DISPOSAL DYNAMICS FROM THE VICINITY OF NEAR RECTILINEAR HALO ORBITS IN THE EARTH-MOON-SUN SYSTEM A Thesis Submitted to the Faculty of Purdue University by Kenza Katiane Boudad In Partial Fulfillment of the Requirements for the Degree of Master of Sciences December 2018 Purdue University West Lafayette, Indiana ii iii THE PURDUE UNIVERSITY GRADUATE SCHOOL STATEMENT OF THESIS APPROVAL Dr. Kathleen C. Howell, Chair School of Aeronautics and Astronautics Dr. Carolin Frueh School of Aeronautics and Astronautics Dr. David A. Spencer School of Aeronautics and Astronautics Approved by: Dr. Weinong Wayne Chen Head of the Graduate Program iv Pour Mamie et Jedda v ACKNOWLEDGMENTS Maman, Papa, merci de votre amour et de votre support. Vous m'encouragez inconditionnellement dans tous mes projets, m^emeceux qui incluent partir vivre au fin fond du Midwest. I would like to express my deepest gratitude to my advisor, Professor Kathleen C. Howell. I have enjoyed taking your classes, and I enjoy working in your research group even more. Thank you for this amazing opportunity. Your insights and your support made this research possible. I am looking forward to continuing studying under your guidance. I would also like to express my gratitude and appreciation to Dr. Diane C. Davis, from a.i. solutions for her guidance during the research and writing process of our conference paper. It was quite the ride, and I have learned so much in these few months. Thank you for your patience! I am incredibly grateful for the support of my friends, especially Mayank Aggarwal, Camille Briquet, Emily Spreen and Danae de Vries. Whether you live next street from me or half the world away, your friendship helped me through the challenging parts of this degree. I would like to thank my committee members, Professor Carolin Frueh and Pro- fessor David A. Spencer. Thank you for taking some time from your busy schedule to review my document and provide feedback during the defense. For providing funding for my studies, I would like to thank the School of Aeronautics and Astronautics. Sup- port provided by the NASA Cooperative Agreement 80NSSC18M0122 is also greatly appreciated. Finally, I would like to thank the members, past and current, of the Multi-Body Dynamics Research group, for their suggestions and assistance. vi TABLE OF CONTENTS Page LIST OF TABLES :::::::::::::::::::::::::::::::::: viii LIST OF FIGURES ::::::::::::::::::::::::::::::::: ix ABSTRACT ::::::::::::::::::::::::::::::::::::: xiii 1 INTRODUCTION :::::::::::::::::::::::::::::::: 1 1.1 Problem Definition ::::::::::::::::::::::::::::: 1 1.2 Previous Contributions ::::::::::::::::::::::::::: 3 1.3 Document Overview :::::::::::::::::::::::::::: 3 2 SYSTEM MODELS :::::::::::::::::::::::::::::::: 7 2.1 Multi-Body Regimes :::::::::::::::::::::::::::: 7 2.2 Circular Restricted Three-Body Problem ::::::::::::::::: 9 2.2.1 Equations of Motion in the Inertial Frame :::::::::::: 9 2.2.2 Equations of Motion in the Rotating Frame ::::::::::: 11 2.2.3 Integral of the Motion ::::::::::::::::::::::: 15 2.2.4 Equilibrium Solutions :::::::::::::::::::::::: 15 2.2.5 Symmetry Properties :::::::::::::::::::::::: 17 2.2.6 Zero Velocity Surfaces ::::::::::::::::::::::: 18 2.3 Bicircular Restricted Four-Body Problem | Planet-Moon ::::::: 22 2.3.1 Equations of Motion - Sun Out Of Plane ::::::::::::: 22 2.3.2 Equations of Motion - Sun In Plane :::::::::::::::: 27 2.3.3 Epoch Definition :::::::::::::::::::::::::: 28 2.3.4 Hamiltonian in Planet-Moon Frame :::::::::::::::: 30 2.3.5 Instantaneous Equilibrium Solutions ::::::::::::::: 31 2.3.6 Instantaneous Zero Velocity Surfaces ::::::::::::::: 35 2.3.7 Hamiltonian Plots ::::::::::::::::::::::::: 37 2.4 Bicircular Restricted Four-Body Problem | Sun-Barycenter :::::: 40 2.4.1 Transformation between frames :::::::::::::::::: 40 2.4.2 Hamiltonian in Sun-B1 frame ::::::::::::::::::: 43 2.4.3 Instantaneous Equilibrium Solutions ::::::::::::::: 45 2.4.4 Instantaneous Zero Velocity Surfaces ::::::::::::::: 47 2.4.5 Hamiltonian Plots ::::::::::::::::::::::::: 48 3 DYNAMICAL SYSTEMS THEORY :::::::::::::::::::::: 51 3.1 State-Transition Matrix :::::::::::::::::::::::::: 51 3.2 Differential Corrections ::::::::::::::::::::::::::: 55 vii Page 3.2.1 Single Shooting Method :::::::::::::::::::::: 57 3.2.2 Multiple Shooting Method ::::::::::::::::::::: 59 3.3 Continuation Schemes ::::::::::::::::::::::::::: 62 3.3.1 Natural Parameter Continuation ::::::::::::::::: 63 3.3.2 NPC Example: BCR4BP Periodic Orbits ::::::::::::: 63 3.3.3 Pseudo-Arclength Continuation :::::::::::::::::: 67 1 2 3 3.3.4 PALC Example: BCR4BP E3, E3 and E3 :::::::::::: 69 3.4 Finite-Time Lyapunov Exponent ::::::::::::::::::::: 72 3.5 Lagrangian Coherent Structures :::::::::::::::::::::: 76 3.5.1 Numerical Computation :::::::::::::::::::::: 77 3.5.2 LCS Example: Flow near the Moon in the BCR4BP ::::::: 81 4 MISSION APPLICATION: DISPOSAL FROM A NEAR RECTILINEAR HALO ORBIT TO HELIOCENTRIC SPACE :::::::::::::::::::::::::: 87 4.1 Mission Definition :::::::::::::::::::::::::::::: 87 4.2 Near Rectinilinar Halo Orbits | CR3BP ::::::::::::::::: 89 4.3 Near Rectinilinar Halo Orbits | BCR4BP :::::::::::::::: 92 4.3.1 Homogeneous 9-revolution Stacking Method ::::::::::: 92 4.3.2 Non-Homogeneous Stacking Method ::::::::::::::: 95 4.4 Mission Success Criteria :::::::::::::::::::::::::: 96 4.4.1 NRHO Departure Criterion :::::::::::::::::::: 98 4.4.2 Escape Criterion ::::::::::::::::::::::::: 100 4.5 Tidal Effects from the Sun :::::::::::::::::::::::: 102 4.6 Results and Analysis ::::::::::::::::::::::::::: 105 4.6.1 Direct Escapes :::::::::::::::::::::::::: 107 4.6.2 Indirect Escapes :::::::::::::::::::::::::: 113 4.6.3 Captures :::::::::::::::::::::::::::::: 119 4.7 Long-Term Behavior of the Disposal Trajectories :::::::::::: 126 4.7.1 Natural Propagation ::::::::::::::::::::::: 126 4.7.2 Additional Maneuver in Heliocentric Space ::::::::::: 128 4.8 Ephemeris Model Validation ::::::::::::::::::::::: 134 4.9 Summary of Results ::::::::::::::::::::::::::: 135 5 SUMMARY AND FUTURE WORK ::::::::::::::::::::: 139 5.1 Summary ::::::::::::::::::::::::::::::::: 139 5.2 Future Work :::::::::::::::::::::::::::::::: 141 REFERENCES ::::::::::::::::::::::::::::::::::: 142 Appendix A: Stability ::::::::::::::::::::::::::::::: 147 Linear Definition :::::::::::::::::::::::::::::::: 148 Stability Index ::::::::::::::::::::::::::::::::: 