital nodes, transfer edges, and spacecraft vehicles used to distribute a single homogeneous product from an asteroid source to a set of customers in Earth orbit.

Figure 5.1 Layout of the supply chain network.

5.2.1.1 Orbital Nodes Each node in the network specifies an orbital location in heliocentric or cislunar space. The network consists of five nodes partitioned into four layers: a source (layer I); a set of arrival and departure parking orbits (layer II); a propellant depot (layer III); and customers (layer IV). Figure 5.1 shows the network as a set of nodes and edges in a layer diagram.

The source (or asteroid) node (AST) describes the asteroid being mined (supply points of the product). The arrival (ARR) and departure (DEP) parking orbits are orbital locations in which orbital insertion and plane change manoeuvres are carried out upon arrival/departure of Earth orbit (transhipment points). The propellant depot (DEPOT) is an orbital facility capable of storing the product returned from the asteroid. This facility acts as both a warehouse and distribution centre, and may possess processing and maintenance capabilities for the product and spacecraft. The customers (CUST)

Scott Dorrington – June 2019 151 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions describe a set of customer spacecraft sharing a single orbit (demand points of the product).

5.2.1.2 Transfer Edges The edges connecting the nodes define transfer trajectories between orbits. The transfers are modelled as impulsive manoeuvres with delta-V burns carried out at the start and end of the transfer edge. The edge costs are a function of the total delta-V of the transfer between the start and end nodes. Section 5.3 outlines the calculation of the delta-Vs for transfers between various orbital nodes.

5.2.1.3 Product The product being transported may be a mixture of asteroid-derived resources at various grades of refinement. As we are using in situ propellant production, we require a portion of this product to be used as propellant to carry out the orbital transfers between the various nodes of the network. This portion of the product is required to be water-based propellant. (Water can be electrolyzed – by means of solar energy – to produce liquid or gaseous H2 and O2 that can be combusted to produce thrust [147].) The total mass of the load of product will decrease over each transfer edge as the propellant mass is consumed. The remainder of the product delivered to the depot and customers may also be propellant, or any other resource available at the asteroid.

5.2.1.4 Spacecraft The transportation of material throughout the network will be conducted by a fleet of spacecraft. The edges of the network can be partitioned into two echelons [221] based on the type of orbital transfer: interplanetary and cislunar. We introduce two classes of spacecraft to operate within these two echelons:

1. Mining spacecraft (MS) – interplanetary spacecraft operating between layers I, II, and III. Responsible for carrying out the heliocentric transfers to/from the asteroid, delivering the mining equipment to the asteroid, and returning the extracted product to cislunar space; and 2. Transport spacecraft (TS) – spacecraft used to transport the returned material around cislunar space (between layers II, III, and IV). Responsible for the distribution of product from the depot to the customers, and retrieval of the product from the arrival parking orbit.

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Routes through the network will require the use of both types of spacecraft. The two spacecraft types are each characterized by a dry mass 푀푠 and exhaust velocity 푣푠. Each spacecraft type may have a different design to achieve its goal. For example, the mining spacecraft is considered to have a larger mass due to the higher performance requirements of the communication system and radiation shielding in interplanetary space.

5.2.2 Expanded Network The network in figure 5.1 shows the basic components of the supply chain network. This network is expanded in figure 5.2 to include multiple source and customer locations (layers I and IV), and a set of candidate facility locations (layers II and III). In this network, the locations of the source and customer nodes may be known a priori from a set of possible trajectories to/from the asteroid, and a list of client spacecraft requesting delivery of asteroid material. These nodes are shown in black to indicate known locations. The location of the depot and the arrival/departure parking orbits are to be selected from a set of discrete candidate facility locations. These nodes are shown in white to indicate facility location decisions. The network in figure 5.2 can be thought of as a directed graph representing the various possible paths through the network. Each selection of candidate facility locations and vehicle route will define a path through the graph. An objective function describing the total edge cost of each path can then be used to compare the various candidates in order to find an optimal solution.

Figure 5.2 Supply chain network with candidate depot and parking orbit locations.

5.2.2.1 Source and Customer Locations We wish to consider multiple mining trips to an asteroid, whose position is constantly moving in an orbit around the Sun. We can handle the time-dependent element of the network by introducing discrete time intervals, representing individual mining trips.

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Each mining trip to the asteroid (indexed by 푡 ) consists of the following four components: 1. Mining operations – the mining spacecraft extracts and processes an initial load of asteroid material; 2. Asteroid-to-Earth transfer – the mining spacecraft returns the extracted material to Earth orbit; 3. On-orbit operations – the mining and transport spacecraft distribute the returned material to customers, then wait until the departure date of the next Earth-to- Asteroid transfer; and 4. Earth-to-Asteroid transfer – the mining spacecraft returns to the asteroid to commence the next mining trip.

Each of these components will have a varying duration dependent on the optimal launch and arrival dates of the Earth-to-Asteroid and Asteroid-to-Earth trajectories. This arrangement of the components was chosen such that the total duration of each mining trip is measured from the start of production of the asteroid material (the beginning of the supply chain).

An additional Earth-to-Asteroid transfer would be required prior to the beginning of the first mining trip (i.e. trip 푡 = 0) to deliver the mining spacecraft and mining equipment to the asteroid. This mission constitutes the initial capital investment required for the mining venture. The cost of deploying the propellant depot in Earth orbit should also be included in this capital investment.

The location of the asteroid source node for each mining trip may be specified from the launch/arrival dates of the trajectories to/from the asteroid. Each location may be ⃗⃗ characterised by the incoming asymptote 푺푖푛푡 of the Asteroid-to-Earth transfer, and the ⃗⃗ outgoing asymptote 푺표푢푡푡 of the Earth-to-Asteroid transfer. The source nodes in figure 5.2 represent the time-varying location of the asteroid over successive mining trips. Solid edges are displayed for a single mining trip only, to indicate that each time window is treated separately.

Each mining trip will extract an initial load of asteroid material 퐿푡 to be returned to a depot in Earth orbit, before being distributed to a set of customer orbits (indexed by 푘). The locations and demands of the customers may be specified from expected future markets of asteroid-derived resources. For example, we could consider two such markets: 1. Water-based propellant delivered to Geostationary Orbit (GEO) (Equatorial orbit of altitude 35,786 km); and

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2. Construction material, propellant, and other consumables delivered to Low Earth Orbit (LEO) (orbit of altitude 200 – 2,000 km).

These markets serve the purpose of refuelling communication satellites, on-orbit manufacturing, the support of space stations, and end-of-life disposal of satellites to the [222]. The mass returned to the depot orbit will be distributed to the customers according to a demand 푑푘 (the fraction of available mass to be delivered to each customer).

5.2.2.2 Candidate Depot and Parking Orbit Locations The locations of the depot and arrival/departure parking orbits are to be selected from a number of discrete candidate facility locations. Each candidate depot location (indexed by 푗) is defined by a discrete orbital radius 푟푗, and a discrete orbit normal vector 풏⃗⃗ 푗 (the specific vector of the orbit) chosen from characteristic orbital planes

(Equatorial 풏⃗⃗ 퐸푞, Ecliptic 풏⃗⃗ 퐸푐푙, or Lunar 풏⃗⃗ 퐿).

Each candidate arrival/departure parking orbit location (indexed by 푖) is defined by a discrete orbital radius (or apoapsis) 푟푖. To reduce the complexity of the network, both arrival and departure parking orbits are set to the same orbital radius. The normal vectors of the arrival and departure orbits are independent, and are characterized by aim vector orientation angles with respect to the incoming/outgoing asymptotes (outlined in section 5.3.1). These arrival and departure orientation angles (휃푚 and 휃푛 ) are each selected from a set of discrete candidate orientations (indexed by 푚 and 푛 , respectively). We may also consider two cases for the shape of the parking orbits: Circular (C); and Elliptical (E).

5.2.2.3 Vehicle Routing We wish to assign routes to the spacecraft in the two echelons to transport the product along a predefined path through the network: AST→ARR→DEPOT→CUST. To ensure reusability of spacecraft over successive mining trips, we require each spacecraft’s route to start and end at the same node. Each spacecraft will be stationed at a home node, carrying out tours to multiple nodes before returning at the end of the tour. Transport spacecraft are stationed at the depot, and may visit the customers, and the arrival parking orbit in their tours. Mining spacecraft are stationed at the asteroid, and may visit the arrival and departure parking orbits, and the depot on their tours. Based on these requirements, we can identify two candidate vehicle routes for the two echelons (displayed in figure 5.3).

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Figure 5.3 Candidate vehicle routes through the network.

In Route 1 (R1) the mining spacecraft delivers the material to the depot via an arrival parking orbit, where Earth Orbit Insertion and plane change manoeuvres are carried out. From here, it is delivered to the customers by a transport spacecraft. The mining spacecraft will remain at the depot until the next departure date, at which time it will transfer to a departure parking orbit where a plane change manoeuvre will place it on the correct orbital plane for the next departure asymptote. This route affords the opportunity for inspection and maintenance operations of the mining and transport spacecraft to be carried out at the depot facility.

In Route 2 (R2) the mining spacecraft delivers the material to an arrival parking orbit. From here, a transport spacecraft retrieves the load and delivers it to the depot, before distributing it to the customers. The mining spacecraft remains in the arrival parking orbit, and performs a plane change manoeuvre to a departure parking orbit to set up for the next departure asymptote. This route may require additional design considerations for the two spacecraft types, allowing for the transfer of the product between the spacecraft.

The customer node shown in figure 5.3 may consist of a constellation of multiple customer spacecraft in a single well-defined orbit. In this case, we are not concerned with the routing of transport spacecraft to individual customers, only the delivery to the orbit containing the set of customer spacecraft. Delivery to multiple customer orbits is assumed to be achieved by an N-to-N distribution method, where each customer orbit is serviced by a separate transport spacecraft [216]. Delivery using a 1-to-N method could also be considered, where a single transport spacecraft makes a delivery tour to multiple

156 Scott Dorrington – June 2019 Chapter 5: Supply Chain Network customer orbits. This would introduce an additional travelling salesman problem to determine the optimal sequencing of customer orbits to visit.

5.2.2.4 Decision Variables We wish to select the location and orientation of the depot and parking orbits from a list of candidate locations. The depot is a large piece of infrastructure that would be very costly to relocate after deployment. As such, a single depot location is to be selected to serve the set of customers over all planed (and future) mining trips. The selected location of the depot orbit is defined by the binary decision variable 푥푗. The variable takes a value of 1 if depot orbit 푗 is selected; otherwise it is zero.

The location of the arrival/departure parking orbits are transhipment points with no physical facilities associated with them. An optimal set of parking orbits is to be selected independently for each mining trip (and for each candidate depot location) to minimize the delta-V of the asteroid to depot (and return) transfers. The selected location of the parking orbit radius is defined by the binary decision variable 푦푖푗푡, taking a value of 1 if parking orbit 푖 is selected for depot orbit 푗 and mining trip 푡; otherwise it is zero.

The optimal orientation angles of the arrival and departure parking orbits are also to be selected for each mining trip and candidate parking orbit. The arrival orientation angle

휃푚 is to be selected from a set of discrete candidate orientation angles (푁 equally spaced angles ranging from −휋/2 to 휋/2). The departure orientation angle 휃푛 is also selected from the same set of candidate orientation angles. These limits are selected to ensure prograde parking orbits.

For Route 1, the selected orientations of the arrival and departure parking orbits are defined using the binary decision variables 푌푚푖푡 (taking a value of 1 if orientation angle

휃푚 is selected for parking orbit radius 푟푖 and mining trip 푡; zero otherwise) and 푍푛푖푡

(taking a value of 1 if orientation angle 휃푛 is selected for parking orbit radius 푟푖 and mining trip 푡; zero otherwise).

For Route 2, the selected departure orientation angle is defined using the decision variable 푈푛푡푖푚 (equal to 1 if departure orientation angle 휃푛 is selected for arrival parking orbit (푖, 푚) and mining trip 푡; zero otherwise). The arrival orientation angle is defined using the decision variable 푊푚푡푖푗 (equal to 1 if arrival orientation angle 휃푚 is selected for parking orbit 푖, depot orbit 푗, and mining trip 푡; zero otherwise).

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The locations of the selected parking and depot orbits (푦푖푗푡 , 푥푗) can be summarized by a single binary decision variable 푋푡푖푗 (taking a value of 1 if parking orbit 푖 and depot orbit 푗 are selected for mining trip 푡; zero otherwise). This variable indicates the combination of selected locations (푖, 푗, 푡) of the parking orbits and depot, for each mining trip. It can also be interpreted as a vehicle flow along edges from asteroid source 푡 to arrival parking orbit 푖 to depot 푗.

The decision variable 푋푡푖푗 forms a three-dimensional array. The components of the decision variable can be used to visualize the selected locations on a three-dimensional grid, with the (푥, 푦, 푧) coordinates corresponding to the indices (푗, 푖, 푡). (A full decision variable including the selected arrival and departure orientation angles would have five dimensions (푗, 푖, 푡, 푚, 푛), that would be difficult to visualize.)

The x-coordinate of the variable represents the candidate depot radii 푟푗 (퐽 discrete radii evenly spaced from a minimum to maximum radius). The y-coordinate represents the candidate arrival/departure parking orbit radii 푟푖 (퐼 discrete radii evenly spaced from a minimum to maximum radius). Three such grids are produced, representing the three candidate orbital planes of the depot location. These may be joined to produce an 퐼 × 3퐽 grid of possible location combinations (shown in figure 5.4). The time component can be visualized by stacking these grids along a third z-coordinate representing successive mining trips (a total of 푇 mining trips).

Figure 5.4 Grid of candidate facility location combinations.

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5.2.3 Problem Formulation We may formulate the asteroid mining location-routing problem by introducing the following notation:

Source/Customer Variables (Inputs)

푡 ∈ {1, 2, … , 푇} Index of mining trip

⃗⃗ 3 푺푖푛푡 ∈ ℝ Incoming asymptote of mining trip 푡

⃗⃗ 3 푺표푢푡푡 ∈ ℝ Outgoing asymptote of mining trip 푡

퐿푡 ∈ [0 퐿푚푎푥] Product mass at beginning of mining trip 푡

푘 ∈ {1, 2, … , 퐾} Index of customer orbits

푟푘 ∈ [푟푚푖푛 푟푚푎푥] Radius of customer 푘

풏⃗⃗ 푘 ∈ {풏⃗⃗ 퐸푞, 풏⃗⃗ 퐸푐푙, 풏⃗⃗ 퐿} Orbit normal of customer 푘

푑푘 ∈ [0 1] ∶ ∑ 푑푘 = 1 Demand fraction of each customer 푘 푘

Depot/Parking Orbit Variables (Candidate Locations)

푗 ∈ {1, 2, … , 3퐽} Index of candidate depot locations

푟푗 ∈ [푟푚푖푛 푟푚푎푥] Orbital radius of candidate depot 푗

풏⃗⃗ 푗 ∈ {풏⃗⃗ 퐸푞, 풏⃗⃗ 퐸푐푙, 풏⃗⃗ 퐿} Orbit normal of candidate depot 푗

푖 ∈ {1, 2, … , 퐼} Index of candidate arrival/departure parking orbits

Orbital radius (or apogee) of candidate arrival/departure 푟푖 ∈ [푟푚푖푛 푟푚푎푥] parking orbit 푖

푚 ∈ {1, 2, … , 푁} Index of candidate arrival aim vector orientation angles

휋 휋 휃 ∈ [− ] Arrival orbit aim vector orientation angle 푚 2 2

3 풏⃗⃗ 푚 ∈ ℝ Arrival orbit normal (found from 휃푚)

푛 ∈ {1, 2, … , 푁} Index of candidate departure aim vector orientation angles

휋 휋 휃 ∈ [− ] Departure orbit aim vector orientation angle 푛 2 2

3 풏⃗⃗ 푛 ∈ ℝ Departure orbit normal (found from 휃푛)

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Decision Variables

1, 푖푓 푑푒푝표푡 표푟푏푖푡 푗 푖푠 푠푒푙푒푐푡푒푑 푥 = { Candidate depot orbit selection 푗 0, 표푡ℎ푒푟푤푖푠푒

1, 푖푓 푝푎푟푘푖푛푔 표푟푏푖푡 푖 푖푠 푠푒푙푒푐푡푒푑 푓표푟 푦푖푗푡 = { 푑푒푝표푡 표푟푏푖푡 푗 푎푛푑 푚푖푛푖푛푔 푡푟푖푝 푡 Candidate parking orbit selection 0, 표푡ℎ푒푟푤푖푠푒

1, 푖푓 푝푎푟푘푖푛푔 표푟푏푖푡 푖 푎푛푑 푑푒푝표푡 푗 푋푡푖푗 = { 푖푠 푠푒푙푒푐푡푒푑 푓표푟 푚푖푛푖푛푔 푡푟푖푝 푡 Full candidate facility selection 0, 표푡ℎ푒푟푤푖푠푒

1, 푖푓 표푟푖푒푛푡푎푡푖표푛 푎푛푔푙푒 푚 푖푠 푠푒푙푒푐푡푒푑 푓표푟 Arrival orbit orientation selection 푌푚푖푡 = { 푝푎푟푘푖푛푔 표푟푏푖푡 푖 푎푛푑 푚푖푛푖푛푔 푡푟푖푝 푡 0, 표푡ℎ푒푟푤푖푠푒 (Route 1)

1, 푖푓 표푟푖푒푛푡푎푡푖표푛 푎푛푔푙푒 푛 푖푠 푠푒푙푒푐푡푒푑 푓표푟 Departure orbit orientation 푍푛푖푡 = { 푝푎푟푘푖푛푔 표푟푏푖푡 푖 푎푛푑 푚푖푛푖푛푔 푡푟푖푝 푡 0, 표푡ℎ푒푟푤푖푠푒 selection (Route 1)

1, 푖푓 표푟푖푒푛푡푎푡푖표푛 푎푛푔푙푒 푛 푖푠 푠푒푙푒푐푡푒푑 푓표푟 Departure orbit orientation 푈푛푡푖푚 = { 푝푎푟푘푖푛푔 표푟푏푖푡 (푖, 푚) 푎푛푑 푚푖푛푖푛푔 푡푟푖푝 푡 0, 표푡ℎ푒푟푤푖푠푒 selection (Route 2)

1, 푖푓 표푟푖푒푛푡푎푡푖표푛 푎푛푔푙푒 푚 푖푠 푠푒푙푒푐푡푒푑 푓표푟 Arrival orbit orientation selection 푝푎푟푘푖푛푔 표푟푏푖푡 푖, 푑푒푝표푡 표푟푏푖푡 푗 푎푛푑 푊 = { 푚푡푖푗 푚푖푛푖푛푔 푡푟푖푝 푡 (Route 2) 0, 표푡ℎ푒푟푤푖푠푒

Orbital Node Locations

퐷퐸푃푂푇 = (푟푗, 풏⃗⃗ 푗) Co - ordinates of candidate depot orbit location 푗

퐴푅푅 = (푟푖, 풏⃗⃗ 푚, 푟푝) Co -ordinates of candidate arrival parking orbit location(푖, 푚)

퐷퐸푃 = (푟푖, 풏⃗⃗ 푛, 푟푝) Co -ordinates of candidate departure parking orbit location(푖, 푛)

⃗⃗ ⃗⃗ 퐴푆푇 = (푺푖푛푡, 푺표푢푡푡) Co - ordinates of asteroid source orbit location for mining trip 푡

퐶푈푆푇 = (푟푘, 풏⃗⃗ 푘) Co - ordinates of customer orbit location 푘

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Edge Cost Functions

⃗⃗ ∆푉푡푖 = 푓(푟푖, 푟푝, 푺푖푛푡) Delta -V of transfer from source 푡 to arrival parking orbit 푖

⃗⃗ ∆푉푖푡 = 푓(푟푖, 푟푝, 푺표푢푡푡) Delta -V of transfer from departure parking orbit 푖 to source 푡

⃗⃗ 휑푡푚푗 = 푓(풏⃗⃗ 푚, 풏⃗⃗ 푗, 푺푖푛푡) Plane change angle from arrival orbit (푖, 푚) to depot 푗

Delta-V of plane change from arrival parking orbit (푖, 푚) to ∆푉푝푐 = 푓(푟푗, 푟푖, 휑푡푚푗) 푡푖푚푗 depot 푗

Delta-V of circularization manoeuvre from arrival orbit 푖 to ∆푉푐푖푟푐푖푗 = 푓(푟푗, 푟푖) depot 푗

Delta-V of transfer from source 푡 to arrival parking orbit 푖 to ∆푉푡푖푗 = 푓(푟푗, 푟푖, 휑푡푚푗, 푺⃗⃗ 푖푛 ) 푡 depot 푗

⃗⃗ 휑푗푛푡 = 푓(풏⃗⃗ 푗, 풏⃗⃗ 푛, 푺표푢푡푡) Plane change angle from depot 푗 to departure orbit (푖, 푛)

Delta-V of transfer from depot 푗 to departure parking orbit 푖 to ∆푉푗푖푡 = 푓(푟푗, 푟푖, 휑푗푛푡) source 푡

∆푉푗푖푚 = 푓(푟푗, 푟푖, 휑푡푚푗) Delta -V of transfer from depot 푗 to arrival parking orbit (푖, 푚)

∆푉푖푗푚 = ∆푉푗푖푚 Delta-V of transfer from arrival parking orbit (푖, 푚) to depot 푗

휑푗푘 = 푓(풏⃗⃗ 푗, 풏⃗⃗ 푘) Plane change angle from depot 푗 to customer 푘

∆푉푗푘 = 푓(푟푗, 푟푘, 휑푗푘) Delta -V of transfer from depot 푗 to customer 푘

∆푉푘푗 = ∆푉푗푘 Delta-V of transfer from customer 푘 to depot 푗

Plane change angle from arrival orbit (푖, 푚) to departure orbit 휑푡푚푛 = 푓(풏⃗⃗ 푚, 풏⃗⃗ 푛) (푖, 푛)

Delta-V of transfer from arrival orbit (푖, 푚) to departure orbit ∆푉푖푚푛 = 푓(푟푖, 휑푡푚푛) (푖, 푛)

Delta-V of transfer from arrival orbit (푖, 푚) to departure orbit ∆푉푖푚푛푡 = 푓(∆푉푖푡, ∆푉푖푚푛) (푖, 푛) to source 푡

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5.3 Orbital Transfers This section outlines the orbital mechanics of the transfer trajectories between various orbits. In all equations, 휇 represents the gravitational parameter of the Earth.

5.3.1 Asteroid to Arrival Orbit ∆푽풕풊 In both routes, the mining spacecraft will approach the Earth on a along an incoming asymptote 푺⃗⃗ 푖푛 . At the periapsis of the approach hyperbola, the spacecraft will perform an Earth Orbit Insertion (EOI) manoeuvre to be captured into an arrival parking orbit. The magnitude of the delta-V of this manoeuvre will be determined by the hyperbolic excess velocity (푉∞ = |푺⃗⃗ 푖푛|), and the periapsis 푟푝 and apoapsis 푟푖 of the desired parking orbit:

2휇 휇 휇 2 ∆푉푡푖 = ∆푉퐸푂퐼 = |√ + 푉∞푡 − √2 [ − ]| . (5.1) 푟푝 푟푝 (푟푝 + 푟푖)

The set of possible orientations of this arrival parking orbit will be restricted such that the normal vector of the arrival orbital plane 풏⃗⃗ 푚 is perpendicular to the incoming asymptote (the orthogonal set of the incoming asymptote). For a given incoming asymptote, the orientation of the arrival parking orbit may be selected by designing the hyperbolic approach via B-plane targeting [223]. The arrival B-plane (shown in figure 5.5) is a plane perpendicular to the incoming asymptote.

Figure 5.5 Geometry of the arrival B-plane.

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We can design the arrival hyperbola by defining an aim vector 푩⃗⃗ that describes the point the spacecraft would pass the Earth if the trajectory was unperturbed by the Earth’s gravity.

We can construct a B-plane from an incoming asymptote vector 푺⃗⃗ and a reference plane characterized by a normal vector 풏⃗⃗ . We define vectors 푻⃗⃗ and 푹⃗⃗ that form the basis of the B-plane:

푻⃗⃗ = 푺⃗⃗ × 풏⃗⃗ , (5.2)

푹⃗⃗ = 푺⃗⃗ × 푻⃗⃗ . (5.3)

The aim vector 푩⃗⃗ is selected from an aim vector orientation angle 휃 (measured clockwise from 푻⃗⃗ ). The normal vector of the arrival trajectory plane (and hence, the arrival parking orbit) 풏⃗⃗ 퐴푟푟 is then perpendicular to both the incoming asymptote and aim vector:

푩⃗⃗ = cos(휃) 푻⃗⃗ + sin(휃) 푹⃗⃗ , (5.4)

풏⃗⃗ 퐴푟푟 = 푩⃗⃗ × 푺⃗⃗ . (5.5)

5.3.2 Asteroid to Arrival to Depot ∆푽풕풊풋 (Route 1) For Route 1, the mining spacecraft is required to transfer from the arrival parking orbit to the depot. This is achieved through a Hohmann transfer, with a combined plane change manoeuvre at the apoapsis of the parking orbit, and a circularization manoeuvre at the periapsis of the transfer orbit. The delta-Vs of these manoeuvres are dependent on the orbital radii and relative inclination of the depot and parking orbits. In the notation introduced in section 5.2.3, the depot is a circular orbit characterised by radius 푟푗, and normal vector 풏⃗⃗ 푗. The arrival parking orbit is a circular orbit characterised by radius 푟푖 and an arrival B-plane aim vector orientation angle 휃푚 that specifies the normal vector of the arrival orbit 풏⃗⃗ 푚 through the process described in the previous section (using Eqs. 5.2 to 5.5).

We may also consider an elliptical parking orbit with apoapsis 푟푖 (equal to the circular radius), and a periapsis 푟푝. Elliptical parking orbits have been shown to significantly reduce the delta-V of planetary capture/departure manoeuvres [224]. To form a comparison between the two parking orbit shapes, we set the periapsis of the elliptical

Scott Dorrington – June 2019 163 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions orbit to coincide with the depot (푟푝 = 푟푗). (This selection simplifies the calculation of the Hohmann transfer between the depot and parking orbits.) Both circular and elliptical parking orbits are then characterised by an apoapsis 푟푖, and orientation angle 휃푚.

In both cases, the arrival B-plane is defined using the normal vector of the depot orbit

풏⃗⃗ 푗 as the reference plane. For each mining trip 푡, the plane change angle 휑푡푚푗 from the candidate arrival orbital plane 푚 to candidate depot orbit 푗 can be found from the dot product of the two normal vectors:

⃗⃗ ⃗⃗ cos(휑푡푚푗) = 풏⃗⃗ 푚 ∙ 풏⃗⃗ 푗 = (푩푚 × 푺푖푛푡) ∙ 풏⃗⃗ 푗 . (5.6)

This value is independent of the radius of the arrival parking orbit, instead depending on ⃗⃗ ⃗⃗ the orientation of the incoming asymptote 푺푖푛푡 , and the arrival aim vector 푩푚 associated with the orientation angle 휃푚.

5.3.2.1 Circular Parking Orbit The transfer between two circular orbits may be modelled as a Hohmann transfer with a combined plane change manoeuvre at the higher orbit, and a circularization manoeuvre at the periapsis of the transfer orbit. For the circular parking orbit to depot transfer (with

푟푖 ≥ 푟푗) these two delta-Vs are given by:

∆푉 = ∆푉 = √푣 2 + 푣 2 − 2푣 푣 cos 휑 , (5.7) 1 푝푐푡푖푚푗 1 2 1 2 푡푚푗 and:

휇 휇 휇 ∆푉2 = ∆푉푐푖푟푐푖푗 = |√2 [ − ] − √ | , (5.8) 푟푗 (푟푖 + 푟푗) 푟푗 where:

휇 휇 휇 푣1 = √ and 푣2 = √2 [ − ] . (5.9) 푟푖 푟푖 (푟푖 + 푟푗)

The plane change manoeuvre is required to be carried out at a point where the two orbits intersect (the ascending or descending node of the arrival parking orbit with respect to the depot orbit).

The total delta-V of the transfer from asteroid 푡 to arrival orbit 푖 (with orientation 푚) to depot 푗 is given by:

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∆푉푡푖푗 = ∆푉퐴푠푡퐷푒푝 + ∆푉푡푖 + ∆푉1 + ∆푉2 , (5.10) where ∆푉퐴푠푡퐷푒푝 is the delta-V required for the departure burn of the Asteroid-to-Earth transfer, and ∆푉푡푖 is found by setting 푟푝 = 푟푖 in Eq. 5.1. A similar process may be used to calculate the delta-V of the return transfer to the asteroid ∆푉푗푖푡 .

Additional phasing manoeuvres (∆푉푝ℎ푎푠푒푡푖 and ∆푉푝ℎ푎푠푒푖푡) may be required in order to rendezvous with the depot upon arrival in the depot orbit, and to arrive at the periapsis of the departure hyperbola at the required departure date for the return transfer to the asteroid.

5.3.2.2 Elliptical Parking Orbit The transfer from an elliptical parking orbit to the depot can also be modelled as a Hohmann transfer with two delta-V manoeuvres. Since the periapsis of the parking orbit is set at the same radius as the depot, the first manoeuvre is a simple plane change manoeuvre:

∆푉 = ∆푉 = 2푉 sin(휑 /2) , 1 푝푐푡푖푚푗 푁퐶 푡푚푗 (5.11) where 푉푁퐶 is the velocity of the parking orbit at the node crossing where the manoeuvre is carried out.

For circular parking orbits, the velocity of the node crossing will be the same as the orbital velocity, and independent of the orientation of the orbit. For an elliptical parking orbit, the node crossing velocity is dependent on the semi-major axis and the right ascension of the ascending/descending node. The plane change manoeuvre should be carried out at the node crossing with the lowest velocity (i.e. the highest orbital radius). Comparing the two options, an elliptical orbit has a lower velocity at its apoapsis than a circular orbit with the same radius.

The second (circularization) manoeuvre is conducted at the periapsis of the Hohmann transfer, placing the spacecraft into the depot orbit. The delta-V of this manoeuvre will be equal to the corresponding manoeuvre in the circular parking orbit case (Eq. 5.8).

For an elliptical parking orbit, the total delta-V of the transfer from the asteroid to arrival to depot is also found using Eq. 5.10, where ∆푉1 is found from Eq. 5.11 and ∆푉푡푖 is found by setting 푟푝 = 푟푗 in Eq. 5.1.

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5.3.3 Arrival to Departure to Asteroid ∆푽풊풎풏풕 (Route 2) In Route 2, after being captured into an arrival parking orbit, the mining spacecraft is required to transfer to a departure parking orbit, before returning to the asteroid along an outgoing hyperbolic trajectory. The delta-Vs of the arrival to departure to asteroid manoeuvres will be dependent on the radii and relative orientations of the arrival and departure parking orbits. Similar to Route 1, we may consider two cases for circular and elliptical parking orbits.

5.3.3.1 Circular Parking Orbit For the circular parking orbit case, the mining spacecraft will first be captured into a circular parking orbit of radius 푟푖, with ∆푉푡푖 equal to that of the circular case of route 1

(Eq. 5.1 with 푟푝 = 푟푖). After the transferal of the load to the transport spacecraft, the mining spacecraft will transfer to a circular departure parking orbit at the same orbital radius.

This transfer is achieved by a simple plane change manoeuvre carried out at one of the node crossings of the two orbital planes. The delta-V is a function of the parking orbit radius 푟푖, and the plane change angle 휑푡푚푛 between the arrival and departure parking orbits:

휇 휑푡푚푛 ∆푉푖푚푛 = 2√ sin ( ) . (5.12) 푟푖 2

The mining spacecraft would then wait in this parking orbit until the launch date of the next Earth-to-Asteroid transfer. At this time, it would perform an Earth departure manoeuvre to place it onto the outgoing asymptote. The delta-V of this departure manoeuvre ∆푉푖푡 is calculated in a similar manner to the Earth Orbit Insertion manoeuvre (Eq. 5.1), with the hyperbolic excess velocity (푉∞ = |푺⃗⃗ 표푢푡|), and the periapsis set to 푟푝 = 푟푖. The total delta-V of the arrival to departure to asteroid transfer is then given by:

∆푉푖푚푛푡 = ∆푉푖푚푛 + ∆푉푖푡 + ∆푉퐴푠푡퐴푟푟 , (5.13) where ∆푉퐴푠푡퐴푟푟 is the delta-V required for the arrival burn at the asteroid of the Earth- to-Asteroid transfer.

The insertion and departure manoeuvres ∆푉푡푖 and ∆푉푖푡 will be carried out at the periapsis of the respective incoming and outgoing hyperbolic trajectories. The

166 Scott Dorrington – June 2019 Chapter 5: Supply Chain Network orientation of the periapsis of a hyperbola may be defined by the angle 훽 between the periapsis direction (defined by the vector 풑⃗⃗ ) and the hyperbolic asymptote 푺⃗⃗ :

1 cos 훽 = 2 . 푟푝푉∞ (5.14) 1 + 휇

For the circular parking orbit case, the periapsis orientations of the incoming and outgoing hyperbolas may be calculated independently, and do not affect the delta-Vs of the manoeuvres.

5.3.3.2 Elliptical Parking Orbit For the elliptical parking orbit case, the orientations of the arrival and departure hyperbolas will determine the orientations of the arrival and departure parking orbits. If the two periapsis vectors do not coincide, the transfer from the arrival to departure parking orbits would require an additional apse rotation manoeuvre, such that the periapsis of the departure parking orbit is oriented to achieve the desired outgoing asymptote. This may increase the total delta-V of the transfer.

The orientations of the two parking orbits may be aligned by selecting a common periapsis vector 풑⃗⃗ directed along the intersection of the two parking orbit planes. This direction may be calculated from the cross product of the parking orbit normal vectors

풏⃗⃗ 퐴푟푟 and 풏⃗⃗ 퐷푒푝 (calculated from the arrival and departure orientation angles, using Eqs. 5.2 to 5.5):

풑⃗⃗ = 풏⃗⃗ 퐷푒푝 × 풏⃗⃗ 퐴푟푟 . (5.15)

The orientation of the arrival hyperbola can then be calculated from the dot product with the arrival orbit normal:

cos 훽 = 풑⃗⃗ ∙ 풏⃗⃗ 퐴푟푟 . (5.16)

In order to achieve the orientation angles of both incoming and outgoing hyperbolas, the periapsis radius 푟푝 is restricted to a single value (found by solving Eq. 5.14 for 푟푝). As a result, the periapsis of the parking orbits, and the delta-Vs of the insertion and departure manoeuvres (calculated from Eq. 5.1) will be dependent on the orientation angles 휃푚 and 휃푛.

Depending on the combinations of orientation angles, the periapsis radius may be larger than the apoapsis 푟푖 of the parking orbits. In this case, the periapsis of the arrival

Scott Dorrington – June 2019 167 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions hyperbola will be at the apoapsis of the arrival parking orbit. The periapsis radius may also take a negative value, indicating that the periapsis is in the opposite direction. If the absolute value of the periapsis is below the radius of the Earth (or a specified minimum altitude), the combination of candidate locations (푖, 푚, 푛) may be disregarded.

The plane change from the arrival to departure parking orbits is conducted by a simple plane change at the apoapsis of the parking orbits, which coincides with the plane crossing. This strategy is known as an apo-twist parking orbit. The delta-V of this manoeuvre is given by: 휑 ∆푉 = 2푉 sin ( 푡푚푛) , (5.17) 푖푚푛 퐴푝표 2 where:

휇 휇 2 [ − ] , 푟 < 푟 √ 푝 푖 푟푖 (푟푝 + 푟푖) 푉퐴푝표 = (5.18) 휇 휇 √2 [ − ] , 푟 ≥ 푟 . 푟 푝 푖 { 푝 (푟푝 + 푟푖)

The total delta-V of the arrival to departure to asteroid transfer can then be found from Eq. 5.13. The geometry of the elliptical parking orbit case is shown in figure 5.6, with manoeuvres shown for the mining spacecraft.

Figure 5.6 Geometry of the elliptical parking orbit case for Route 2 showing the orbital manoeuvres of the mining spacecraft.

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5.3.4 Depot to Customer Orbit ∆푽풋풌 Both routes require transfers from the depot to the customers. The depot and customer orbits are each assumed to be circular orbits that are independent of the asteroid location. Transfers between the orbits are achieved by a Hohmann transfer with a combined plane change manoeuvre carried out at the higher orbit. This is similar to Eqs. 5.7 to 5.9, with the parking orbit radius replaced with the customer radius, and the plane change angle 휑푗푘 (calculated from the dot product of the normal vectors of the two orbits as shown in Eq. 5.6). An additional delta-V may be added to account for phasing manoeuvres to rendezvous with the individual satellites in the customer orbit ∆푉푝ℎ푎푠푒푘.

The return transfer from customer to depot will have the same delta-V (∆푉푘푗 = ∆푉푗푘).

5.3.5 Arrival to Depot ∆푽풊풋풎 (Route 2) For Route 2, the transfer from arrival to depot is also modelled as a Hohmann transfer with combined plane change at apoapsis (Eqs. 5.7 to 5.9). For the elliptical orbit case, the velocities of the plane change manoeuvre are replaced with the apoapsis velocity of the parking orbit defined by Eq. 5.18. As described above, the plane change angle is calculated from the respective normal vectors, and the return transfer is also equal

(∆푉푗푖푚 = ∆푉푖푗푚).

5.4 Asteroid Mining Location-Routing Problem In section 5.2 we formulated the asteroid mining supply chain as a two-echelon, four- layer supply chain network. We wish to solve a location-routing problem to determine the optimal location and orientation of the depot and parking orbits, and the optimal route of spacecraft to achieve this delivery.

Four-layer problems are considered amongst the most difficult of the location-routing problems classified by Laporte [202], and have been covered by a number of studies [225]. Solutions to difficult location-routing problems often make use of heuristic algorithms that decompose the problem into simpler sub-problems that are solved sequentially [226]. This problem involves the following decisions: 1. Allocation decisions – allocation of customers to depots, and allocation of sources to depots; 2. Routing decisions – design of tours between facilities for the vehicles in the two echelons; and 3. Location decisions (at layers II and III) – where to locate the depot and ARR/DEP parking orbits.

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We make use of an ARL algorithm (allocation-routing-location) [202], solving first the allocation, then identifying potential routes, then solving for facility locations. As we are only considering the location of a single depot, the allocation section may be simplified. Each customer is to be allocated to a single depot, and the depot is allocated to a single source location for each mining trip. Furthermore, there are only a small number of possible routes that can be assigned to the two echelons. The two routes described in section 5.2.2.3 were found through enumeration.

