A Study of Eccentric Orbit Circularization using Low-Thrust Propulsion by Brenton John Du®y B.S. in Aerospace Engineering, Dec. 2005, North Carolina State University A Thesis submitted to the Faculty of The School of Engineering and Applied Science of The George Washington University in partial satisfaction of the requirements for the degree of Master of Science May 18, 2008 Thesis directed by Dr. David F. Chichka Assistant Professor of Engineering and Applied Science The School of Engineering and Applied Science of The George Washington University certi¯es that Brenton John Du®y has passed the Final Examination for the degree of Master of Science as of May 9, 2008. This is the ¯nal and approved form of the thesis. A Study of Eccentric Orbit Circularization using Low-Thrust Propulsion Brenton John Du®y Thesis Review Committee: Dr. David F. Chichka Assistant Professor of Engineering and Applied Science Dr. James D. Lee Professor of Engineering and Applied Science Dr. Helmut Haberzettl Associate Professor of Physics °c Copyright by Brenton John Du®y 2008 Acknowledgements I would like to express my thanks to Dr. David F. Chichka for all of his support and insight during this study. Our discussions were invaluable toward the progression of the study and I am sincerely grateful for all of his help. I would also like to thank Dr. Chichka, Dr. James D. Lee and Dr. Helmut Haberzettl for serving on my thesis committee and providing their expertise in reviewing my work. iv Abstract This study considers the optimal control problem of circularizing an orbit via a con- tinuous and variable low-thrust maneuver. In addition, constraints are applied to avoid disrupting scienti¯c experiments being conducted at apoapsis. The dynamics are modeled using Lagrange's planetary equations for the classic orbit elements with control variables de¯ned by the speci¯c thrust components. The objective is to max- imize the raise in periapsis while minimizing the fuel consumption. Several variations of the problem are considered including an unconstrained maneuver, a terminal point constraint maneuver, and a multi-point constraint maneuver. The results show a trade-o® between adding energy by increasing the semi-major axis and circularizing by decreasing the eccentricity. In comparing the low-thrust performance to equivalent impulsive maneuvers, the results demonstrate that the low-thrust systems can provide improved performance over the impulsive maneuvers. In addition, the optimization of multi-orbit maneuvers has a strong dependence on the number of orbits whereby the optimal thrust pro¯le changes between the orbits. This can be mitigated with some loss in performance by adding interior apoapsis constraints to each orbit. v Table of Contents Acknowledgements iv Abstract v Table of Contents viii List of Figures xi List of Tables xii List of Symbols xiii 1 Introduction 1 1.1 Magnetospheric Multiscale Mission ................... 1 1.2 Orbital Mechanics ............................. 2 1.2.1 Classic Orbit Elements ...................... 2 1.2.2 Circularization .......................... 5 1.3 Low-Thrust Propulsion .......................... 7 1.4 Notation .................................. 8 2 Optimal Control Theory for Low-Thrust Propulsion 10 2.1 Problem Formulation ........................... 