151 viii LIST OF TABLES Table Page 2.1 Characteristic quantities of the Earth-Moon system ::::::::::::: 12 2.2 Collinear equilibrium points of the CR3BP :::::::::::::::::: 16 2.3 Nondimensional parameters of the BCR4BP ::::::::::::::::: 23 2.4 Minimum and maximum z amplitudes of the instantaneous equilibrium solutions over one synodic month in the Earth-Moon-Sun BRC4BP for Ω = 45◦ ::::::::::::::::::::::::::::::::::::: 35 2.5 Characteristic quantities of the Sun-B1 system :::::::::::::::: 40 4.1 Characteristic of the Earth-Moon 9:2 synodic resonant NRHO ::::::: 92 4.2 Characteristic of the Earth-Moon-Sun 9:2 2 Tsyn-perodic orbit for θS0 = 0 : 94 4.3 Relationship between the disposal outcome and the influencing factors : 138 ix LIST OF FIGURES Figure Page 2.1 N-body problem :::::::::::::::::::::::::::::::: 8 2.2 Relationship between the inertial and the rotating frames :::::::::: 11 2.3 Nondimensional 3-body problem in the rotating frame ::::::::::: 13 2.4 General location of the five CR3BP equilibrium points ::::::::::: 18 2.5 Cross-sections of the zero velocity surfaces and forbidden regions (in grey) at z = 0 in the Earth-Moon CR3BP ::::::::::::::::::::: 20 2.6 3D view of zero velocity surfaces in the Earth-Moon CR3BP :::::::: 21 2.7 Sample Sun orbit as seen in the Planet-Moon frame, for i = 15◦ and Ω = 120◦23 2.8 Nondimensional BCR4BP in the P1 − P2 rotating frame :::::::::: 24 2.9 Moon configurations in the BCR4BP Earth-Moon-Sun model, viewed in the Earth-Moon rotating frame :::::::::::::::::::::::: 29 2.10 Earth-Moon-Sun set of equilibria E1 in the BCR4BP SOOP model :::: 33 2.11 Earth-Moon-Sun set of equilibria E2 in the BCR4BP SOOP model :::: 33 2.12 Instantaneous Earth-Moon-Sun E3 in the BCR4BP SOOP Model over one ◦ ◦ synodic month for Ω = 90 . At θS = −74 , multiple solutions for E3(θS) exist and are indicated by the orange markers. The grey line on the right plot indicates the Sun location at that instant. :::::::::::::::: 34 2.13 Instantaneous Earth-Moon-Sun E4 (left) and E5 (right) in the BCR4BP SOOP Model over one synodic month for Ω = 90◦ :::::::::::::: 35 2.14 Earth-Moon-Sun Instantaneous ZVC over a trajectory with Ω = 0◦ :::: 37 2.15 Earth-Moon-Sun Hamiltonian plot with Ω = 0◦ ::::::::::::::: 38 2.16 Earth-Moon-Sun Hamiltonian plot (bottom) and instantaneous ZVC (top) along a propagated trajectory, with Ω = 0◦ :::::::::::::::::: 39 2.17 Definition of the Sun angle θS and the Planet-Moon angle θM ::::::: 43 2.18 Comparison of the Hamiltonian and Jacobi constant in the Earth-Moon- Sun BCR4BP :::::::::::::::::::::::::::::::::: 45 x Figure Page 2.19 Instantaneous equilibrium solutions in the Sun-Earth-Moon BCR4BP sys- tem with Ω = 45◦ :::::::::::::::::::::::::::::::: 47 2.20 Instantaneous zero velocity curves in the Sun-B1 frame ::::::::::: 48 ◦ 2.21 Sun-Earth-Moon Sun-B1 Hamiltonian plot with Ω = 0 ::::::::::: 49 ◦ 2.22 Sun-Earth-Moon Sun-B1 Hamiltonian plot with Ω = 0 ::::::::::: 50 3.1 Definition of the segments in the multiple shooting algorithm :::::::: 60 3.2 Tsyn-periodic family evolving over the parameter mS ::::::::::::: 65 3.3 Comparison between (a) the natural parameter continuation and (b) the pseudo-arclength