This reduces the problem down to a multi-facility location problem for each of the identified vehicle routes. Each location problem consists of three sub-problems of selecting: 1. The optimal orientation of the arrival and departure parking orbits, for each candidate parking orbit and depot orbit, for each mining trip; 2. The optimal orbital radius of the arrival and departure parking orbits for each candidate depot orbit, for each mining trip; and 3. The optimal radius and orbital plane of the depot orbit.

These problems are interdependent, and must be solved in a strict order. In both routes we first solve the optimal orientation angles, then the optimal parking orbit radii, then the optimal depot location.

5.4.1 Optimal Orientation Angles

5.4.1.1 Route 1 For Route 1, the orientation angles of the arrival and departure parking orbits are selected to minimize the delta-Vs of the asteroid to depot, and the depot to asteroid transfers, respectively. This has the effect of maximizing the mass delivered to the depot, and minimizing the propellant mass required to return to the asteroid. These orientation angles are independent and can be determined by separate optimization problems that can be solved sequentially.

For each mining trip 푡, and for each candidate depot and parking orbit location (푖, 푗), we ∗ may determine the optimal arrival orientation angle 휃푚푡푖푗 that minimizes ∆푉푡푖푗. From Eqs. 5.1 and 5.8 we see that the delta-Vs of the Earth Orbit Insertion and circularization manoeuvres are independent of the orientation of the arrival parking orbit. To minimize the total delta-V of the asteroid to arrival to depot transfer (Eq. 5.10), to find the arrival orientation angle that minimizes the delta-V of the plane change manoeuvre ∆푉 (Eq. 5.7 for the circular case, or Eq. 5.11 for the elliptical case). 푝푐푡푖푚푗

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∗ The optimal orientation angle 휃푚푡푖푗 could be found from the minimum plane change manoeuvre by solving:

휕 ∆푉푝푐 = 0 . (5.19) 휕휃푚

However, this expression is not explicitly formulated as a function of 휃푚. While an analytical expression could be formed through a detailed consideration of the hyperbolic geometry, we may instead solve the problem numerically by formulating an integer programming problem with the decision variable 푌푚푖푡 (described in section 5.2.2.4):

min 푔 = ∑ ∆푉푝푐 푌푚푖푡 , ∀푡, 푖, 푗 푌 푡푖푚푗 (5.20) 푚 subject to:

∑ 푌푚푖푡 = 1 , ∀푡, 푖, 푗 (5.21) 푚

푌푚푖푡 ∈ {0,1} , ∀푚, 푡, 푖, 푗 . (5.22)

In simple terms, at each candidate depot and parking orbit location, and for each mining trip (i.e. at each location (푡, 푖, 푗)), the value of the plane change manoeuvre ∆푉 is 푝푐푡푖푚푗 calculated for each of the candidate orientation angles 휃푚, forming a 1 × 푀 vector. The optimal orientation is found from the index corresponding to the minimum value of the ∆푉 vector. 푝푐푡푖푚푗

A similar problem may be formulated using decision variable 푍푛푖푡 to determine the ∗ optimal departure orientation angle 휃푛푡푖푗 that minimizes ∆푉푗푖푡:

min 푔 = ∑ ∆푉푗푖푡푍푛푖푡 , ∀푡, 푖, 푗 푍 (5.23) 푛 subject to:

∑ 푍푛푖푡 = 1 , ∀푡, 푖, 푗 (5.24) 푛

푍푛푖푡 ∈ {0,1} , ∀푛, 푡, 푖, 푗 . (5.25)

For circular parking orbits, the plane change delta-V from arrival parking orbit to depot is only dependent on the plane change angle 휑푡푚푗. Selecting an orientation angle of zero will minimize the plane change angle, and hence the delta-V of the plane change

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∗ manoeuvre (i.e. for circular parking orbits 휃푚푡푖푗 = 0 , ∀ 푡, 푖, 푗 ). The same can be said for the departure orientation angle.

For an elliptical parking orbit, due to the declination of the arrival asymptote, the periapsis (and hence apoapsis) of the arrival parking orbit will be either above or below the depot orbital plane. Selecting an orientation angle such that the apse line coincides with the plane crossing will minimize the velocity at the plane change manoeuvre 푉푁퐶. This is known as an apo-twist manoeuvre, where the plane change occurs at the apoapsis [224].

It should be noted that this orientation may not always be achievable. The locus of possible periapsis locations will form a circle about the incoming asymptote, (a cross- section of the arrival hyperboloid). For asymptotes with high declinations, the locus of periapsis locations could be entirely above or below the depot orbital plane. In this case, the plane crossing velocity will always be higher than the velocity at apoapsis.

Figure 5.7 shows the variation of plane change delta-V (i.e. the ∆푉 vector) with 푝푐푡푖푚푗 orientation angle in the circular and elliptical parking orbit cases, for a single set of depot and parking orbit radii evaluated at 푟푗 = 42,164 km and 푟푖 = 384,399 km. The geometry of the optimal circular and elliptical parking orbit orientations are illustrated in figure 5.8.

Figure 5.7 Plane change delta-V as a function of arrival orientation angle for circular and elliptical parking orbits.

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Figure 5.8 Geometry of optimal arrival parking orbit orientations for Route 1. Top: Circular parking orbit (min. plane change angle). Bottom: Elliptical parking orbit (periapsis coincides with plane crossing).

5.4.1.2 Route 2

For Route 2, the orientation angle of the arrival parking orbit 휃푚 will determine the plane change angle 휑푡푚푗 of the transfer from the arrival orbit to the depot. The mass delivered to the depot orbit will be dependent on the delta-V of the arrival to departure to asteroid transfer ∆푉푖푚푛푡, which determines the mass of propellant required for the mining spacecraft to return to the asteroid. This propellant mass is required to be extracted before the remaining product is delivered to the depot. The plane change angle from arrival to departure parking orbits 휑푡푚푛 will be dependent on the combination of the orientation angles of the arrival and departure parking orbits 훩푚푛 = (휃푚, 휃푛), and can be calculated from the dot product of the two orbit normal vectors (as in Eq. 5.6).

As these two angles are interdependent, the optimal orientation angles must be solved simultaneously with the facility location problem.

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5.4.2 Facility Location Problem For each of the two identified routes, we solve a multi-facility location problem to determine the optimal locations of a depot and parking orbits. The facility locations are selected using the decision variable 푋푡푖푗 describing the indices (푖, 푗) of the selected candidate locations for each mining trip 푡 (outlined in section 5.2.2.4). This variable is chosen such that it optimizes an objective function, subject to a number of constraints.

5.4.2.1 Maximizing Sellable Mass The first objective function we consider is to maximize the total sellable mass 푆푀 delivered to the customers. This describes the mass delivered to the customers after the subtraction of the propellant mass required for the transport spacecraft to return to the depot, and for the mining spacecraft to return to the asteroid. For Route 2, propellant mass is also required to rendezvous with the mining spacecraft in the arrival orbit. This mass is borrowed from the propellant depot stockpile and returned prior to delivery to the customers.

To define the objective function in terms of the variables described in section 5.2.3, we introduce two types of mass cost functions: Propellant mass (푃) – the mass of propellant consumed over a transfer edge (or over a path consisting of multiple edges); and Delivered mass (퐷푀) – the mass delivered to a node. The cost functions and objective functions for the two vehicle routes are defined using the following notation:

Vehicle and Routing

푠 ∈ {푠1, 푠2} = {푀푆, 푇푆} Index of spacecraft type

푀푠 ∈ ℝ Dry mass of spacecraft type 푠

휐푠 ∈ ℝ Exhaust velocity of spacecraft type 푠

푅 ∈ {푅1, 푅2} Vehicle routing choice

푆ℎ푎푝푒 ∈ {퐶, 퐸} Shape of parking orbits

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Edge Cost and Objective Functions (Route 1)

Propellant mass for next trip from depot 푗 to departure

푠1 푃푗푖푡 = 푀푠1[푒푥푝(∆푉푗푖푡/휐푠1) − 1] parking orbit 푖 to source 푡 , using spacecraft type 푠1 (mining spacecraft)

Propellant mass for next trip from customer 푘 to depot 푠2 푃푘푗 = 푀푠2[푒푥푝(∆푉푘푗/휐푠2) − 1] 푗, using spacecraft type 푠2 (transport spacecraft)

Mass delivered from source 푡 to arrival orbit 푖 to depot 푗 , using 푠1 퐷푀푡푖푗 = (퐿푡 + 푀푠1)푒푥푝[−∆푉푡푖푗/휐푠1] − 푀푠1 spacecraft type 푠1

Sellable mass delivered to

customer 푘 via Route 푅1 , using

푅1 푠1 푠1 spacecraft 푠1 and 푠2 , after 푆푀푡푖푗푘 = [푑푘(퐷푀푡푖푗 − 푃푗푖푡 ) + 푀푠2]푒푥푝[−∆푉푗푘/휐푠2 ] subtraction of propellant required − 푀 − 푃푠2 푠2 푘푗 for return trips. Evaluated at the optimal parking orbit orientation

∗ ∗ angles (휃푚푡푖푗 , 휃푛푡푖푗 )

Edge Cost and Objective Functions (Route 2)

Propellant mass required for next trip from arrival 푠1 ( ) ( ) 푃푖푚푛푡 = 푀푠1[푒푥푝(∆푉푖푚푛푡/휐푠1) − 1] orbit 푖, 푚 to departure orbit 푖, 푛 then to source

푡, using spacecraft type 푠1

Propellant mass to transfer from depot 푗 to arrival 푠2 푃푗푖푚 = 푀푠2[푒푥푝(∆푉푗푖푚/휐푠2) − 1] orbit (푖, 푚), using spacecraft type 푠2

Propellant mass to return from customer 푘 to depot 푠2 푃푘푗 = 푀푠2[푒푥푝(∆푉푘푗/휐푠2) − 1] 푗, using spacecraft type 푠2

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Mass delivered from source 푡 to

푠1 퐷푀푡푖 = (퐿푡 + 푀푠1)푒푥푝[−∆푉푡푖/휐푠1] − 푀푠1 arrival orbit 푖, using spacecraft type

푠1

푠 푠1 푠1 Mass delivered from source 푡 to 퐷푀푡푖푗푚푛 = (퐷푀푡푖 − 푃푖푚푛푡 + 푀푠2)푒푥푝[−∆푉푖푗푚/휐푠2 ] arrival orbit (푖, 푚), to depot orbit 푗 − 푀푠2 using spacecraft type 푠1 and 푠2

Sellable mass delivered to customer 푅2 푠 푠2 푆푀 = [푑푘(퐷푀푡푖푗푚푛 − 푃 ) 푡푖푗푘푚푛 푗푖푚 푘 via route 푅2 , using spacecraft 푠2 + 푀푠2]푒푥푝[−∆푉푗푘/휐푠2 ] − 푀푠2 − 푃푘푗 푠1 and 푠2 , after subtraction of propellant required for return trips

In order to solve the facility location problem, we must first calculate all elements of the 푅 objective functions 푆푀푡푖푗푘 at each location (푡, 푖, 푗, 푘) . Algorithm D.1 and D.2 in Appendix D outline the method used in calculating the objective function for Route 1 and Route 2, respectively. The delta-Vs are calculated with the equations described in section 5.3.

5.4.2.1.1 Facility Location Problem in the form of 푿풕풊풋 The selection of both the depot and parking orbit locations can be stated in a single optimization problem using the decision variable 푋푡푖푗 as:

푅 max 푓 = ∑ ∑ ∑ ∑ 푆푀푡푖푗푘 푋푡푖푗 푋 (5.26) 푡 푗 푖 푘 subject to:

∑ ∑ 푋푡푖푗 = 1 , ∀푡 (5.27) 푖 푗

1 ∑ ∑ 푋 ∈ {0,1} , ∀푗 푇 푡푖푗 (5.28) 푡 푖

∑ 푑푘 = 1 (5.29) 푘

푋푡푖푗 ∈ {0,1} , ∀푡, 푖, 푗 . (5.30)

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Equation 5.27 is an allocation constraint, ensuring that one parking orbit and one depot orbit is selected for each mining trip. Equation 5.28 is a relocation constraint, ensuring the depot location remains constant over all mining trips. Equation 5.29 is a demand constraint, ensuring the sum of customer demands equals 1. Equation 5.30 is an integer constraint on the elements of the decision variable. We could also apply a further constraint to ensure the parking orbit is at a higher altitude than the depot orbit, by only considering locations {(푖, 푗) | 푟푖 ≥ 푟푗} . This would greatly decrease the number of candidate locations to consider.

5.4.2.1.2 Facility location problem in the form of 풙풋 and 풚풊풋풕 The facility location problem (Eqs. 5.26 to 5.30) can be solved using the MATLAB function intlinprog that solves the Mixed-Integer Linear Programming problem. This method can be applied for single-trip missions, where the relocation constraint (Eq. 5.23) may be ignored. For multi-trip missions, this constraint cannot be expressed in the format required by the function (as a linear constraint in the form 퐴푥 ≤ 푏).

We may resolve this by splitting the selection of the parking orbit and depot locations into two sub-problems to be solved sequentially. We replace the single decision variable

푋푡푖푗 with two decision variables 푥푗 and 푦푖푗푡, describing the locations of the depot and parking orbits, respectively (described in section 5.2.2.4).

푅 The objective function that we wish to optimize 푆푀푡푖푗푘 is a 4-dimensional array, defining the sellable mass at the candidate locations (푡, 푖, 푗, 푘). The strategy that we employ to solve the location problem is to successively reduce the dimension of this function, by evaluating it at the optimal locations along its various dimensions. Elements of the objective function that do not meet the constraints may be disregarded.

For Route 1, at each candidate parking orbit and depot location (푖, 푗), and for each

푅1 mining trip 푡, the objective function 푆푀푡푖푗푘 is evaluate using the optimal orientation

∗ ∗ angles (휃푚푡푖푗 , 휃푛푡푖푗 ) found from the previous optimization problems.

The dimension of the objective function can be reduced by summing over all customers

푅1 푘, giving the objective function in the form of 푆푀푡푖푗 (the sellable mass delivered to all

푅1 푅1 customers): 푆푀푡푖푗 = ∑푘 푆푀푡푖푗푘 , ∀푖, 푗, 푡.

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For each candidate depot location 푗, and for each mining trip 푡, we find the optimal parking orbit radius (defined by 푦푖푗푡) that maximizes the sellable mass delivered to all customers:

푅1 max ℎ = ∑ 푆푀 푦푖푗푡 , ∀푡, 푗 푦 푡푖푗 (5.31) 푖 subject to:

∑ 푦푖푗푡 = 1 , ∀푡, 푗 (5.32) 푖

푦푖푗푡 ∈ {0,1} , ∀푖, 푗, 푡 . (5.33)

We then evaluate the objective function at each of these optimal parking orbit radii,

푅1 giving the objective function in the form 푆푀푡푗 . The total sellable mass delivered to each candidate depot location 푗 can then be found by summing over all mining trips:

푅1 푅1 푆푀푗 = ∑푡 푆푀푡푗 , ∀푗 .

We can then find the optimal depot orbit radius (defined by 푥푗) that maximizes the total sellable mass over all mining trips:

푅1 max ℎ = ∑ 푆푀 푥푗 푥 푗 (5.34) 푗 subject to:

∑ 푥푗 = 1 (5.35) 푗

푥푗 ∈ {0,1} , ∀푗 . (5.36)

Each of these optimization problems can be solved by first calculating the coefficients of the equations at each candidate location, and finding the index corresponding to the maximum value of the coefficients. The selected locations of the parking orbits and depot orbits are then fully defined by the elements of the variables 푥푗 and 푦푖푗푡 . The equivalent 푋푡푖푗 variables could also be formed from these variables.

The process of solving the optimal orientation angles, parking orbit radii, and depot orbit for Route 1 is summarized in Algorithm D.1 in Appendix D. This algorithm may be applied to both the circular and elliptical parking orbit cases, with calculation of the delta-Vs using the appropriate equations described in section 5.3.

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As discussed in the above section, for Route 2 the optimal orientation of the parking orbits must be solved simultaneously with the facility location. For Route 2, the sellable

푅2 mass is expressed in the form 푆푀푡푖푗푘푚푛 (i.e. sellable mass at location (푡, 푖, 푗, 푘, 푚, 푛)). In order to solve the facility location problem, we need to calculate the objective

푅2 function in the form 푆푀푡푖푗푘. This is achieved by evaluating the objective function at the ∗ ∗ optimal orientation angles (휃푚푡푖푗 , 휃푛푡푖푚 ). These are found by solving two optimization problems sequentially.

We first find the optimal departure orientation angle. For each mining trip 푡, and for each arrival parking orbit location and orientation (푖, 푚), we may find the optimal ∗ orientation of the departure parking orbit 휃푛푡푖푚 that maximizes the mass delivered to the

푠1 parking orbit 퐷푀푡푖푚푛 after the subtraction of the propellant mass needed to return to the

푠1 푠1 푠1 asteroid 푃푖푚푛푡 (i.e. maximizing (퐷푀푡푖푚푛 − 푃푖푚푛푡) ). This may be stated as an optimization problem using the decision variable 푈푛푡푖푚 (equal to 1 if orientation angle

휃푛 is selected for parking orbit (푖, 푚) and mining trip 푡; zero otherwise):

푠1 푠1 max ℎ = ∑(퐷푀 − 푃 ). 푈푛푡푖푚 , ∀푡, 푖, 푚 푈 푡푖푚푛 푖푚푛푡 (5.37) 푛 subject to:

∑ 푈푛푡푖푚 = 1 , ∀푡, 푖, 푚 (5.38) 푛

푈푛푡푖푚 ∈ {0,1} , ∀푛, 푡, 푖, 푚 . (5.39)

For the circular parking orbit case, the departure manoeuvre ∆푉푖푡 (and therefore the delivered mass) is independent of the arrival orientation, and this problem is equivalent

푠1 to minimizing the propellant mass 푃푖푚푛푡 required for the mining spacecraft to return to the asteroid. This can be achieved by minimizing the plane change angle 휑푡푚푛.

To find the optimal arrival orientation angle, we evaluate the objective function at this

푅2 optimal departure orientation angle to form 푆푀푡푖푗푘푚. Summing this over all customers

푅2 푅2 then gives the total sellable mass delivered to the customers: 푆푀푡푖푗푚 = ∑푘 푆푀푡푖푗푘푚. We then solve another optimization problem to find the optimal arrival orientation angle ∗ 휃푚푡푖푗 using the decision variable 푊푚푡푖푗 (equal to 1 if orientation angle 휃푚 is selected for parking orbit 푖, depot orbit 푗, and mining trip 푡; zero otherwise):

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푅2 max ℎ = ∑ 푆푀 푊푚푡푖푗 , ∀푡, 푖, 푗 푊 푡푖푗푚 (5.40) 푚 subject to:

∑ 푊푚푡푖푗 = 1 , ∀푡, 푖, 푗 (5.41) 푚

푊푚푡푖푗 ∈ {0,1} , ∀푚, 푡, 푖, 푗 . (5.42)

푅2 We then evaluate 푆푀푡푖푗푘푚 at this optimal orientation angle, to give the objective

푅2 function in the form 푆푀푡푖푗푘. We may then follow the same procedure as that described for Route 1 (solving Eqs. 5.31 to 5.36) to find the optimal parking orbit radii and depot orbit radius. This process is summarized in algorithm D.2 in Appendix D (valid for both circular and elliptical parking orbit cases).

5.4.2.2 Maximizing Net Present Value The objective functions described above were developed to maximize the total sellable mass delivered to the customer orbit. This, in effect, is equivalent to minimizing the total transportation costs of delivery to the customers. While this approach is a common objective in transportation problems, facility location problems often consider the fixed costs of establishing the facilities in their candidate locations. We could, instead, consider maximizing the total net present value (NPV), taking into account the capital costs involved in establishing the infrastructure for an asteroid mining industry, the revenues gained from the sale of the returned asteroid material, and the return dates of the successive mining trips.

The capital costs would include the production and launch costs of constructing and deploying the depot and transport spacecraft to the candidate depot radius, as well as the construction and launch costs of deploying the mining spacecraft and mining equipment on its first trip to the asteroid. Additionally, an initial stockpile of propellant could also be needed to be deployed with the depot. Until such a time that there is an established asteroid resource supply chain and on-orbit manufacturing capabilities, the initial setup of this infrastructure is expected to be constructed and launched from Earth.

In the current space launch industry, launch vehicles have a fixed launch cost, with the payload mass capabilities dependent on the altitude of the desired orbit. These price and mass capabilities can be used to produce a specific launch cost expressing the cost per

180 Scott Dorrington – June 2019 Chapter 5: Supply Chain Network unit mass to place a payload into an orbit of a specified altitude. The specific launch cost would be a function of the launch vehicle mass, and the exponential of the delta-V required to reach orbital radius 푟푗 . The capital cost of the deployment of each component of the infrastructure would then be the product of the component mass, and the specific launch cost to the desired orbital radius.

The revenues generated from each mining trip would be the product of the sellable mass, and a specific sale price (price per unit mass) of the material at the orbital radius of the customer. The dates of these revenues will correspond to the delivery dates to the customers. The delivery date 휏푡푖푗 of each mining trip 푡 (measured with respect to a reference epoch 휏0) will be dependent on the arrival date 퐴퐷푡 of the Asteroid-to-Earth transfer, the time of flight 푇푂퐹푝 of the arrival hyperbola with periapsis 푟푝 , and the 푅 delivery time 퐷푇푡푖푗 of the selected delivery route:

푅 휏푡푖푗 = (퐴퐷푡 + 푇푂퐹푝 − 푉∞푅푆푂퐼) + 퐷푇푡푖푗 − 휏0 . (5.43)

The time of flight of the arrival hyperbola is found through Kepler’s equation for hyperbolic orbits, evaluated at the edge of the Sphere of Influence (radius 푅푆푂퐼 and true anomaly 휗):

휇2 (푒2 − 1)3/2푇푂퐹 = 푒 sinh 퐹 − 퐹 , (5.44) ℎ3 푝 where:

√푒2 − 1 sin 휗 퐹 = sinh−1 [ ] , (5.45) 1 + 푒 cos 휗

1 ℎ2 휗 = cos−1 [ ( − 1)] , (5.46) 푒 휇푅푆푂퐼

2 푟푝푉∞ 푒 = 1 + . (5.47) 휇

The delivery time will be determined by the times of flight of the various transfers in the delivery route (half the period of the arrival parking orbit, half the period of the arrival to depot transfer, and half the period of the depot to customer transfer). These may be calculated from the various radii, with the periapsis 푟푝 differing for the circular and elliptical cases:

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3/2 3/2 3/2 푅 휋 푟푝 + 푟푖 푟푖 + 푟푗 푟푗 + 푟푘 퐷푇푡푖푗 = [( ) + ( ) + ( ) ] . (5.48) √휇 2 2 2

This gives the minimum possible delivery time, assuming that delivery occurs upon arrival into the customer orbit. Additional time may be required to conduct phasing manoeuvres to rendezvous and dock with the customers in their orbits.

The NPV is obtained by discounting the future revenues to the time of the initial capital investment (at 휏0), at a discount rate ʀ that reflects the risk of the investment. Initially, asteroid mining would be considered a highly risky venture. Discount rates of 30% have been proposed in previous economic studies [169]. We can formulate a new objective function expressing the total NPV using the following notation:

Parameters

퐾 ∈ ℤ+ Number of transport spacecraft (equal to number of customers)

푀퐷 ∈ ℝ Dry mass of depot

푀푃 ∈ ℝ Initial mass of propellant in depot

푃퐶 ∈ ℝ Specific production cost of spacecraft ($/kg)

퐿퐶푗 = 푓(푟푗) Specific launch cost ($/kg) to depot radius 푟푗

푆푃푘 = 푓(푟푘) Specific sale price ($/kg) at customer radius 푟푘

Capital cost ($) of deploying mining spacecraft on initial trip to 퐶푀 = 푓(푀푠 , 푺⃗⃗ 표푢푡 ) 0 1 0 asteroid

ʀ ∈ [0 1] Discount rate

Delivery date (since initial capital investment) of mining trip 푡, 휏푡푖푗 ∈ ℝ using depot orbit 푗 and parking orbit 푖

The facility location problem can then be stated as:

푅 −휏푡푖푗 max 푓 = ∑ ∑ ∑ ∑ 푆푃푘푆푀푡푖푗푘 (1 + ʀ) 푋푡푖푗 푋 푡 푗 푖 푘 (5.49)

−(푀푠1 + 푀푠2 + 푀퐷)푃퐶 − 퐶푀0 − (퐾. 푀푠2 + 푀퐷 + 푀푃) ∑ 퐿퐶푗 푥푗 , 푗

182 Scott Dorrington – June 2019 Chapter 5: Supply Chain Network subject to the same constraints as those stated in the previous section (Eqs. 5.27 to 5.30).

This problem can be solved as a series of optimization problems in terms of the variables 푦푖푗푡 and 푥푗, in a process similar to that described above.

5.4.3 Problem Scenarios The optimization problem in Eqs. 5.26 to 5.30 describes the generic location-routing problem with multiple mining trips, and multiple customers. If only single-trip missions or single-customer missions are considered, the problem may be simplified. We may define a number of optimization problems describing different scenarios for an asteroid mining operation. The problems are classified by the following components:

Table 5.1 Classification of problem scenarios.

Component Options

Number of trips Single-trip (ST) Multi-trip (MT)

Number of customers Single-customer (SC) Multi-customer (MC)

We can identify four main optimization problems based on the number of trips and the number of customers. These problems are stated in terms of the decision variable 푋푡푖푗 in table 5.2 for the case of maximizing the total sellable mass. Similar formulations could be generated for the case of maximizing total NPV. For each problem, both vehicle routing choices should be compared.

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Table 5.2 Optimization problems for different scenarios.

Problem Scenario Optimization Problem Constraints

푅 ∑ ∑ 푋1푖푗 = 1 max 푓 = ∑ ∑ 푆푀1푖푗1 푋1푖푗 P1 ST/SC 푋 푗 푖 푗 푖 푋푡푖푗 ∈ {0,1} , ∀푡, 푖, 푗

∑ ∑ 푋1푖푗 = 1 푗 푖 푅 max 푓 = ∑ ∑ ∑ 푆푀1푖푗푘 푋1푖푗 P2 ST/MC 푋 ∑ 푑 = 1 푗 푖 푘 푘 푘 푋푡푖푗 ∈ {0,1} , ∀푡, 푖, 푗

∑ ∑ 푋푡푖푗 = 1 , ∀푡 푗 푖 푅 P3 MT/SC max 푓 = ∑ ∑ ∑ 푆푀푡푖푗1 푋푡푖푗 1 푋 ∑ ∑ 푋 ∈ {0,1} , ∀푗 푡 푗 푖 푇 푡푖푗 푡 푖 푋푡푖푗 ∈ {0,1} , ∀푡, 푖, 푗

∑ ∑ 푋푡푖푗 = 1 , ∀푡 푗 푖 max 푓 1 푋 ∑ ∑ 푋푡푖푗 ∈ {0,1} , ∀푗 P4 MT/MC 푅 푇 = ∑ ∑ ∑ ∑ 푆푀푡푖푗푘 푋푡푖푗 푡 푖 푡 푗 푖 푘 ∑ 푑푘 = 1 푘 푋푡푖푗 ∈ {0,1} , ∀푡, 푖, 푗

5.5 Numerical Example Consider the problem of maximizing total sellable mass for an asteroid mining mission consisting of a single-trip, and single-customer (P1). We wish to determine the optimal orbital location of parking orbits and propellant depot to supply a customer in Geostationary Orbit (Equatorial orbit of radius 42,164 km). The candidate facility locations consist of 200 candidate depot radii, evenly spaced from 6,578 km (200 km altitude) to 384,399 km (1 Lunar Distance); 200 candidate parking orbit radii, evenly spaced from 6,578 km to 768,798 km (2 Lunar Distances); and 100 candidate orientation angles, equally spaced from −휋/2 to 휋/2. The upper limit of the parking orbit radius was selected to ensure the parking orbit is within the Earth’s Sphere of Influence. Three orbital planes for the depot were considered: Equatorial, Ecliptic, and Lunar.

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Incoming and outgoing asymptotes were obtained from optimal Asteroid-to-Earth and

Earth-to-Asteroid trajectories to the near-Earth asteroid (459872) 2014 EK24. A Lambert solver was used together with Earth and asteroid ephemeris data to find the delta-Vs of the transfers over a range of launch and arrival dates. The delta-Vs were calculated to depart from a circular parking orbit in the same plane as the depot, with departure manoeuvres following the same strategy as the mining spacecraft in the elliptical case of Route 1 (i.e. using Eq. 5.10). Two successive local minima in the resulting delta-V plots were used. The delta-V contour plots and methods used in calculating these trajectories are presented in the next chapter. The details of these trajectories are listed in tables 5.3 and 5.4. Table 5.5 lists the parameters used in the example.

Table 5.3 Asteroid-to-Earth Transfer (Arrival at Earth).

Parameter Value Description

LD 23/10/2016 Launch Date AD 04/08/2017 Arrival Date TOF 284.2 Days Time of Flight

∆푉퐴푠푡퐷푒푝 1.757 km/s Asteroid departure burn

푉∞ 2.1607 km/s Incoming hyperbolic excess velocity at Earth Declination and Right Ascension of Arrival (Dec, RA) (49.92, 148.2) (deg) Asymptote

⃗⃗ [-1.1824, 0.7331, 1.6533] Incoming asymptote (Heliocentric Ecliptic 푺푖푛 km/s Frame)

Table 5.4 Earth-to-Asteroid Transfer (Departure from Earth).

Parameter Value Description

LD 12/02/2018 Launch Date AD 23/11/2018 Arrival Date TOF 284.2 Days Time of Flight

∆푉퐴푠푡퐴푟푟 1.32885 km/s Asteroid arrival burn

푉∞ 3.0480 km/s Outgoing hyperbolic excess velocity at Earth Declination and Right Ascension of (Dec, RA) (-49.07, 141.4) (deg) departure Asymptote

⃗⃗ [-1.1561, 1.2459, -2.3028] Outgoing asymptote (Heliocentric Ecliptic 푺표푢푡 km/s Frame)

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Table 5.5 Parameters used in example.

Parameter Value Description

퐿푡 30,000 kg Product mass at beginning of mining trip

푟푘 42,164 km Radius of customer orbit Orbit normal vector of customer orbit 풏⃗⃗ 풏⃗⃗ = [0, 0.3978, 0.9175] 푘 퐸푞 (Equatorial) Number of candidate depot and parking 퐽, 퐼 200 orbit radii Number of candidate arrival and departure 푁 100 orientation angles

푀푠1 1,000 kg Mass of spacecraft 푠1 (Mining spacecraft)

푀푠2 500 kg Mass of spacecraft 푠2 (Transport spacecraft) Exhaust velocity of spacecraft (equal for 푣 4.42 km/s 푠 both)

5.5.1 Results Applying algorithms D.1 and D.2 produces the sellable mass at each candidate depot and parking orbit location (푖, 푗) , evaluated at the optimal arrival and departure orientation angles (i.e. solving the optimal orientation problems at each location). For the single mining trip case, the sellable mass can be plotted as a contour plot over a grid of candidate depot and parking orbit radii (discussed in section 5.2.2.4 and illustrated in figure 5.4). The contours connect candidate facility locations that deliver equal amounts of sellable mass to the customer. The facility location problem can be solved graphically by finding the global maxima in the contour plots.

5.5.1.1 Optimal Parking Orbit Radii Figure 5.9 shows contour plots of the sellable mass delivered to the customer for the case of an Equatorial depot orbit. Results are shown for both routes and parking orbit shapes. The vertical dashed line in these contour plots indicates the orbital radius of the customer (푟푗 = 푟푘). The sloped line indicates the depot and parking orbit being at the same radius (푟푖 = 푟푗 ). The solid line shows the parking orbit radius that gives the maximum sellable mass at each candidate depot orbit radius (the solutions to the optimization problem Eqs. 5.31 to 5.34). (This can be interpreted graphically as a ridge in the contour plot with the partial derivative in the 푟푗 direction equal to zero or constrained by the maximum limit.)

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From the contour plots in figure 5.9, we can see that for the elliptical parking orbit cases of both routes, at each candidate depot orbit radius, the sellable mass increases monotonically with the parking orbit radius. For these cases, at all candidate depot orbits, the optimal arrival and departure parking orbits should have an apoapsis set at the maximum value permissible by the constraints (in this case 768,798 km, or 2 Lunar Distances).

For the circular parking orbit cases, the solid green line describing the optimal parking orbit radius at each candidate depot radius follows a ridge in the contour plot, passing through a saddle point. For Route 2, this saddle point is located at a higher parking orbit radius (as well as a higher depot orbit radius). As a result, at some candidate depot radii, the optimal parking orbit radii are constrained by the maximum value.

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Figure 5.9 Contour plots of sellable mass over depot and parking orbit radii for an Equatorial depot orbit, shown for the various routes and parking orbit shapes.

5.5.1.2 Optimal Depot Orbit Radius Figure 5.10 shows the values of the total sellable mass evaluated at the optimal parking orbit radius, for each candidate depot orbit radius (i.e. the coefficients 푆푀푗 of Eq. 5.34), plotted for all three candidate depot orbital planes, in each of the route and parking orbit cases. The sellable mass is expressed as a fraction of the initial load extracted at the asteroid (30,000 kg). This allows the results to be applied to different values of the initial load.

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Figure 5.10 Maximum sellable mass at each depot orbit radius, shown for the different routes and parking orbit cases.

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From the plots in figure 5.10, we can see that the Equatorial plane gives the highest total sellable mass at all candidate depot radii, for each case of the route and parking orbit shapes. For this depot orientation, there is a peak in the maximum total sellable mass at a depot radius equal to that of the customer.

For the circular parking orbit cases with non-Equatorial depot orbits (Ecliptic and Lunar), the total sellable mass increases monotonically with the candidate depot orbit radius. For an Equatorial depot, the total sellable mass decreases from its peak value (at the customer orbital radius), to a minimum value (located at the saddle point seen in figure 5.9a), before increasing with a similar trend to that of the other depot orbit cases.

The total sellable mass at the maximum depot orbit radius does not exceed the value of that at the peak. As such, the optimal depot location for the circular parking orbit cases is in an Equatorial orbit with the orbital radius equal to that of the customer. For the circular case of Route 1, if the maximum depot orbit radius is extended to 2 Lunar Distances (the same limit placed on the parking orbits), the value of the sellable mass at the maximum depot radius is found to be greater than the peak value at the customer orbit radius. However, this maximum value is still lower than that found for the elliptical case, which has an optimal depot orbit located at the customer orbit.

For the circular case of Route 2, and for the elliptical cases of both routes, the same decreasing then increasing trend is seen in the total sellable mass at candidate depot orbit radii above the customer radius. In these cases, the decrease occurs at a slower rate, and the maximum value does not exceed that of the peak, even if the maximum depot radius is extended to 2 Lunar Distances.

From these results, we can conclude that the optimal placement of a propellant depot that delivers the maximum total sellable mass to customers is an Equatorial orbit with the orbital radius equal to the customer orbit. The optimal route choice is Route 2 with an elliptical parking orbit with apoapsis 768,798 km (the maximum limit), and periapsis 6,884 km (an altitude of 504 km). This location and routing combination results in a maximum total sellable mass of 12,908 kg of product delivered to the customers. This amounts to 43.03% of the mass initially extracted from the asteroid (30,000 kg). The remainder is used as propellant mass in the delivery.

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5.5.2 Sensitivity Analysis A sensitivity analysis was performed to investigate the effects of varying the spacecraft parameters on the maximum total sellable mass and the optimal location of the depot orbit. The parameters of interest were: 1. Specific Impulse; 2. Dry mass of the mining spacecraft; 3. Dry mass of the transport spacecraft; 4. Orbital radius of the customer; and 5. Incoming/Outgoing asymptotes of the trajectories.

Each of these parameters was varied over a range of values, and the global maximum total sellable mass was calculated for each of the routes and parking orbit shapes, for the case of an Equatorial depot orbit. The results are shown in figures 5.11 to 5.15, where the maximum total sellable mass is expressed as a fraction of the initial load extracted at the asteroid (30,000 kg).

5.5.2.1 Specific Impulse Figure 5.11 shows the effects of varying the specific impulse of the spacecraft (equal for both mining and transport spacecraft) over a range from 200 to 3,100 seconds (exhaust velocities ranging from 1.96 to 30.411 km/s).

In all cases, the maximum total sellable mass fraction increases monotonically with specific impulse. There is a minimum specific impulse value that results in zero sellable mass. At these critical values, the propellant mass required for the mining spacecraft to return to the asteroid is equal to the mass delivered to the parking orbit, and there is no additional propellant that can be sold to the customers.

Figure 5.11 Maximum total sellable mass as a fraction of the initial load expressed for varying values of specific impulse.

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At specific impulses of 3,000 seconds the total sellable mass fraction increases to 82.83%, 87.17%, 83.01%, and 88.69% for the R1 Circular, R1 Elliptical, R2 Circular, and R2 Elliptical cases, respectively. These specific impulses are typically achieved by Solar Electric Proulsion systems using Xenon or Argon as propellant. In the formulation presented in section 5.4.2.1, the sellable mass is designed with in situ propellant production at the asteroid to allow for reusability over multiple mining trips. Propellants common to electric propulsion are not expected to be present in asteroids in high enough quantities to be considered viable. Very long stay-times may be required to process the required propellant for the return trip. The low thrust trajectories required by Solar Electric Propulsion may also affect the calculation of the delta-Vs of the transfers.

As the specific impulse approaches infinity, the maximum total sellable mass fraction will approach unity. The circular and elliptical cases also appear to approach one another, for both routes. This asymptotic behaviour can be more clearly seen by plotting the sellable mass fractions as a function of exp (1/휐푒) (shown in figure 5.12). As seen from the cost and objective functions defined in section 5.4.2.1, the delivered and propellant masses scale exponentially with the inverse of the exhaust velocity: 퐷푀 ∝ exp (−∆푉/휐푒) and 푃 ∝ exp (∆푉/휐푒).

Figure 5.12 Maximum total sellable mass fraction expressed as a function of 퐞퐱퐩 (ퟏ/흊풆).

5.5.2.2 Mining Spacecraft Dry Mass

Figure 5.13 shows the effects of varying the dry mass of the mining spacecraft 푀푠1 from 500 to 10,000 kg, while keeping the dry mass of the transport spacecraft constant at 500 kg, and resetting the exhaust velocity to 4.42 km/s.

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Figure 5.13 Maximum total sellable mass fraction for varying values of mining spacecraft dry mass.

