Optimization of interplanetary trajectories with deep space maneuvers - Model development and application to a Uranus orbiter mission Thesis Report Sjoerd Molenaar August 2009 Optimization of interplanetary trajectories with deep space maneuvers - Model development and application to a Uranus orbiter mission Thesis Report Sjoerd Molenaar August 2009 Astrodynamics & Satellite Systems Faculty of Aerospace Engineering Delft University of Technology Kluyverweg 1, 2629 HS Delft The Netherlands Preface This report concludes the work that has been performed for my Master of Science thesis research at the Faculty of Aerospace Engineering of Delft University of Technology in The Netherlands. During the course of the study my interest in space engineering, in particular interplanetary trajectories, started to grow. When I was searching for a thesis subject, I looked at previous interplanetary missions. I found that Uranus has never been (or will be in the near future) the main target for a scientific mission. Uranus is considered a boring planet, but I found out that there are many interesting features at Uranus, its rings and its moons. Therefore I chose Uranus to be the target planet for this thesis research. At the end of the literature study I did prior to this thesis research, which was focused on the Uranus mission, the concept of deep space maneuvers came up. The addition of these maneuvers that are performed between planetary encounters is a recent topic in the field of optimizing interplanetary trajectories. I considered it a challenge to add these maneuvers to already existing software used at the Faculty. Readers who are primarily interested in the planet Uranus, previously flown outer planet missions and the layout of the spacecraft are referred to part I of this report. If the reader is more interested in the orbital mechanics required for interplanetary trajectories with deep space maneuvers, part II of the report is of most value. The software package and optimization method used are the subject of part III of the report. Finally, if the reader would like to view the results obtained in this thesis, together with the conclusions and recommendations, please refer to part IV. I would like to express my gratitude to my thesis supervisor Ron Noomen for his help during my thesis work. I appreciated the critical questions and suggestions during the weekly meetings. I would also like to thank all the students on the ninth floor for the wonderful time I have had doing my literature study and thesis research there. It was well worth it to travel from Amsterdam to Delft and back almost every day. Last, but certainly not least, I would like to thank my parents, my brother, my friends and family for supporting me during this period of hard work. Sjoerd Molenaar v Contents Preface vi Glossary xix Abstract xxii 1 Introduction 1 1.1 Document overview . .2 2 Mission definition 5 2.1 Assumptions for the Uranus orbiter mission . .5 2.2 Time and reference systems . .6 2.2.1 Julian epochs . .6 2.2.2 Heliocentric reference frame . .7 3 Uranus and its surroundings 11 3.1 The planet Uranus . 11 3.1.1 The giant planets . 12 3.1.2 Planet data . 12 3.1.3 Planetary interior . 13 3.1.4 Configuration . 13 3.1.5 Atmosphere and meteorology . 14 3.1.6 Magnetic field . 15 3.2 The ring system of Uranus . 17 3.3 The moons of Uranus . 17 3.4 Possible scientific objectives for a mission to Uranus . 20 vii viii Contents 3.5 Science orbit . 20 4 Previous missions to the outer planets 23 4.1 Previous mission to Uranus: Voyager 2 . 23 4.1.1 Mission overview . 23 4.1.2 Spacecraft overview . 24 4.1.3 Trajectory . 25 4.2 Jupiter mission: Galileo . 25 4.2.1 Mission overview . 25 4.2.2 Spacecraft overview . 26 4.2.3 Trajectory . 27 4.3 Saturn mission: Cassini-Huygens . 28 4.3.1 Mission overview . 28 4.3.2 Spacecraft overview . 29 4.3.3 Trajectory . 30 4.4 Pluto Mission: New Horizons . 31 4.4.1 Mission overview . 31 4.4.2 Spacecraft overview . 32 4.4.3 Trajectory . 32 4.5 Implications for a mission to Uranus . 33 5 Characteristics of the Uranus orbiter spacecraft 35 5.1 Chemical propulsion systems . 35 5.1.1 Propellant types . 36 5.1.2 Characteristics . 36 5.2 Generating electrical power . 37 5.3 Payload science instruments . 39 5.4 Preliminary layout of the spacecraft . 40 Contents ix 6 Principles of orbital mechanics 45 6.1 The many-body problem . 45 6.1.1 Newton's laws of motion and gravitational law . 45 6.1.2 A system of n bodies . 