10 2.2 Necessary Conditions for Optimality . 16 2.2.1 First-Order Conditions ...................... 20 2.2.2 Second-Order Conditions ..................... 23 2.3 Control Law ................................ 24 vi 2.4 Numerical Solution to the Terminally Constrained TPBVP . 26 2.5 Interior Point Constraints ........................ 31 2.5.1 First-Order Necessary Conditions for a Single Orbit . 32 2.5.2 First-Order Necessary Conditions for Multiple Orbits . 38 2.5.3 Second-Order Necessary Conditions . 40 2.5.4 Numerical Solution to the Interior Constrained MPBVP . 40 2.6 Performance Measures .......................... 44 3 Algorithm Validation 45 3.1 Maximize Semi-Major Axis ........................ 46 3.2 Minimize Eccentricity ........................... 49 4 Optimizing the Circularization Maneuver 54 4.1 Unconstrained Apoapsis ......................... 54 4.1.1 Single Orbit ............................ 55 4.1.2 Multiple Orbits .......................... 59 4.2 Terminally Constrained Apoapsis .................... 65 4.2.1 Single Orbit ............................ 66 4.2.2 Multiple Orbits .......................... 70 4.3 Interior and Terminally Constrained Apoapsis . 76 4.3.1 Single Orbit ............................ 76 4.3.2 Multiple Orbits .......................... 79 5 Conclusions 86 5.1 Low-Thrust Maneuver vs. Impulsive Maneuver . 87 5.2 Dependence on Number of Orbits .................... 88 References 91 vii Appendix A Co-State Di®erential Equations 94 Appendix B State and Co-State Perturbation Equations 96 Appendix C TPBVP Second Variation M-Matrix 109 Appendix D MPBVP Second Variation M-Matrix 111 viii List of Figures 1.2.1 In-Plane Orbit Elements ........................ 3 1.2.2 Classic Orbit Elements ......................... 4 2.1.1 Speci¯c Thrust Components ...................... 11 3.1.1 Maximize Semi-Major Axis: Optimization Parameters . 47 3.1.2 Maximize Semi-Major Axis: Orbit Elements . 47 3.1.3 Maximize Semi-Major Axis: Speci¯c Thrust Pro¯le . 48 3.1.4 Maximize Semi-Major Axis: Thrust Direction . 49 3.2.1 Minimize Eccentricity: Optimization Parameters . 50 3.2.2 Minimize Eccentricity: Orbit Elements . 51 3.2.3 Minimize Eccentricity: Apsidal Radii . 51 3.2.4 Minimize Eccentricity: Speci¯c Thrust Pro¯le . 52 3.2.5 Minimize Eccentricity: Thrust Direction . 53 4.1.1 UCA Single Orbit Maneuver: Orbit Elements . 55 4.1.2 UCA Single Orbit Maneuver: Apsidal Radii . 56 4.1.3 UCA Single Orbit Maneuver: Thrust Pro¯le . 57 4.1.4 UCA Single Orbit Maneuver: Thrust Direction . 57 4.1.5 UCA Single Orbit Maneuver: Argument of Periapsis . 58 4.1.6 UCA Multi-Orbit Maneuver: Semi-Major Axis . 59 4.1.7 UCA Multi-Orbit Maneuver: Eccentricity . 60 4.1.8 UCA Multi-Orbit Maneuver: Periapsis Radius . 60 4.1.9 UCA Multi-Orbit Maneuver: Apoapsis Radius . 61 4.1.10 UCA Multi-Orbit Maneuver: Thrust Pro¯les . 61 4.1.11 UCA Multi-Orbit Maneuver: Total Thrust . 62 ix 4.1.12 UCA Multi-Orbit Maneuver: Thrust Direction . 62 4.1.13 UCA Multi-Orbit Maneuver: Argument of Periapsis . 63 4.1.14 UCA Multi-Orbit Maneuver: Trajectories . 64 4.2.1 TCA Single Orbit Maneuver: Orbit Elements . 67 4.2.2 TCA Single Orbit Maneuver: Apsidal Radii . 67 4.2.3 TCA Single Orbit Maneuver: Thrust Pro¯le . 68 4.2.4 TCA Single Orbit Maneuver: Thrust Direction . 69 4.2.5 TCA Single Orbit Maneuver: Argument of Periapsis . 