In each case, the maximum total sellable mass fraction shows a negative linear relation to the mining spacecraft dry mass. The circular case for Route 1 shows a piecewise linear relation, with a steeper gradient at lower mining spacecraft dry masses. This is due to a change in the optimal location of the depot orbit from the peak at the customer orbit, to the maximum permissible depot orbit (as discussed in section 5.1.2). In all other cases, the optimal depot location is at the customer radius.

The circular and elliptical cases of Route 2 both show a shallower gradient with respect to the varying mining spacecraft dry mass. This implies that Route 2 has a higher resilience to increases in dry mass that may occur during the design phase of the systems of the mining spacecraft. The linear relation of the elliptical case of Route 2 continues at higher values of the mining spacecraft dry mass, with the sellable mass decreasing to zero at 21,000 kg.

5.5.2.3 Transport Spacecraft Dry Mass

Figure 5.14 shows the effects of varying the dry mass of the transport spacecraft 푀푠2 from 50 to 10,000 kg, while keeping the dry mass of the mining spacecraft constant at 1000 kg (and exhaust velocity 4.42 km/s).

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Figure 5.14 Maximum total sellable mass fraction for varying values of transport spacecraft dry mass.

The circular and elliptical cases of Route 1 both show a constant value of sellable mass as a function of transport spacecraft dry masses. In Route 1, the transport spacecraft delivers the product from a circular depot orbit to a circular customer orbit. As the optimal location of the depot is selected at the same location as the customer, this transfer requires no delta-V, and the transport spacecraft has no effect on the sellable mass. In practice, there will always be a delta-V required for phasing manoeuvres and rendezvousing with the customers (∆푉푝ℎ푎푠푒푘 discussed in section 5.3.4), resulting in a small negative gradient with respect to transport spacecraft dry mass.

The circular and elliptical cases of Route 2 both show a negative linear relation with approximately the same gradient.

5.5.2.4 Customer Orbital Radius Figure 5.15 shows the effect of varying the customer radius from 6,578 km (200 km altitude) to 384,399 km (1 Lunar Distance).

Figure 5.15 Maximum sellable mass variation with customer orbit.

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The results show an increasing amount of sellable mass that can be delivered to customers in higher Earth orbits. In each of these cases, the optimal depot location was found to coincide with the customer radius.

For the elliptical case of Route 1, the sellable mass increases with customer radius to a peak at an altitude of 100,00 km (around twice the Geostationary altitude), before decreasing with radius. This increasing then decreasing trend is due to the components of ∆푉푡푖푗. At a fixed parking orbit radius, the Earth Orbit Insertion ∆푉푡푖 increases with depot radius, while the Hohmann transfer from arrival parking orbit to depot decreases with increasing depot radius.

5.5.2.5 Periapsis of Elliptical case for Route 1 In section 5.3.2, the elliptical case for Route 1 was defined as having a periapsis equal to the depot orbit. This assumption simplifies the calculation of the delta-Vs, with a simple plane change manoeuvre at the node crossing.

As seen from the results in the elliptical parking orbit cases, lowering the periapsis of the arrival parking orbit significantly increases the maximum total sellable mass delivered to the customers. This is due to a reduction in the delta-Vs of the Earth orbit insertion and plane change manoeuvres for elliptical parking orbits. An elliptical parking orbit has a higher velocity at periapsis, and lower velocity at apoapsis in comparison to a circular orbit with orbital radius equal to the apoapsis radius. This improves both the Earth orbit insertion manoeuvre, which is made more efficient at higher velocities, and the plane change manoeuvre, which is made more efficient at lower velocities.

We may further improve the efficiency of the Route 1 elliptical case by lowering the periapsis for all candidate locations to a minimum altitude of 200 km. The updated contour plot of total sellable mass is shown in figure 5.16. This amendment increases the maximum total sellable mass to 12,669 kg (or 42.22% of the initial load). This amount is only slightly lower than the elliptical case of Route 2.

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Figure 5.16 Contour plot of sellable mass for the elliptical case of Route 1, with a lowered periapsis.

5.5.2.6 Incoming/Outgoing Asymptotes The above analysis was conducted with a single set of incoming and outgoing asymptotes calculated for two successive heliocentric transfers to asteroid (459872)

2014 EK24. The sets of incoming and outgoing asymptotes will vary with different launch dates and target asteroids. The magnitude and relative orientations of these asymptotes are expected to have a large influence on the plane change angles between the various transfers, and hence the total delta-V and sellable mass.

To test the effects of the trajectories on the optimal route and sellable mass, the analysis was repeated with sets of incoming and outgoing asymptotes calculated from return trips to three additional near-Earth asteroids: (141424) 2002 CD, (21062) 1991 JW, and

(341843) 2008 EV5. These asteroids were chosen due to their large size and large synodic period, allowing for multiple successive low delta-V trajectories with a cadence of around 6 months close to the asteroid’s opposition. The details of these trajectories are listed in table 5.6 along with the total sellable mass of each of the routing cases.

For each of the new sets of trajectories, the same optimal routing strategy and depot location was found as that in the previous analysis (Route 2 with an elliptical parking orbit, and a depot located at the customer orbit). While these results imply that this supply chain network layout is the optimal design, a wider range of return trajectories to numerous asteroids should be analysed to ensure that the selected depot location is a globally optimal solution, prior to the deployment of an actual propellant depot.

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Table 5.6 Trajectory details and sellable masses calculated for three additional asteroids.

Asteroid Asteroid-to-Earth Earth-to-Asteroid SM/L

LD = 07/03/2036 LD = 27/03/2037 R1C: 0.0787

(141424) AD = 28/10/2036 AD = 08/09/2037 R1E: 0.1997

2002 CD ∆푉퐴푠푡퐷푒푝= 2.1468 km/s ∆푉퐴푠푡퐴푟푟= 2.5515 km/s R2C: 0.1126

⃗푺⃗ 푖푛= [-2.4307, -2.4418, -1.9491] ⃗푺⃗ 표푢푡 = [0.5519, 1.6594, -3.0596] R2E: 0.3315

LD = 24/09/2027 LD = 12/11/2028 R1C: 0.1007

(21062) AD = 16/05/2028 AD = 14/08/2029 R1E: 0.2590

1991 JW ∆푉퐴푠푡퐷푒푝= 0.7921 km/s ∆푉퐴푠푡퐴푟푟= 0.4462 km/s R2C: 0.1425

⃗푺⃗ 푖푛= [-0.7673, -2.2844, 4.2022] ⃗푺⃗ 표푢푡 = [2.0189, 2.9694, 4.3841] R2E: 0.4524

LD = 15/08/2023 LD = 20/06/2024 R1C: 0.1425

(341843) AD = 23/12/2023 AD = 27/12/2024 R1E: 0.3534

2008 EV5 ∆푉퐴푠푡퐷푒푝= 0.2586 km/s ∆푉퐴푠푡퐴푟푟= 1.3851 km/s R2C: 0.1921

⃗푺⃗ 푖푛= [-0.9057, -2.3072, -3.6429] ⃗푺⃗ 표푢푡 = [-0.2257, 2.5060, -3.1487] R2E: 0.5263

The results also show that the combination of incoming and outgoing asymptotes at Earth have a large effect on the total sellable mass that can be delivered to customers – ranging from 33.15 to 52.63% of the initial load for the asteroids tested in this analysis. These trajectories were selected individually to minimize the delta-Vs of the heliocentric transfers. In the next chapter, it will be shown that further improvements that can be made to the sellable mass by using a time-expanded network to consider numerous candidate Earth-to-Asteroid and Asteroid-to-Earth trajectories, and selecting combinations that maximize the total sellable mass (rather than minimizing the total delta-V).

5.6 Discussion

5.6.1 Need for Stockpiling The results of the numerical example indicate that to maximize the total sellable mass delivered to a single customer, the depot should be placed in the same orbit as the customer. This placement has the effect of removing the delta-V of the depot to customer transfer. The result calls into question the need for an orbital depot at all for this objective. The mining spacecraft could transfer the product to the transport

Scott Dorrington – June 2019 197 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions spacecraft (as proposed in Route 2), or deliver the product directly to customer spacecraft, without the need for an orbital depot.

The absence of a propellant depot with a stockpile of product in reserve would result in an intermittent supply of product to customers. Due to the duration of the heliocentric transfers, the time from the start of a mining trip to delivery would be at least one year (including mining time at the asteroid), requiring a significant lead time in predicting customer demands. The time of delivery would also be restricted by the optimal arrival dates of the Asteroid-to-Earth transfers. Stockpiling of the product would guarantee a constant, timely supply of product, with the ability to respond to varying demands and market uncertainty.

The results indicate that the optimal location for an orbital depot to supply Geostationary satellites is in the Geostationary orbit. This location may not in fact be practical, as orbit assignments in the Geostationary orbit are limited and highly sort after by communication and weather satellites. The increased traffic flow of transport and mining spacecraft from this orbit, and the exchange and possible processing of materials at the depot may also pose an increased collision risk to Geostationary satellites. The orbital depot may instead be placed in an orbit above the Geostationary orbit. This position would reduce the risk of impact from space debris, and would also be advantageous for providing services of removing existing orbital debris in the graveyard orbit.

5.6.2 Determining Specific Sale Price The numerical example illustrated the methods outlined in this chapter for the objective of maximizing the total sellable mass delivered to the customer. This example was chosen for its simplicity, however in a future asteroid mining industry, maximizing net present value will likely be the major objective.

One issue that was not addressed in section 5.4.2.2 is the determination of the specific sale price at which to offer the product to customers. Determining the specific sale price at various customer orbits would require consideration of a number of factors, including the costs of delivering the product, and the supply and demand for the product in these orbits. The total supply of product in a given orbit would be specified by the sellable mass, and the number and frequency of mining trips. The demand in that orbit would be influenced by the supply, and the specific sale price. As there is no established market for asteroid resources, the supply and demand relationship is difficult to predict.

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Initially, there may be an over or under-supply of asteroid-derived resources. Over time, the sale price would naturally settle to a market equilibrium.

A 2016 study by United Launch Alliance (ULA) provided an initial pricing proposal for an orbital water-based propellant market [148]. Prices were estimated from the launch costs of delivering the product from both the Earth, and the lunar surface (noting that near-Earth asteroid sources could also be considered). The study showed the cost of delivery from Earth increasing with customer altitude, while the cost of delivery from the Moon decreasing with customer altitude (measured from Earth’s surface).

The results from the sensitivity analysis (figure 5.15) show that the sellable mass delivered to customers increases with customer radius, indicating that the cost of delivery to higher orbits is less than that to lower orbits. From this, we could expect that the specific sale price should decrease with increasing customer radius, in agreement with the proposal by ULA. From these relations, we might expect that customers at lower altitudes would be best served by launches from Earth, while customers at higher altitudes would be best served by delivery from asteroids. The critical altitude, above which asteroid mining is the best solution, would be determined by the total capital cost of the asteroid mining approach.

The specific sale price could be more accurately assigned by adapting the optimization problem in section 5.4.2.2 to include factors of supply and demand. An initial sale price could be estimated by determining the sale price that returns a zero NPV at the end of a desired payback period. A more detailed estimate could be obtained by reformulating the location problem as a spatial price equilibrium problem [227]. This problem is to determine the optimal facility location, and production levels to maximize profits, taking into account product price variation with location, and multiple competing suppliers.

5.6.3 Wait-Time Constraints In both formulations of the objective function, additional constraints on the depot and parking orbit radii should be placed such that the mining spacecraft can complete its delivery route and return to the periapsis of the departure hyperbola in time for the departure date of the next Earth-to-Asteroid transfer.

푅 The minimum return time 푀푅푇푡푖푗 that the mining spacecraft can complete its delivery and return to the departure manoeuvre can be calculated by the orbital periods of the

Scott Dorrington – June 2019 199 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions various transfers, measured from the periapsis passage of the arrival hyperbola to the periapsis passage of the departure hyperbola. For Route 2, the minimum return time will be equal to half the period of the arrival parking orbit plus half the period of the departure parking orbit. For Route 1, the mining spacecraft must transfer to the depot before departing back to the asteroid, and the minimum return times will differ for the two parking orbit shapes:

푟 + 푟 3/2 휋 3/2 푖 푗 [푟푖 + ( ) ] , 푓표푟 푅1 퐶 √휇 2 3/2 푅 2휋 푟푝 + 푟푖 푀푅푇푡푖푗 = [( ) ] , 푓표푟 푅1 퐸 (5.50) √휇 2 3/2 휋 푟푝 + 푟푖 [( ) ] , 푓표푟 푅2 . {√휇 2

The total wait-time of the mining spacecraft at the depot orbit (Route 1) or departure parking orbit (Route 2) will be defined by the difference between this minimum return time and the duration between the launch and arrival dates of the heliocentric transfers. This wait-time must be positive for the candidate depot and parking orbit radii to be acceptable. In terms of the decision variable 푋푡푖푗, this wait-time constraint may be stated as:

∗ ∗ 푅 ∑ ∑(퐿퐷푡푖 − 퐴퐷푡푖 − 푀푅푇푡푖푗)푋푡푖푗 ≥ 0 , ∀푡 (5.51) 푗 푖 where:

∗ 퐴퐷푡푖 = 퐴퐷푡 + 푡표푓푖 − 푉∞푖푛푅푆푂퐼 , (5.52)

∗ 퐿퐷푡푖 = 퐿퐷푡 − 푡표푓푖 + 푉∞표푢푡푅푆푂퐼 . (5.53)

From the maximum limits of the depot and parking orbit radii used in the numerical example, the minimum return times for each of these cases are calculated as 64.04, 50.43, and 27.80 days, respectively. As the time between the arrival and departure dates of the trajectories in tables 5.2 and 5.3 was 192 days, all candidate locations satisfied these constraints. The elliptical case of Route 2 showed the lowest minimum return time, and hence maximum wait-time.

These wait-time constraints should also be applied in the process of selecting the launch and arrival dates of the successive Asteroid-to-Earth and Earth-to-Asteroid transfers. A minimum wait-time should be imposed to avoid having short duration delivery times.

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Short duration wait-times may however be preferential to longer wait-times between launch opportunities. For example, the next optimal launch date could occur shortly after the optimal arrival date. It may be beneficial to select a shorter parking orbit at the cost of a higher delta-V to ensure the next launch date is met rather than having to wait (around 6 months) for the next departure date.

5.6.4 Scheduled Maintenance A future asteroid mining industry will likely consist of numerous mining spacecraft, each carrying out multiple mining trips to various asteroids. Over the course of several interplanetary missions, it is highly likely that a number of these mining spacecraft may encounter failures in critical systems that may result in the loss of a mining spacecraft.

The profitability of the mining venture will be determined by the average number of successful mining trips each mining spacecraft can perform during its design life. To reduce the risk of lost revenue from system failures in the mining spacecraft, regular inspection and maintenance tasks should be scheduled during the wait-time in Earth orbit between the arrival of the Asteroid-to-Earth transfer and the departure of the next Earth-to-Asteroid transfer.

For Route 1, the mining spacecraft would spend the duration of its wait-time at the propellant depot. This affords the opportunity of incorporating inspection and maintenance facilities into the design of the propellant depot. For the selected routing strategy (Route 2), the mining spacecraft will spend the duration of the wait-time in a highly elliptical Earth orbit. Inspection may be carried out by the transport spacecraft prior to retrieving the product from the mining spacecraft. If a maintenance task is required, the mining spacecraft may return to the propellant depot. Following maintenance, the mining spacecraft would return to the asteroid following the departure parking orbit of Route 1.

The additional transfer of the mining spacecraft to the depot, and the lower efficiency of the departure transfer associated with Route 1 will result in a reduced sellable mass for any mining trip that requires maintenance. Any maintenance tasks that require replacement components launched from Earth would also incur an additional capital expenditure. The lost profit and increased cost would be acceptable for the increased reliability of the mining spacecraft that would increase the average number of successful trips for each mining spacecraft.

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5.6.5 Computational Complexity The procedures for calculating the optimal parking orbit and depot orbit radii are outlined in algorithms D.1 and D.2 (in Appendix D), for the two candidate distribution routes respectively. These algorithms use a set of nested for loops to calculate the elements of the cost functions and objective functions at each of the candidate locations (푡, 푖, 푗, 푘, 푚, 푛).

The organization of these algorithms was chosen to minimize the number of operations, by avoiding the repeated calculation of functions that do not change with respect to the various dimensions of the objective function. For example, the calculation of the delta- V and propellant masses for the depot to customer transfers are independent of the parking orbit, mining trip, orientation angles, and route. These delta-Vs are calculated a single time at the beginning of each algorithm.

The computational complexity of each algorithm can be described by the number of operations as a function of the dimensions of the various input parameters using Big-O notation. The complexity of algorithm D.1 is 푂(퐽퐾 + 푇퐽퐼(푁 + 퐾)), while algorithm D.2 has complexity 푂(퐽(푇퐼 + 퐾) + 푇퐼푁2 + 푇퐼푁퐽퐾).

The values for the number of mining trips 푇 and customers 퐾 will be defined as inputs to the algorithm, depending on the mining scenario. These values are expected to be small, as the customers of asteroid-derived products will likely share similar orbits, and the number of mining trips will be constrained by the design life and reliability of the mining spacecraft. Treating these values as constants, the complexity of the algorithms can then be expressed as 푂(퐽 + 퐽퐼푁) and 푂(퐽퐼 + 퐼푁2 + 퐽퐼푁).

In algorithm D.1, the calculation of the arrival and departure orientation angles are dependent only on the delta-Vs of the transfers, and independent of the spacecraft parameters (such as mass and specific impulse). For the purposes of the sensitivity analysis, it is convenient to separate the calculation of the orientation angles, and the objective functions into two separate nested for loops. The orientation angles need only be calculated once, while the objective functions can be calculated many times for various spacecraft parameter settings, without the need to re-calculate the optimal orientation angles.

Algorithm D.2 has a higher complexity due to the interdependence of the arrival and departure orientation angles that scales with the square of the number of candidate orientation angles 푁2. In algorithm D.2, the optimal orientation angles are selected to

202 Scott Dorrington – June 2019 Chapter 5: Supply Chain Network minimize the propellant required for various transfers. These values are dependent on the spacecraft parameters, and the algorithm must be run in its entirety for each variation of the spacecraft parameters.

The numerical example presented in section 5.5 used values of 퐼 = 퐽 = 200 and 푁 = 100. The average total computational time for the circular and elliptical cases for Route 1 was 90.14 and 187.819 seconds, respectively. The majority of this time was in calculating the optimal orientation angles, with only 0.168 seconds required for the calculation of the objective functions and optimal parking orbit and depot orbit radii. The average total computation time for the circular and elliptical cases for Route 2 was 158.633 and 118.045 seconds, respectively.

5.6.6 Improvements to the Model The results of the numerical example showed that the shape of the parking orbits had a significant impact on the performance. Further investigation of parking orbit shapes, and alternative transfer methods such as bi-elliptic transfers, lunar gravity assist manoeuvres, low-thrust trajectories, and ballistic transfers through Lagrange points could further increase the efficiency.

In the formulation presented in this chapter, the load extracted from the asteroid was assumed to be constant. In reality, the load extracted from the asteroid may be a variable function depending on the demand of the customer, the production rate and stay-time at the asteroid, and the maximum capacity of the storage tank. To account for variable extraction loads, the initial load extracted from the asteroid 퐿푡 could be set as an 푅 additional decision variable. As the expressions for 푆푀푡푖푗푘 are a nonlinear function of the initial load, the objective function would need to be expanded. The resulting optimization problem would be a quadratic programming problem.

While the methods outlined in this chapter focus on asteroid mining, the method could easily be adapted to consider delivery from the lunar surface. This would provide a baseline against which to compare the feasibility of each asteroid target. Only those asteroids that can deliver product at a lower cost than from the Moon should be considered as targets for asteroid mining (unless those resources are not present on the Moon).

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5.7 Conclusion This chapter outlined the formulation of a location-routing problem for an asteroid mining supply chain network. A solution method was presented that allows the determination of the optimal location and orientation of an orbital propellant depot, arrival and departure parking orbits, and the optimal routing of spacecraft through the network in order to maximize either the total sellable mass delivered to customers, or the total net present value of the mining venture.

A numerical example was presented to illustrate the method for the case of maximizing total sellable mass delivered to a single customer in Geostationary orbit with a single mining trip. The analysis was repeated for all routing strategies, using return trajectories to several near-Earth asteroids. The results showed that in all cases, the maximum sellable mass is achieved with Route 2, where the mining spacecraft remains in a highly elliptical parking orbit before returning to the asteroid, while a small transport spacecraft retrieves the load of material and delivers it to an orbital propellant depot and customers. The optimal depot location was found to be at the same orbital radius and orbital plane as the customer orbit. The optimal parking orbit is an elliptical orbit with apoapsis altitude 762,420 km, and periapsis altitude 504 km.

The same optimal depot location and routing strategy was found when repeating the analysis with return trajectories to several near-Earth asteroids. This layout of the supply chain network is capable of delivering between 33.15 – 52.63% of the mass extracted from the asteroid to customers in Geostationary orbit (depending on the trajectories to/from the target asteroid). The remaining fraction is consumed as propellant mass in the delivery.

A sensitivity analysis showed that this routing strategy was also the most resilient to variations in the mining spacecraft dry mass that may occur during the design phase.

Suggestions on improvements to the model were presented to account for the supply- demand relationship in a future water-based propellant market.

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6 COMBINATORIAL TRAJECTORY OPTIMIZATION

6.1 Introduction The subject of spacecraft trajectory optimization deals with the determination of a spacecraft’s trajectory that satisfied initial and terminal conditions defined by the mission, while minimizing some quantity of performance [228]. For impulsive transfers between two planetary bodies (a body-to-body transfer), the terminal conditions are defined by the positions of the two planets at the launch and arrival dates of the transfer. Constraints are further added such that the duration of the transfer (the time of flight) is equal to the difference between the launch and arrival dates. The body-to-body transfer problem can then be described as an optimization problem of selecting the launch and arrival dates (decision variables) that minimize/maximize an objective function describing the mission performance, subject to time of flight (and other) constraints.

In Chapter 3, it was noted that an asteroid mining campaign would require numerous interplanetary transfers between the Earth and a target asteroid. Furthermore, several different types of transfers will be involved, dependent on the direction of the transfer (Earth-to-Asteroid or Asteroid-to-Earth), encounter type (flyby or rendezvous), and whether the spacecraft is launched from the Earth’s surface or departing from or arriving at an orbital propellant depot.

The trajectory of a spacecraft in an asteroid mining campaign can be described by a “flight itinerary”, listing the launch and arrival dates of each consecutive transfer. The flight itinerary optimization problem can then be described as that of finding a flight itinerary that minimizes/maximizes an objective function measuring the performance of the mining campaign. This can be viewed as solving a sequence of body-to-body

Scott Dorrington – June 2019 205 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions transfer problems, with additional constraints on the durations between consecutive transfers.

In this chapter, methods are developed for solving the flight itinerary optimization problem to maximize the total net present value of mining an individual asteroid candidate. The chapter will first review methods currently used in the determination of optimal launch and arrival dates of body-to-body transfers, describing characteristics of “porkchop” plots that can be used to visualize the variation of the objective function over a range of launch and arrival dates. A set of objective functions are then developed to compute the total delta-Vs of the various transfer types involved in an asteroid mining campaign. A computer program is developed to identify numerous optimal launch opportunities for each transfer type over a period from 2020 to 2050.

Methods will then be presented to construct flight itineraries as combinations of Earth- to-Asteroid and Asteroid-to-Earth transfers, selected from the sets of identified optimal launch opportunities. These combinations are represented using network graphs, with nodes representing candidate trajectories, and edges representing the durations between consecutive transfers. Edge costs are developed to represent mission performance parameters such as the delta-Vs of the transfers, and the stay-times at the asteroid, and wait-times between consecutive mining trips. Candidate flight itineraries are constructed by enumerating paths through the network. Optimal flight itineraries can then be found by employing graph theory algorithms to solve a shortest path problem, where the objective function is a sum of the edge costs in the path. Several objective functions are developed describing the total sellable mass, total profit, and total net present value.

6.2 Lambert’s Problem Finding trajectories between two celestial bodies requires the solution to Lambert’s problem – a two-point boundary value problem commonly studied in astrodynamics that aims to find a trajectory between two points with a desired time of flight [229].

6.2.1 Transfer Geometry Consider the transfer between the Earth (orbit 1) and an arbitrary asteroid (orbit 2). Let point 푃1 be the departure position in the initial orbit defined by the state vector 풙⃗⃗ 1 = 푇 [풓⃗ 1, 풗⃗⃗ 1] , and point 푃2 be the arrival position in the final orbit defined by the state 푇 vector 풙⃗⃗ 2 = [풓⃗ 2, 풗⃗⃗ 2] . These state vectors may be found from ephemeris data of the two bodies at a given launch and arrival date.

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Figure 6.1 Left: Geometry of the transfer. Right: Unique properties of an example transfer orbit.

There is a continuity of possible transfer trajectories between 푃1 and 푃2, each having a distinct transfer time (or time of flight 푇푂퐹). Each of these possible transfers will share a number of properties that are independent of the transfer time. These properties are defined by the geometry of the transfer – the triangle △ 퐹푃1푃2, containing the terminal points 푃1, 푃2, and the common focus of the two orbits 퐹 (the Solar System barycentre). This geometry is shown in the left of figure 6.1.

All trajectories between the points will have a common transfer angle 휃, chord length 푐 (distance between the points), and semi-perimeter 푠 (half the perimeter of the triangle). They will also share a common orbital plane containing the three vertices of the triangle. As such, all trajectories will have the same inclination 푖푡푥 with respect to the Ecliptic, and the same plane change angles with respect to the initial and final orbits.

Each of the transfers will also have unique properties dependent on the transfer time, distinguished by the terminal velocity vectors 풗⃗⃗ 푡푥1 and 풗⃗⃗ 푡푥2 of the transfer orbit at the terminal positions. These vectors will determine the flight path angles 훾1 and 훾2 describing the orientations of these vectors from the local radial direction, and the incoming and outgoing asymptotes 푺⃗⃗ 푖푛 and 푺⃗⃗ 표푢푡 describing the relative velocities of the transfer with respect to the target and departure bodies, respectively. These vectors, along with the geometry of the departure/arrival hyperbolas, are used to calculate the delta-Vs of the transfers between the two bodies.

The unique properties of the transfers are shown in the right of figure 6.1 for a single transfer orbit. Table 6.1 summarizes the common and unique properties of transfers between the two points.

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Table 6.1 Properties of transfer trajectories.

Transfer Geometry

푃1, 푃2 Terminal points 퐹, 퐹′ Focus, and empty focus

풓⃗ 1, 풓⃗ 2 Terminal vectors

풗⃗⃗ 1, 풗⃗⃗ 2 Terminal velocity vectors Common Transfer Parameters Unique Transfer Parameters

풓⃗ 1 ∙ 풓⃗ 2 휃 = cos−1 Transfer angle 푇푂퐹 Time of flight |풓⃗ 1||⃗풓 2| Terminal velocity 푐 = |풓⃗ − 풓⃗ | Chord 풗⃗⃗ , 풗⃗⃗ 2 1 푡푥1 푡푥2 vectors of transfer Incoming and 푠 = (푟 + 푟 + 푐)/2 Semi-perimeter 푺⃗⃗ , 푺⃗⃗ 1 2 푖푛 표푢푡 outgoing asymptote ( ) ̂ −1 풓⃗ 1 × 풓⃗ 2 ∙ 풌 Inclination of 푖푡푥 = cos [ ] 훾1, 훾2 Flight path angles ‖풓⃗ 1 × 풓⃗ 2‖ transfer

6.2.2 Lambert Solving Algorithms

For a set of terminal points 푃1 and 푃2, Lambert’s problem requires finding the transfer trajectory with the desired time of flight. The time of flight (푡2 − 푡1) of a partial elliptical orbit can be expressed through Kepler’s equation as:

1 1 1 √휇(푡 − 푡 ) = 푎3/2 [ (퐸 − 퐸 ) − 푒 sin (퐸 − 퐸 ) cos (퐸 + 퐸 )] , (6.1) 2 1 2 2 1 2 2 1 2 2 1 where 휇 is the gravitational parameter of the Sun. In this form, the time of flight equation has four independent variables: the semi-major axis 푎, eccentricity 푒, and the initial and final eccentric anomalies 퐸1, 퐸2 of the orbit. The number of independent variables can be reduced using a geometric property of conic trajectories know as Lambert’s theorem [230]:

Lambert’s Theorem

The transfer time of a body moving between two points on a conic trajectory is a function only of the sum of the distances of the two points from the origin of the force, the linear distance between the points, and the semi-major axis of the conic.

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Using this property, the time of flight equation for elliptical orbits can be expressed as a function of the semi-major axis 푎, and the semi-perimeter 푠 and chord length 푐 of the transfer geometry [229]:

3/2 √휇(푡2 − 푡1) = 푎 [(훼 − sin 훼) − (훽 − sin 훽)] , (6.2) where the angles 훼 and 훽 are calculated as:

훼 푠 2 sin = ( ) , (6.3) 2 2푎

훽 푠 − 푐 2 sin = ( ) . (6.4) 2 2푎

For a set of two terminal points, the semi-perimeter and chord length are constant, and the time of flight equation has only one independent variable – the semi-major axis. The time of flight equations (Eqs. 6.2 – 6.4) are only valid for elliptical orbits. There are a corresponding set of equations for hyperbolic trajectories making use of the hyperbolic anomaly 퐻 in place of the , and the hyperbolic trigonometric functions sinh and cosh.

These two sets of equations can be reconciled into a single set by making use of the universal anomaly 휒 that is defined for both hyperbolic and elliptical orbits:

√휇∆푡 = 휒3푆(푧) + 퐴휒√퐶(푧) , (6.5) where:

휒2 푧 = = ∆퐸2 . (6.6) 푎 and:

푟 푟 퐴 = sin 휃 √ 1 2 . (6.7) 1 − cos 휃

The “Stumpff” functions 푆, and 퐶 are defined by an infinite series expansion of 푧 [231]:

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∞ 푧푘 1 푧 푧2 푧3 푆(푧) = ∑(−1)푘 = − + − + ⋯ (6.8) (2푘 + 3)! 6 120 5040 362880 푘=0

∞ 푧푘 1 푧 푧2 푧3 퐶(푧) = ∑(−1)푘 = − + − + ⋯ (6.9) (2푘 + 2)! 2 24 720 40320 푘=0

These can be expressed as piecewise functions dependent on the sign of 푧:

푧 − sin 푧 √ √ 3 , 푧 > 0 (√푧)

sinh √−푧 − √−푧 푆(푧) = , 푧 < 0 (6.10) 3 (√푧)

1 , 푧 = 0 { 6

1 − cos √푧 , 푧 > 0 푧 cosh √−푧 − 1 퐶(푧) = , 푧 < 0 (6.11) −푧 1 , 푧 = 0 . { 2

A Lambert solving algorithm numerically solves Lambert’s problem by iterating through different values of the independent variable until the required time of flight is met. There are a number of algorithms using different formulations of the time of flight equation, and iterative root finding methods. The computer programs developed in thesis will make use of the universal anomaly method that has been shown to be one of the simplest and fastest methods. A full discussion on Lambert solving algorithms is presented by Battin [9].

Figure 6.2 shows an example of the time of flight as a function of 푧 (defined in Eq. 6.6) using the universal anomaly formulation for an arbitrary transfer geometry. For a required time of flight (shown with the horizontal red line), the corresponding variable 푧∗ can be found from an initial estimate through a root finding method such as the Newton-Raphson method. The Keplerian elements of the transfer can then be found through analytical expressions using the functions 퐴, 푆, and 퐶 described above (see Curtis [231] for algorithms describing these computations).

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Figure 6.2 Time of flight as a function of 풛 for one particular transfer geometry [58].

As can be seen in figure 6.2, for larger values of the time of flight, multiple solutions may exist corresponding to transfer angles greater than 360o. These solutions are multi- revolution transfers that pass through the terminal points numerous times. Each solution is distinguished by 푁 – the number of complete revolutions.

6.3 Porkchop Plots Preliminary trajectory design for body-to-body transfers commonly make use of “porkchop” plots. A porkchop plot is the colloquial name given to contour plots of various trajectory parameters plotted over a grid of launch date/arrival dates (or alternatively launch date/times of flight). Porkchop plots allow for a visual representation of the locations of multiple optimal trajectories between bodies.

Optimal trajectories are found from local minima in the porkchop plots. This is equivalent to solving an optimization problem to find the set of launch and arrival dates that minimizes an objective function. When listing optimal launch dates, it is important to note which objective function is being minimized (or maximized), as different objective functions may produce different shaped plots, leading to a different set of optimal launch and arrival dates.

Porkchop plots have been extensively studied for the case of Earth-to-Mars and Mars- to-Earth transfers, with a list of optimal launch and arrival dates compiled for the periods of 2009–2024 [232] and 2026–2045 [233]. Porkchop plots for the Earth-to-Mars 2 transfer most commonly display contours of the (퐶3 = 푉∞표푢푡) of

Scott Dorrington – June 2019 211 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions the departure hyperbola from Earth. The 퐶3 energy is often desired to be minimized as it determines the payload capacity of launch vehicles launching into hyperbolic departure trajectories. Porkchop plots may alternatively display the hyperbolic excess velocity of the arrival hyperbola 푉∞푖푛, or a combination of the two parameters, expressing the total delta-V of the transfer. Constraints on the declination of the outgoing asymptote are also commonly applied to the porkchop plots, reflecting the performance capabilities of the launch vehicle and launch site. These constraints may be plotted as additional contours overlaid on the porkchop plot.

6.3.1 Launch Opportunities Figure 6.3 shows a contour plot of the total delta-V of the Earth-to-Mars transfer over a timeframe ranging from 2012 to 2020. It can be seen that local minima in the plot occur in periodic intervals across the range. Each of these intervals specifies a “launch opportunity". For Earth-to-Mars transfers, optimal launch opportunities occur each synodic period (every 26 months), when the relative orientations of the Earth and Mars are favourable (close to the Hohmann transfer). A detailed plot of the 2018 launch opportunity is shown at the bottom of figure 6.3.

Due to the slight eccentricity and inclination in the orbits of the two bodies, the magnitudes and relative positions of the local minima do not repeat exactly each synodic period. The relative positions of Earth and Mars return to their original orientation every 7 to 8 synodic periods (15 to 17 years), leading to a repeating cycle in the magnitude of the type I and type II minima [232]. This cyclic behaviour is shown in the top of figure 6.3.

The synodic period is determined by the relative orbital periods of the two bodies. Mars has a 26 month synodic period with respect to Earth. Asteroids with semi-major axes close to that of Earth can have long synodic periods on the order of decades. For example, asteroid 2008 EV5 has a semi-major axis of 0.96 AU, leading to a 15 year synodic period. The porkchop plots for transfers to these asteroids show multiple launch opportunities over a single synodic period, created by resonances in the relative positions of the two bodies.

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Figure 6.3 Porkchop plot showing 2018 optimal Earth-to-Mars transfers [234].

6.3.2 Type I and Type II Transfers Each launch opportunity has a bi-lobed contour plot with two local minima on either side of a central ridge. (It is from this shape that porkchop plots received their name.) The central ridge follows the contour of trajectories with 180o transfer angles. For orbits that are coplanar, it has been shown that the optimal (minimum delta-V) transfer is a

Scott Dorrington – June 2019 213 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions two-impulse 180o transfer with the apses tangential to the initial and final orbits [87] 10. This is known as a Hohmann transfer.

For orbits that are non-coplanar (as is the case for Mars and the majority of near-Earth asteroids), 180o transfers are only possible along the line of nodes, where the two orbital planes intersect. For trajectories with 180o transfer angles that do not originate at one of the two nodes, the arrival position of the target body will be above or below the ecliptic, and the only 1800 transfer possible between these points will require a 90o inclination, with high plane change delta-Vs associated (the same can be said for 0/360o transfers.) This geometry is illustrated in figure 6.4.

Figure 6.4 The 180o transfer for Earth-to-Mars trajectories showing the requiring 90o inclination [233].

The 1800 transfer ridge separates the contour plot into two distinct regions. Type I transfers are those below the ridge with transfer angle 0° ≤ 휃 ≤ 180°, and Type II transfers are those above the ridge with transfer angles 180° ≤ 휃 ≤ 360°. The two regions may be connected by a “neck region”, where nodal transfers occur. The

10 For transfers in which the ratio of the two orbital radii are large (푟2/푟1 > 11.95 or 푟2/푟1 < 0.084), a three-impulse “bi-elliptic” transfer is more efficient than the Hohmann transfer. As we are interested in asteroids within the main asteroid belt (푎 < 3 AU), we will not consider the bi-elliptic transfer.

214 Scott Dorrington – June 2019 Chapter 6: Combinatorial Trajectory Optimization locations of the type I and type II minima are usually close to the neck region, as seen in the bottom of figure 6.3.

6.3.3 Multi-revolution Transfers From figure 6.3, it can be seen that the majority of low delta-V trajectories are restricted to a narrow band following the contours of times of flight between 100 and 700 days (approximately equal to the orbital period of Mars at 687 days). For type I transfers, as the time of flight decreases, the solution to Lambert’s problem may require a hyperbolic transfer trajectory (negative 푧 value seen in figure 6.2). These transfers will have large velocities at the terminal points, hence large delta-Vs.

For type II transfers, the delta-V is also seen to increase with the time of flight. This increase is caused by the transfer angle approaching 360o, leading to large plane change angles. If the time of flight is increased further, the transfer angle will exceed 360o. For these trajectories, multiple solutions can exist to Lambert’s problem, associated with single-revolution (0° ≤ 휃 ≤ 180°) or multi-revolution (360° ≤ 휃 ≤ 540°) trajectories. The multi-revolution transfers are known as type III transfers, and a similar set of trajectories with larger transfer angle ( 540° ≤ 휃 ≤ 720° ) are known as type IV transfers. (Additional sets of transfers may also exist with each additional full revolution.)

For single-revolution trajectories with a fixed transfer angle, as the time of flight increases, the transfer trajectory is required to become highly elliptical. This can create large difference in the flight path angles at the terminal points, leading to a large delta- V. This is illustrated in figure 6.5, showing single and multi-revolution trajectory solutions for a single set of launch and arrival dates computed for the transfer between Earth and asteroid 101955 Bennu.

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Figure 6.5 Comparison of single- and multi-revolution transfers with the same launch and arrival dates.