46 6.1.3 Motion of a body with respect to another body . 47 6.2 The two-body problem . 48 6.2.1 Conic sections . 48 6.2.2 Elliptical orbits . 49 6.2.3 Hyperbolic orbits . 51 6.2.4 Sphere of influence . 53 7 Orbit perturbations 57 7.1 Radially non-symmetric mass distribution . 57 7.2 Aerodynamic forces . 59 7.3 Attraction forces from other bodies . 59 7.4 Solar radiation pressure . 61 7.5 Electromagnetic forces . 62 7.6 Implications . 64 8 High thrust interplanetary trajectories 65 8.1 Two-dimensional interplanetary trajectories . 65 8.1.1 Example: Hohmann transfer to Uranus . 67 8.1.2 Three-dimensional interplanetary trajectories . 68 8.2 Lambert's problem . 68 8.3 Calculating the ∆V needed for launch and capture . 74 8.4 Unpowered gravity assists . 76 8.4.1 Principle of gravity assists . 76 8.4.2 Heliocentric energy increase . 77 8.5 Powered gravity assists . 80 x Contents 9 Deep Space Maneuvers 85 9.1 Basic principles of deep space maneuvers . 85 9.2 Trajectory model . 87 ¯ 9.2.1 Randomly generating V1L ................... 89 ¯ 9.2.2 Random rotation of V1in at an unpowered swingby . 90 9.2.3 Consequences on the size of the search space . 92 10 Propagation along a Kepler orbit 95 10.1 Introducing the Kepler problem . 95 10.1.1 Kepler's equation for elliptical orbits . 95 10.1.2 Kepler's equation for hyperbolic orbits . 97 10.1.3 Kepler's equation for parabolic orbits . 98 10.2 Position and velocity at a certain point in time . 100 10.2.1 Solution of Kepler's problem for elliptical orbits . 101 10.2.2 Solution of Kepler's problem for hyperbolic orbits . 101 10.2.3 Solution of Kepler's problem for parabolic orbits . 103 10.3 Universal variable formulation . 103 10.3.1 The Sundman transformation . 103 10.3.2 Kepler's equation using the universal variable formulation . 105 10.3.3 Solving the universal variable form of Kepler's problem . 108 11 Optimization 113 11.1 Overview of optimization methods . 113 11.2 The simple GA . 114 11.2.1 The working principle of a simple GA . 114 11.2.2 Structure of the simple GA . 115 11.3 Some advanced genetic operators . 120 11.3.1 Elitism . 120 11.3.2 Immigration . 120 11.3.3 Constraint handling . 120 11.4 Random number generator . 121 Contents xi 12 GALOMUSIT 123 12.1 History and development of GALOMUSIT . 123 12.2 Original version of GALOMUSIT . 124 12.2.1 Program structure . 124 12.2.2 Generation of individuals . 125 12.2.3 Creating new generations . 126 12.2.4 Multi-objective optimization . 127 12.2.5 Convergence . 127 12.3 Adaptations made to GALOMUSIT for this thesis . 127 12.3.1 Adaptations prior to the thesis research . 128 12.3.2 Setting up the program . 129 12.3.3 Generating individuals for DSM trajectories . 129 12.3.4 New subroutines added to GALOMUSIT . 130 12.3.5 New output files . 132 13 Benchmark runs for GALOMUSIT 133 13.1 Verification of results produced by previous users of GALOMUSIT 133 13.1.1 Pluto flyby with gravity assist at Jupiter . 133 13.1.2 Trajectory to Neptune with multiple swingbys . 134 13.2 Verification of GAMOLUSIT for an Earth-DSM-Mars transfer . 135 13.2.1 Problem description and results from literature . 135 13.2.2 Results for the EdM test case using GALOMUSIT . 137 13.2.3 Additional tests for the Earth-DSM-Mars test case . 143 13.3 Verification of GALOMUSIT for an Earth-Venus-DSM-Mars transfer146 13.3.1 Search space and results from literature . 146 13.3.2 Results for the EVdM test case using GALOMUSIT . 147 13.3.3 Additional tests for the EVdM test case . 152 xii Contents 14 Results for the Uranus orbiter mission without DSMs 157 14.1 Direct transfer from Earth to Uranus . 157 14.2 Results for trajectories from Earth to Uranus with swingbys . 159 14.2.1 Mission specific settings in GALOMUSIT . 160 14.2.2 Initial GA settings in GALOMUSIT . 161 14.2.3 Results for 10% elitism, 5% immigration and 0.1% mutation 162 14.2.4 Additional tests . 164 14.3 Sequences when a Venus swingby is allowed after an Earth swingby 166 14.3.1 Results for 15% elitism, 15% immigration . 167 15 Results for trajectories with DSMs 169 15.1 Earth-Uranus transfer without swingbys . 169 15.1.1 Results of an EdU transfer with a complete search space . 169 15.1.2 Results of an EdU transfer with a constrained search space 172 15.2 Trajectories to Uranus with one swingby . 179 15.2.1 No DSM on trajectories to Uranus with one swingby . 180 15.2.2 DSM on the first leg of trajectories to Uranus with one swingby181 15.2.3 DSM on the second leg of trajectories to Uranus with one swingby .