69 4.2.6 TCA Multi-Orbit Maneuver: Semi-Major Axis . 70 4.2.7 TCA Multi-Orbit Maneuver: Eccentricity . 71 4.2.8 TCA Multi-Orbit Maneuver: Periapsis Radius . 71 4.2.9 TCA Multi-Orbit Maneuver: Apoapsis Radius . 72 4.2.10 TCA Multi-Orbit Maneuver: Thrust Pro¯les . 73 4.2.11 TCA Multi-Orbit Maneuver: Total Thrust . 73 4.2.12 TCA Multi-Orbit Maneuver: Thrust Direction . 74 4.2.13 TCA Multi-Orbit Maneuver: Argument of Periapsis . 74 4.2.14 TCA Multi-Orbit Maneuver: Trajectories . 75 4.3.1 ITCA Single Orbit Maneuver: Orbit Elements . 77 4.3.2 ITCA Single Orbit Maneuver: Apsidal Radii . 77 4.3.3 ITCA Single Orbit Maneuver: Thrust Pro¯le . 78 4.3.4 ITCA Single Orbit Maneuver: Thrust Direction . 78 4.3.5 ITCA Single Orbit Maneuver: Argument of Periapsis . 79 4.3.6 ITCA Multi-Orbit Maneuver: Semi-Major Axis . 80 4.3.7 ITCA Multi-Orbit Maneuver: Eccentricity . 80 4.3.8 ITCA Multi-Orbit Maneuver: Periapsis Radius . 81 4.3.9 ITCA Multi-Orbit Maneuver: Apoapsis Radius . 81 x 4.3.10 ITCA Multi-Orbit Maneuver: Thrust Pro¯les . 82 4.3.11 ITCA Multi-Orbit Maneuver: Total Thrust . 83 4.3.12 ITCA Multi-Orbit Maneuver: Thrust Direction . 83 4.3.13 ITCA Multi-Orbit Maneuver: Argument of Periapsis . 84 4.3.14 ITCA Multi-Orbit Maneuver: Trajectories . 84 xi List of Tables 5.1 Variation in the Orbit Elements .................... 86 5.2 Performance Results ........................... 87 5.3 Normalized Performance Results .................... 89 xii List of Symbols (_ ) Total Time Derivative d=d( ), @=@( ) Total Derivative, Partial Derivative a Semi-Major Axis e Eccentricity 2 g0 Standard Sea Level Acceleration of Gravity (= 9:81m=s ) h Angular Momentum Vector i Orbit Inclination m, M Mass of Satellite, Mass of Central Body p n Mean Motion (= ¹=a3) q Number of Orbits for Multi-Orbit Maneuvers r Position Magnitude ra Apoapsis Radius rp Periapsis Radius rs Desired Scienti¯c Apoapsis Radius t; t0, tf , tn Time; Initial, Final, and Interior Times u Vector of Control Variables v Velocity Magnitude x; x0 Vector of State Variables; Prescribed Initial Conditions z Error Vector A, B, C, D Second Variation Sub-Matrices AU Canonical Distance Unit E Speci¯c Orbit Energy E; E0, Ef , En Eccentric Anomaly; Initial, Final, and Interior Eccentric Anomalies xiii F (x; u; t) Vector of System Dynamic Equations G Gravitational Constant H(x; u; ¸; t) Hamiltonian Function Isp Speci¯c Impulse J, J~ Performance Index, Augmented Performance Index L(x; u; t) Fuel Consumption Penalty Function M Error Matrix MR Mass Ratio of Final Mass over Initial Mass R Radial Speci¯c Thrust S Circumferential Speci¯c Thrust T Normal Speci¯c Thrust ± First Variation Operator ±2 Second Variation Operator ´n Vector of Interior Constraint Lagrange Multipliers for t = tn ¸ Co-States (Dynamic Equation Lagrange Multipliers) ¹ Gravitational Parameter (= G(M + m)) º Vector of Terminal Constraint Lagrange Multipliers º True Anomaly (only for Chapter 1) »n Vector of Interior Constraints for t = tn Á Objective Function à Vector of Terminal Constraints ! Argument of Periapsis ¢v Required Change in Velocity © Transition Matrix ­ Á + ºT à ­ Longitude of the Ascending Node (only for Chapter 1) xiv Chapter 1 - Introduction As the science of astrodynamics and celestial physics progresses, so too must the technology and engineering for designing our spacecraft of exploration.