6.4 Delta-Vs of Hyperbolic Departure/Arrival Trajectories The objective functions used in evaluating the optimal launch and arrival dates will be calculated from the total delta-V of the transfer trajectory. In this thesis, four different types of heliocentric transfers are considered: 1. Earth-to-Asteroid flyby (launch from Earth); 2. Earth-to-Asteroid rendezvous (launch from Earth); 3. Asteroid-to-Earth rendezvous (to Earth orbit); and 4. Earth-to-Asteroid rendezvous (from Earth orbit).

These four trajectory types are distinguished by the transfer direction (Earth-to-Asteroid or Asteroid-to-Earth), the encounter type (flyby or rendezvous), and the launching condition (launching from Earth surface or departing from an Earth parking orbit).

Each of these mission types will have different objective functions 푓1 to 푓4 depending on the design of the departure/capture manoeuvres. For each asteroid, a separate set of optimal launch and arrival dates will be generated for each mission type over the 2020 to 2050 timeframe. (Ideally, a new set of optimal launch dates should be calculated for each customer orbital plane.)

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6.4.1 Earth-to-Asteroid flyby (launch from Earth) The flyby exploration mission described in chapter 3 will be initially launched from Earth’s surface by a launch vehicle into a parking orbit in LEO. From here, the spacecraft will conduct a single impulsive departure manoeuvre to achieve a hyperbolic trajectory that will place it onto a heliocentric transfer orbit that intercepts the target asteroid.

As the spacecraft is not required to rendezvous with the asteroid, the optimization criterion will be to minimize the Earth departure manoeuvre. The delta-V of the departure manoeuvre from a circular parking orbit can be calculated as a function of the hyperbolic excess velocity 푉∞표푢푡 and the radius 푟푝 of the parking orbit:

2휇 휇 2 퐸 퐸 푓1 = ∆푉1 = √푉∞표푢푡 + − √ . (6.12) 푟푝 푟푝

In order to maintain consistency, it is assumed that the spacecraft is launched into a circular parking orbit with an altitude of 500 km (the nominal orbit of the Electron launch vehicle [235]).

The departure manoeuvre could be accomplished in a number of successive manoeuvres, each raising the apoapsis of the parking orbit, with a final manoeuvre to complete the hyperbolic insertion. These manoeuvres are additive, with the total delta-V remaining the same. Figure 6.6 shows the geometry of this departure manoeuvre from a circular orbit, with the intermediate elliptical orbits of successive delta-V manoeuvres. The orientation of the periapsis of the departure manoeuvre with respect to the outgoing asymptote is calculated by the angle 훽:

1 cos 훽 = 2 . 푟푝푉∞ (6.13) 1 + 휇

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Figure 6.6 Geometry of the hyperbolic departure from a circular orbit.

Depending on the size of the spacecraft, there is likely to be excess payload capacity of the launch vehicle that is not utilized. Launching into a circular LEO parking orbit affords the opportunity of reducing the launch cost by offering ridesharing to smaller satellites to utilize the excess payload capacity (or, if the spacecraft is small enough, launching as a secondary payload on another launch).

The Earth departure delta-V could alternatively be provided by the upper stage of the launch vehicle, placing the spacecraft into an elliptical parking orbit, or directly onto a hyperbolic escape trajectory. This would require less propellant to be carried by the spacecraft; however, a larger launch vehicle would be required, with a higher launch cost. The targeted outgoing asymptote would require the spacecraft to be the primary payload of a dedicated launch. Due to the specialized hyperbolic trajectory, the excess capacity is unlikely to be attractive to ridesharing customers. This option is likely to be more expensive, and hence is not considered.

6.4.2 Earth-to-Asteroid rendezvous (launch from Earth) For trajectories that are required to rendezvous with the asteroid (such as the sampling and mining missions), an additional delta-V manoeuvre is required upon arrival at the asteroid. Due to the negligible gravitational field of the asteroid, the arrival hyperbola can be neglected and the magnitude of the arrival manoeuvre at the asteroid is calculated simply as the magnitude of the hyperbolic excess velocity of the arrival hyperbola:

∆푉2 ≈ 푉∞푖푛 . (6.14)

The objective function to minimize will be the total delta-V of the transfer:

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푓2 = ∆푉푇표푡푎푙 = ∆푉1 + ∆푉2 , (6.15) where ∆푉1 is given by Eq. 6.12.

The delta-V calculations for both the flyby and rendezvous transfers do not consider any plane change manoeuvres, as it assumes that the departure parking orbit is orientated in the same plane as the outgoing asymptote (perpendicular to the B-plane of the outgoing asymptote). This is often the case with porkchop plots generated for planetary missions.

Figure 6.7 Geometry of the launch trajectory [236].

The targeted outgoing asymptote can be achieved by designing the inclination and right ascension of the ascent trajectory of the launch vehicle. The of the parking orbit after launch will be dependent on the latitude of the launch site, and the launch (direction of the ascent path measured clockwise from local north of the launch site). The right ascension can be controlled by the launch time. Figure 6.7. shows the geometry of the ascent trajectory. The design of ascent trajectories is beyond the scope of this thesis. For details on designing the launch azimuth to target an outgoing asymptote, the reader is directed to Sergeyevsky et al. [236].

6.4.3 Asteroid-to-Earth rendezvous (to Earth orbit) Planetary missions typically are only designed as one-way missions that are launched from Earth. Missions requiring the spacecraft to return to Earth, such as sample return missions or human exploration missions also assume that the returning vehicle will re- enter the Earth’s atmosphere on a hyperbolic trajectory. As a result, delta-V calculations

Scott Dorrington – June 2019 219 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions used in the generation of porkchop plots do not generally include any plane change manoeuvres, instead only considering the magnitude of the hyperbolic excess velocity.

For an asteroid mining mission, during each mining trip, the mining spacecraft will return a shipment of asteroid material to an orbital propellant depot, from which it will be delivered to customers. As a result, the total delta-V of the Asteroid-to-Earth transfers must include additional plane change and orbit adjusting manoeuvres in order to rendezvous with the depot.

In chapter 5, two routing strategies were discussed for the design of the arrival parking orbit. The first strategy (Route 1) is one in which the mining spacecraft rendezvous with the propellant depot. The second strategy (Route 2), is one in which the mining spacecraft remains in a highly elliptical arrival parking orbit, transferring to a departure parking orbit to target the outgoing asymptote of the next Earth-to-Asteroid transfer.

While Route 2 was found to be more efficient, the calculation of the delta-Vs required both the incoming and outgoing asymptotes to be defined. The delta-Vs for Route 1, on the other hand, are only dependent on the incoming asymptote. Route 1 is therefore more practical to apply to porkchop plots, wherein the optimal launch dates of the Asteroid-to-Earth and Earth-to-Asteroid transfers are considered separately.

Route 1 consists of four delta-V manoeuvres. The first manoeuvre is the asteroid departure, which may be approximated as the hyperbolic excess velocity of the outgoing asymptote at the asteroid:

∆푉1 ≈ 푉∞표푢푡 . (6.16)

Following the heliocentric transfer, the mining spacecraft will approach the Earth on a hyperbolic trajectory along the incoming asymptote. The spacecraft will then perform an Earth orbit insertion manoeuvre at the periapsis of the hyperbola to be captured into an arrival parking orbit with periapsis 푟푝 and apoapsis 푟푎. This delta-V is independent of the orientation of the arrival orbit, instead only dependent on the periapsis and apoapsis and the hyperbolic excess velocity 푉∞푖푛 of the incoming asymptote:

2휇 휇 휇 2 퐸 퐸 퐸 (6.17) ∆푉2 = √푉∞푖푛 + − √2 ( − ) . 푟푝 푟푝 푟푝 + 푟푎

The mining spacecraft will then perform a plane change manoeuvre at one of the node crossings with respect to the depot orbital plane. As noted in chapter 5, setting the

220 Scott Dorrington – June 2019 Chapter 6: Combinatorial Trajectory Optimization periapsis radius equal to the depot orbit allows for the manoeuvre to be achieved by a simple plane change manoeuvre (where the periapsis and apoapsis are unchanged after the manoeuvre). This delta-V can be calculated from the simple plane change manoeuvre:

∆푉3 = ∆푉푝푐 = 2푉푁퐶 sin(휑/2) . (6.18) where 푉푁퐶 is the velocity of the parking orbit at the node crossing, and 휑 is the plane change angle between the arrival parking orbit and the depot orbit. As noted in chapter 5, this delta-V is dependent on the orientation of the aim vector, which should be selected to minimize the plane change delta-V.

The mining spacecraft is then in an elliptical parking orbit in the same plane as the depot orbit. A final circularization delta-V is applied at the periapsis (equal to the depot radius) to place the mining spacecraft into the depot orbit:

휇퐸 휇퐸 휇퐸 ∆푉4 = √2 ( − ) − √ . (6.19) 푟퐷푒푝표푡 푟퐷푒푝표푡 + 푟푎 푟퐷푒푝표푡

In chapter 5, the depot orbit and apoapsis were both treated as independent variables. To simplify calculations, all delta-Vs are calculated with an apoapsis of 푟푎 = 768,798 km (2 lunar distances), and a depot orbit at 푟퐷푒푝표푡 = 42,364 km (200 km above Geostationary orbit).

The total delta-V of the Asteroid-to-Earth (to Earth orbit) transfer is then calculated as:

푓3 = ∆푉푇표푡푎푙 = ∆푉1 + ∆푉2 + ∆푉3 + ∆푉4 . (6.20)

Figure 6.8 shows the geometry of this capture manoeuvre.

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Figure 6.8 Geometry of capture manoeuvre of the Asteroid-to-Earth (to Earth orbit) transfer.

6.4.4 Earth-to-Asteroid rendezvous (from Earth orbit) During the first mining mission, the Earth-to-Asteroid transfer will be launched from Earth’s surface. On subsequent mining trips, the mining spacecraft will depart from the depot orbit. The departure manoeuvres (and hence the delta-Vs) will be the same as those described for the previous section, with the ordering reversed, and the hyperbolic excess velocity replaced with that of the outgoing hyperbola 푉∞표푢푡.

The delta-Vs for the Earth-to-Asteroid and Asteroid-to-Earth transfers are used as the objective functions to select the optimal launch dates. The actual delta-V of the arrival and departure manoeuvre will be recalculated in section 6.8 to account for the combination of successive Asteroid-to-Earth and Earth-to-Asteroid transfers.

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6.5 Procedure to Generate Porkchop plots Porkchop plots are commonly produced and studied on an individual basis for the targets of planetary missions, often focusing on only single launch opportunities. Porkchop plots have been produced for transfers to Mars, Venus, Jupiter, Neptune, and (for example, [237]). In this thesis, it is desired to identify optimal launch and arrival dates of transfers to a large number of near-Earth asteroid targets (identified in chapter 7), over an extensive time period. In this section, a generic method is outlined for generating porkchop plots to asteroids from ephemeris data.

Step 1: Generate Search Space The location of optimal trajectories will be described by the decision variables 푥 = (퐿퐷, 퐴퐷) describing the launch date 퐿퐷 and arrival date 퐴퐷 of the trajectories. The search space over which to visualize the various objective functions 푓(푥) will be defined by a two-dimensional grid of 퐿퐷/퐴퐷 coordinates. This grid is generated from a set of discrete launch dates over the range of the :

퐿퐷 = [퐿퐷푠푡푎푟푡, … , 퐿퐷푒푛푑] . (6.21)

For this thesis, the launch window is defined as ranging from 2020 to 2050 (inclusive). This 31 year time frame should produce sufficient numbers of launch opportunities to conduct multiple return trips to the asteroid.

A set of discrete arrival dates 퐴퐷 is then generated with the start arrival date equal to the start launch date, and the end arrival date equal to the end launch date plus the maximum time of flight:

퐴퐷 = [퐿퐷푠푡푎푟푡, … , 퐿퐷푒푛푑 + 푇푂퐹푚푎푥] . (6.22)

The spacing of the vectors will affect the resolution of the porkchop plot. A five day resolution is sufficient for the initial identification of local minima.

푳푫/푨푫 Grid The 퐿퐷 and 퐴퐷 vectors are used to produce a 2-dimensional matrix defining a grid of possible launch and arrival dates. For each grid point, the time of flight is calculated from the difference between the launch and arrival dates:

푇푂퐹(푖, 푗) = 퐴퐷(푗) − 퐿퐷(푖) , (6.23)

Scott Dorrington – June 2019 223 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions where 푖, 푗 are the indices of the launch and arrival dates, respectively.

The times of flight will form sloping contours in the 퐿퐷/퐴퐷 grid. In order to be feasible, trajectories must arrive at the target body after they depart the initial body. As such, grid points with times of flight less than zero may be immediately disregarded. Long duration times of flight will also be undesirable as they will lead to long times between the launch and delivery of material. A maximum time of flight of 2.5 years is applied to the search grid.

Figure 6.9 shows the search space of trajectories with coordinates 푥 = (퐿퐷, 퐴퐷) describing the launch and arrival dates. The time of flight constraints are shown with grey regions. The remaining feasibility region (shown in white) is defined by:

2 {푥 ∈ ℝ | 0 ≤ 푥2 − 푥1 ≤ 푇푂퐹푚푎푥} . (6.24)

Figure 6.9 LD/AD grid showing time of flight contours and constraints.

푳푫/푻푶푭 Grid Over long launch date ranges, the feasibility region in the 퐿퐷/퐴퐷 grid can be long and narrow, making it is difficult to resolve features in the porkchop plots (as can be seen in the previous Earth-to-Mars porkchop plot in figure 6.3). The porkchop plots may alternatively be displayed on a grid of 퐿퐷/푇푂퐹. The coordinates of trajectories in this

224 Scott Dorrington – June 2019 Chapter 6: Combinatorial Trajectory Optimization plot are defined by the decision variables 푦 = (퐿퐷, 푇푂퐹). This is a simple linear transformation from the 퐿퐷/퐴퐷 grid: 푦 = (푥1, 푥2 − 푥1).

Step 2: Compute Transfer Geometry For each of the grid points (푖, 푗) within the feasibility region, the terminal state vectors

풙⃗⃗ 1 and 풙⃗⃗ 2 are found from ephemeris data of the two bodies at the launch and arrival dates. This is first applied for the Earth-to-Asteroid transfer, then repeated for the Asteroid-to-Earth transfer.

From these vectors, a number of geometric properties of the transfer can be defined, including the transfer angle 휃, and inclination. These properties may be used to further constrain the feasibility region of the search space, reducing the number of points for which to solve Lambert’s problem (discussed in section 6.5.4).

As noted above, the contours of 0/360o and 180o transfer angle typically define ridges of high delta-V in the porkchop plot. It is convenient to plot these contours on the 퐿퐷/푇푂퐹 grid (shown in figure 6.10) to visually segment the search space into alternating regions of type I and type II transfers (this may include multi-revolution type III and type IV transfers):

푇푦푝푒 퐼 = {푦 ∈ ℝ2| 0 + 2휋푁 ≤ 휃 ≤ 휋 + 2휋푁, 푓표푟 푁 = 0,1} , (6.25)

푇푦푝푒 퐼퐼 = {푦 ∈ ℝ2| 휋 + 2휋푁 ≤ 휃 ≤ 2휋 + 2휋푁, 푓표푟 푁 = 0,1} . (6.26)

Figure 6.10 LD/TOF grid showing contours of transfer angle separating the type I and type II transfers.

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Step 3: Solve Lambert’s Problem A Lambert solving algorithm is then applied to each feasible grid point to return the terminal velocity vectors 풗⃗⃗ 푡푥1, 풗⃗⃗ 푡푥2 of the transfer trajectory:

[풗⃗⃗ 푡푥1, 풗⃗⃗ 푡푥2] = 푙푎푚푏푒푟푡(풓⃗ 1, 풓⃗ 2, 푇푂퐹(푖, 푗)) . (6.27)

From these vectors, the incoming and outgoing asymptotes of the hyperbolic trajectories are computed at the two bodies:

⃗⃗ 푺푖푛 = 풗⃗⃗ 2 − 풗⃗⃗ 푡푥2 , (6.28)

⃗⃗ 푺표푢푡 = 풗⃗⃗ 푡푥1 − 풗⃗⃗ 1 . (6.29)

The transfer trajectory generated by Lambert’s problem is independent of the delta-Vs of the particular transfer. As such, it only needs to be generated once for each grid point, from which the delta-Vs of each of the different Earth-to-Asteroid transfers can be computed (푓1(푥), 푓2(푥), and 푓4(푥) defined in the previous section). For each transfer, a 2-dimensional delta-V matrix is generated specifying the computed objective function at each grid point (푖, 푗).

Step 4: Repeat for Asteroid-to-Earth transfer Steps 2 and 3 are then repeated for the Asteroid-to-Earth transfer using the same search space grid, and the 푓3(푥) objective function.

Step 5: Locate Minima Initial estimates of optimal trajectories are found from local minima in each of the delta- V matrices. These are found by applying an image search algorithm that compares the values of all neighbouring grid points to determine if the point is a local minima. Due to the finite resolution of the search grid, some points may be mis-identified as local minima, while other regions may not be identified. For each of the transfer types, a porkchop plot is generated and visually inspected, adding additional initial estimates close to local minima that were missed.

Each of the initial estimates are used as an initial guess in an iterative optimization algorithm to find better approximations of the optimal launch and arrival dates. The algorithm searches a limited region surrounding the point to identify any point that has a lower objective function. A range of 25 days either side of the initial estimate is used to

226 Scott Dorrington – June 2019 Chapter 6: Combinatorial Trajectory Optimization ensure that the solution is not attracted to a more efficient neighbouring basin of attraction.

6.5.1 Programming Implementation The process for producing a porkchop plot is summarized in algorithm 6.1. The algorithm uses a double-nested for loop to cycle through the grid points. This strategy has been implemented by other programs used to generate porkchop plots [238]. A computer program was written in MATLAB to implement this algorithm, producing a set of optimal launch and arrival dates for each of the four transfer types for a given asteroid candidate.

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The launch and arrival dates are read from ephemeris data of the asteroid and Earth over the time period of interest. Ephemeris data are represented using the “SPICE” file format developed by NASA's Navigation and Ancillary Information Facility (NAIF). SPICE data sets, known as "kernels", or "kernel files", are commonly used for representing trajectory and attitude data for spacecraft, planetary bodies, and scientific instruments [239]. SPK (Spacecraft Planet Kernels) files contain information on spacecraft or planetary ephemeris as a function of time.

SPK files for individual asteroids were generated using a web-based interface to the JPL Horizons system [240]. The program makes use of the NAIF MATLAB SPICE Toolkit (MICE) [241] that contains of a library of subroutines that allows for input/output, and manipulation of SPICE kernel data in the MATLAB environment. Ephemeris data for the Earth (and other major planets) is read from DE405 ephemeris files generated by the JPL Horizons system [10].

All calculations in the program are carried out in the standard units used in astrodynamics; namely kilometres for distance, and seconds for time. The launch and arrival date vectors are generated using ephemeris time (ET), defined in seconds past a reference epoch. These dates are converted to Julian Ephemeris date (JED) for displaying in the porkchop plots (using units of days).

The state vectors of the Earth and asteroid are defined from the Solar System Barycentre, in the J2000 Ecliptic coordinate frame (with z-axis normal to the Ecliptic plane, the x-axis defined by the vernal , and the y-axis defined by the right-hand rule, to complete the orthogonal basis).

Initial estimates for the location of the local minima are found from the local minima of the delta-V matrix using the MATLAB function imextrema [242]. These are used as initial guesses for an optimization program to refine the estimate. This program uses the SNOPT nonlinear optimization algorithm [243, 244]. The implementation of the SNOPT optimization program is adapted from a similar porkchop plot program produced by Eagle [245], available from the MATLAB file exchange. This implementation also allows for constraints to be applied on the magnitude of the 퐶3 and declination of the outgoing asymptote, the time of flight, and the hyperbolic excess velocity of the incoming asymptote.

The program has been adapted to include the objective functions defined in section 6.4, and an additional constraint for the transfer angle. Eagle’s original program also

228 Scott Dorrington – June 2019 Chapter 6: Combinatorial Trajectory Optimization contains subroutines for computing the primer vector history of the resulting trajectories (discussed in the next section).

6.5.2 Example Porkchop Plots Figure 6.11 shows an example porkchop plot produced for the Earth-to-Asteroid (launch from Earth) transfer to asteroid 101955 Bennu. The plot only displays trajectories with a total delta-V less than 10 km/s. Contours of 0/360o and 180o transfers are plotted in red. Local optimal trajectories are displayed with black and red asterisks, showing the initial guesses and final optimal locations, respectively. These were computed with only time of flight constraints added. Lines are shown connecting the initial guesses to the optimal solutions. A total of 162 launch opportunities were identified over the 31 year launch range. Following a manual inspection of the porkchop plot, an additional 13 initial guesses were added in basins of attraction that were missed by the initial delta-V matrix search. Figure 6.12 shows a detailed view of the plot. In this region, several basins of attraction can be seen for type I, type II, and type III transfers, separated by the transfer angle contours.

Figure 6.11 Example porkchop plot for the Earth-to-Asteroid (launch from Earth) transfer to asteroid 101955 Bennu.

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Figure 6.12 Detailed view of the porkchop plot showing several basins of attraction of type I, II and III transfers.

The goal of the program is to identify a wide range of launch opportunities to be considered for potential flight itineraries between the Earth and asteroid (discussed in the next section). Due to the presence of multiple basins of attraction, the search range over which the optimization algorithm is run should be kept small to avoid initial guesses being attracted to adjacent basins of attraction.

Figure 6.13 shows a plot of the differences in the launch and arrival dates between the initial guess and the converged optimal solution. This plot was generated for a search range of ±50 days in both launch date and arrival date. From the plot, it can be seen that the majority of initial guesses were fairly close to the true local minima, within ±25 days. Several points can be seen at the edges of the search range. These points are being attracted to adjacent basins with lower delta-Vs. For these points, the optimal solution is replaced with the initial guess. While these points may not be local minima, they are kept to ensure that no basins of attraction are missed.

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Figure 6.13 Distances in LD and AD from the initial guess to the converged solution.

6.5.3 Computation Time The porkchop plots were generated using a resolution of 5 days in both the launch and arrival date vectors. This provides sufficient resolution for the initial identification of local minima in the delta-V matrices. For launch dates ranging from 1st January 2020 to 31st December 2050, and time of flight from 0 to 912.5 days, the search grid has a total of 5,542,455 elements. This reduces to a total of 412,230 feasible grid points after applying the time of flight constraints.

The calculation of the transfer geometry took on average 1 minute for each of the Earth- to-Asteroid and Asteroid-to-Earth transfers. The solving of Lambert’s problem, and calculation of the delta-Vs took a further 6.9 for the Earth-to-Asteroid transfers, and 5.8 minutes for the Asteroid-to-Earth transfers. The Earth-to-Asteroid porkchops are expected to take longer, as there are three delta-Vs to compute, compared to a single delta-V for the Asteroid-to-Earth transfer.

The total computation time for both Earth-to-Asteroid and Asteroid-to-Earth transfers was approximately 15 minutes, on an Intel® Core™ i5-4570 processor at 3.20 GHz. The short runtime of the program is ideal for processing the large number of asteroids identified in chapter 7. Additional time is required for the physical inspection of the initial estimates to ensure no basins were missed.

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6.5.4 Search Space Pruning The porkchop plot displayed in figure 6.11 shows trajectories with total delta-V less than 10 km/s. As can be seen from the plot, these trajectories only make up a fraction of the grid points in the feasibility region. The computation time of the program can be reduced by eliminating or “pruning” grid points that are expected to produce high delta- Vs.

Inclination As noted in section 6.3.2, the high delta-V ridge following the 180o transfer angle contours is produced by a large inclination approaching 90o. As the inclination of the transfer can be computed from the transfer geometry, without the need to solve Lambert’s problem, grid points with inclinations above a set threshold value could be disregarded, reducing the number of grid points in the feasibility region. Figure 6.14 shows a plot of the transfer inclinations and delta-Vs for the Earth-to-Asteroid (launch from Earth) transfer for asteroid 101955 Bennu.

From this plot, it can be seen that the minimum delta-V increases with the transfer angle, and all transfers with inclinations above 30o have delta-Vs greater than 10 km/s. If a limiting inclination of 30o is applied, a total of 31,069 grid points can be removed, hence reducing the number of computations of Lambert’s problem by 7.54%.

Figure 6.14 Delta-V vs Inclination.

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Characteristic transfers The total delta-V of a transfer is primarily determined by the magnitudes of the hyperbolic asymptotes and the inclination of the transfer. While the accurate velocities of the transfer orbit would require solving Lambert’s problem, an order of magnitude estimate for the terminal velocity vectors can be generated from characteristic transfer computed from the geometry of the transfer. One such example is the Fundamental ellipse (shown in figure 6.15).

Figure 6.15 The Fundamental ellipse [229].

The Fundamental ellipse is defined as the minimum eccentricity ellipse between two points, with its apse line parallel to the chord. The semi-major axis and eccentricity can be computed from the distances of the terminal points and the chord length:

푟 + 푟 푎 = 1 2 , (6.30) 퐹 푐 |푟 − 푟 | 푒 = 2 1 . (6.31) 퐹 푐

From these elements, the terminal velocity vectors of the transfer can be computed through analytical expressions (see Battin [229]), and used to compute the hyperbolic asymptotes and delta-Vs without the need to solve Lambert’s problem.

Figure 6.16 shows a plot of the actual delta-V for the 퐸퐴퐿 transfer compared to the delta-V computed from the Fundamental ellipse. These delta-Vs estimates have been computed from the hyperbolic excess velocity, and do not include the delta-Vs of arrival into the specified parking orbits. As a result, the delta-Vs do not match those calculated from the Lambert solution. However, they do show a general trend of increasing values,

Scott Dorrington – June 2019 233 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions with all points having estimated delta-Vs above 35 km/s having actual delta-Vs above 10 km/s. There were a total of 45,083 grid points above this value, which if removed would reduce the number of Lambert computations by 10.94%.

Figure 6.16 Delta-V computed with the Fundamental ellipse vs actual delta-V.

These pruning methods have only been tested on a few asteroids, after the full delta-V is computed. More testing would be required to ensure that they can be used to prune the grid points without a priori knowledge of the actual delta-Vs.

6.5.5 Primer Vector Analysis While the SNOPT optimization algorithm succeeds in locating local minima in the delta-V plot, the solution trajectories may not in fact be locally optimal. The optimality of an impulsive transfer can be checked using Primer vector theory [246].

The primer vector 풑 of a trajectory is defined as the adjoint vector to the velocity, that is parallel to the thrust vector applied by the spacecraft [246]. For constant thrust trajectories, the primer vector is defined by the thrust direction history throughout the transfer (generated by solving an optimal control problem).

For impulsive transfers, the primer vector is aligned with the thrust direction at any time an impulsive manoeuvre is applied. For the two-impulse body-to-body transfer, the primer vector is a unit vector pointing in the direction of the outgoing asymptote at the launch date and the incoming asymptote at the arrival date:

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∆푽ퟎ ̂ 푝0 = 푝(푡0) = = 푺⃗⃗ 표푢푡 , (6.32) ‖∆푽ퟎ‖

∆푽풇 ̂ 푝푓 = 푝(푡푓) = = 푺⃗⃗ 푖푛 . (6.33) ‖∆푽풇‖

At any time between the impulses (along the coasting or non-thrust arc), the primer vector can be found by propagating the initial condition using the 6x6 state transmission matrix 훷(푡, 푡0) in a similar manner to the propagation of the state vector (position and velocity) [247]:

푝(푡) 푝(푡0) [ ] = 훷(푡, 푡0) [ ] . (6.34) 푝̇(푡) 푝̇(푡0)

For an impulsive trajectory to be considered locally optimal, the primer vector 풑 and its derivative 풑̇ must meet the following set of necessary conditions of optimality [246, 248]: 1. 풑 and 풑̇ are continuous; 2. 푝 = 1 at impulse times; 3. 푝 ≤ 1 on non-thrust arcs separating impulses; and 4. 푝̇ = 0 at all interior impulses (not at the initial or final times).

Figure 6.17 shows the magnitude of the primer vector and its derivative for one example trajectory identified by the program (using functions developed by Eagle [245]). This transfer is seen to meet all the necessary conditions (the final condition is not necessary, as there are no interior impulses), and is therefore locally optimal according to primer vector theory.

Only a small number of the identified launch opportunities were found to meet these necessary conditions. The majority displayed primer vectors that exceed unity at some point along the trajectory (an example of which is shown in figure 6.18).

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Figure 6.17 Example primer vector of a locally optimal trajectory.

Figure 6.18 Example primer vector of a non-optimal trajectory.

Primer vector theory provides some directions that can be applied to improving upon a given trajectory. The first method is to add coasting arcs before and/or after the two impulses (i.e. adjusting the launch and arrival dates). The direction in which to move the end points can be determined by the magnitude of the derivative at the end points [245]. However, as this adjusting of the launch and arrival dates was performed by the SNOPT optimization algorithm, the trajectory cannot be improved using this method.

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The second method is to apply an additional impulsive manoeuvre (a deep space manoeuvre) at some time 푡푚 between the initial and final impulses (i.e. adding an interior impulse). If the magnitude of the primer vector is greater than unity at any time, then there exists a three-impulse trajectory which has lower cost (delta-V) than the reference two-impulse trajectory [249].

The time and magnitude of the deep space manoeuvre are required to be found such that the necessary conditions of optimality are met (i.e. the magnitude of the primer vector is unity and the magnitude of the derivative is zero). This requires an additional optimization algorithm to be applied to find a primer vector history that meets these conditions, while maximizing the reduction in cost (delta-V) from the reference two- impulse trajectory. As a first approximation, the greatest improvement in the cost occurs at the time the primer vector magnitude is greatest.

While the addition of a deep space manoeuvre can improve the delta-V of a trajectory, the propagation of the primer vector, and the computation of the optimal time and magnitude of the additional manoeuvre comes at an increased computation time. For this reason, it is not applied in the program in this thesis, and is instead left for future work.

6.6 Flight Itinerary Optimization Problem For a given mining campaign, the flight itinerary will consist of numerous combinations of the four types of body-to-body transfers discussed in the previous section. Consider a mining campaign consisting of 푇 mining trips, with no exploration missions. The flight itinerary will begin with the deployment of the mining spacecraft on an Earth-to- Asteroid transfer launched from Earth (trip 푡 = 0). Following this initial deployment, each mining trip 푡 consists of an Asteroid-to-Earth and Earth-to-Asteroid transfer, both arriving at and departing from Earth orbit.

The selected trajectories of the Earth-to-Asteroid and Asteroid-to-Earth transfers for each mining trip 푡 can be described by the decision variables 푥푡 and 푦푡, respectively.

These decision variables describe the set of launch and arrival dates of the transfer: 푥푡 =

(퐿퐷푡, 퐴퐷푡) and 푦푡 = (퐿퐷푡, 퐴퐷푡).

As discussed in chapter 5, each mining trip 푡 begins and ends at the arrival dates of the Earth-to-Asteroid transfers. As such, the flight itinerary of each mining trip 푡 can be

Scott Dorrington – June 2019 237 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions described by the decision vector 푋푡 composed of a combination of three decision variables:

푋푡 = (푥푡−1, 푦푡, 푥푡) . (6.35)

The full flight itinerary of the mining campaign can similarly be described by a decision vector 푋 composed of an alternating sequence of these decision variables:

푋 = (푥0, 푦1, 푥1, 푦2, 푥2, … , 푥푇−1, 푦푇, 푥푇) . (6.36)

For mining campaigns including exploration missions, additional decision variables 푥퐹퐵 and/or 푥푆 describing the trajectories selected for the flyby and sampling missions may be inserted in the flight itinerary prior to the deployment mission.

As with the body-to-body transfer problem, a set of time constraints must be placed on the decision variables to ensure that the flight itinerary is feasible. At a minimum, these constraints require that the launch date of each transfer occurs after the arrival date of the preceding transfer. Further stringent constraints can be added such that the duration between each consecutive transfer (the stay-time at the asteroid or the wait-time at Earth) are bounded by a lower and upper limit.

The flight itinerary optimization problem can be defined as that of finding the set of launch and arrival dates of the various transfers to maximize an objective function 퐽(푋), subject to stay and wait-time constraints.

The selection of the optimal launch and arrival dates of the transfers discussed in the previous section can be considered as a flight itinerary consisting of a single transfer, with the objective function describing the total delta-V of the transfer. In this case, each of the components of the decision variables 푋 = 푥 = (퐿퐷, 퐴퐷) were treated as continuous variables over a range of values, with a search space of dimension ℝ2. Using this process, a flight itinerary consisting of 푇 mining trips would have a search space of dimension ℝ4푇+2.

This method has been applied to porkchop plots including a planetary flyby mission such as Earth-Venus-Mars transfers – a flight itinerary consisting of two transfers [238]. In this study, the full factorial search through an ℝ4 grid of launch and arrival dates took 140 hours to compute. (The problem was simplified by computing separate ℝ2 grids for each transfer, and combining them.) For mining campaigns consisting of numerous transfers, using a full search through a grid of all feasible trajectories for would therefore be highly computationally expensive.

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To reduce the complexity of the problem, a heuristic method is applied in which each of the transfers is to be selected from the sets of optimal (minimum delta-V) trajectories identified in the previous sections.

For a given asteroid, let 퐹퐵, 퐸퐴퐿, 퐴퐸, and 퐸퐴 be the sets of optimal trajectories for the four transfer types, identified by the process described in the preceding sections:

퐹퐵 = {1,2, … , 퐾} Set of optimal Earth-to-Asteroid Flyby trajectories;

퐸퐴퐿 = {1,2, … , 퐻} Set of optimal Earth-to-Asteroid (launch from Earth) trajectories;

퐴퐸 = {1,2, … , 퐽} Set of optimal Asteroid-to-Earth (to Earth orbit) trajectories; and

퐸퐴 = {1,2, … , 퐼} Set of optimal Earth-to-Asteroid (from Earth orbit) trajectories.

For each of the transfer legs in a given mining campaign, a single trajectory is selected from the corresponding sets of candidates. In this case, the decisions variables 푥푡 and 푦푡 are integer values describing the indices of the selected candidate trajectories for each trip 푡:

푥0 ∈ 퐸퐴퐿 Index of selected 퐸퐴퐿 trajectory for trip 푡 = 0;

푦푡 ∈ 퐴퐸 Index of selected 퐴퐸 trajectory for trip 푡; and

Index of selected 퐸퐴 trajectory for trip 푡. 푥푡 ∈ 퐸퐴

The full flight itinerary of the mining campaign can be described by the sequence of decision variable of length 2푇 + 1 (shown in Eq. 6.36). The selection of the optimal flight itinerary may be solved as an integer programming problem, following a similar method to that used in chapter 5. However, in this chapter, a graph theory approach will be applied.

6.7 Trajectory Combinations using Network Graphs The set of possible combinations of the decision variables making up a flight itinerary can be enumerated and visualized using a network graph (shown in figure 6.19). In this graph, the various transfer legs in the flight itinerary are laid out in equal intervals along the y-axis. For each of these transfer legs, a set of nodes is added listing all possible candidate trajectories. These nodes are laid out along the x-axis, with the x-coordinate representing the dates of the candidate trajectories (the coordinate uses the mid-point of the trajectory – the average of the launch and arrival dates).

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Edges are added between the sets of nodes, representing the possible combinations of successive transfers in the flight itinerary. These edges are divided into two categories. “Stay edges” represent the possible stay-times at the asteroid during which mining operations are carried out. “Wait edges” represent the possible wait-times at Earth, during which the product is distributed to the customers, and the mining spacecraft is prepared for the next departure back to the asteroid. These edges are added between pairs of candidate nodes, with constraints placed on the maximum and minimum stay and wait-times. This method has been employed in previous studies in representing dynamical space transportation networks, where the graph is known as a “time- expanded network” [213, 250].

Figure 6.19 Time-expanded network of possible combinations of selected trajectories.

For each of the edges in the graph, a number of edge cost functions can be defined, describing a metric “distance” or “length” of the edges. The stay-times and wait-times described above, are one such metric defining the duration of each edge.

Possible flight itineraries may be represented as paths through the graph, originating at one of the EAL nodes and terminating at one of the EA nodes (an example path is highlighted in red in the graph). The flight itinerary of each path can be determined by a node list 푋 containing the indices of nodes contained in the path (the same as that defined in Eq 6.31). The flight itinerary may alternatively be expressed as an edge list 퐸퐿 containing the indices of the edges contained in the path:

퐸퐿 = (푠1, 푤1, 푠2, 푤2, … , 푠푇, 푤푇 ) , (6.37) where 푠 and 푤 indicate stay and wait edges.

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The benefit of this representation is that the total objective function of the flight itinerary can be expressed as the sum of edge costs contained in the path, and an optimal flight itinerary can be found by applying graph theory algorithms to solve a shortest or longest path problem.

6.7.1 Objective Functions As discussed in chapter 3, there are several figures of merit that could be used as objective functions to assess the performance of the mining campaign for each candidate flight itinerary. These include the total delta-V, the total mission duration, the total sellable mass (SM), the total profit, and the total Net Present Value (NPV). In order to apply shortest path algorithms, the objective functions must be expressed as linear functions of the edge costs of the graph.

The delta-Vs of the transfers are defined as functions of each node ∆푉(푥푡) or ∆푉(푦푡). These may be converted to edge costs by assigning the delta-V of the start node to each edge (i.e. assigning the delta-V of the starting Earth-to-Asteroid transfer to each stay edge and the delta-V of the start Asteroid-to-Earth transfer to each wait edge). The total duration of each edge can similarly be found by adding the time of flight of the starting transfer to the stay or wait-time of the edge.

The total mission delta-V and mission duration can then be found by the sum of the edge costs contained in the path. As each of the edge cost functions are defined to be positive, Dijkstra’s algorithm [251] can be applied to solve the shortest path problem to find flight itineraries that minimize the total delta-V or mission duration.

As described in chapter 5, the sellable mass is defined as the mass of asteroid material that can be delivered to the customers, after subtracting the propellant mass required to deliver it to the depot, and for the mining spacecraft to return to the asteroid. This sellable mass will determine the total revenues of each mining trip, and hence the profit and Net Present Value.

While the initial extraction mass can be calculated as a function of the stay-time at the asteroid, the propellant mass consumed over the edges is dependent on the delta-Vs of the Asteroid-to-Earth and following Earth-to-Asteroid transfers. As such, the sellable mass is a function of a combination of stay and wait edges 푆푀(푠, 푤), and hence cannot be expressed as a linear edge cost.

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6.7.2 Two-Step Graph The sellable mass can be defined as a linear edge cost if the combinations of stay and wait edges are combined into a single two-step edge, defining a single mining trip (i.e. the flight itinerary of a single trip 푋푡 = (푥푡−1, 푦푡, 푥푡)). As can be seen from the graph in figure 6.19, each mining trip consists of a two-step path originating at one of the EAL nodes and terminating at one of the EA nodes (or two adjacent sets of EA nodes for the second trip).

Figure 6.20 shows a two-step graph, where the “trip edges” define candidate mining trips consisting of a stay and wait edge. Between each set of nodes (ℎ, 푖), there may be numerous two-step paths passing through separate intermediate AE trajectories. These are added as additional edges between sets of nodes, producing a multi-graph. A simpler directed graph can be produced by only selecting a single edge between each set of terminal nodes (the intermediate path that maximizes one of the trip edge costs defined below). Using this new graph, the sellable mass (and hence the revenue, profit, and NPV) of each trip can be expressed as a linear edge cost for each trip, allowing for the use of graph theory algorithms to find flight itineraries that maximize these objective functions.

Figure 6.20 A two-step graph of the network.

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6.8 Defining the Graphs A network graph may be defined mathematically as 퐺 = (푉, 퐸) , where 푉 are the vertices or nodes, and 퐸 are the edges of the graph. The vertices of the graph will be composed of the subsets of trajectory nodes 퐹퐵, 퐸퐴퐿, 퐴퐸, and 퐸퐴, with the ordering dependent on the arrangement of the graph.

The edges of the graph will similarly be composed of subsets of edges between the adjacent node sets. For the full graph (shown in figure 6.19), these consist of three subsets of edges:

푆퐸1 = {1,2, … , 푁푆퐸1} Set of stay edges between 퐸퐴퐿 and 퐴퐸 nodes (for trip 푡 = 1);

푆퐸 = {1,2, … , 푁푆퐸} Set of stay edges between 퐸퐴 and 퐴퐸 nodes (for trips 푡 > 1);

푊퐸 = {1,2, … , 푁푊퐸} Set of wait edges between 퐴퐸 and 퐸퐴 nodes (for trips 푡 ≥ 1); where 푁 indicates the number of elements of each set.

For the two-step graph (shown in figure 6.20), two sets of edges are defined:

Set of trip edges between the 퐸퐴퐿 and 퐸퐴 nodes (for trip 푡 = 푇퐸1 = {1,2, … , 푁푇퐸 } 1 1); and

Set of trip edges between adjacent 퐸퐴 and 퐸퐴 nodes (for trips 푇퐸 = {1,2, … , 푁푇퐸} 푡 > 1).

Graphs specific to each mining campaign can be defined by first defining the sets of edges and associated edge costs, and later constructing graphs dependent on the ordering of transfer legs in the mining campaign.

6.8.1 Adjacency Matrices and Edge Costs Edges between two adjacent sets of nodes may be defined using an 푀 × 푁 adjacency matrix 퐀 (where 푀 and 푁 define the lengths of the two sets). The elements of the matrix describe the presence or absence of an edge connecting pairs of nodes from the two sets:

1, 푖푓 푎푛 푒푑푔푒 푐표푛푛푒푐푡푠 푛표푑푒 푚 푡표 푛표푑푒 푛 퐀(푚, 푛) = 푎 = { (6.38) 푚푛 0, 표푡ℎ푒푟푤푖푠푒.

Edge costs can similarly be defined by matrices of the same size, where the elements define a metric length of the edge between the sets of nodes. The elements of the adjacency matrix can be determined by applying constraints to the edge cost matrices.

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Stay Edges (풕 = ퟏ)

The stay edges 푆퐸1 for the first mining trip (푡 = 1) are defined between the 퐸퐴퐿 and 퐴퐸 trajectories. For each edge (ℎ, 푗) between each candidate 퐸퐴퐿 trajectory ℎ, and each candidate 퐴퐸 trajectory 푗, the following edge costs are defined 11:

Table 6.2 Edge cost functions for (풕 = ퟏ) stay edges.

Edge Cost Definition

푆푇1(ℎ, 푗) = 퐿퐷푗 − 퐴퐷ℎ

Stay-time where 퐿퐷푗 is the launch date of 퐴퐸 trajectory 푗, and

퐴퐷ℎ is the arrival date of 퐸퐴퐿 trajectory ℎ. 퐷 (ℎ, 푗) = 푇푂퐹 + 푆푇 (ℎ, 푗) Duration 푆1 ℎ 1 where 푇푂퐹ℎ is the time of flight of 퐸퐴퐿 trajectory ℎ. ∆푉 (ℎ, 푗) = ∆푉 Delta-V 푆1 ℎ where ∆푉ℎ is the delta-V of 퐸퐴퐿 trajectory ℎ. 퐿 (ℎ, 푗) = 푚푖푛(푀 ∗ 푆푇(ℎ, 푗), 푀 ) Extraction 1 푅 푚푎푥 where 푀 is the mining rate (kg/day), and Mass 푅 푀푚푎푥 is the maximum capacity of the mining spacecraft.

An 퐻 × 퐽 adjacency matrix 푺ퟏ is defined to describe feasible stay edges limited by a minimum 푆푇푚푖푛 and maximum 푆푇푚푎푥 stay-time: 1, 푆푇 ≤ 퐿퐷 − 퐴퐷 ≤ 푆푇 푺 (ℎ, 푗) = { 푚푖푛 푗 ℎ 푚푎푥 (6.39) ퟏ 0, 표푡ℎ푒푟푤푖푠푒.

Each non-zero element of the adjacency matrix will describe a separate candidate stay edge, indexed by 푑. The total number of stay edges in the set 푆퐸1 will be given by

∑ℎ ∑푗 푺ퟏ(ℎ, 푗).

Stay Edges (풕 > ퟏ) The stay edges 푆퐸 for the subsequent mining trips (푡 > 1) are defined between the 퐸퐴 and 퐴퐸 trajectories. For each edge (푖, 푗) between each candidate 퐸퐴 trajectory 푖, and each candidate 퐴퐸 trajectory 푗, the following edge costs are defined:

11 The subscript 1 is used to differentiate edge costs that differ for the first mining trip.

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Table 6.3 Edge cost functions for (풕 > ퟏ) stay edges.

Edge Cost Definition

푆푇(푖, 푗) = 퐿퐷푗 − 퐴퐷푖

Stay-time where 퐿퐷푗 is the launch date of 퐴퐸 trajectory 푗, and

퐴퐷푖 is the arrival date of 퐸퐴 trajectory 푖. 퐷 (푖, 푗) = 푇푂퐹 + 푆푇(푖, 푗) Duration 푆 푖 where 푇푂퐹푖 is the time of flight of 퐸퐴 trajectory 푖. ∆푉 (푖, 푗) = ∆푉 Delta-V 푆 푖 where ∆푉푖 is the delta-V of 퐸퐴 trajectory 푖. Extraction 퐿(푖, 푗) = 푚푖푛(푀 ∗ 푆푇(푖, 푗), 푀 ) Mass 푅 푚푎푥

An 퐼 × 퐽 adjacency matrix 퐒 is defined to describe feasible stay edges limited by the stay-time:

1, 푆푇 ≤ 퐿퐷 − 퐴퐷 ≤ 푆푇 퐒(푖, 푗) = { 푚푖푛 푗 푖 푚푎푥 (6.40) 0, 표푡ℎ푒푟푤푖푠푒.

Each non-zero element of the adjacency matrix will describe a separate candidate stay edge, indexed by 푠. The total number of stay edges in the set 푆퐸 will be given by

∑푖 ∑푗 퐒(푖, 푗).

Wait Edges The wait edges 푊퐸 for all mining trips (푡 ≥ 1) are defined between the 퐴퐸 and 퐸퐴 trajectories. For each edge (푗, 푖) between each candidate 퐴퐸 trajectory 푗 , and each candidate 퐸퐴 trajectory 푖, the following edge costs are defined:

Table 6.4 Edge cost functions for wait edges.

Edge Cost Definition

푊푇(푗, 푖) = 퐿퐷푖 − 퐴퐷푗

Wait-time where 퐿퐷푖 is the launch date of 퐸퐴 trajectory 푖, and 퐴퐷푗 is the arrival date of 퐴퐸 trajectory 푗.

퐷푊(푗, 푖) = 푇푂퐹푗 + 푊푇(푖, 푗) Duration where 푇푂퐹푗 is the time of flight of 퐴퐸 trajectory 푗.

∆푉푊(푗, 푖) = ∆푉푗 Delta-V where ∆푉푗 is the delta-V of 퐴퐸 trajectory 푗.

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A 퐽 × 퐼 adjacency matrix 퐖 is defined to describe feasible wait edges limited by a minimum 푊푇푚푖푛 and maximum 푊푇푚푎푥 wait-time: 1, 푊푇 ≤ 퐿퐷 − 퐴퐷 ≤ 푊푇 퐖(푗, 푖) = { 푚푖푛 푖 푗 푚푎푥 (6.41) 0, 표푡ℎ푒푟푤푖푠푒.

Each non-zero element of the adjacency matrix will describe a separate candidate wait edge, indexed by 푤. The total number of wait edges in the set 푊퐸 will be given by

∑푗 ∑푖 퐖(푗, 푖).

Trip Edges (t=1)

The trip edges 푇퐸1 for the first mining trip (푡 = 1) are defined as two-step paths between the 퐸퐴퐿 , 퐴퐸 and 퐸퐴 trajectories. For each trip edge (ℎ, 푖) between each candidate 퐸퐴퐿 trajectory ℎ, and each candidate 퐸퐴 trajectory 푖, there may be numerous paths through intermediate 퐴퐸 trajectories 푗.

These paths can be enumerated by creating a sub-graph defined by the vertices 푉 = (퐸퐴퐿, 퐴퐸, 퐸퐴), and an adjacency matrix 퐀 describing the sets of stay and wait edges. The matrix 퐀 is an 푁 × 푁 adjacency matrix (where 푁 = 퐻 + 퐼 + 퐽) constructed as a block matrix from the smaller adjacency matrices:

ퟎ 푺ퟏ ퟎ 퐀 = (ퟎ ퟎ 퐖) , (6.42) ퟎ ퟎ ퟎ where ퟎ are zero matrices of appropriate dimensions.

The number of two-step paths through this graph can be determined by a property of 푘 adjacency matrices that the element (푎푖푗) gives the number 푘-step paths from 푖 to 푗 [252].

Using this theorem, 퐀2(푖, 푗) will give the number of 2-step paths between the any two nodes (푖, 푗), where 퐀2 is the square of the adjacency matrix:

ퟎ 푺ퟏ ퟎ ퟎ 푺ퟏ ퟎ ퟎ ퟎ 푺ퟏ퐖 퐀2 = (ퟎ ퟎ 퐖) (ퟎ ퟎ 퐖) = (ퟎ ퟎ ퟎ ) . (6.43) ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ

The various blocks of this matrix will give the connectivity between each of the three sets of nodes contained in the vertices list. For the first mining trip, we can extract the matrix 퐌1 = 푺ퟏ퐖 defining the connectivity between the 퐸퐴퐿 and 퐸퐴 nodes. The elements 퐌1(ℎ, 푖) will give the number of possible two-step paths between ℎ ∈ 퐸퐴퐿

246 Scott Dorrington – June 2019 Chapter 6: Combinatorial Trajectory Optimization and 푖 ∈ 퐸퐴. This can be converted to an adjacency matrix 퐁1 defining the trip edges between 푇퐸1 by replacing all non-zero elements of 퐌1 with ones.

For each trip edge (ℎ, 푖) between each candidate 퐸퐴퐿 trajectory ℎ, and each candidate 퐸퐴 trajectory 푖, the set of intermediate 퐴퐸 trajectories of the two-step path can be found th th by comparing the ℎ row of the 푺ퟏ and the 푖 column of 퐖, finding the indices that have non-zero values in each (i.e. the nodes that have both stay edges from node ℎ and wait edges to node 푖).

For each of these trip edges (ℎ, 푗, 푖), the following edge costs are defined:

Table 6.5 Edge cost functions for Trip edges.

Edge Cost Definition

푇푇1(ℎ, 푖) = 퐿퐷푖 − 퐿퐷ℎ Trip-time where 퐿퐷ℎ is the launch date of 퐸퐴퐿 trajectory ℎ, and 퐿퐷푖 is the launch date of 퐸퐴 trajectory 푖.

∆푉푇1(ℎ, 푖, 푗) = ∆푉ℎ + ∆푉푗

Delta-V where ∆푉ℎ is the delta-V of 퐸퐴퐿 trajectory ℎ, and ∆푉푗 is the delta-V of 퐴퐸 trajectory 푗.

−∆푉푗 휈 Delivered 퐷푀1(ℎ, 푖, 푗) = (퐿(ℎ, 푗) + 푀푑푟푦)푒 푒 − 푀푑푟푦 Mass where ∆푉푗 is the delta-V of 퐴퐸 trajectory 푗 (re-computed using algorithm B.1)

Sellable Mass 푆푀1(ℎ, 푖, 푗) = 퐷푀1(ℎ, 푖, 푗) − 푀푃(푗)

푅1(ℎ, 푖, 푗) = 푆푀1(ℎ, 푖, 푗)푐푠 Revenue where 푐푠 is the specific sale price ($/kg) of asteroid material at the depot orbit.

Profit 푃1(ℎ, 푖, 푗) = −퐶0(ℎ) + 푅1(ℎ, 푖, 푗) −휏 푁푃푉1(ℎ, 푖, 푗) = −퐶0(ℎ) + 푅1(ℎ, 푖, 푗) ∗ (1 + 푟) Net Present where 휏 = (퐴퐷 − 푡 )/365.25 is the time of delivery measured from Value 푗 0 a reference date 푡0.

The functions above make use of node costs defined over a single transfer node, rather than an edge:

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Table 6.6 Node cost functions defined over single transfer node.

Node Cost Definition

Propellant ∆푉ℎ 휈 Mass (EAL 푀푃(ℎ) = 푀푑푟푦 (푒 푒 − 1) transfer)

Propellant ∆푉푗 휈 Mass (AE 푀푃(푗) = 푀푑푟푦 (푒 푒 − 1) transfer)

Capital 퐶0(ℎ) = (푐푝푟표푑 + 푐푙푎푢푛푐ℎ)푚푑푟푦 + (푐푙푎푢푛푐ℎ + 푐푝)푀푃(ℎ)

Trip Edges (풕 > ퟏ) The trip edges 푇퐸 for the subsequent mining trip (푡 > 1) are defined as two-step paths between the 퐸퐴 , 퐴퐸 and 퐸퐴 trajectories. For each trip edge (푖, 푖′) between each candidate 퐸퐴 trajectory 푖, and each candidate 퐸퐴 trajectory 푖′ (with 푖 ≠ 푖′), a similar process can be used to define the number of two-step paths through the intermediate 퐴퐸 trajectories 푗.

The adjacency matrix 퐀 defining the paths between the nodes will be the same, with the exception that the 푺ퟏ matrix (defined only for 푡 = 1) is replaced with 퐒 (defined for 푡 >

1). The elements 퐌푡(푖, 푖′) of the connectivity matrix 퐌푡 = 퐒퐖 will define the number of two-step paths between each 퐸퐴 nodes 푖 and adjacent 퐸퐴 node 푖′. This can be used to create the adjacency matrix 퐁푡 defining the trip edges 푇퐸.

For each of trip edge between (푖, 푗, 푖′), the following edge costs are defined:

Table 6.7 Edge cost functions over trip edges.

Edge Cost Definition

푇푇(푖, 푖′) = 퐿퐷푖′ − 퐿퐷푖 Trip-time where 퐿퐷푖 is the launch date of 퐸퐴 trajectory 푖, and 퐿퐷푖′ is the launch date of 퐸퐴 trajectory 푖′.

∆푉푇(푖, 푗, 푖′) = ∆푉푖 + ∆푉푗

Delta-V where ∆푉푖 is the delta-V of 퐸퐴 trajectory 푖, and ∆푉푗 is the delta-V of 퐴퐸 trajectory 푗.

−∆푉푗 휈 Delivered 퐷푀푡(푖, 푗, 푖′) = (퐿(푖, 푗) + 푀푑푟푦)푒 푒 − 푀푑푟푦 Mass where ∆푉푗 is the delta-V of 퐴퐸 trajectory 푗 (re-computed using algorithm B.1)

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Sellable Mass 푆푀푡(푖, 푗, 푖′) = 퐷푀푡(푖, 푗, 푖′) − 푀푃(푗)

푅푡(푖, 푗, 푖′) = 푆푀푡(푖, 푗, 푖′)푐푠 Revenue where 푐푠 is the specific sale price ($/kg) of asteroid material at the depot orbit.

Profit 푃푡(푖, 푗, 푖′) = 푅푡(푖, 푗, 푖′) −휏 푁푃푉푡(푖, 푗, 푖′) = 푅푡(푖, 푗, 푖′) ∗ (1 + 푟) Net Present where 휏 = (퐴퐷 − 푡 )/365.25 is the time of delivery measured from Value 푗 0 a reference date 푡0.

The sellable masses 푆푀1(ℎ, 푗, 푖) and 푆푀푡(푖, 푗, 푖′) for each mining trip can be computed by applying algorithms B.1 and B.2 defined in chapter 5 (for delivery Route 1 and 2, respectively), using a single set of depot and parking orbit radii12. In the above section, the 퐸퐴 and 퐴퐸 trajectories were computed using delta-Vs determined for the Route 1 delivery strategy, where the mining spacecraft delivers the asteroid material to the propellant depot. In applying algorithm B.2, the incoming and outgoing aim orientation angles are computed to maximize the total sellable mass. In this process, the total delta- V of the 퐸퐴 and 퐴퐸 are re-computed. These could alternatively be recomputed using the Route 2 delivery strategy.

The edge cost functions for the trip edges in tables 6.5 and 6.7 are each expressed as a function of the start and end 퐸퐴 trajectory nodes, as well as the intermediate 퐴퐸 trajectory nodes (i.e. 푓(푖, 푗, 푖′) ). These can be used to construct three-dimensional matrices containing the edge cost values of all paths in the multi-graph (containing multiple paths between each set of end nodes). The multi-graph can be reduced to a standard directed graph by selecting only a single intermediate 퐴퐸 trajectory between each set of terminal nodes that maximizes one of the edge cost functions:

푓(푖, 푖′) = 푓(푖, 푗표푝푡, 푖′) , (6.44) where 푗표푝푡 is the index of the optimal 퐴퐸 trajectory. This is similar to the dimension reducing process employed in chapter 5. To maintain consistency, each of the edge costs should be computed using the same intermediate 퐴퐸 trajectory 푗표푝푡.

12 Note that the indices 푖 and 푗 used in this chapter do not correspond to the indices used in the algorithms presented in chapter 5.

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6.8.2 Constructing Graphs with Arbitrary Trip Numbers The full network graph for a 푇-trip mining campaign can be constructed as a block matrix from the elements, as seen in Eq. 6.42. For each mining campaign, two graphs can be produced describing the full graph containing stay and wait edges (as shown in figure 6.19), and the two-step graph containing trip edges (as shown in figure 6.20).

Full Graph

For the full graph 퐺1 = (푉1, 퐸1), the vertices will consist of alternating sets of 퐸퐴 and 퐴퐸 nodes, with the length dependent on the number of mining trips:

푉1 = {퐸퐴퐿, 퐴퐸, 퐸퐴, 퐴퐸, 퐸퐴, … , 퐸퐴, 퐴퐸, 퐸퐴} . (6.45)

The edges will be defined by:

퐸1 = {푆퐸1, 푊퐸, 푆퐸, 푊퐸, … , 푆퐸, 푊퐸} . (6.46)

The adjacency matrix 퐀 of the graph can be constructed as an 푁 × 푁 block matrix, where 푁 = 2푇 + 1 is the number of node sets in the vertices. Each element 퐀(푚, 푛), will be the adjacency matrix associated with the edges between the node sets 푉1(푚) and

푉1(푛). As all the edge sets connect each set of nodes to the following set of nodes, the only non-zero elements of the matrix will be the super-diagonal elements (the elements above and to the right of the diagonal):

ퟎ 푺ퟏ ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ 푾 ⋯ ퟎ ퟎ ퟎ

ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ

퐀 = ⋮ ⋱ ⋮ . (6.47) ퟎ ퟎ ퟎ ퟎ 푺 ퟎ ퟎ ퟎ ퟎ ⋯ ퟎ ퟎ 푾 (ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ )

Two-Step Graph

For the two-step graph 퐺2 = (푉2, 퐸2), consisting of trip edges, the vertices can be similarly expressed as a repeating set of 퐸퐴 nodes:

푉2 = {퐸퐴퐿, 퐸퐴, 퐸퐴, … , 퐸퐴} , (6.48) and the edges as:

퐸2 = {푇퐸1, 푇퐸, … , 푇퐸} . (6.49)

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The adjacency matrix 퐁 of the two-step graph can be constructed as a (푇 + 1) × (푇 + 1) block matrix, where 푇 is the total number of mining trips:

ퟎ 퐁ퟏ ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ 퐁 ⋯ ퟎ ퟎ ퟎ 풕 ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ

퐁 = ⋮ ⋱ ⋮ . (6.50) ퟎ ퟎ ퟎ ퟎ 퐁풕 ퟎ ퟎ ퟎ ퟎ ⋯ ퟎ ퟎ 퐁풕 (ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ )

6.8.3 Edge Cost Matrices Weighed adjacency matrices expressing the total edge costs of the graph can also be constructed in a similar manner, using the edge cost matrices of the corresponding adjacency matrices. Functions that are expressed over the stay and wait edges can be formed by replacing the blocks of the adjacency matrix 퐀 in the full graph 퐺1. For example, the total delta-V can be expressed as:

ퟎ ∆푉푆1 ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ ∆푉 ⋯ ퟎ ퟎ ퟎ 푊 ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ

∆푉푡표푡푎푙 = ⋮ ⋱ ⋮ , (6.51) ퟎ ퟎ ퟎ ퟎ ∆푉푆 ퟎ ퟎ ퟎ ퟎ ⋯ ퟎ ퟎ ∆푉푊 (ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ ) where ∆푉푆1 is an 퐻 × 퐽 matrix, ∆푉푆 is an 퐼 × 퐽 matrix, and ∆푉푊 is a 퐽 × 퐼 matrix, with the elements defined by the delta-V functions in tables 6.2, 6.3, and 6.4, respectively.

Functions that are defined over trip edges (containing combinations of stay and wait edges) can be expressed by replacing the blocks of the adjacency matrix 퐁 in the two- step graph 퐺2. For example, the total profit 푃푡표푡푎푙 can be expressed as:

ퟎ (−푪ퟎ + 퐑ퟏ) ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ 퐑 ⋯ ퟎ ퟎ ퟎ 풕 ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ 푃푡표푡푎푙 = ⋮ ⋱ ⋮ , (6.52) ퟎ ퟎ ퟎ ퟎ 퐑풕 ퟎ ퟎ ퟎ ퟎ ⋯ ퟎ ퟎ 퐑풕 ( ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ ) where 푪ퟎ and 퐑ퟏ are 퐻 × 퐼 matrices expressing the capital and revenues for each (푡 =

1 ) trip edge between nodes ℎ and 푖 , and the 퐑풕 matrices are each 퐼 × 퐼 matrices expressing the revenues of each (푡 > 1) trip edge between nodes 푖 and 푖′. In this format,

Scott Dorrington – June 2019 251 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions only a single edge cost can be expressed for each trip edge, evaluated at the optimal intermediate 퐴퐸 trajectories 푗표푝푡 (as discussed above).

The total NPV can be expressed in a similar manner:

ퟎ 퐍퐏퐕ퟏ ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ 퐍퐏퐕 ⋯ ퟎ ퟎ ퟎ 풕 ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ

푁푃푉푡표푡푎푙 = ⋮ ⋱ ⋮ , (6.53) ퟎ ퟎ ퟎ ퟎ 퐍퐏퐕풕 ퟎ ퟎ ퟎ ퟎ ⋯ ퟎ ퟎ 퐍퐏퐕풕 (ퟎ ퟎ ퟎ ퟎ ퟎ ퟎ ) where 퐍퐏퐕ퟏ is an 퐻 × 퐼 matrix, and 퐍퐏퐕풕 are each 퐼 × 퐼 matrices.

The expressions (−푪ퟎ + 퐑ퟏ) and 푵푷푽ퟏ for the profit and NPV for the first mining trip contain the revenue of the first trip, from which the capital cost is extracted. As the capital is only incurred once during a flight itinerary, it is not included in the profit and NPV expressions for the subsequent missions, so that it is only counted once in the total path cost.

6.8.4 Shortest Path Problem Each potential flight itinerary through the graph will have a total edge cost (or “distance”) computed from the sum of the edges in the path. Flight itineraries that optimize a given objective function can be found by applying a shortest path algorithm that finds the set of edges that minimizes the total edge cost, giving the shortest distance.

There are several graph theory algorithms that have been developed to solve shortest path problems on directed graphs. The most widely used is Dijkstra’s algorithm [251] that finds the shortest path in a directed graph with non-negative edge costs between a starting node and all reachable nodes, returning path details of the single optimal solution. The Floyd-Warshall algorithm [253] finds the shortest paths between all pairs of nodes in a graph (the all-pairs shortest path problem), however does not save details of the paths.

Path details for the all-pairs shortest path problem can be computed by applying Dijkstra’s algorithm between each starting 퐸퐴퐿 node, and all reachable 퐸퐴 nodes (identified by applying a depth first search algorithm). This can be used to save details on numerous shortest paths, from which a global optimal flight itinerary can be selected.

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It also allows for additional constraints to be placed on the solution, such as having a minimum number of trip edges.

Figure 6.21 shows an example of the two-step graph for a 10-trip mining campaign to asteroid (459872) 2014 EK24, with blue lines showing trip edges between the successive sets of 퐸퐴 nodes. A tree is highlighted in red showing all shortest paths between the first 퐸퐴퐿 node to all reachable nodes. This graph and solutions were generated using functions from the MATLAB graph and network algorithms package [254]. As the delta-V and trip time are all positive edge costs, Dijkstra’s algorithm can be applied to find flight itineraries that minimize the total delta-V or total trip duration.

In this graph, each of the edges have similar valued delta-Vs and duration edge costs. As would be expected, the single-source shortest path solution in this graph is one with only a single trip edge. The all-pairs shortest path solutions can be used to find the minimum delta-V or duration flight itinerary with exactly 10 mining trips.

Figure 6.21 Two-step graph for a 10-trip mining campaign to asteroid (459872) 2014 EK24, with trip edges shown in blue, and all shortest path itineraries from the first starting 푬푨푳 node highlighted in red.

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6.8.5 Longest Path Problem Flight itineraries that maximize a given objective function (such as total sellable mass, profit, or NPV) require solving a longest path problem. This can be found by solving an equivalent shortest path problem in a graph having negative edge costs (i.e. minimizing the negative of the cost function). To achieve this, the entire edge cost matrix is multiplied by -1 (some of the profit and NPV edge costs are originally defined as negative, which would become positive during this process).

As Dijkstra’s algorithm can only handle positive edge costs, other shortest path algorithms must be applied to find the longest path. Problems can also arise in applying this method if there exist negative cycles in the graph, that would lead to infinitely negative total edge costs. As all the edges in both the full and two-step graphs are defined with positive durations, no such cycles exist (the graphs are directed acyclic graphs, DAGs).

An all-pairs longest path problem is solved in a similar process to that described above, using the Bellman-Ford algorithm [255, 256] that can handle mixed (both positive and negative) edge costs, containing no negative cycles. After computation, the resulting total edge costs are then converted back to positive values.

As noted in table 6.7, the NPV cost function is required to be defined with respect to a reference date 푡0, making the NPV cost matrix, defined in Eq. 6.53, only applicable to a single starting 퐸퐴퐿 node. In order to solve the all-pairs longest path problem, the NPV cost matrix is required to be re-computed with the reference date set at the launch date of each 퐸퐴퐿 starting node.

6.8.6 Computational Complexity The worst-case computational time complexity of the Bellman-Ford algorithm for computing shortest paths from a single source node to all other nodes in a weighted directed graph has been shown to be of order 푂(|푉| ∙ |퐸|), where |푉| is the total number of nodes and |퐸| is the total number of edges in the graph [252]. The total computation time for generating the all-pairs shortest/longest paths from each starting 퐸퐴퐿 node is therefore of the order 푂(퐻 ∙ |푉| ∙ |퐸|).

The total number of nodes in the graph will be dependent on the number of trajectories in the 퐸퐴퐿 and 퐸퐴 sets, increasing with the number of trips:

254 Scott Dorrington – June 2019 Chapter 6: Combinatorial Trajectory Optimization

|푉| = 퐻 + 푇 ∙ 퐼 . (6.54)

The total number of edges in the graph will also increase with the number of trips:

|퐸| = |퐸1| + 푇 ∙ |퐸푡| (6.55) where |퐸1| is the number of trip edges for the first trip (푡 = 1), and |퐸푡| is the number of trip edges for subsequent trips (푡 > 1). These will depend on both the number of nodes in the adjacent sets, and the time constraints placed on the stay and wait-times.

The graph shown in figure 6.21 was produced for 푇 = 10 mining trips, with 퐻 = 151 퐸퐴퐿 trajectories, 퐼 = 242 퐸퐴 trajectories, and 퐽 = 382 퐴퐸 trajectories, giving a total of |푉| = 2571 nodes, and |퐸| = 67,641 edges. The total computation time required to generate and solve this graph was found to be 3.8 minutes. This runtime was typical of the asteroid candidates considered in the next chapter, giving a total processing time for each asteroid of around half an . A few of the asteroids with more frequent launch opportunities (around 700 in each set) had significantly larger graphs, taking over 1.5 hours to solve the longest path problem, however the majority were solved in less than 5 minutes.

6.9 Conclusion This chapter has reviewed numerical porkchop plot methods that can be used to identify optimal (minimum delta-V) launch opportunities for body-to-body transfers. These methods are widely used in trajectory design studies; however, often use objective functions that do not include plane change manoeuvres in the total delta-Vs. Objective functions were developed to compute the total delta-Vs (including plane change manoeuvres) for four types of heliocentric transfers between the Earth and a target asteroid, distinguished by the direction of the transfer (Earth-to-Asteroid or Asteroid-to- Earth), encounter type (flyby or rendezvous), and launching/arrival location (Earth’s surface or a parking orbit). These objective functions were applied to a porkchop plot program, developed in MATLAB, to compute the total delta-Vs of the four transfer types over a grid of launch and arrival dates ranging from 2020 to 2050. Separate sets of local optimal launch opportunities were obtained by finding local minima in the resulting delta-V matrices, and applying an optimization algorithm.

Graph theory methods were then applied to construct network graphs enumerating all potential flight itineraries for multiple return-trip missions, from combinations of these

Scott Dorrington – June 2019 255 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions optimal launch opportunities. Objective functions describing the total delta-V, mission duration, sellable mass, profit, and NPV were derived as linear functions of edge costs in the network, allowing for optimal flight itineraries to be identified by applying an all- pairs shortest/longest path algorithm.

256 Scott Dorrington – June 2019 Chapter 7: Asteroid Candidates

7 ASTEROID CANDIDATES

When considering missions to asteroids, there are numerous objects to choose from, and a systematic target selection process is required. In this chapter, lists of asteroid candidates will be generated from two databases containing orbital elements and photometric properties. For each candidate, methods described in the previous chapters will be applied to find optimal launch opportunities and flight itineraries that maximize the mission’s total net present value. These will be used to assess the economic feasibility of each asteroid candidate with realistic trajectories over multiple return trip missions.

7.1 Candidate Selection Studies As noted in chapter 2, the majority of past asteroid missions were designed as flyby missions, with their targets selected due to their proximity to the nominal trajectory of the primary mission (for example, [45]). For dedicated missions, targets were selected based on scientific interest and mission considerations, such as delta-V.

The Hayabusa2 and OSIRIS-REx missions were both designed to return samples from carbonaceous asteroids. Both missions conducted studies and surveys to find appropriate targets with low delta-V trajectories. To aid in the design of the spacecraft and rendezvous operation sequence, the target selection process for the Hayabusa2 mission required that the target asteroid have known physical and dynamical properties, such as rotation period, spin vector, geometric albedo, and taxonomic class.

To find appropriate targets, a series of observation campaigns were performed between 2000 and 2015, producing spectral data for 74 near-Earth asteroids (NEAs) – including 43 sets of color indices, 41 light curves, and 5 detailed spectra [257]. These observations were processed to produce rotation rates, geometric , and taxonomic classes (both Tholen and Bus-DeMeo). Initial delta-V estimates were obtained from Benner [8] (using the Shoemaker-Helin equations – discussed in chapter 2).

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The near-Earth asteroid 162173 Ryugu was selected as being one of the most accessible C-type asteroids (known at the time) reachable with realistic delta-Vs (computed from a two-impulse Lambert solution) [258]. 162173 Ryugu was selected as the primary target before the completion of this study, and the list produced by Hasegawa et al. [257] was used for backup targets – the most promising of which were identified as (153591) 2001

SN263 and (341843) 2008 EV5.

Several other studies have been conducted to produce lists of asteroids that could be visited for future asteroid missions, including potential targets for human exploration, and asteroid mining missions.

Benner [8] maintains a list of delta-V estimates for NEAs, calculated using the Shoemaker-Helin equations. These delta-Vs are often used as initial estimates for asteroid missions (including the analysis for the Hayabusa2 mission, discussed above).

Elvis et al. [259] conducted a study of the distribution of delta-Vs from Benner, to find potential asteroid targets for human missions to NEAs. Of the 6,699 NEAs listed at the time (as of March 2010), 65 asteroids were identified with “ultra-low” delta-V less than 4.5 km/s. These asteroids were found to have Earth-like orbits with long synodic periods (> 20 years), low inclinations (< 7o), and large magnitudes (H > 22) (small diameters < 140 m). Other factors were also identified as being important for target selection consideration for human missions, such as the large rotation rates of small asteroids, the presence of binary companions orbiting the main asteroid, and the need for long launch windows that allow a return to Earth in the case of an aborted mission.

Sánchez & McInnes [2] applied analytical delta-V estimates to a statistical distribution of NEAs over the orbital elements space developed by Bottke et al. [260]. Trajectories were modelled as two-impulse transfers aligned with the asteroid’s line of nodes. An analytical expression was developed to define a contour in the 푎-푒 orbital element space specifying a maximum eccentricity 푒푚푎푥 , below which all asteroids can be captured with a delta-V less than a specified value:

2 2 1 1 푉∞ (7.1) 푒푚푎푥 = √1 − (3 − − ) , 4푎 푎 휇푠 where 푉∞ is the hyperbolic excess velocity of the incoming asymptote for a capture delta-V ∆푉푐푎푝 and parking orbit radius 푟푝:

258 Scott Dorrington – June 2019 Chapter 7: Asteroid Candidates

2 2휇퐸 2휇퐸 (7.2) 푉∞ = √(∆푉푐푎푝 + √ ) − , 푟푝 푟푝

where 휇푠 and 휇퐸 are the gravitational parameters of the Sun and Earth. This was used to identify an orbital element space for which asteroids could potentially be captured with delta-Vs less than the escape velocity of the moon (∆푉푐푎푝 = 2.37 km/s).

More detailed trajectory studies make use of porkchop plot methods, described in the previous chapter, that solve Lambert’s problem over a wide range of launch and arrival dates. Dong et al. [151] produced sets of optimal launch opportunities for flyby, rendezvous, and sample return missions over the period from 2015 to 2025 for a list of 60 asteroids identified by Perozzi et al. [152] to be of scientific interest.

A more comprehensive list has been generated by the Trajectory Browser tool [154], with launch opportunities for a large number of NEAs from 2010 to 2040. These trajectories can also include gravity assist manoeuvres with the Earth, Venus, or Mars in searching for an optimal trajectory. The tool contains a user interface that can be used to filter through the launch opportunities for single-trip flyby or rendezvous missions, or round-trip sample return missions. The NASA Near-Earth Object Human Space Flight Accessible Targets Study (NHATS) [261] provides a similar list of candidate asteroids that have been derived with human constraints such as shorter mission durations.

In all of these numerical studies, round-trip sample return missions are designed with the assumption that the spacecraft will re-enter the atmosphere on the return journey. As such, a final Earth Orbit Insertion manoeuvre is not included in the total mission delta-V used in the objective function. The trajectories are also assumed to launch from Earth directly onto an outgoing hyperbolic trajectory, with departure delta-Vs calculated from a 200 km parking orbit. This type of trajectory is comparable to the Earth-to-Asteroid (launch from Earth) transfer described in chapter 6 (although a different parking orbit radius was used).

7.2 Data Sources & Processing In this study, asteroid candidates are to be selected from the entire list of known near- Earth asteroids. The Minor Planet Center Orbital Elements (MPCORB) database [60] is used as the primary dataset for asteroid candidates, containing lists of orbital elements

Scott Dorrington – June 2019 259 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions for all catalogued asteroids, updated on a daily basis. This data is supplemented by photometric and physical data for a limited number of asteroids, available in the Asteroid Lightcurve Database (LCDB) [99]. Initial delta-V estimates are calculated from the orbital elements using the Shoemaker-Helin equations (outlined in Appendix A). These data sources are combined and processed to produce lists of candidate asteroids with known, assumed, and potential C-complex taxonomic classes.

7.2.1 MPC Orbital Elements Data MPCORB data is available in a text file containing a single row for each asteroid, with fields specifying the number (if assigned), name or provisional designation, absolute magnitude 퐻, slope parameter 퐺, orbital elements (푎, 푒, 푖, 휔, 훺), mean anomaly 푀 at a given epoch, mean motion 푛, orbit uncertainty parameter 푈, and a 4-hexdigit flag specifying the orbital group [60]. Additional metadata is available detailing the conditions of the observations used to fit the orbital elements, such as the number of observations and oppositions, the length of , and residuals. This chapter uses MPCORB data obtained 16th May 2019.

Data from the MPCORB text file was read into a MATLAB script, making use of the Fortran 77 format specified in the MPCORB documentation [262]. Additional orbital parameters were then calculated from the orbital elements, including periapsis 푞 , apoapsis 푄, period 푇, synodic period 푇푠푦푛, and Earth and Jovian Tisserand parameters

푇퐸 and 푇퐽 (using Eqs. 2.1 to 2.6 defined in chapter 2). Diameter estimates were calculated (using Eq. 2.10) from the absolute magnitude with an assumed albedo of 0.058 for C-type asteroids [98]. Delta-V estimates were also calculated using equations for the Hohmann-like transfer (both periapsis and apoapsis), and Shoemaker-Helin equations (detailed in Appendix A).

7.2.2 Asteroid Lightcurve Data The asteroid LCDB contains a mixture of direct and indirect observational data. Data obtained directly from photometric observations include the rotation period, light curve amplitude, absolute magnitude, binary status, color indices, and diameter (observed from stellar or /radar). Indirect data that is obtained by calculation and/or assumption include the taxonomic class (from color indices), orbital group/family, geometric albedo 푝푉 , and diameter. Additional data fields are used to

260 Scott Dorrington – June 2019 Chapter 7: Asteroid Candidates indicate whether each of the data values have been obtained directly, calculated, or assumed.

Pravec & Harris [98] note that there is a complex relationship between taxonomic class, albedo, and diameter. If direct data are available for either diameter or albedo, the other parameter is computed using the absolute magnitude in Eq. 2.10. If neither of these values are known, diameters are calculated based on an assumed albedo based on the taxonomic class (either observed or assumed based on orbital group).

Many asteroids in the LCDB contain multiple values for each of these parameters, obtained from different sources. These are listed in a detailed text file (“LC_DAT_PUB.txt”) containing multiple lines of data for each reference. This study makes use of the summary file (“LC_SUM_PUB.txt”) containing a single line for each asteroid, with values judged to be the most accurate by the authors [99]. LCDB data used in this chapter was last updated 31st January 2019.

Updating Taxonomic Classes Data from the LCDB summary text file was read into the MATLAB script, making use of the format specified in the LCDB documentation [99]. When searching through the detailed taxonomic classifications of asteroid candidates for the Hayabusa2 mission (from Hasegawa et al. [257] discussed above), several inconsistencies were discovered in the classifications listed in the summary file. For instance, asteroid 162173 Ryugu (the selected target of the Hayabusa2 mission) is listed as an assumed S-type asteroid in the LCDB summary file, despite all of the classifications listed in the detailed file being C-complex (listed as a C-type by Müller et al. [263], and as a C/B-type by Hasegawa et al. [257]). Similarly, asteroids (162567) 2000 RW37, (341843) 2008 EV5, 2010 JV34, and 101955 Bennu (the target for the OSIRIS-REx mission) were all listed in the summary file as assumed S-type, despite having multiple C-complex classifications in the details file. To ensure their inclusion in the C-complex candidates list (discussed below), the taxonomy classifications of these asteroids were manually updated to supersede the values listed in the summary file.

Two of the asteroids were also found to have ambiguous classifications. Asteroid (164202) 2004 EW is listed as a C-type asteroid in the LCDB summary file, while the details file lists it as C/X [264], Xn [257], or X-type [265]. This asteroid has been updated to a known X-type. Asteroid 2010 JV34 is listed as an assumed S-type in the summary file, while listed as a Q/C type [257] in the details file. This has been updated

Scott Dorrington – June 2019 261 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

to a known Q-type (following the judgement of Hasegawa et al. [257] when considering its albedo value).

There were also found to be three C-complex asteroids [(141424) 2002 CD, (363505)

2003 UC20, and 2005 JU108] listed by Hasegawa et al. [257] not present in the LCDB. These asteroids were added. Table 7.1 lists the taxonomic classes for the Hayabusa2 candidates assigned by Hasegawa et al. [257] and those listed in the LCDB summary files. Notes are made on changes that have been adapted for this work.

There may be other cases in which these ambiguities occur, and future amendments to the program should search through the detailed text file to flag C-complex asteroids that may have been missed.

Table 7.1 Changes made to the taxonomic class.

Hasegawa et al. [257] H LCDB [99] Asteroid Bus- Changes Made (mag) RP (hr) Tholen (summary) DeMeo

(14402) 1991 DB 18.7 2.261 (C) (Ch) C –

(65679) 1989 UQ 19.5 7.748 C/B C B –

(141424) 2002 CD 20.5 – B B – Added as B-type

(153591) 2001 SN263 16.9 3.426 B/C Cb B –

(162173) Ryugu 19.3 7.631 C/B (C) (S) Updated to C-type

(162567) 2000 RW37 19.9 2.439 (C) – (S) Updated to C-type

(164202) 2004 EW 20.8 – C/B Xn C Updated to X-type.

(341843) 2008 EV5 20.0 3.725 (C) {B} (S) Updated to B-type

(363505) 2003 UC20 18.1 – B/C Cb – Added as Cb-type

2005 JU108 19.7 5.34 (C) – – Added as C-type

2010 JV34 20.8 – Q/C – S Updated to Q-type

7.2.3 Trajectory Browser For each of the identified asteroid candidates (listed below), the Trajectory Browser tool was used to find global minimum delta-V trajectories for a one-way rendezvous mission over a time period from 2010 to 2040. The results were saved as a csv file, and read into the MATLAB program for use as a comparison to the Shoemaker-Helin delta-Vs and those calculated using the porkchop plot program. As noted above, these delta-Vs may differ from those produced by the porkchop plot program due to the use of different

262 Scott Dorrington – June 2019 Chapter 7: Asteroid Candidates parking orbit radii, and the inclusion of gravity assist manoeuvres. There were several asteroids identified in the candidates list that were not found using the Trajectory Browser database. It is unknown if this is due to no solution being available, or the incompleteness of the database (no asteroids discovered after 2017 appear to be in the database).

7.2.4 Combining Data As the data sets for the MPCORB and LCDB contain different numbers of asteroids, a data matching method was applied to combine both data sets into a single set. The LCDB data contains a much larger number of asteroids, however the majority of these are main belt asteroids, with only a small number being NEAs. The family/group association data was used to find the row indices of all NEAs in the LCDB dataset. For each of these NEAs, a string search was performed to find the number, or provisional designation (or name) in the MPCORB data, returning the corresponding row index.

Asteroids were first searched by number, as this is a unique field for each asteroid. Difficulties were found to arise with asteroids having similar names (for example asteroids 3361 Orpheus and 4197 Morpheus), and asteroids whose numbers correspond to calendar years used as part of provisional designations (for example ). These searches resulted in multiple matches that required a series of further checks to confirm the correct index matching between MPCORB and LCDB data.

If a match was found, the LCDB data was copied to new vectors with the same ordering as the MPCORB data, and the LCDB diameter values were used to replace those estimated from the assumed albedo. Non-matches were recorded as empty data.

There were also several NEAs found in the LCDB that did not match any asteroids in the MPCORB data set. The orbital elements of these asteroids were extracted from a separate text file containing MPC orbital elements data on all asteroids (including main belt and Kuiper belt asteroids).

Scott Dorrington – June 2019 263 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions

7.3 Asteroid Candidate Groups The taxonomic classes in the combined data set were used to segment the asteroids into several categories including known C-complex, assumed C-complex, known non-C- complex, assumed non-C-complex, and unknown taxonomic class. This classification is shown in figure 7.1.

Figure 7.1 Segmentation of asteroids based on taxonomic class.

7.3.1 Known & Assumed C-Complex Asteroids Of the 19,880 NEAs in the MPCORB database, there are only 219 asteroids with known taxonomic classes. These include 25 known C-complex asteroids, and a further 194 asteroids with known non-C-complex taxonomic classes (110 of which are known S- complex asteroids). The known C-complex asteroids were selected as a first set of asteroid candidates to be considered. An additional 12 asteroids listed as assumed C- complex asteroids (indicated by an ‘A’ in the field describing the source of the taxonomic class assignment in the LCDB) are selected as a second set of asteroid candidates. Asteroids with known non-C-complex taxonomic classes were excluded as candidates.

Tables 7.2 and 7.3 list the main orbital, photometric, and trajectory parameters of asteroid candidates with known and assumed C-complex taxonomic classes. These candidates are arranged in order of increasing magnitude (decreasing diameter). To more clearly show where assumed values have been used in the LCDB, albedo and taxonomic classes that have been assumed (indicated by an ‘A’ in the source field), are specified in brackets.

264 Scott Dorrington – June 2019 Chapter 7: Asteroid Candidates

Table 7.2 Known C-complex asteroids.

Orbital Elements Photometric Properties Trajectory Parameters Asteroid Designation 풂 풆 풊 U H 풑푽 D Class ΔVSH ΔVTB TE Tsyn (AU) (deg) (mag) (m) (km/s) (km/s) (yrs) (162173) 1.190 0.190 5.88 0 19.3 0.07 870 C 4.73 4.31 2.97 4.36 Ryugu (101955) 1.126 0.204 6.03 0 20.9 0.04 492 B 5.17 4.76 2.95 6.14 Bennu (65679) 0.915 0.265 1.30 2 19.5 (0.06) 701 B 5.38 4.45 2.94 6.99 1989 UQ (341843) 0.958 0.083 7.44 1 20.0 0.12 400 C 5.48 4.48 2.98 15.13 2008 EV5 (363505) 0.781 0.337 3.80 0 18.1 0.03 609.49 CB 5.54 6.52 2.94 2.23 2003 UC20 (141424) 0.980 0.177 6.88 2 20.5 (0.06) 438.34 B 5.84 5.00 2.96 32.36 2002 CD (416151) 1.112 0.306 4.58 1 20.7 (0.2) 215 C 6.02 5.74 2.90 6.81 2002 RQ25 (7350) 1.356 0.391 7.26 0 17.2 (0.2) 1080 C 6.08 5.83 2.86 2.73 1993 VA (153591) 1.987 0.478 6.68 0 16.9 (0.05) 2630 B 6.13 5.22 2.96 1.56 2001 SN263 (14402) 1.715 0.402 11.42 0 18.7 0.12 600 C 6.18 6.47 2.93 1.80 1991 DB (162567) 1.248 0.250 13.75 1 19.9 0.17 311 C 6.28 6.16 2.90 3.54 2000 RW37 (25330) 1.540 0.371 14.33 0 16.6 0.04 3200 B 6.42 7.79 2.88 2.10 1999 KV4

2.124 0.463 6.53 0 19.7 (0.06) 633.59 C 6.60 6.07 3.04 1.48 2005 JU108 (175706) 1.054 0.350 1.99 1 18.4 0.04 1900 B 6.68 5.39 2.87 13.22 1996 FG3 (14827) 2.840 0.666 1.98 0 18.3 (0.06) 907 C 6.89 5.73 2.86 1.26 Hypnos (65706) 2.398 0.555 9.64 0 16.4 (0.06) 2790 C 7.03 5.94 2.96 1.37 1992 NA (3671) 2.199 0.542 13.53 0 16.4 0.16 1540 C 7.18 7.45 2.88 1.44 Dionysus (16064) 2.850 0.589 4.54 0 16.7 0.03 4100 C 7.26 6.66 3.07 1.26 Davidharvey 2012 KP24 1.498 0.370 18.47 6 26.4 0.1 20 C 7.30 7.96 2.83 2.20 (108519) 1.603 0.271 16.40 1 17.9 (0.06) 1460 C 7.41 7.86 2.96 1.97 2001 LF (446804) 1.733 0.370 19.48 1 19.5 (0.06) 701 C 7.81 8.61 2.88 1.78 1999 VN6 (2100) 0.832 0.436 15.75 0 16.1 0.08 2780 C 9.63 8.29 2.78 3.15 Ra-Shalom

2.888 0.594 28.61 0 17.6 (0.06) 1700 C 10.78 9.01 2.75 1.26 2004 JR1 (3200) 1.271 0.890 22.26 0 14.6 0.25 3600 B 15.36 1.74 3.31 Phaethon (1580) 2.197 0.488 52.10 0 14.7 0.09 4200 C 17.01 2.05 1.44 Betulia

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Table 7.3 Assumed C-complex asteroids.

Orbital Elements Photometric Properties Trajectory Parameters Asteroid Designation 풂 풆 풊 U H 풑푽 D Class ΔVSH ΔVTB TE Tsyn (AU) (deg) (mag) (m) (km/s) (km/s) (yrs) (68278) 1.436 0.114 2.62 0 18.4 (0.2) 621 (C) 5.85 5.09 3.07 2.39 2001 FC7 (18109) 1.881 0.368 0.81 0 17.0 (0.2) 1110 (C) 6.18 4.70 3.08 1.63 2000 NG11 (6456) 2.193 0.409 8.22 0 15.9 (0.18) 1990 (C) 7.27 6.20 3.13 1.44 Golombek (370307) 1.979 0.492 16.18 0 16.9 (0.18) 1310 (C) 7.27 9.25 2.86 1.56 2002 RH52 (175114) 2.249 0.664 5.72 0 16.6 (0.2) 1420 (C) 7.42 6.35 2.68 1.42 2004 QQ (162566) 2.638 0.574 13.86 0 15.7 (0.2) 2150 (C) 7.88 7.16 2.96 1.30 2000 RJ34 (192642) 2.645 0.770 6.80 0 16.3 (0.2) 1630 (C) 8.43 8.17 2.44 1.30 1999 RD32 (40263) 1.495 0.161 25.84 0 17.7 (0.2) 857 (C) 10.34 2.84 2.21 1999 FQ5 (408980) 1.430 0.722 15.05 2 18.6 (0.2) 566 (C) 10.62 2.30 2.41 2002 RB126 (68350) 1.670 0.248 29.57 0 15.9 (0.2) 1960 (C) 11.04 2.78 1.86 2001 MK3 (52762) 2.419 0.651 33.88 0 14.8 0.05 6720 (C) 11.90 2.37 1.36 1998 MT24 (154555) 1.184 0.580 36.83 1 16.7 (0.2) 1360 (C) 14.58 2.26 4.46 2003 HA

7.3.2 Potential C-Complex Asteroids The remaining 19,649 asteroids have either an assumed non-C-complex taxonomic class (1330 asteroids), or no LCDB data at all (18,319 asteroids). The assumed taxonomic class and albedo assigned in the LCDB is based on the most prevalent asteroids in the respective orbital groups. For NEAs, these are listed as assumed S-type asteroids, with assumed albedo of 0.2 or 0.18. S-complex asteroids are presumed to make up around 22% of the total NEA population, and C-complex asteroids a further 10% (based on the de-biased distribution of Stuart & Binzel [36] – discussed in chapter 2). As there is no spectral data from which to assign a taxonomic class, these remaining asteroids can be considered to have a probabilistic taxonomic class, with a 22% probability of being S- complex, and a 10% probability of being C-complex. These asteroids are considered as a third set of potential C-complex asteroid candidates.

Due to the large number of objects, it is necessary to filter this list to provide a more reasonable number that can be studied in further detail. It is expected that asteroids with

266 Scott Dorrington – June 2019 Chapter 7: Asteroid Candidates orbital elements close to that of the Earth will have low delta-Vs and mission durations. To test this hypothesis, three sub-groups of these candidates were formed based on constraints of orbital parameters.

Co-orbitals The first group to be considered are the co-orbital asteroids, with orbital periods close to that of the Earth (defined here as 푇 = 1 ± 0.05 years). These asteroids experience a 1:1 mean motion resonance with the Earth, demonstrating a number of distinctive co-orbital motions such as Tadpole, Horseshoe, Quasi-satellite, and Passing (discussed in chapter 2). A total of 24 co-orbital asteroids were identified, many of which have been discussed in literature, including the only known Earth Trojan: 2010 TK7 [80], and the five known Quasi-satellites: (469219) 2016 HO3, (164207) 2004 GU9, (277810) 2006

FV35, 2013 LX28, and 2014 OL339 [266]. (Asteroid (469219) 2016 HO3 was recently re- named 469219 Kamoʻoalewa, but is still listed by its provisional designation in the MPCORB database.)

Arjunas The second group of asteroids to be considered are the Arjuna asteroids – those with very Earth-like orbits, defined by the region: 0.985 < 푎 < 1.013 AU; 푒 < 0.1; and 푖 < 10° (and not belonging to the co-orbitals group). (This inclination range has been extended from 8.56o in the original definition of de la Fuente Marcos [267]).

Low Delta-V The third group to be considered are asteroids with “ultra-low” Shoemaker-Helin delta- Vs < 4.5 km/s (as defined by Elvis et al. [259]). There were found to be a total of 273 low delta-V asteroids (as compared to the 65 identified by Elvis et al. [259]). This list was further reduced to include the 14 lowest delta-Vs (< 4 km/s) and the 18 largest asteroids, with diameters over 100 m (퐻 ≤ 23.7 using an assumed albedo of 0.058).

MPCORB and LCDB data for these candidate groups are listed in tables 7.4 to 7.6.

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Table 7.4 Co-orbital asteroids.

Orbital Elements Photometric Properties Trajectory Parameters Asteroid Designation 풂 풆 풊 U H 풑푽 D Class ΔVSH ΔVTB TE Tsyn (AU) (deg) (mag) (m) (km/s) (km/s) (yrs) (255071) 1.001 0.632 2.64 4 18.2 (0.058) 1264.19 10.56 8.51 2.55 532.18 2005 UH6 (138852) 1.001 0.298 21.50 3 20.2 (0.2) 271.00 (S) 9.83 9.74 2.78 623.61 2000 WN10 (419624) 1.003 0.075 14.52 1 20.5 (0.058) 438.34 7.65 5.92 2.93 211.65 2010 SO16 (85770) 0.998 0.345 33.18 3 20.5 (0.058) 438.34 16.31 2.57 351.27 1998 UP1 2010 TK7 0.999 0.190 20.90 3 20.8 (0.058) 381.78 10.45 8.43 2.83 1073.07 (164207) 1.001 0.136 13.65 2 21.1 (0.058) 332.51 7.28 6.15 2.93 500.63 2004 GU9 (441987) 1.000 0.370 11.63 0 21.5 (0.2) 149.00 (S) 8.15 6.81 2.82 2761.41 2010 NY65 (277810) 1.001 0.377 7.10 4 21.6 (0.058) 264.13 7.64 6.01 2.84 507.62 2006 FV35 2013 LX28 1.002 0.452 49.98 3 21.8 (0.058) 240.89 19.50 2.15 412.41 2005 QQ87 1.001 0.302 33.97 3 22.7 (0.058) 159.15 13.78 2.58 1106.44 2014 OL339 0.999 0.461 10.19 4 22.9 (0.058) 145.15 8.80 7.49 2.75 798.83 2016 CA138 1.000 0.048 27.72 3 23.3 (0.058) 120.73 12.63 2.77 4732.41 (469219) 1.001 0.104 7.78 1 24.3 (0.2) 41.00 (S) 5.71 5.04 2.97 597.20 2016 HO3 2017 XQ60 1.000 0.214 27.15 7 24.4 (0.058) 72.75 13.03 2.74 3638.25 2017 YQ5 0.997 0.169 16.96 7 24.7 (0.058) 63.36 8.93 2.89 224.85 2018 AN2 1.001 0.154 22.07 7 24.8 (0.058) 60.51 9.89 2.83 469.53 2018 XW2 0.998 0.302 19.73 7 25.5 (0.058) 43.83 10.46 2.79 435.42 2016 CO246 0.999 0.126 6.33 3 25.8 (0.058) 38.18 5.57 4.62 2.97 452.60 2017 SL16 1.000 0.153 8.68 3 25.8 (0.058) 38.18 6.09 2.95 2052.15 2019 HS2 0.997 0.214 19.58 7 26.6 (0.058) 26.41 10.01 2.84 262.91 2019 AE3 0.998 0.100 14.81 6 27.2 (0.058) 20.04 8.09 2.92 359.71 2015 XX169 1.003 0.185 7.60 3 27.4 (0.058) 18.27 6.03 4.90 2.95 212.15 2018 PN22 0.997 0.039 4.38 4 27.5 (0.058) 17.45 4.86 2.99 235.28 2019 GM1 1.002 0.070 6.81 6 27.5 (0.058) 17.45 5.43 2.98 372.47

Table 7.5 Arjuna asteroids.

Orbital Elements Photometric Properties Trajectory Parameters Asteroid Designation 풂 풆 풊 U H 풑푽 D Class ΔVSH ΔVTB TE Tsyn (AU) (deg) (mag) (m) (km/s) (km/s) (yrs) (459872) 1.008 0.070 4.80 1 23.3 (0.2) 65.00 (S) 4.91 3.85 2.99 82.80 2014 EK24 2012 FC71 0.988 0.088 4.94 7 25.2 (0.058) 50.33 5.03 4.07 2.99 53.52 2005 CN61 0.989 0.068 9.55 5 25.3 (0.058) 48.06 6.30 4.71 2.97 59.41 2018 ER1 1.009 0.092 6.42 5 25.6 (0.058) 41.86 5.32 2.98 77.81 2019 HM 0.996 0.007 5.76 6 25.8 (0.058) 38.18 5.34 2.99 172.20 2018 PK21 0.988 0.081 1.20 5 25.9 (0.058) 36.46 4.37 2.99 54.73 2010 HW20 1.011 0.050 8.18 5 26.1 (0.058) 33.25 5.75 4.27 2.98 60.14 2012 LA11 0.986 0.096 5.13 4 26.1 (0.058) 33.25 5.10 3.90 2.98 48.31 2009 SH2 0.991 0.094 6.81 3 24.9 (0.2) 31.00 (S) 5.54 4.09 2.98 75.32 2003 YN107 0.989 0.014 4.32 2 26.5 (0.058) 27.66 4.83 3.89 2.99 58.14

268 Scott Dorrington – June 2019 Chapter 7: Asteroid Candidates

2014 QD364 0.986 0.041 4.01 4 27.2 (0.058) 20.04 4.68 3.66 2.99 46.31 2013 BS45 0.992 0.084 0.77 0 25.9 (0.2) 20.00 (S) 4.39 3.87 2.99 79.82 2019 GF1 0.994 0.050 1.46 5 27.4 (0.058) 18.27 4.24 3.00 108.61 2014 UR 0.995 0.016 8.27 0 26.6 (0.2) 14.00 (S) 6.05 4.14 2.98 120.72 2016 GK135 0.986 0.087 3.21 6 28.1 (0.058) 13.24 4.66 3.68 2.99 47.62 2008 KT 1.011 0.085 1.98 6 28.2 (0.058) 12.64 4.50 3.72 2.99 60.96 2008 UC202 1.011 0.069 7.45 6 28.3 (0.058) 12.07 5.53 4.09 2.98 62.56 2006 JY26 1.010 0.083 1.44 3 28.4 (0.058) 11.53 4.44 3.73 2.99 65.59 2017 YS1 0.991 0.053 5.55 5 28.9 (0.058) 9.16 5.14 2.99 76.31

Table 7.6 Low Delta-V asteroids.

Orbital Elements Photometric Properties Trajectory Parameters Asteroid Designation 풂 풆 풊 U H 풑푽 D Class ΔVSH ΔVTB TE Tsyn (AU) (deg) (mag) (m) (km/s) (km/s) (yrs) 2006 RH120 1.033 0.025 0.59 1 29.5 (0.2) 3.00 (S) 3.84 3.50 3.00 20.95 2012 TF79 1.050 0.038 1.00 3 27.4 (0.058) 18.27 3.84 3.56 3.00 14.22 2009 BD 1.062 0.052 1.27 0 28.1 (0.2) 7.00 (S) 3.85 3.36 3.00 11.65 2008 HU4 1.071 0.056 1.39 1 28.3 (0.058) 12.07 3.86 3.36 3.00 10.17 2007 UN12 1.054 0.060 0.24 3 28.7 (0.058) 10.04 3.90 3.38 3.00 13.22 2010 UE51 1.055 0.060 0.62 2 28.3 (0.058) 12.07 3.90 3.34 3.00 12.92 2011 UD21 0.979 0.030 1.06 3 28.5 (0.058) 11.01 3.95 3.31 3.00 30.22 2016 RD34 1.046 0.035 1.96 2 27.6 (0.058) 16.67 3.95 3.00 15.23 2017 FJ3 1.133 0.118 0.96 5 29.9 (0.058) 5.78 3.96 3.00 5.84 2015 YO10 1.122 0.100 2.43 6 26.6 (0.058) 26.41 3.97 3.87 3.00 6.31 2005 LC 1.134 0.103 2.80 6 26.8 (0.058) 24.09 3.98 3.67 3.00 5.82 2010 VQ98 1.023 0.027 1.48 3 28.2 (0.058) 12.64 3.99 3.32 3.00 29.66 2011 MD 1.056 0.037 2.45 0 28.0 (0.2) 7.00 (S) 3.99 3.31 3.00 12.69 2014 UV210 1.159 0.134 0.60 3 26.9 (0.2) 12.00 (S) 4.00 3.51 3.00 5.03 2012 EC 1.152 0.137 0.91 0 23.3 (0.058) 120.73 4.03 3.61 2.99 5.24 2011 CG2 1.177 0.159 2.76 0 21.4 (0.2) 156.00 (S) 4.19 4.40 2.99 4.60 2001 BB16 0.856 0.172 2.03 1 23.2 (0.058) 126.42 4.23 3.83 2.99 3.80 2013 WA44 1.100 0.060 2.30 2 23.7 (0.2) 54.00 (S) 4.24 3.52 3.00 7.48 2005 RK3 1.247 0.185 3.72 0 23.7 (0.058) 100.42 4.26 4.02 2.99 3.55 2000 AE205 1.165 0.138 4.46 2 23.0 (0.058) 138.62 4.27 4.09 2.99 4.89 2003 SM84 1.125 0.082 2.80 3 22.7 (0.058) 159.15 4.29 4.31 3.00 6.16 2001 QJ142 1.062 0.086 3.10 2 23.7 (0.058) 100.42 4.30 3.83 2.99 11.57 2013 GN3 1.227 0.185 3.79 3 23.6 (0.058) 105.15 4.30 4.71 2.99 3.79 2010 CE55 1.290 0.221 2.53 1 22.2 (0.058) 200.36 4.32 4.12 2.99 3.15 2005 YA37 1.280 0.228 2.24 2 22.3 (0.058) 191.34 4.37 4.63 2.98 3.23 2019 JF1 1.267 0.226 1.60 7 23.7 (0.058) 100.42 4.38 2.98 3.35 2009 QE34 1.289 0.232 2.63 5 23.6 (0.058) 105.15 4.40 4.05 2.98 3.16 2003 EZ16 1.175 0.139 5.81 2 22.6 (0.058) 166.65 4.43 4.63 2.99 4.65 2018 UM1 1.144 0.174 1.99 4 23.6 (0.058) 105.15 4.44 2.98 5.48 2009 RT1 1.155 0.106 4.15 5 23.6 (0.058) 105.15 4.46 4.04 3.00 5.13 (89136) 1.356 0.253 1.91 0 20.2 (0.2) 260.00 (S) 4.47 4.58 2.99 2.73 2001 US16 2015 TJ1 1.229 0.221 2.22 2 22.6 (0.058) 166.65 4.50 3.84 2.97 3.76

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7.3.3 Distribution of Candidate Groups Figure 7.2 shows the distribution of semi-major axis and eccentricity for each of the asteroid candidate groups. As per their definition, the Arjuna asteroids show orbital elements very close to that of the Earth (located at 푎 = 1 AU and 푒 ≈ 0). The co-orbital asteroids occupy a thin region at 푎 ≈ 1 AU, with eccentricities ranging from 0.039 to 0.632, and inclinations ranging from 2.64o to 49.98o. The low delta-V asteroids also show orbital elements close to that of the Earth, tending to follow the 푄 = 0.983 AU and 푞 = 1.017 AU contours used to divide the NEA regions. The 푒푚푎푥 contour of Sánchez & McInnes [2] has been shown in red to indicate a boundary in the orbital element space, below which asteroids can potentially be captured with delta-Vs less than that of the escape velocity of the moon (found using Eqs. 7.1 and 7.2, with ∆푉푐푎푝 =

2.37 km/s and 푟푝 = 200 km). The upper eccentricity of the low delta-V asteroids appears to follow a similar shape to the contours generated by these equations

(computed with a lower ∆푉푐푎푝 = 1 km/s, shown with a red dashed line).

Figure 7.2 Orbital element distribution of the asteroid candidate groups.

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The known and assumed C-complex asteroids show a wider range of orbital elements, dispersed throughout the entire NEA region. Only around half of the known and assumed C-complex candidates are shown in this figure, with the remainder having semi-major axes greater than 1.5 AU. The Shoemaker-Helin delta-Vs of these asteroids tend to increase at further distances from Earth.

Figure 7.3 shows the distribution of Shoemaker-Helin delta-Vs and diameters for these asteroid candidate groups. From this figure, it can be seen that many of the low delta-V asteroids are significantly smaller than the other candidates, some less than 10 meters in diameter.

Figure 7.3 Distribution of Shoemaker-Helin delta-V and diameter for the asteroid candidate groups.

7.4 Launch Opportunities For each candidate asteroid, methods described in chapter 6 were applied to produce sets of optimal (minimum delta-V) launch opportunities for each of the four transfer types over the period from 2020 to 2050. This section describes trends that were observed in the distribution of Earth-to-Asteroid (launch from Earth) (EAL) transfers. The Earth-to-Asteroid and Asteroid-to-Earth trajectories from Earth parking orbits (EA

Scott Dorrington – June 2019 271 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions and AE trajectories) were found to have similar trends, with larger delta-Vs due to the inclusion of plane change manoeuvres upon departure/arrive from Earth orbit. In most cases, the EAL and EA trajectories were found to occur at similar launch and arrival dates. In cases where the optimal EAL trajectories occurred at high declinations of the departure asymptote, the corresponding EA trajectories had higher delta-Vs as they account for plane change manoeuvres. These lead to different shaped porkchop plots, with different sets of launch opportunities.

7.4.1 Known and Assumed C-complex Figure 7.4 shows the distribution of delta-Vs for all identified launch opportunities of the EAL transfers for the known and assumed C-complex asteroids. In this figure, asteroids are evenly spaced along the x-axis, arranged in order of increasing Shoemaker-Helin delta-V (shown with red open circles). Delta-V estimates taken from the Trajectory Browser database (where available) are displayed with green triangles for comparison.

Figure 7.4 Delta-V distribution of 푬푨푳 launch opportunities for known and assumed C-complex asteroids.

Figure 7.5 shows a corresponding plot of the distribution of launch dates (Julian Ephemeris Date, JED) of each opportunity along the x-axis, with a colour axis

272 Scott Dorrington – June 2019 Chapter 7: Asteroid Candidates indicating the delta-V values. In both these figures, launch opportunities have been clustered using a density-based clustering algorithm (MATLAB implementation of the DBSCAN algorithm of Ester et al. [268]) to consider all opportunities within 25 days of each other (Euclidean distance in the launch date/arrival date space) as a single launch opportunity, returning the minimum value of each cluster. This has been performed to more clearly show the spacing of separate launch opportunities. (The porkchop plot program identified numerous opportunities that were not considered true local minima, but were kept to improve the range of flight itineraries considered in the time-expanded network.) Only launch opportunities with delta-Vs less than 10 km/s are shown.

Figure 7.5 Launch date and delta-V distribution of 푬푨푳 launch opportunities for known and assumed C-complex asteroids.

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As seen in figure 7.4, each asteroid candidate shows numerous EAL launch opportunities with delta-Vs ranging from the upper 10 km/s limit to minimum values that tend to increase in agreement with the Shoemaker-Helin delta-Vs. In many cases, the minimum EAL delta-Vs show comparable values to those obtained by the Trajectory Browser tool. However, the Trajectory Browser returns lower delta-Vs in many cases through the use of gravity assist trajectories. The Shoemaker-Helin equations also tend to over or under-estimate the minimum delta-V values for the different asteroid candidates; however, show a general trend in increasing values.

The values of the delta-Vs were found to follow the trends in the relative mean longitude of the asteroid with respect to Earth, cycling over the synodic period with the lowest values occurring at opposition (0o separation) and maximum values occurring at conjunction (180o separation).

Asteroids with short synodic periods are found to have only a few launch opportunities each synodic period, leading to a mixture of high and low delta-V opportunities interspersed throughout the entire launch date range (e.g. asteroid 101955 Bennu as seen in figure 7.5).

Asteroids with higher delta-V values show more sparsely spaced launch opportunities, caused by 퐸퐴퐿 delta-Vs exceeding the upper 10 km/s limit that are not included (e.g. asteroid 2100 Ra-Shalom). These asteroids were also found to have few or even no launch opportunities for the other transfers types, from which it is not possible to form flight itineraries for return trip missions.

Asteroids with long synodic periods show more frequent launch dates, with long-term periodic variations in delta-V values (as seen for asteroid (341843) 2008 EV5 in figure 7.5). Figure 7.6 shows a plot of the geocentric Earth-to-Asteroid distance, relative mean longitude (absolute value), and delta-Vs of all 퐸퐴퐿 launch opportunities to asteroid

(341843) 2008 EV5 (푇푠푦푛 = 15.13 years) over the 2020 to 2050 timeframe.

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Figure 7.6 Variation of range, relative mean longitude, and delta-V of 푬푨푳 opportunities

to asteroid (341843) 2008 EV5.

7.4.2 Arjunas Figures 7.7 and 7.8 show the delta-V and launch date distributions of EAL launch opportunities to the Arjuna asteroids. These asteroids show similar cyclic variations in delta-V values over long synodic periods (of the order of decades). Due to their Earth- like orbital parameters, these asteroids show significantly lower delta-V values close to their oppositions.

From figure 7.7, it can be seen that the majority of these asteroids have oppositions (indicated by minimum delta-V values) that occur early in the launch date range, with delta-Vs tending to increase over time. As noted by Elvis et al. [259], these asteroids spend much of their time effectively behind the Sun, as viewed from Earth, and can only be detected in the dusk or dawn sky as they approach Earth. This means that the asteroids tend to be discovered close to or after their opposition.

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Figure 7.7 Launch date and delta-V distribution of 푬푨푳 launch opportunities for the Arjuna asteroids.

Figure 7.8 Delta-V distribution of 푬푨푳 launch opportunities for the Arjuna asteroids.

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7.4.3 Co-orbitals Figures 7.9 and 7.10 show the delta-V and launch date distributions of EAL launch opportunities to the co-orbital asteroids. These asteroids show a wide range of minimum delta-V values, that tend to increase with the orbital inclination.

Due to their Earth-like orbital periods, these asteroids show extremely long synodic periods ranging from 211.65 to 4732.4 years. However, due to their unique motions, these asteroids do not have mean longitudes varying over a full 360o, and the synodic period as calculated by Eq. 2.5 does not give a good indication of the period of resonant motion.

For example, the Earth Trojan asteroid 2010 TK7 has a relative mean longitude varying from 26o to 60o, over a period of approximately 200 years (measured from peak-to-peak distances in the relative mean longitude), despite having an estimated synodic period of

푇푠푦푛 = 1073.1 years. The high inclination of this asteroid (푖 = 20.9°), also leads to consistently large delta-Vs over 9 km/s.

Figure 7.9 Delta-V distribution of 푬푨푳 launch opportunities for the co-orbital asteroids.

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Figure 7.10 Launch date and delta-V distribution of 푬푨푳 launch opportunities for the co-orbital asteroids.

o The Quasi-satellite (469219) 2016 HO3, on the other hand, always remains within 6 relative separation from the Earth, varying over a period of around 40 years (despite an estimated 푇푠푦푛 = 597.2 years), with consistently low delta-Vs of around 5 km/s. The close proximity of Quasi-satellites would also provide an operational advantage in reducing the power requirements and time-delay of communications with Earth.

Figure 7.11 shows the range and relative mean longitude of asteroid (469219) 2016

HO3, and the delta-V distribution of the EAL opportunities.

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Figure 7.11 Range, relative mean longitude, and 푬푨푳 delta-Vs to the

Quasi-satellite (469219) 2016 HO3.

7.4.4 Low Delta-Vs Figures 7.12 show the delta-V and launch date distributions of EAL launch opportunities to the low delta-V asteroids. These asteroids were all found to have very low minimum delta-Vs, consistent with those estimated by the Shoemaker-Helin equations.

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Figure 7.12 Launch date and delta-V distribution of 푬푨푳 launch opportunities for the low delta-V asteroids.

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7.5 Net Present Value Results For each asteroid candidate, a time-expanded network was generated from the sets of launch opportunities to enumerate all possible flight itineraries for a mining campaign consisting of up to 10 return trips, using the following settings: • 10 ≤ Stay time ≤ 530.5 days; • 10 ≤ Wait time ≤ 530.5 days; • Mining rate = 800 kg/day; and • Maximum capacity = 160,000 kg (maximum volume of rocket fairing).

Edge cost functions describing mass parameters, capital costs, revenues, and net present values (NPVs) were computed using the same system and specific cost parameters as those used in the parametric model of chapter 3 (using a 20% discount rate and a sale price at 90% of the launch cost, described in section 3.6.1).

An all-pairs longest path problem was then solved to find the maximum NPV flight itineraries from each starting 퐸퐴퐿 node to all reachable nodes (as described in the previous chapter). For each flight itinerary, an average single-trip mission duration was computed from the total mission duration and the number of return trips 푁 in the flight itinerary. An average one-way delta-V was also computed as half the average delta-V of the return trips in the flight itinerary.

To form a comparison to the parametric model developed in chapter 3, analytical NPV estimates were also generated over a range of one-way delta-Vs from 2 to 10 km/s, using an average trip duration of 2.05 years (two 6-month transfers, a 6-month wait- time, and a stay-time computed from the mining rate and maximum capacity).

7.5.1 All-pairs Longest Path Solutions Figure 7.13 shows plots of the average one-way delta-V, average trip duration, and total NPV for the all-pairs maximum NPV flight itinerary solutions for four representative asteroids: 2010 EU51 and 2019 GM1, having the largest NPV solutions for the low delta-

V and co-orbital groups; (459872) 2014 EK24, having the largest diameter of the Arjuna group; and 162173 Ryugu, having the lowest Shoemaker-Helin delta-V of the known and assumed C-complex groups (and being the selected target of the Hayabusa2 mission). Figure 7.13 contains two sets of plots for the four representative asteroids. In both sets, the x-axis displays the average one-way delta-V. In the top set, the y-axis displays average trip duration and the colour axis displays NPV; while in the bottom set,

Scott Dorrington – June 2019 281 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions the y-axis displays NPV and the colour axis displays average trip duration. Analytical NPV estimates from the parametric model, and the average trip duration used to calculate them are displayed with dashed lines for comparison. The negative NPV regions have been shown in grey.

From this figure, it can be seen that a large number of flight itineraries were generated from the time-expanded networks, with a wide range of average delta-Vs. The solutions are observed to form clusters, likely due to flight itineraries sharing similar sub-sets of trajectory sequences.

From the top set of plots, it can be seen that there is a bifurcation in the average trip duration at larger average delta-V values, with the high and low duration branches having flight itineraries mostly composed of trajectories that depart/arrive close to the asteroid’s apoapsis and periapsis, respectively. (Flight itineraries in between these branches have a mixture of apoapsis and periapsis transfers.) This effect is most prominently seen in the known and assumed C-complex asteroids, having larger semi- major axes (see for example, asteroid 162173 Ryugu in the top set of plots in figure 7.13).

In each of these plots, the estimated trip duration was found to be at the lower end of the range of trip durations, close to those of the highest NPV solutions, making it a good approximation for computing the analytical NPV estimates. As seen from the bottom set of plots, in most cases, the central clusters of the solutions tend to follow the trends predicted by the parametric model, with higher average delta-V flight itineraries having lower NPVs. The known and assumed C-complex asteroids, having larger Shoemaker- Helin delta-Vs, all tended to show clusters at the higher average delta-V end, with mostly negative NPVs, all below that predicted by the parametric model. The other representative asteroids show clusters centred at moderate delta-Vs, with higher average NPVs.

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Figure 7.13 Distribution of average one-way delta-V, average trip duration, and total NPV for the all-pairs maximum NPV flight itineraries for four representative asteroids.

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7.5.2 Global Optimal NPV Solutions As expected, the global optimal NPV solutions occurred in flight itineraries with a combination of low average delta-V and trip durations. A total of 41 asteroids were found to have positive NPVs, with the best result, asteroid 2010 UE51 having an NPV of $359.4M and a total profit of $1.67Bn. Table 7.7 lists the details of the global optimal NPV solutions for asteroids with positive NPVs (arranged in order of decreasing NPV). Flight itinerary details include the launch date of the first 퐸퐴퐿 trajectory, the total mission duration, total number of trips, and the average duration and average one-way delta-V of the trips. Economic parameters include the total NPV and corresponding profit, and the payback period at which the mission achieves a positive NPV.

The results show that many of the highest NPV solutions occur for asteroids having small diameters. The NPVs of these asteroids will be recomputed in later sections to account for their limited water content. However, the results in table 7.7 (and other negative NPV solutions not shown) were used in the following sections to identify trends in the NPV with trajectory and orbital parameters, that are independent of the asteroid’s size.

Table 7.7 Flight itinerary results for candidates with positive NPVs.

Asteroid Details Economic Parameters Flight Itinerary Pay- Avg. Avg. Asteroid D NPV Profit back Launch Date Dur. N one- Group Dur. Desig. (m) ($M) ($M) Period (yrs) way ∆푽 (yrs) (yrs) (km/s) 2010 UE51 Low DVs 12.07 359.42 1671.82 1.32 12 Jul 2022 27.38 9 3.04 4.39 2011 UD21 Low DVs 11.01 339.31 977.40 1.97 13 Oct 2039 10.95 5 2.19 4.41 2010 VQ98 Low DVs 12.64 335.59 829.12 1.52 12 Nov 2038 11.94 5 2.39 4.86 2006 Low DVs 3.00 320.55 1483.15 1.44 14 May 2027 21.57 8 2.70 4.29 RH120 2019 GM1 Co-orb. 17.45 277.45 1584.87 2.93 28 Mar 2032 17.49 10 1.75 4.49 2012 TF79 Low DVs 18.27 257.42 1241.60 1.42 16 Oct 2025 23.38 9 2.60 5.17 2018 PN22 Co-orb. 17.45 251.41 1376.89 3.00 30 Jan 2021 22.46 10 2.25 5.09 2019 GF1 Arjunas 18.27 244.36 1571.88 3.30 25 Mar 2020 29.13 10 2.91 5.36 2016 RD34 Low DVs 16.67 237.28 1377.20 1.57 14 Mar 2030 20.33 8 2.54 4.57 2018 PK21 Arjunas 36.46 231.20 1328.48 1.86 29 Jul 2020 28.97 10 2.90 5.36 (469219) Co-orb. 41.00 228.02 1547.65 3.67 26 Sep 2022 18.66 10 1.87 5.19 2016 HO3 2009 BD Low DVs 7.00 224.00 1204.49 1.81 10 Aug 2032 16.78 7 2.40 4.93 2007 UN12 Low DVs 10.04 220.43 1089.91 2.15 17 Apr 2033 16.20 5 3.24 3.84 2012 EC Low DVs 120.73 196.33 831.54 1.87 21 Aug 2036 12.71 5 2.54 5.18 2008 HU4 Low DVs 12.07 169.39 1525.57 1.99 28 Apr 2024 25.95 9 2.88 4.86 2011 MD Low DVs 7.00 162.87 1676.39 3.84 18 Feb 2021 29.64 10 2.96 4.93 2001 QJ142 Low DVs 100.42 147.64 1470.68 1.97 04 Apr 2022 26.82 10 2.68 4.93 2017 FJ3 Low DVs 5.78 145.41 1498.51 1.94 31 Mar 2021 26.91 9 2.99 4.23 2015 YO10 Low DVs 26.41 135.63 815.94 1.97 17 Jan 2033 16.00 6 2.67 4.94 2018 ER1 Arjunas 41.86 123.33 987.27 3.55 31 Mar 2020 27.05 10 2.71 5.65

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2019 HM Arjunas 38.18 111.43 1093.61 5.98 12 Apr 2020 26.48 10 2.65 5.72 2013 WA44 Low DVs 54.00 109.54 1181.30 4.61 16 Sep 2026 23.67 8 2.96 5.24 2005 LC Low DVs 24.09 103.80 597.77 1.97 31 May 2038 11.95 4 2.99 4.85 2001 BB16 Low DVs 126.42 99.42 601.75 5.67 11 May 2037 12.69 5 2.54 5.22 2017 YS1 Arjunas 9.16 97.96 1136.72 5.00 22 Dec 2020 28.02 9 3.11 5.77 2013 BS45 Arjunas 20.00 94.07 1368.02 5.47 09 Feb 2020 30.53 10 3.05 5.52 2003 SM84 Low DVs 159.15 85.04 557.35 5.82 18 Sep 2038 11.83 4 2.96 5.81 (459872) Arjunas 65.00 76.53 923.07 7.13 16 Jan 2020 28.01 10 2.80 6.17 2014 EK24 2016 Arjunas 13.24 58.31 934.74 5.54 07 Aug 2020 29.64 9 3.29 6.06 GK135 2016 Co-orb 38.18 48.96 1093.53 8.50 07 Feb 2020 26.14 10 2.61 5.65 CO246 2014 Low DVs 12.00 48.35 1470.46 7.79 29 Dec 2022 27.90 10 2.79 4.93 UV210 (141424) Known C 438.34 46.88 518.99 5.96 06 Oct 2033 15.71 6 2.62 6.00 2002 CD 2015 Co-orb 18.27 39.51 896.06 10.66 17 Oct 2021 19.13 10 1.91 6.10 XX169 2015 TJ1 Low DVs 166.65 38.96 1169.64 10.00 07 Jan 2028 21.94 6 3.66 3.83 2019 JF1 Low DVs 100.42 36.98 1250.46 9.87 22 Aug 2026 22.95 7 3.28 3.94 2018 UM1 Low DVs 105.15 27.17 1008.59 12.58 10 Nov 2027 22.79 8 2.85 5.20 2014 UR Arjunas 14.00 26.80 931.94 10.43 15 Apr 2020 26.46 10 2.65 6.14 2014 Arjunas 20.04 7.06 885.87 13.95 31 Aug 2020 29.97 10 3.00 6.33 QD364 2005 YA37 Low DVs 191.34 4.35 779.75 15.97 04 Apr 2032 15.97 5 3.19 3.98 2010 CE55 Low DVs 200.36 1.57 735.83 15.88 06 Jun 2032 15.88 5 3.18 4.96 2009 RT1 Low DVs 105.15 0.00 291.43 6.91 07 Sep 2043 6.91 2 3.46 5.23

7.5.3 Trends in Average Trip Parameters Figures 7.14 show the global optimum NPV flight itineraries for each asteroid in the candidate groups, plotted against the average one-way delta-V. The corresponding average trip duration for each of these flight itineraries is plotted in figure 7.15.

Figure 7.14 Global optimal NPV vs average one-way delta-V.

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Figure 7.15 Average trip duration and average one-way delta-V.

The asteroid candidates show average one-way delta-Vs ranging from 3.83 to 9.71 km/s, and NPVs from –$382.2M to $359.42M. The NPVs tended to follow the trend formed by the analytical estimates, with lower average delta-V missions having higher NPVs. All asteroids were found to have NPVs below the prediction line, indicating that the parametric model slightly over-estimates the NPV. This is likely due to the under- estimation of the average trip durations, as seen in figure 7.15.

As predicted by their Shoemaker-Helin delta-V values, the low delta-V group showed the lowest average one-way delta-Vs and the highest NPVs of the candidate groups, with 25 of the 32 asteroids showing positive NPVs, and the remainder showing moderate negative NPVs.

The co-orbitals were found to have average delta-Vs covering the entire range, with the largest NPVs occurring in asteroids with low eccentricities and inclinations (< 7.8o), and small geocentric ranges and relative mean longitudes. Other co-orbitals with large negative NPVs were found to have large eccentricities or inclinations, leading to larger delta-Vs.

The Arjuna asteroids were found to have average delta-V around the median of the range (over the 2nd and 3rd Quartiles), with around half having positive and half having negative NPVs. The known and assumed C-complex asteroids showed the largest average delta-V values, with mostly negative NPVs, with the exception of asteroid

(141424) 2002 CD (NPV = $46.87M). Asteroid 2008 EV5 also showed only a relatively small negative NPV of –$3.77M. From figure 7.15, it can be seen that the higher delta-

286 Scott Dorrington – June 2019 Chapter 7: Asteroid Candidates

V asteroids were found to have significantly longer average trip durations, due to longer transfer times at larger semi-major axes.

7.5.4 NPV Trends with Orbital Parameters The above results compared the NPVs to trajectory parameters calculated from the flight itineraries. As noted in chapter 6, these required numerical calculations that took around half an hour to compute for each asteroid candidate. In this section, NPV results are compared to orbital parameters and ephemeris data, to see if any indications can be made in predicting positive NPVs from more easily calculable parameters.

Orbital Elements Figure 7.16 shows the NPV results plotted against the three main orbital elements (semi-major axis, eccentricity, and inclination). From this figure, it can be seen that there is a trend in increasing NPVs as the orbital elements approach those of the Earth. In each plot, a limiting value can be assigned, above which all asteroid candidates were found to have negative NPVs (shown with vertical dotted lines). These limiting values can be used to define an orbital elements space containing all asteroid candidates found to have positive NPVs: • 0.85 ≤ 푎 ≤ 1.3 AU; • 푒 ≤ 0.25; and • 푖 ≤ 8.5°.

This orbital elements space approximately corresponds to the region limited by the 푒푚푎푥 contour shown in figure 7.2, below which all asteroids with ∆푉푆퐻 < 4.5 km/s were found.

While this region contains all 41 asteroids found to have positive NPVs, it does not necessarily mean that all asteroids within the region will have positive NPV (17 of the asteroids with negative NPVs were also found to occupy this region). This region can instead be used as a general filter to exclude candidates at further orbital distances. For example, the majority of the known and assumed C-complex asteroids are seen to have large semi-major axes and eccentricities, falling outside this orbital region. These would be expected to have large delta-Vs and mission durations, and hence lower NPVs. The only co-orbital asteroids to have negative NPVs were also found to be located outside this orbital region.

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Figure 7.16 NPV trends with semi-major axis, eccentricity, and inclination.

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7.5.5 Tisserand Parameter

A similar trend is also seen in the Earth Tisserand parameter 푇퐸, computed as a function of all three orbital elements (using Eq. 2.6). Figure 7.17 displays the NPV results plotted against the Earth Tisserand parameter, expressed in terms of the normalized encounter velocity 푈. This figure shows that asteroids closer to Earth (low encounter velocities, or Tisserand parameters close to 3) have lower NPVs. From this figure, a similar orbital parameter constraint can be defined to exclude asteroids with likely negative NPVs:

• 푈 ≤ 0.25; (equating to 2.9375 ≤ 푇퐸 ≤ 3.0625).

Figure 7.17 NPV trends in Earth Tisserand parameter (expressed in terms of normalized encounter velocity 푼).

Shoemaker-Helin Delta-V As defined in Appendix A, the Shoemaker-Helin delta-V is also computed from the three main orbital elements. Figure 7.18 shows the NPVs plotted against the Shoemaker-Helin delta-Vs. This figure shows a similar trend as that seen for the computed average one-way delta-Vs (in figure 7.14), with decreasing NPVs at higher delta-V values. However, while in the previous figure, the values closely followed the line predicted by the parametric model, this figure shows more deviation towards lower NPVs.

This is due to the Shoemaker-Helin equations tending to under-estimate the total delta- V in comparison to those computed from the numerical methods, shifting the values to

Scott Dorrington – June 2019 289 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions the left of the figure. Figure 7.19 shows a comparison of the Shoemaker-Helin delta-Vs average one-way delta-Vs (a dashed line is shown to indicate equal values).

Figure 7.18 NPV trends with Shoemaker-Helin delta-V.

Figure 7.19 Comparison of Shoemaker-Helin delta-Vs to average one-way delta-Vs.

Despite these differing values, figures 7.14 and 7.18 both show similar cut-off delta-V values above which all asteroid candidates are found to have negative NPVs. This can be used to add a further constraint to the orbital parameter space for filtering out asteroid candidates with likely negative NPVs:

• ∆푉푆퐻 ≤ 6.5 km/s.

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Synodic Period and Relative Mean Longitude The trend in semi-major axis can be more clearly seen when represented in terms of the synodic period, shown in figure 7.20. The synodic period is computed from the reciprocal of the period (using Eq. 2.5), which is solely dependent on semi-major axis. As noted above, this equation leads to extremely large predicted synodic periods for the co-orbital asteroids, that does not accurately represent the period of their co-orbital motion. For these asteroids, the synodic period has been measured with the peak-to- peak distances in the relative mean longitudes, derived from ephemeris data.

Figure 7.20 NPV trends with synodic period.

For the known and assumed C-complex, and low delta-V groups, this figure shows a trend in increasing NPV with increasing synodic period. As demonstrated in figure 7.6, low delta-V launch opportunities tend to occur at dates close to the asteroid’s opposition, with values increasing over the synodic period. For asteroids with longer synodic periods, the relative mean longitude varies at a slower rate, resulting in smaller increases in the average delta-V between consecutive launch opportunities. These asteroids would therefore be more likely to allow for multiple return trip missions with low average delta-Vs (provided the asteroid’s opposition occurs during the launch date range).

For asteroids with short synodic periods (of the order of 1 – 2 years), low average delta- V flight itineraries would likely be composed of launch opportunities occurring close to the opposition dates in successive synodic periods. This would lead to long wait-times at Earth, and extended stay-times at the asteroid, further increasing the duration of each

Scott Dorrington – June 2019 291 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions mining trip. Shorter duration trips would require selecting higher delta-V trajectories. Both of these cases would result in lower NPVs.

This trend is not seen in the co-orbital or Arjuna asteroids, with longer synodic periods. For these asteroids, the NPVs were found to be dependent on the variation in relative mean longitudes over the launch date range. Figure 7.21 shows the NPVs plotted against the average absolute separation of the asteroid from Earth. (The absolute value of the relative mean longitude, such as those shown in figures 7.6 and 7.11, averaged over the entire launch date range.)

Figure 7.21 NPV trends with average absolute separation from Earth.

In this figure, the known and assumed C-complex, and low delta-V asteroids show values close to 90o. These asteroids have shorter synodic periods, allowing their relative mean longitudes to cycle from 0 to 180o multiple times throughout the launch date range, leading to average values of around 90o. The Arjunas are seen to have an almost linear trend, with decreasing NPVs as their average separation increases towards 1800. This would be expected, as asteroids spending the majority of the launch date range close to conjunction (180o separation) would have higher delta-V launch opportunities, than those close to opposition (0o separation).

Figure 7.22 shows the trajectories of two Arjuna asteroids with the highest (left) and lowest (right) NPVs over the duration of the 2020 to 2050 launch date range, plotted in an Earth-Sun co-rotating reference frame. These asteroids both show interior (푎 < 1 AU) Passing motions, with the long-term average mean motion rotating anti-clockwise, and short-term epicyclic motion caused by oscillations in the heliocentric distance and

292 Scott Dorrington – June 2019 Chapter 7: Asteroid Candidates orbital velocity as the asteroid passes through periapsis and apoapsis in each orbit. (Exterior Passing asteroids, with 푎 > 1 AU, show similar trajectories, with the long- term motion rotating clockwise.)

The Arjuna asteroids with positive NPVs were all found to have relative mean longitudes beginning close to opposition, with the average separation increasing towards conjunction (or beyond) over the duration of the launch date range (similar to that shown for asteroid 2019 GF1 in the left of figure 7.22). Arjunas with negative NPVs, on the other hand, were all found to have higher average separations, centred around the conjunction (similar to that shown for asteroid 2005 CN61 in the right of figure 7.22).

These results are expected as asteroids having oppositions close to the beginning of the launch date range have lower delta-V launch opportunities early in the launch date range. From this observation, we can also predict that some of the poor performing asteroids could have higher NPVs in future launch date ranges (beyond 2050), as they approach their oppositions.

Figure 7.22 Trajectories of two Arjuna asteroids having the highest (left) and lowest (right) NPV results, over the period 2020 to 2050, shown in an Earth-Sun co-rotating reference frame.

Similar trends were also found for the co-orbital asteroids, with those having lower average separations showing the highest NPVs. These occur for asteroids following either Quasi-satellite motions, or Horseshoe motion during their close approach to Earth. Co-orbitals with higher separations, showing negative NPVs, occur in those following Tadpole orbits, or Horseshoe motion close to conjunction. An exception to

Scott Dorrington – June 2019 293 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions this general trend is seen in the Quasi-satellite (164207) 2004 GU9, who’s low NPV can be explained by its high inclination (푖 = 13.65°).

Figure 7.23 shows the trajectories of two co-orbitals asteroid: the Quasi-satellite

(469219) 2016 HO3, having the lowest average separation and second highest NPV of the co-orbitals group; and (441987) 2010 NY65, following a Tadpole orbit, with average separation close to 60o, having the lowest NPV of the group.

Figure 7.23 Trajectories of two co-orbital asteroids having the positive (left) and negative (right) NPV results, over the period 2020 to 2050, shown in an Earth-Sun co-rotating reference frame.

Summary of Orbital Parameter Space The limits on each of the orbital parameters identified above are summarized in table 7.8. Applying these constraints to the 19,880 NEAs in the MPCORB database results in a total of 949 asteroids that could potentially produce positive NPVs.

Table 7.8 Summary of orbital parameters space for potentially positive NPVs.

Parameter Limit Semi-major axis 0.85 ≤ 푎 ≤ 1.3 AU Eccentricity 푒 ≤ 0.25 Inclination 푖 ≤ 8.5°

Earth Tisserand parameter 2.9375 ≤ 푇퐸 ≤ 3.0625

Shoemaker-Helin delta-V ∆푉푆퐻 ≤ 6.5 km/s

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7.6 NPV Adjusted for Finite Water Content in Small Asteroids The global optimal NPV results displayed in table 7.7 show that a number of the positive NPV asteroids were found to have small diameters (< 30 m), the smallest of which, asteroid 2006 RH120, being just 3 m in size. (This asteroid was noted in chapter 2 as having previously been captured into a temporary orbit around Earth between 2006 and 2007.) This is likely due to the fact that low delta-V asteroids tend to have smaller diameters (as demonstrated in figure 7.3).

These NPV results were generated from time-expanded networks allowing for up to 10 return trip missions, with the mining spacecraft capable of extracting a maximum capacity of 160,000 kg of water in each trip. However, the smaller asteroids are unlikely to contain sufficient water resources to reach this maximum capacity, with the total water content depleted after a few trips. For these asteroids, additional constraints should be placed on the total number of return trips, to find the total NPV of the mining mission, accounting for the finite resource mass in the asteroid.

The total water content 푀푤 of an asteroid can be estimated from its size and bulk properties, using the equations defined in chapter 4:

3 4 퐷 (7.3) 푀 = 휀 푐 (1 − 푃)휌 휋 ( ) , 푤 푟 푤 푔 3 2 where 퐷 is the diameter, 휌푔 is the average grain density, 푃 is the porosity, 푐푤 is the H2O concentration (by weight), and 휀푟 is the recovery efficiency of the mining process.

The maximum possible water content would be achieved if the asteroid was completely monolithic (having zero porosity), having the maximum estimated grain density (3 3 g/cm ) and H2O concentration (10 wt%), and having a 100% recovery efficiency in the mining operations.

Figure 7.24 shows the estimated total mass and water content for all asteroid candidates as a function of diameter. Dotted lines are shown to indicate the total cumulative capacity that can be extracted by the mining spacecraft over 푁 successive mining trips.

From this figure, it can be seen that asteroids with diameters 퐷 ≥ 21.68 m, have sufficient water content for 10 complete return trips, with some of the larger asteroids having sufficient water content to make tens of millions of trips. Asteroids with diameters 퐷 < 10.06 m, only have enough water content to partially fill the resources tank in a single trip.

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For all asteroid candidates with diameters smaller than 21.68 m, new NPV results were generated from time-expanded networks having maximum trip numbers 푁푚푎푥 determined from the total estimated water content 푀푤 and maximum capacity of the spacecraft 푀푚푎푥:

푀푤 (7.4) 푁푚푎푥 = ⌊ ⌋ , 푀푚푎푥 where ⌊ ⌋ is the floor function, rounding down to the nearest integer.

For asteroids with 퐷 < 10.06 m (푁푚푎푥 = 0), a single trip is allowed, however the maximum capacity was reduced to the total water content of the asteroid. (Ideally, an additional partial trip could be added for the other asteroids, however this would require a separate maximum capacity to be defined for the final trip, requiring changes to the edge cost functions in the graph formulation. This final maximum capacity would be determined by the remaining water content in the asteroid, that may be dependent on the total mass extracted in all previous trips, making it path dependent. This addition is left for future work.)

Figure 7.24 Total water content in asteroid candidates.

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Figure 7.25 shows the NPV results for all asteroid candidates plotted against diameter both before (top) and after (bottom) these constraints were added to the total number of trips. From this figure, it can be seen that the inclusion of this constraint lowers the NPVs calculated for the smaller asteroids. This constraint was also found to remove many of the trends seen in the orbital parameters (figure 7.14 to 7.21). (The results above were analysed prior to the inclusion of this constraint to identify trends due to orbital parameters, independent of asteroid size.)

Figure 7.25 NPV trends with asteroid diameter. Top: NPV results for a full 10 trips. Bottom: Maximum trip numbers for small asteroids constrained by total water content.

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The adjusted NPV results show that all of the smallest asteroids (퐷 < 10 m), for which only a partial trip can be completed, had negative NPVs. Of the remaining small asteroids (10 ≤ 퐷 ≤ 21.68 m), the larger candidates saw only small reductions in the total number of trips, with relatively small reductions in total NPV. For example, asteroid 2013 BS45 had an original NPV of $94.07M with 10 trips, which reduced to $91.32M after being constrained to 7 trips. This relatively small reduction is due to the fact that the majority of the NPV comes from the revenues generated in the first few trips, with the later trips having an ever-reducing contribution to the total NPV (as noted in chapter 3). Most of the small (10 ≤ 퐷 ≤ 21.68 m) asteroids with positive NPVs were still capable of generating positive values after the constraints were applied.

From these results, an additional size constraint can be added to the orbital constraints defined above for the filtering of asteroid candidates with expected negative NPVs:

• 퐷 > 10 m (corresponding to 퐻 < 28.71 for 푝푉 = 0.058).

(Smaller asteroids may still be useful as targets for early demonstration missions.)

Applying this further constraint to the orbital parameter space defined in table 7.8 reduces the total number of potentially positive NPV asteroids to 885.

Constraints Due to Limited Launch Data Range It should be noted that the NPV results of some asteroids may have been restricted by the finite launch date range considered in the time-expanded network. As seen from figure 7.15, flight itineraries have average trip durations of the order of 2 – 4 years, resulting in average durations for full 10 trip mission of over 20 years. Flight itineraries beginning later in the 2020 to 2050 launch date range are less likely to be able to fit in a full 10 trip mission (particularly for asteroids having sparsely spaced launch opportunities).

This effect is demonstrated in figure 7.26, showing a time-expanded network for asteroid (459872) 2014 EK24, with up to 10 trips (trip edges are shown in blue). Two longest path trees are highlighted in yellow and green, enumerating optimal flight itineraries starting from two 퐸퐴퐿 trajectories occurring early and late in the launch date range. The global optimal NPV flight itinerary is highlighted in red. The tree beginning early in the launch date range shows many flight itineraries capable of the full 10 trips. The tree beginning later in the range is restricted to a maximum of 4 trips.

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This constraint can be removed in future work by extending the launch date range for the 퐸퐴 and 퐴퐸 trajectories to 2070 or beyond (20 years after the final 퐸퐴퐿 launch date). This may identify larger NPV solutions beginning at later dates (particularly for long synodic period asteroids with oppositions occurring after 2050). This extension would however significantly increase the size and complexity of the time-expanded networks, and therefore the computation time.

Figure 7.26 Time-expanded network for a 10-trip mission to the Arjuna-type asteroid

(459872) 2014 EK24. Two longest path trees beginning early and late in the launch date range are highlighted in yellow and green. The global optimal NPV flight itinerary is highlighted in red.

7.7 Ideal Candidates for Asteroid Mining Missions After applying the maximum trip constraints accounting for the limited water content of small asteroids, a total of 35 asteroids were found to have positive NPVs, details of which are listed in table 7.9.

Asteroid mining missions to each of these asteroid candidates, following these optimal flight itineraries, would be considered economically viable, generating positive NPVs, and even larger profits.

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Figure 7.27 shows the total NPVs and profits of each of the asteroids in table 7.9, plotted against their capital costs (dotted lines are shown to connect the NPV and profit of each asteroid).

Table 7.9 NPV results adjusted for finite water content in smaller asteroids.

Asteroid Details Economic Parameters Flight Itinerary Pay- Avg. Avg. Asteroid D NPV Profit back Launch Date Dur. N one- Group Dur. Desig. (m) ($M) ($M) Period (yrs) way ∆푽 (yrs) (yrs) (km/s) 2012 TF79 Low DVs 18.27 254.34 609.75 1.44 12 Oct 2040 10.13 4 2.53 5.30 2016 RD34 Low DVs 16.67 231.39 409.50 1.32 26 Aug 2047 2.88 2 1.44 3.37 2018 PK21 Arjunas 36.46 231.20 1328.48 1.86 29 Jul 2020 28.97 10 2.90 5.36 (469219) Co-orb 41.00 228.02 1547.65 3.67 26 Sep 2022 18.66 10 1.87 5.19 2016 HO3 2019 GF1 Arjunas 18.27 225.87 919.63 3.30 25 Mar 2020 12.13 5 2.43 4.07 2018 PN22 Co-orb 17.45 217.79 763.90 3.00 30 Jan 2021 10.47 5 2.09 4.21 2019 GM1 Co-orb 17.45 210.03 811.75 3.51 30 Mar 2026 9.50 5 1.90 4.55 2012 EC Low DVs 120.73 196.33 831.54 1.87 21 Aug 2036 12.71 5 2.54 5.18 2010 UE51 Low DVs 12.07 173.08 327.33 1.32 26 Oct 2022 1.32 1 1.32 5.05 2001 QJ142 Low DVs 100.42 147.64 1470.68 1.97 04 Apr 2022 26.82 10 2.68 4.93 2011 UD21 Low DVs 11.01 140.57 288.84 1.37 07 Oct 2041 1.37 1 1.37 3.03 2010 VQ98 Low DVs 12.64 139.75 307.98 1.52 12 Nov 2038 1.52 1 1.52 4.14 2015 YO10 Low DVs 26.41 135.63 815.94 1.97 17 Jan 2033 16.00 6 2.67 4.94 2007 UN12 Low DVs 10.04 134.95 300.92 1.51 23 Apr 2048 1.51 1 1.51 4.16 2018 ER1 Arjunas 41.86 123.33 987.27 3.55 31 Mar 2020 27.05 10 2.71 5.65 2019 HM Arjunas 38.18 111.43 1093.61 5.98 12 Apr 2020 26.48 10 2.65 5.72 2013 WA44 Low DVs 54.00 109.54 1181.30 4.61 16 Sep 2026 23.67 8 2.96 5.24 2005 LC Low DVs 24.09 103.80 597.77 1.97 31 May 2038 11.95 4 2.99 4.85 2001 BB16 Low DVs 126.42 99.42 601.75 5.67 11 May 2037 12.69 5 2.54 5.22 2013 BS45 Arjunas 20.00 91.32 1051.75 5.47 09 Feb 2020 19.51 7 2.79 5.01 2003 SM84 Low DVs 159.15 85.04 557.35 5.82 18 Sep 2038 11.83 4 2.96 5.81 (459872) Arjunas 65.00 76.53 923.07 7.13 16 Jan 2020 28.01 10 2.80 6.17 2014 EK24 2008 HU4 Low DVs 12.07 71.80 269.65 1.97 03 May 2045 1.97 1 1.97 3.75 2016 Co-orb 38.18 48.96 1093.53 8.50 07 Feb 2020 26.14 10 2.61 5.65 CO246 (141424) Known C 438.34 46.88 518.99 5.96 06 Oct 2033 15.71 6 2.62 6.00 2002 CD 2015 TJ1 Low DVs 166.65 38.96 1169.64 10.00 07 Jan 2028 21.94 6 3.66 3.83 2019 JF1 Low DVs 100.42 36.98 1250.46 9.87 22 Aug 2026 22.95 7 3.28 3.94 2014 Low DVs 12.00 27.22 198.27 1.87 08 Jan 2049 1.87 1 1.87 5.39 UV210 2018 UM1 Low DVs 105.15 27.17 1008.59 12.58 10 Nov 2027 22.79 8 2.85 5.20 2015 Co-orb 18.27 6.65 444.23 9.85 17 Aug 2024 9.85 5 1.97 5.94 XX169 2014 Arjunas 20.04 5.90 700.33 13.95 31 Aug 2020 21.95 7 3.14 6.09 QD364 2005 YA37 Low DVs 191.34 4.35 779.75 15.97 04 Apr 2032 15.97 5 3.19 3.98 2016 Arjunas 13.24 4.20 275.75 5.54 07 Aug 2020 5.54 2 2.77 4.12 GK135 2010 CE55 Low DVs 200.36 1.57 735.83 15.88 06 Jun 2032 15.88 5 3.18 4.96 2009 RT1 Low DVs 105.15 0.00 291.43 6.91 07 Sep 2043 6.91 2 3.46 5.23

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Figure 7.27 Total NPVs and profits of positive NPV asteroids.

7.7.1 Asteroids with Maximum NPV and Profit The majority of these candidates were found to be from the low delta-V group, with a few Arjunas and co-orbitals (including the Quasi-satellite (469219) 2016 HO3), and a single known C-complex asteroid (141424) 2002 CD (identified by Hasegawa et al. [257] as a B-type asteroid).

These asteroids had positive NPVs ranging from as little as $2647.8 up to a maximum of $254.34M. The capital costs of these missions were fairly consistent, ranging from $381.3M to $400M. These values are relatively comparable to the costs of previous asteroid missions such as Hayabusa2 mission, that had around half the cost and half the mass of the mining spacecraft considered in this analysis [269].

The NPV is a useful metric in the comparison and evaluation of decision alternatives, accounting for the time-value of money. However, future asteroid mining companies, and their stakeholders, would likely be more interested in the actual cash-flows and

Scott Dorrington – June 2019 301 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions profits generated from the mission. The asteroids in this table can all generate positive profits ranging from $198.27M to $1.55Bn, with returns on investment between 50.4 to 394.1% of the initial capital costs.

Maximum NPV Candidate

The best asteroid candidate in terms of NPV was found to be asteroid 2012 TF79. A 4- trip mission to this asteroid, launching in October 2040 could be completed in 10.13 years for a total profit of $609.75M, with an initial capital investment of $394.02M. This mission would break-even, producing a positive NPV after its first trip (payback period of 1.44 years). A cash-flow diagram of this mission is displayed in figure 7.28 showing the times of the revenue streams and the cumulative NPV and profit throughout the mission.

Figure 7.28 Cash-flow diagram for 4-trip mission the maximum NPV candidate,

asteroid 2012 TF79.

Maximum Profit Candidate The best asteroid candidate in terms of total profit was found to be the Quasi-satellite asteroid (469219) 2016 HO3 (having the fourth highest NPV). While this mission would produce a higher total profit of $1.55Bn, it would require more than twice the total number of trips than the previous example (a total of 10 trips), with a total mission duration of 18.66 years. This mission would require a lower capital investment, at $392.7M, however, would be a considerably more complex mission. A cash-flow diagram for this mission is shown in figure 7.29.

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Figure 7.29 Cash-flow diagram for 10-trip mission to the maximum profit candidate,

asteroid (469219) 2016 HO3.

7.7.2 Asteroid with Maximum Expected NPV and Profit As discussed in chapter 3, there is a large uncertainty in the presence of resources in individual asteroids. This uncertainty can be accounted for by the calculating the expectation values of the profit:

(7.5) 〈푃〉 = 푝푠(푃) + (1 − 푝푠)(−퐶0) , and NPV:

(7.6) 〈푁푃푉〉 = 푝푠(푁푃푉) + (1 − 푝푠)(−퐶0) , where 푝푠 is the probability of a successful mission.

The majority of the asteroids in table 7.7 were from candidate groups with unknown taxonomic class, for which a value of 푝푠 = 0.025 can be assigned, while the known C- complex asteroid (141424) 2002 CD is assigned a value 푝푠 = 0.25. (These values are defined in chapter 3 from the estimated 10% of NEAs being C-complex asteroids [36], and 25% of C-complex asteroids having high water content [259].) The total expected NPV and profits of these asteroids are shown in figure 7.30.

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Figure 7.30 Total expected NPVs and expected profits of the positive NPV asteroids.

Due to the increased certainty, the known C-complex asteroid has significantly larger values for both expected NPV and expected profit than the other asteroids, making it the ideal candidate considering the uncertainty in the presence of water resources. However, the expected value of this asteroid is still negative, indicating that it would still be highly risky to send a mining mission to this asteroid with the current level of knowledge of its composition.

This uncertainty could be reduced in future with further detailed photometric and spectroscopic observations, or through the use of precursor prospecting missions. Figure B.2 in Appendix B shows observation conditions for asteroid (141424) 2002 CD, calculated from ephemeris data and a phase curve generated using the two parameter (퐻-퐺) model [93], showing predicted visible observation opportunities (푉 < 22 mag) between November 2029 and June 2042.

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An improved knowledge of the composition of B-type asteroids could also be gained when the OSIRIS-REx mission returns a sample of the B-type asteroid 101955 Bennu in September 2023 [57].

Maximum Expected NPV and Profit Candidate The optimal NPV flight itinerary to asteroid (141424) 2002 CD was found to be a 6-trip mission launching in October 2033, with a total duration of 15.71 years. This would require a capital investment of $384.6M, generating a total profit of $519M. Figure 7.31 shows a cash-flow diagram for this mission (assuming there is water found to be present in the asteroid). A flight itinerary listing the launch and arrival dates of the consecutive Earth-to-Asteroid and Asteroid-to-Earth transfers is shown in table 7.10.

Figure 7.31 Cash-flow diagram for a 6-trip mission to the known B-type asteroid (141424) 2002 CD (maximum expected NPV and profit candidate).

Table 7.10 Optimal NPV flight itinerary for mission to asteroid (141424) 2002 CD.

Earth-to-Asteroid Asteroid-to-Earth Trip Launch Date Arrival Date Launch Date Arrival Date 1 06 Oct 2033 19 Mar 2034 09 Oct 2034 28 Mar 2035 2 01 Jun 2035 16 Mar 2036 25 Mar 2037 05 Oct 2037 3 30 Mar 2038 09 Aug 2038 06 Mar 2039 24 Sep 2039 4 22 Feb 2040 28 Aug 2040 23 Feb 2041 09 Sep 2042 5 12 Jan 2043 23 Jul 2043 10 Jan 2044 28 Jun 2045 6 14 Apr 2046 13 Feb 2047 05 Dec 2047 22 Jun 2049

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7.8 Conclusion In this chapter, two data sets of asteroid orbital elements and physical properties were combined and processed to identify asteroid candidates with known and assumed C- complex taxonomic classifications. From the remaining asteroids with unknown taxonomic class, additional candidate lists were generated for those with low Shoemaker-Helin delta-V, orbital elements close to that of the Earth (Arjunas), and asteroids with a 1:1 mean motion resonance with the Earth (co-orbitals).

For each of these asteroid candidates, methods developed in chapter 6 were applied to identify sets of optimal launch opportunities and produce time-expanded networks enumerating all potential flight itineraries with up to 10 return trip missions, from which optimal NPV flight itineraries were identified by solving an all-pairs longest path problem.

Initial results identified 41 asteroids with positive NPVs. These results were compared to the orbital parameters of the asteroid candidates to identify limiting values above which all of the asteroids had negative NPVs. This produced an orbital parameters space (summarized in table 7.8) that could be applied to filter the large list of 19,880 asteroids in the MPCORB database to just 949 asteroids that could potentially produce positive NPVs.

It was noted that a number of the asteroid candidates identified to have positive NPVs had small diameters that would be unlikely to contain sufficient water content to complete the total number of return trips necessary to reach this positive NPV. For these asteroids, new results were produced in which the total number of trips was limited by the total estimated water content and the maximum capacity of the mining spacecraft. These results produced a list of 35 asteroids with positive NPVs, all with diameters larger than 10 m. Applying this additional size constraint to the orbital parameters space reduced the total number of potential candidates to 885 asteroids. This is a reasonably small number of asteroids, for which the numerical methods presented in this chapter could be applied in future to identify additional asteroids with positive NPVs.

Asteroid mining missions to each of the 35 identified asteroids would generate positive NPVs, with capital costs of $381.3M to $400M, and generating large profits ranging from $198.27M to $1.55Bn (a return on investment of 50.4 to 394.1%). Of these candidates, asteroids 2012 TF79 and (469219) 2016 HO3 were identified as producing the highest NPV and profit, respectively. The known B-type asteroid (141424) 2002 CD

306 Scott Dorrington – June 2019 Chapter 7: Asteroid Candidates was identified as having the largest expected NPV and profit, accounting for the uncertainty of resources present in the asteroid. However, it was noted that this expected NPV was still negative, indicating that additional information on the composition would still be required before an asteroid mining mission could commence.

These results from this chapter confirm the economic feasibility of asteroid mining mission with realistic trajectories over multiple return trip missions.

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8 CONCLUSION

8.1 Summary of Chapters The aim of this thesis was to evaluate the feasibility of asteroid mining from a multi- disciplinary approach, considering the trajectory design, mission architecture, supply chain logistics, economics, and mining operations. The various chapters of this thesis developed methods for evaluating different aspects of the space mission architecture. The main content and conclusions drawn from these chapters will be summarized in the following sections.

Chapter 1: Introduction Chapter 1 presented an introduction to the thesis, outlining the motivation for asteroid mining missions, and the limitations in the current feasibility studies that often focus on the market value of the minerals contained in the asteroid. The chapter also posed a series of questions that were aimed to be answered throughout the work.

Chapter 2: Asteroid Properties and Resources Chapter 2 presented a literature review of the current state of knowledge of the orbital, physical, and compositional properties of asteroids. This was intended to introduce a number of concepts and datasets that were used throughout the thesis.

The near-Earth asteroids were identified as the main candidates that should be considered for asteroid mining missions, reachable with significantly lower delta-Vs than the main belt or Mars-crossing asteroids.

The main resource of interest was identified as being water in the form of hydrated minerals that is expected to be present in C-complex asteroids, having compositions similar to carbonaceous chondritic meteorites. Probabilistic estimates of the distribution of C-complex asteroids and water content in carbonaceous chondrites suggested that only around 2.5% of the near-Earth asteroids are expected to have rich sources of water.

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Known C-complex asteroids were therefore identified as the ideal candidates for asteroid mining, with around 25% of them expected to have rich sources of water.

Chapter 3: Parametric Economic Analysis Chapter 3 outlined a generalized space mission architecture describing the elements of an asteroid mining operation. Alternative approaches were discussed for the mining/extraction of asteroid material, and the source of propellant. From these alternatives, two main mission alternatives were identified: 1. Whole asteroid return with Earth-based propellant supply (EBPS); and 2. In situ processing with in situ propellant processing (ISPP).

Additional alternatives were presented for the ISPP case with reserve propellant, and a reference mission in which the same mass of resources is supplied directly from Earth.

A parametric economic model was produced to calculate the capital costs, revenues, profits, and net present values (NPVs) of the different mission alternatives as a function of a set of system, mission, and specific cost parameters. This model was used to evaluate the different mission alternatives to identify the conditions under which each were economically feasible (producing positive NPVs).

The results concluded that single-trip missions using chemical propellant supplied from Earth are only feasible for asteroids with very low delta-Vs less than 1.8 km/s. Electric propulsion is feasible up to 4.5 km/s, while ISPP is feasible up to larger delta-Vs of 8.8 km/s.

It was found that missions using ISPP that return small shipments of resources over multiple trips could produce larger NPVs than a single-trip mission retrieving a whole large asteroid using propellant sourced from Earth. These missions could produce billions of dollars in profits over a 20-year design life, recovering their initial capital investment after the first or second trip. However, these missions were not capable of producing positive expected NPVs, when accounting for the uncertainty in the presence of water resources in asteroids (particularly for asteroids of unknown taxonomic class). Positive expected NPVs can be achieved by supplying reserve propellant allowing the mining spacecraft to return to Earth in the event of a failed mission.

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Chapter 4: Parametric Mining Rate Model Chapter 4 presented a similar parametric model used to calculate the achievable mining rate as a function of spacecraft parameters, geometric parameters of the asteroid orebody, and specific energies of mining operations.

A numerical example was presented to compute mining rates as a function of power available to the rover, with asteroid geological parameters taken from asteroid 25143 Itokawa. It was found that solar array areas of 5 – 10 m2 could produce 1.22 – 2.24 kW of power with mining rates of 217 – 409 kg/day, and mining rates of 800 kg/day could be achieved with 5 kW. This mining rate was selected for use in subsequent analysis.

It was also noted that the energy requirements for the chemical processing of regolith were an order of magnitude larger than those required for fragmentation and excavation, therefore the spacecraft power should also be an order of magnitude greater than that of the rover.

Chapter 5: Supply Chain Network Chapter 5 developed methods to model the distribution or asteroid-derived resources to customer spacecraft as a supply chain network. The network featured nodes defining the orbital locations of an asteroid source, an orbital propellant depot, arrival and departure parking orbits, and customer spacecraft. Edges between nodes represented transfer trajectories modelled as impulsive manoeuvres. Two classes of spacecraft vehicles were assigned to transport the product through various edges in the network.

A location-routing problem was formulated to identify the optimal location of nodes and vehicle routes through the network, with the objective of maximizing either the total sellable mass delivered to customers, or the NPV of the asteroid mining venture. A numerical example was presented to illustrate the method for the case of maximizing total sellable mass delivered from a near-Earth asteroid to a single customer in Geostationary Orbit.

The results showed that the maximum sellable mass is achieved with a delivery route in which the mining spacecraft remains in a highly elliptical parking orbit before returning to the asteroid, while a small transport spacecraft retrieves the load of material and delivers it to an orbital propellant depot and customers. A sensitivity analysis showed that this routing strategy was also the most resilient to variations in the mining spacecraft dry mass that may occur during the design phase. However, it was noted that

310 Scott Dorrington – June 2019 Chapter 8: Conclusion an alternative route in which the mining spacecraft delivers the material directly to the propellant depot would be simpler to implement, removing the need to transfer material between spacecraft, and could allow for scheduled maintenance of the mining spacecraft between mining trips.

The optimal location of a propellant depot was found to be at the same orbital radius and orbital plane as the customers. This depot location and vehicle routing can deliver between 33.15 – 52.63% of the mass extracted from the asteroid in a single trip (depending on the trajectories to/from the target asteroid), with the ability to conduct numerous return trips to the asteroid. It was noted that further testing would be required on a large number of asteroids and trajectories to ensure this is a globally optimal solution before a propellant depot is deployed into this orbit.

Chapter 6: Combinatorial Trajectory Optimization Chapter 6 developed a graph theory approach to the combinatorial optimization of trajectories for multiple return trip missions to asteroids. A porkchop plot program was developed to identify sets of optimal (minimum delta-V) launch opportunities between 2020 and 2050 for four types of heliocentric transfers, using a Lambert solver and asteroid ephemeris data.

These launch opportunities were used to construct time-expanded network graphs with edges representing combinations of successive transfers between the Earth and asteroid, and paths through the network enumerating all potential flight itineraries (lists of launch and arrival dates) for multiple return trip missions. Objective functions describing the total delta-V, mission duration, sellable mass, profit, and NPV were derived as linear functions of edge costs in the network, allowing for optimal flight itineraries to be identified by applying an all-pairs shortest/longest path algorithm. A mathematical formulation was presented to compute these edge costs and construct graphs with arbitrary numbers of return trip missions.

Chapter 7: Asteroid Candidates Chapter 7 combined two datasets of asteroid orbital and physical properties to generate lists of asteroid candidates having known and assumed C-complex taxonomic classes, low Shoemaker-Helin delta-Vs (< 4.5 km/s), orbital elements close to Earth (Arjunas), and asteroids with a 1:1 mean motion resonance with the Earth (co-orbitals). For each

Scott Dorrington – June 2019 311 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions asteroid candidate, optimal NPV flight itineraries were identified using the methods developed in chapter 6.

The results found that the low delta-V asteroids returned the largest NPVs. Trends in NPV were identified from the orbital parameters, noting that asteroids with long synodic periods produce the largest NPVs, so long as their opposition dates fall within the considered launch date range. These orbital trends were used to define an orbital parameter space that can be used to identify asteroid candidates that could potentially produce positive NPVs.

It was found that many of the asteroids with positive NPVs had small diameters that would be unlikely to contain sufficient water content to complete the total number of return trips necessary to reach this positive NPV. For these asteroids, new results were produced in which the total number of trips was limited by the total estimated water content and the maximum capacity of the mining spacecraft.

The results produced a list of 35 asteroids with positive NPVs, all with diameters larger than 10 m. Asteroid mining missions to each of these asteroids, following the identified optimal flight itineraries would be considered economically viable missions. These missions could be achieved with capital costs of around $400M, generating large profits ranging from $198.27M to $1547.7M (or $1.55Bn) – a return on investment of 50.4 to

394.1%. Of these candidates, asteroids 2012 TF79 and (469219) 2016 HO3 were identified as producing the highest NPV and profit, respectively. The known B-type asteroid (141424) 2002 CD was identified as having the largest expected NPV and profit, accounting for the uncertainty of resources present in the asteroid.

Answers to Questions Chapter 1 posed a series of questions that were aimed to be answered throughout the work. These are answered in table 8.1, summarizing the major conclusions drawn from the chapters.

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Table 8.1 Answers to questions of asteroid mining.

Question Answer The mining rate can be expressed as a function of How much asteroid available power. Mining rates of 217 – 800 kg/day material can be extracted could be achievable with power levels of 1.22 – 5 kW, during a given stay-time with up to 146 tonnes of material extracted during a at an asteroid? 6-month stay-time. A portion of the extracted material is used as propellant in returning the resources to Earth orbit, distributing it to the customers, and returning to the asteroid. The portion is dependent on the delta-Vs of How much of this mass the trajectories, the orbital locations of the propellant can be delivered to depot and customer spacecraft, the routing of the customer spacecraft in spacecraft, and the design of the arrival and departure Earth orbit? parking orbits. Numerical examples showed that around 33 to 53% of the extracted mass can be delivered to customers for sale (dependent on the trajectories). The best location for a propellant depot was found to What is the best location be at the same orbital plane and radius as the for a propellant depot in customer spacecraft. The optimal routing strategy Earth orbit to stockpile was one in which the mining spacecraft remains in a returned resources for highly elliptical parking orbit before returning to the distribution to customer asteroid, while a small transport spacecraft retrieves spacecraft? the load of material and delivers it to an orbital propellant depot and customers. A list of 35 asteroids were found that can generate positive NPVs. These were mostly those with low Which asteroids are the Shoemaker-Helin delta-Vs less than 4.5 km/s, and best targets? asteroids in very Earth-like orbits, with long synodic periods. The asteroids should be larger than 10 m in diameter to ensure sufficient water content. Total profits for the 35 identified asteroids ranged from $198.27M to $1.55Bn, with capital costs ranging from $381.3M to $400M (a return on investment of How much profit can be 50.4 to 394.1%). made from mining a given asteroid target? The 18.27 m diameter asteroid 2012 TF79 showed that largest NPV, with total profits of $609.75M generated over a 10.13 year, 4-trip mission, from a $394.02M capital investment. The 35 identified asteroids had payback periods How many return trips ranging from 1.32 to 15.97 years. In most cases, these are required to recover missions could recover their initial capital investment the capital investment? after the first or second trip.

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8.2 Feasibility of Asteroid Mining The results of this thesis demonstrated the economic viability of asteroid mining mission to 35 identified asteroid candidates, with realistic trajectories over multiple return trip missions, at least in the deterministic case. However, it was found that knowledge of the presence of water in the form of hydrated minerals in asteroids was still too uncertain to produce positive expected NPVs. At this current level of knowledge, asteroid mining missions would likely lose money in more cases than not.

Further scientific investigations through photometric and spectroscopic observations, and spacecraft missions should be conducted to produce additional data and an improved understanding of the chemical composition of C-complex asteroids before commercial asteroid mining should commence. These investigations could be conducted by asteroid mining companies; however, these would require additional capital investment that would reduce the total profits and NPVs of the mission.

8.3 Discussion

Alternative Objectives In this thesis, NPV was used as the primary metric for assessing the economic feasibility of asteroid mining missions. This metric is commonly used in capital budgeting of projects, as it accounts for the time-value of money, favouring short-term returns on investment. This is an appropriate metric assuming that asteroid mining is conducted as a purely profit seeking commercial venture.

Asteroid mining missions could alternatively be conducted by non-profit entities such as space agencies to supply propellants and other resources on an as-needed basis in the support of scientific or human exploration missions (a concept known as in situ resource utilization, ISRU). In this case, alternative objectives could be to maximize the total mass of material delivered, or to maximize the opportunity cost – the savings in the total mission cost that could be achieved by using asteroid-sourced materials compared to traditional Earth-sourced approaches (such as the reference mission described in chapter 3). These objective functions would likely lead to alternative optimal flight itineraries and asteroid candidates.

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Asteroids – A Limited Resource Beyond the technical and economic considerations, the mining of asteroids and other planetary bodies presents numerous legal, ethical, social, and environmental issues that should be discussed prior to the first asteroid mining mission taking place.

Despite the perceived limitless supplies of resources in asteroid, often quoted by asteroid mining companies, the resources present in near-Earth asteroids are in fact a limited resource. As noted in chapter 2, it is estimated that only 2.5% of near-Earth asteroids are expected to possess rich supplies of water. This amounts to only 460 asteroids amongst the known population of near-Earth asteroids.

Presuming asteroid mining missions are demonstrated to be successful, and indeed profitable, it is expected that this limited supply of asteroids will eventually be depleted. Further mining missions would then be required to target asteroids further out in the main asteroid belt, and perhaps even to the outer Solar System. These resources may too be consumed in an ever-expanding human presence in the Solar System.

While this would not likely happen in the near-term, it would be an inevitability if human exploration and exploitation of the Solar System are not constrained. This concern has been noted by some researchers, suggesting that we should even now begin discussing how much of the Solar System’s resources should be left in its natural state. Elvis & Milligan [270] suggest that only one eighth of the total Solar System’s resources should be exploited by all future generations, to provide insurance against depleting the entire resources in the expectation of exponential growth.

8.4 Future Work

Selection of Exploration Strategy A major conclusion drawn from this thesis is the inability to produce positive expected NPVs accounting for the uncertainty of the presence of water resources. In chapter 3 it was noted that the use of precursor exploration missions (either flyby and/or sampling missions) could lower the potential losses from an unsuccessful mining mission at the expense of additional capital, and that the optimal number and sequencing of missions should be selected to maximize the total expected NPV.

At the time of writing, a decision tree has been produced (although not included in the thesis) to enumerate the decisions and outcomes of several exploration strategies. However, more work is required to present a formulation of the computation of the

Scott Dorrington – June 2019 315 The Trajectory Optimization & Space Logistics of Asteroid Mining Missions expectation value over an arbitrary number of return trips, similar to those presented in section 3.5 (the decision tree shows a recursive structure with the number of outcomes increasing exponentially with additional trips).

Additional work is also required to assign appropriate conditional probabilities to the likelihood of each outcome (for example, the probability of a failed mining mission, given that the exploration mission provided evidence for hydrated minerals). These should be quantified using a Bayesian inference approach.

Improved Parametric Modelling The parametric models presented in chapters 3 and 4 allowed for the variation of system parameters individually. However, there are likely to be many inter-dependencies between these parameters that should be accounted for in future. For example, the mining rate was found to be dependent on the available power of the rover. Increasing the size of the solar panels would increase the mining rate, thereby reducing the required stay-time at the asteroid. However, the increased solar panel size would also increase spacecraft dry mass, which would increase the total capital cost of the mission. An improved parametric model should account for these inter-dependencies to identify the optimal mission alternatives.

While chapter 3 computed the NPVs over a range of delta-Vs and specific impulses, the results were computed for a single set of subsystem and cost parameter values. Similarly, the numerical example presented in chapter 4 was also computed for a single set of values for the mining subsystems and asteroid geology. Future work should repeat these analyses with a detailed sensitivity analysis to test the results over a range of possible parametric values. This would identify the critical design drivers that may aid in the system design and sizing of future mining spacecraft and equipment, as well as testing the affects of economic drivers such as the expected reduction in future launch costs.

Extended Launch Date Range In chapter 7 it was noted that the limited launch date range (2020 to 2050) considered in the time-expanded networks make it less likely for flight itineraries starting later to be able to complete the full number of return trips, and may restrict the total NPV achievable for the asteroid candidates. In future analysis of asteroid candidates, launch

316 Scott Dorrington – June 2019 Chapter 8: Conclusion opportunities should be extended to at least 20 years after the final launch date of the first Earth-to-Asteroid transfer.

The chapter also identified an orbital parameter space containing asteroids that could potentially have positive NPVs. These should be investigated in future work, along with additional asteroids that are discovered on a daily basis.

Computing Expected NPVs from Time-Expanded Networks In chapter 3, the expected NPV for multi-trip missions was calculated with the assumption that additional mining spacecraft would be deployed to new asteroid candidates in the event of a failed mission. The ISPP case with reserve propellant was able to show positive expected NPVs at low delta-V due to the lower capital cost of completing the next mission.

In chapter 7, the expectation value of the known B-type asteroid (141424) 2002 CD was only computed from the identified optimal NPV flight itinerary, with a failed mission resulting in the total loss of the capital investment. Future analysis could adapt the time- expanded network for the reserve propellant ISPP case, and compute contingency flight itineraries to other asteroid candidates that could be carried out in the event of a mission failure in the first trip of the optimal flight itinerary. This may reveal that positive expected NPVs can be achieved at the current level of uncertainty.

Generalized Interplanetary Supply Chain Network In chapter 5, a supply chain network was presented to model the distribution of asteroid resources to customers in Earth orbit. While this network focused on Earth orbits, it could be extended to include additional nodes around other planets, allowing for the optimization of parking orbits and vehicle routing for more complex space mission architectures.

In order to remain competitive with Earth-sourced resources, the specific sale price was set at a fraction of the cost of the launch cost to the same orbit. Considering the large costs required to deliver materials to the Moon or Mars, higher sale prices could be set, allowing for larger revenue streams with each trip. The time-expanded networks developed in chapter 6 could be easily adapted to launch the first mining trip from Earth, with all subsequent trips between the asteroid and Mars. This would likely lead to much larger NPVs than delivering it to customers in Earth orbit. Optimal asteroid candidates for these missions are expected to be in Mars-crossing orbits.

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[223] W. Kizner, A method of describing miss distances for lunar and interplanetary trajectories, Planetary and Space Science, 7 (1961) 125-131. [224] R.W. Luidens, B.A. Miller, Efficient planetary parking orbits with examples for Mars, in: N.L.R. Center (Ed.), United States, 1966, [225] M.T. Melo, S. Nickel, F. Saldanha-da-Gama, Facility location and supply chain management – A review, European Journal of Operational Research, 196 (2009) 401- 412. [226] G. Nagy, S. Salhi, Location-routing: Issues, models and methods, European Journal of Operational Research, 177 (2007) 649-672. [227] R.L. Tobin, T.L. Friesz, Spatial competition facility location models: Definition, formulation and solution approach, Annals of Operations Research, 6 (1986) 47-74. [228] B.A. Conway, Spacecraft trajectory optimization, Cambridge University Press, 2010. [229] R.H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, revised edition, American Institute of Aeronautics and Astronautics, 1999. [230] J.F. Jordan, Application of Lambert's Theorem to the Solution of Interplanetary Transfer Problems, Jet Propulsion Laboratory, California Institute of Technology, 1964. [231] H.D. Curtis, Orbital mechanics for engineering students, Butterworth-Heinemann, 2013. [232] L. George, L.D. Kos, Interplanetary mission design handbook: Earth-to-mars mission opportunities and mars-to-earth return opportunities 2009-2024, (1998). [233] L.M. Burke, R.D. Falck, M.L. McGuire, Interplanetary Mission Design Handbook: Earth-to-Mars Mission Opportunities 2026 to 2045, (2010). [234] R.C. Woolley, C.W. Whetsel, On the nature of Earth-Mars porkchop plots, (2013). [235] Rocket Lab, Electron - Frequent and reliable access to space for small satellites, in, 2019, https://www.rocketlabusa.com/electron/ [236] A.B. Sergeyevsky, G.C. Snyder, R.A. Cunniff, Interplanetary mission design handbook. Volume 1, part 2: Earth to Mars ballistic mission opportunities, 1990-2005, (1983). [237] K.M. Hughes, J.W. Moore, J.M. Longuski, Preliminary analysis of ballistic trajectories to neptune via gravity assists from venus earth mars jupiter and , in: AAS/AIAA Astrodynamics Specialist Conference, 2013, [238] T. Ishimatsu, J. Hoffman, O. de Weck, METHOD FOR RAPID INTERPLANETARY TRAJECTORY ANALYSIS USING∆ V MAPS WITH FLYBY OPTIONS, a a, 2 (2011) 2. [239] NASA Navigation and Ancillary Information Facility, SPICE An Observation Geometry System for Space Science Missions, in, 2019, https://naif.jpl.nasa.gov/naif/ [240] JPL Solar System Dynamics Group, Advanced Horizons. Asteroid & Comet SPK File Generation Request, in, 2019, https://ssd.jpl.nasa.gov/x/spk.html [241] N.N.a.A.I. Facility, The SPICE Toolkit, in, 2015, https://naif.jpl.nasa.gov/naif/toolkit.html

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[242] T. Ursell, Find Image Extrema, in, MATLAB File Exchange, 2013, https://au.mathworks.com/matlabcentral/fileexchange/41955-find-image-extrema [243] P.E. Gill, W. Murray, M.A. Saunders, SNOPT: An SQP algorithm for large-scale constrained optimization, SIAM review, 47 (2005) 99-131. [244] P.E. Gill, W. Murray, M.A. Saunders, User’s guide for SNOPT version 7: Software for large-scale nonlinear programming, (2006). [245] D. Eagle, Ballistic Interplanetary Trajectory Design and Optimization, in, 2015, https://au.mathworks.com/matlabcentral/fileexchange/39446-ballistic-interplanetary- trajectory-design-and-optimization?s_tid=prof_contriblnk [246] D.J. Jezewski, Primer vector theory and applications, (1975). [247] D.J. Jezewski, H.L. Rozendaal, An efficient method for calculating optimal free- space n-impulse trajectories, AIAA journal, 6 (1968) 2160-2165. [248] D.F. Lawden, Optimal trajectories for space navigation, Butterworths, 1963. [249] M. Handelsman, P. Lion, Primer vector on fixed-time impulsive trajectories, Aiaa Journal, 6 (1968) 127-132. [250] M.R. Silver, O.L. De Weck, Time‐expanded decision networks: A framework for designing evolvable complex systems, Systems Engineering, 10 (2007) 167-188. [251] E.W. Dijkstra, A note on two problems in connexion with graphs, Numerische mathematik, 1 (1959) 269-271. [252] K. Thulasiraman, M.N. Swamy, Graphs: theory and algorithms, Wiley Online Library, 1992. [253] R.W. Floyd, Algorithm 97: shortest path, Communications of the ACM, 5 (1962) 345. [254] MathWorks, MATLAB. Graph and Network Algorithms, in, 2019, https://au.mathworks.com/help/matlab/graph-and-network-algorithms.html [255] R. Bellman, On a routing problem, Quarterly of applied mathematics, 16 (1958) 87-90. [256] L.R. Ford Jr, Network flow theory, in, Rand Corp Santa Monica Ca, 1956, [257] S. Hasegawa, D. Kuroda, K. Kitazato, T. Kasuga, T. Sekiguchi, N. Takato, K. Aoki, A. Arai, Y.-J. Choi, T. Fuse, Physical properties of near-Earth asteroids with a low delta-v: Survey of target candidates for the Hayabusa2 mission, Publications of the Astronomical Society of Japan, 70 (2018) 114. [258] Y. Tsuda, M. Yoshikawa, M. Abe, H. Minamino, S. Nakazawa, System design of the Hayabusa 2—Asteroid sample return mission to 1999 JU3, Acta Astronautica, 91 (2013) 356-362. [259] M. Elvis, J. McDowell, J.A. Hoffman, R.P. Binzel, Ultra-low delta-v objects and the human exploration of asteroids, Planetary and Space Science, 59 (2011) 1408-1412. [260] W.F. Bottke Jr, A. Morbidelli, R. Jedicke, J.-M. Petit, H.F. Levison, P. Michel, T.S. Metcalfe, Debiased orbital and absolute magnitude distribution of the near-Earth objects, Icarus, 156 (2002) 399-433. [261] Center for Near Earth Objects Studies, Accessible NEAs, in, 2019, https://cneos.jpl.nasa.gov/nhats/

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[262] Minor Planet Center, Export Format for Minor-Planet Orbits, in, 2019, https://minorplanetcenter.net/iau/info/MPOrbitFormat.html [263] T. Müller, J. Ďurech, M. Ishiguro, M. Mueller, T. Krühler, H. Yang, M.-J. Kim, L. O’Rourke, F. Usui, C. Kiss, Hayabusa-2 mission target asteroid 162173 Ryugu (1999 JU3): Searching for the object’s spin-axis orientation, Astronomy & Astrophysics, 599 (2017) A103. [264] C.A. Thomas, J.P. Emery, D.E. Trilling, M. Delbó, J.L. Hora, M. Mueller, Physical characterization of Warm Spitzer-observed near-Earth objects, Icarus, 228 (2014) 217-246. [265] D. Perna, M.A. Barucci, M. Fulchignoni, M. Popescu, I. Belskaya, S. Fornasier, A. Doressoundiram, C. Lantz, F. Merlin, A spectroscopic survey of the small near-Earth asteroid population: Peculiar taxonomic distribution and phase reddening, Planetary and Space Science, 157 (2018) 82-95. [266] C. de la Fuente Marcos, R. de la Fuente Marcos, Asteroid (469219) 2016 HO3, the smallest and closest Earth quasi-satellite, Monthly Notices of the Royal Astronomical Society, 462 (2016) 3441-3456. [267] C. de la Fuente Marcos, R. de la Fuente Marcos, Geometric characterization of the Arjuna orbital domain, Astronomische Nachrichten, 336 (2015) 5-22. [268] M. Ester, H.-P. Kriegel, J. Sander, X. Xu, A density-based algorithm for discovering clusters in large spatial databases with noise, in: Kdd, 1996, pp. 226-231, [269] W. Zukerman, Hayabusa 2 will seek the origins of life in space, in, 2010, [270] M. Elvis, T. Milligan, How much of the Solar System should we leave as Wilderness?, Acta Astronautica, (2019).

332 Scott Dorrington – June 2019 Chapter 10: Appendices

10 APPENDICES

APPENDIX A ANALYTICAL DELTA-V ESTIMATES ...... 334

APPENDIX B TWO PARAMETER H-G PHASE CURVE MODEL ...... 340

APPENDIX C COST ESTIMATE FOR MINING SPACECRAFT ...... 343

APPENDIX D LOCATION-ROUTING PROBLEM ALGORITHMS ...... 345

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APPENDIX A ANALYTICAL DELTA-V ESTIMATES

A.1 Hohmann Transfer Equations

Let 푟1 and 푟2 be the orbital radii of two circular orbits around a body with gravitational parameter 휇. A Hohmann transfer between the two orbits (shown in figure A.1) will be an elliptical orbit with periapsis at 푟1 and apoapsis at 푟2 (for the case of 푟2 > 푟1).

Figure A.1 Hohmann Transfer

The transfer orbit will have semi-major axis 푎푡푥 equal to the average of the two radii:

푟1 + 푟2 푎 = . (A.1) 푡푥 2

From this, the specific energy of the transfer is given by:

휇 휀푡푥 = − . (A.2) 2푎푡푥

The Hohmann transfer is a two-impulse transfer, requiring two delta-V manoeuvres at the beginning and end of the transfer. The first manoeuvre transfers the spacecraft from the initial circular orbit to the transfer orbit. The delta-V of the first manoeuvre ∆푉1 is calculated as the difference between the initial circular velocity, and the velocity of the transfer orbit at the radius 푟1:

휇 휇 ∆푉1 = |√2 ( + 휀푡푥) − √ | . (A.3) 푟1 푟1

334 Scott Dorrington – June 2019 Chapter 10: Appendices

The second manoeuvre transfers the spacecraft from the transfer orbit to the final circular orbit. The delta-V of the second manoeuvre ∆푉2 is calculated as the difference between the velocity of the transfer orbit at radius 푟2 and the final circular velocity:

휇 휇 ∆푉2 = |√2 ( + 휀푡푥) − √ | . (A.4) 푟2 푟2

Both delta-V manoeuvres are tangential to their respective circular orbits. For transfers to outer planets (푟2 > 푟1), the manoeuvres are both in the prograde direction, increasing the energy of the orbit at both stages. For transfers to inner planets (푟2 < 푟1 ), the manoeuvres are both in the retrograde direction, decreasing the energy of the orbit at both stages.

The time of flight of the transfer can be found from half the orbital period:

푎3 푇푂퐹 = 휋√ 푡푥 . (A.5) 휇

A.2 Circular-to-Elliptical Transfer Equations Consider an asteroid with orbital elements 푎 (expressed in km) and 푒 (dimensionless).

We set the circular radius of the first orbit at the orbit of the Earth 푟1 = 1 AU, and the radius of the second orbit to either the periapsis 푟2 = 푎(1 − 푒) or apoapsis 푟2 = 푎(1 + 푒) of the asteroid (shown in figure A.2).

Figure A.2 Left: Periapsis transfer. Right: Apoapsis transfer

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The delta-Vs of the Periapsis and Apoapsis transfers can then be found by the following equations:

Table A.1 Equations for the coplanar circular-to-elliptical transfer.

Periapsis Transfer Apoapsis Transfer

푟푝 = 푎(1 − 푒) 푟푎 = 푎(1 + 푒)

휇 휇 휇 휇 푣푝 = √2 ( − ) 푣푎 = √2 ( − ) 푟푝 2푎 푟푎 2푎

푟1 + 푎(1 − 푒) 푟1 + 푎(1 + 푒) 푎 = 푎 = 푡푥 2 푡푥 2 −휇 −휇 휀푡푥 = 휀푡푥 = 2푎푡푥 2푎푡푥

휇 휇 휇 휇 ∆푉1 = |√2 ( + 휀푡푥) − √ | ∆푉1 = |√2 ( + 휀푡푥) − √ | 푟1 푟1 푟1 푟1

휇 휇 ∆푉2 = |푣푝 − √2 ( + 휀푡푥)| ∆푉2 = |푣푎 − √2 ( + 휀푡푥)| 푟푝 푟푎

For the case of 푒 = 0, both of these sets of equations are equal, and equivalent to a Hohmann transfer.

336 Scott Dorrington – June 2019 Chapter 10: Appendices

A.3 Shoemaker-Helin Equations The Shoemaker-Helin equations [7] make use of a figure of merit 퐹 describing the total delta-V of the transfer in velocity units normalized to the Earth’s orbital velocity around the sun:

퐹 = 푈퐿 + 푈푅 , (A.6) where:

푈퐿= Earth departure manoeuvre (normalized velocity units); and

푈푅= Asteroid arrival manoeuvre (normalized velocity units).

The delta-V is then converted to km/s using the following formula:

∆푉푆퐻 = 30퐹 + 0.5 [푘푚/푠] . (A.7)

The magnitude of the Earth departure and asteroid arrival manoeuvres are calculated based on the semi-major axis of the asteroids, with a different set of equations used for Apollo, Amor, and Aten class asteroids.

Apollo & Amor Asteroids (a > 1 AU) Apollo and Amor asteroids have semi-major axis 푎 > 1 AU. As such, their apoapsis 푄 is also beyond 1 AU. For these asteroids, the Shoemaker-Helin equations assume a Hohmann-like 180o transfer departing from the Earth’s circular orbit and arriving at the asteroid’s apoapsis (in agreement with the general rules for optimal circular-to-elliptical transfers).

Each of the two delta-V manoeuvres incorporates a combined plane change manoeuvres removing half the relative inclination at each end of the transfer.

The delta-V of the departure manoeuvre from Earth 푈퐿 is calculated as the difference between the circular velocity in Earth orbit 푈0 and the velocity of the departure hyperbola:

2 2 푈퐿 = √푈푡 + 푆 − 푈0 , (A.8) where 푆 = 푉퐸푠푐 is the escape velocity of the Earth’s surface (11.2 km/s), and 푈푡 = 푉∞ is the hyperbolic excess velocity, given by:

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2 2푄 푖 푈2 = 3 − − 2√ cos . (A.9) 푡 푄 + 1 푄 + 1 2

Eq. A.9 is derived from the relative encounter velocity of an inclined elliptical orbit with respect to the Earth [74]. The encounter is assumed to take place at the asteroid’s node crossing, when the Earth is also positioned along the line of nodes. The distance between the Earth and asteroid at this point is the minimum orbit intersect distance (MOID). The inclination of the transfer is assumed to be half the inclination of the asteroid’s orbit (hence the 푖/2 term in Eq. A.9).

The delta-V of the arrival manoeuvre 푈푅 is calculated from a combined plane change manoeuvre at the asteroid’s apoapsis:

푖 푈 = √푈2 + 푈2 − 2푈 푈 cos , (A.10) 푅 푐 푟 푟 푐 2 where 푈푐 and 푈푟 are the velocities of the transfer trajectory and asteroid at the rendezvous location. These equations differ for Amor and Apollo asteroids (defined in table A.2).

Aten Asteroids (a < 1 AU) Aten type asteroids have semi-major axis 푎 < 1 퐴푈. In their paper, Shoemaker & Helin [7] note that for Aten type asteroids (referred to as 1976 AA -type), minimum delta-V missions occur with rendezvous at periapsis, while shorter duration missions occur for rendezvous at apoapsis.

For these asteroids, the Shoemaker-Helin equations employ a different strategy in which the semi-major axis of the transfer is set to 1 AU, and is tangent at apoapsis with the asteroid’s orbit. Rather than follow this strategy, in this thesis, two sets of delta-Vs will be calculated for rendezvous at periapsis and apoapsis, with the minimum delta-V returned. This strategy is employed for both Atens and Atiras (not discussed in their original paper as there were none known at the time).

Table A.2 summarizes the Shoemaker-Helin equations for Amor, Apollo, and Aten asteroids.

338 Scott Dorrington – June 2019 Chapter 10: Appendices

Table A.2 Summary of Shoemaker-Helin equations for Amor, Apollo, and Aten asteroids.

ퟐ ퟐ 푼푳 = √푼풕 + 푺 − 푼ퟎ

푖 푈 = √푈2 + 푈2 − 2푈 푈 cos 푅 푐 푟 푟 푐 2

퐹 = 푈퐿 + 푈푅

∆푉푆퐻 = 30퐹 + 0.5 [푘푚/푠]

For Amor asteroids: For Apollo asteroids:

2 2푄 푖 2 2푄 푖 푈2 = 3 − − 2√ cos 푈2 = 3 − − 2√ cos 푡 푄 + 1 푄 + 1 2 푡 푄 + 1 푄 + 1 2

3 2 2 2 푖 3 2 2 2 푈2 = − − √ ∙ cos 푈2 = − − √ 푐 푄 푄 + 1 푄 푄 + 1 2 푐 푄 푄 + 1 푄 푄 + 1

3 1 2 푎 3 1 2 푎 푖 푈2 = − − √ (1 − 푒2) 푈2 = − − √ (1 − 푒2) ∙ cos 푟 푄 푎 푄 푄 푟 푄 푎 푄 푄 2

For Aten asteroids (apoapsis): For Aten asteroids (periapsis):

푖 2 2푞 푖 푈2 = 2 − 2√2푄 − 푄2 ∙ cos 푈2 = 3 − − 2√ cos 푡 2 푡 푞 + 1 푞 + 1 2

3 2 3 2 2 2 푈2 = − 1 − √2 − 푄 푈2 = − − √ 푐 푄 푄 푐 푞 푞 + 1 푞 푞 + 1

3 1 2 푎 푖 3 1 2 푎 푖 푈2 = − − √ (1 − 푒2) ∙ cos 푈2 = − − √ (1 − 푒2) ∙ cos 푟 푄 푎 푄 푄 2 푟 푞 푎 푞 푞 2

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APPENDIX B TWO PARAMETER H-G PHASE CURVE MODEL

The two-parameter (퐻-퐺) model [93] is an empirical model of the shape of an asteroid’s phase curve:

(B.1) 퐻(훼) = 퐻 − 2.5 log10[(1 − 퐺)훷1(훼) + 퐺훷2(훼)] .

The phase curve can be used to compute the apparent visual magnitude 푉 of an asteroid as viewed from Earth as a function of its position:

(B.2) 푉 = 퐻(훼) + 5 log10(푟∆) , where:

퐻 is the reduced magnitude at zero phase angle (훼 = 0);

G is the slope parameter;

푟 is the heliocentric (Sun-to-Asteroid) distance (AU);

∆ is the geocentric (Earth-to-Asteroid) distance (AU); and

훷1 , 훷2 are functions describing the single and multiple scattering of the asteroid’s surface.

The phase function 훷1(훼) can be calculated by:

(B.3) 훷1(훼) = 푊훷1푆 + (1 − 푊)훷1퐿 , where: 훼 푊 = 푒푥푝 [−90.56 tan2 ] , (B.4) 2 퐶 sin 훼 훷 = 1 − 1 , (B.5) 1푆 0.119 + 1.341 sin 훼 − 0.754 sin2 훼

훼 퐵1 훷 = 푒푥푝 [−퐴 (tan ) ] , (B.6) 1퐿 1 2

훼 퐵1 훷 = 푒푥푝 [−퐴 (tan ) ] , (B.7) 1퐿 1 2

퐴1 = 3.332, 퐵1 = 0.631, 퐶1 = 0.986 . (B.8)

The phase function 훷2(훼) can be calculated by:

340 Scott Dorrington – June 2019 Chapter 10: Appendices

(B.9) 훷2(훼) = 푊훷2푆 + (1 − 푊)훷1퐿 , where:

퐶2 sin 훼 (B.10) 훷 = 1 − , 2푆 0.119 + 1.341 sin 훼 − 0.754 sin2 훼

훼 퐵2 훷 = 푒푥푝 [−퐴 (tan ) ] , (B.11) 2퐿 2 2

퐴2 = 1.862, 퐵2 = 1.218, 퐶2 = 0.238 . (B.12)

Figure B.1 shows an example phase curve generated for asteroid (141424) 2002 CD, with a magnitude of 퐻 = 20.5, and an assumed slope parameter of 퐺 = 0.15.

Figure B.1 Phase curve for asteroid (141424) 2002 CD generated from the H-G model.

This phase curve can be used to predict the visibility of an asteroid from its ephemeris data. Figure B.2 shows the calculated geocentric Earth-to-Asteroid distance (range), phase angle, solar elongation angle, and predicted visual magnitude of asteroid (141424) 2002 CD over the period from 2018 to 2050. This data predicts that the asteroid will have observation opportunities between November 2029 and June 2042 (highlighted in red), when its visible magnitude is below a limiting value, set at 푉 = 22, and its solar elongation angle is greater than 10o.

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Figure B.2 Predicted observation conditions for asteroid (141424) 2002 CD over the period from 2018 to 2050, with visual observation opportunities shown in red.

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APPENDIX C COST ESTIMATE FOR MINING SPACECRAFT

Table C.1 shows the estimated production cost (theoretical first unit T1) of a 1000 kg spacecraft using the USCM8 model. This table is adapted from table 11-9 of Wertz et al. [188]. This cost includes production, assembly, and launch operations.

The subsystem mass and power allocations are based on the assumed breakdown for a planetary mission, shown in table C.2 (adapted from table 14-8 of Wertz et al. [188]).

Table C.1 USCM8 Spacecraft Bus Recurring T1 CERs in FY2020 Thousands of Dollars [188].

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Table C.2 Subsystem component breakdown (adapted from Wertz et al. [188]).

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APPENDIX D LOCATION-ROUTING PROBLEM ALGORITHMS

Algorithms D.1 and D.2 summarize the process of computing the elements of the total sellable mass at candidate orbital locations, and solving the location-routing problem for routes 1 and 2.

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