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Multi-Body Trajectory Design Strategies Based on Periapsis Poincaré Maps

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Multi-Body Trajectory Design Strategies Based on Periapsis Poincaré Maps MULTI-BODY TRAJECTORY DESIGN STRATEGIES BASED ON PERIAPSIS POINCARÉ MAPS A Dissertation Submitted to the Faculty of Purdue University by Diane Elizabeth Craig Davis In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy August 2011 Purdue University West Lafayette, Indiana ii To my husband and children iii ACKNOWLEDGMENTS I would like to thank my advisor, Professor Kathleen Howell, for her support and guidance. She has been an invaluable source of knowledge and ideas throughout my studies at Purdue, and I have truly enjoyed our collaborations. She is an inspiration to me. I appreciate the insight and support from my committee members, Professor James Longuski, Professor Martin Corless, and Professor Daniel DeLaurentis. I would like to thank the members of my research group, past and present, for their friendship and collaboration, including Geoff Wawrzyniak, Chris Patterson, Lindsay Millard, Dan Grebow, Marty Ozimek, Lucia Irrgang, Masaki Kakoi, Raoul Rausch, Matt Vavrina, Todd Brown, Amanda Haapala, Cody Short, Mar Vaquero, Tom Pavlak, Wayne Schlei, Aurelie Heritier, Amanda Knutson, and Jeff Stuart. I thank my parents, David and Jeanne Craig, for their encouragement and love throughout my academic career. They have cheered me on through many years of studies. I am grateful for the love and encouragement of my husband, Jonathan. His never-ending patience and friendship have been a constant source of support. Finally, I owe thanks to the organizations that have provided the funding opportunities that have supported me through my studies, including the Clare Booth Luce Foundation, Zonta International, and Purdue University and the School of Aeronautics and Astronautics through the Graduate Assistance in Areas of National Need and the Purdue Forever Fellowships. iv TABLE OF CONTENTS Page LIST OF TABLES ......................................................................................................................... vii LIST OF FIGURES ...................................................................................................................... viii ABSTRACT ................................................................................................................................. xvii 1. INTRODUCTION .................................................................................................................... 1 1.1. Problem Definition ................................................................................................................ 1 1.2. Previous Work ....................................................................................................................... 3 1.2.1. Understanding the Tidally-Influenced Environment ...................................................... 3 1.2.2. Designing Trajectories in the Tidally-Influenced Environment ..................................... 5 1.2.3. Adding the Influence of an Additional Smaller Primary ................................................ 7 1.3. Scope of the Present Work .................................................................................................... 8 2. BACKGROUND .................................................................................................................... 11 2.1. The Circular Restricted Three-Body Problem .................................................................... 11 2.1.1. Characteristic Quantities and Nondimensionalization ................................................. 13 2.1.2. The CR3BP Equations of Motion ................................................................................. 14 2.1.3. Frame Transformations ................................................................................................ 16 2.1.4. The Jacobi and Tisserand Constants ............................................................................. 17 2.1.5. Equilibrium Solutions ................................................................................................... 19 2.1.6. Hill‘s Restricted 3-Body Problem ................................................................................ 20 2.1.7. Zero Velocity Curves ................................................................................................... 22 2.1.8. Planar Periodic Motion near the Libration Points ........................................................ 23 2.1.9. The State Transition Matrix and Differential Corrections ............................................ 25 2.1.10. The Monodromy Matrix and Stable and Unstable Manifolds .................................... 27 v Page 2.2. Poincaré Maps ..................................................................................................................... 29 2.2.1. Selection of the Surface of Section............................................................................... 31 2.3. Models and Constants ......................................................................................................... 34 2.3.1. The Saturnian System, the Cassini Mission, and Representative Constants ................ 34 2.3.2. Numerical Integration ................................................................................................... 36 3. UNDERSTANDING THE TIDALLY-INFLUENCED ENVIRONMENT: INDIVIDUAL TRAJECTORIES .................................................................................................. 37 3.1. Tidal Acceleration by Quadrants ........................................................................................ 37 3.1.1. Coriolis Acceleration and the Stability of Retrograde Orbits ....................................... 44 3.2. The Tidal Kick Function ..................................................................................................... 45 3.3. Orbital Element Changes Over Time .................................................................................. 50 4. UNDERSTANDING THE TIDALLY-INFLUENCED ENVIRONMENT: PERIAPSIS POINCARÉ MAPS ................................................................................................... 54 4.1. Short-Term Trajectory Behavior ......................................................................................... 54 4.1.1. Interpretation in Terms of Quadrants ........................................................................... 56 4.1.2. Periapse Passages Prior to Escape ................................................................................ 57 4.1.3. Manifold Trajectories Associated with L1 and L2 Lyapunov Orbits ............................ 58 4.1.4. Initial Condition Maps for Changing Parameters ......................................................... 60 4.1.5. Change in rp Over One Revolution............................................................................... 68 4.1.6. Retrograde Behavior ..................................................................................................... 70 4.1.7. Out-of Plane Behavior .................................................................................................. 72 4.1.8. Relationship to Weak Stability Boundary Theory ........................................................ 75 4.2. Long-Term Trajectory Behavior ......................................................................................... 76 4.2.1. Periapse Profiles ........................................................................................................... 77 4.2.2. Long-Term Periapsis Poincaré Maps ........................................................................... 83 4.2.3. Periapsis Poincaré Maps for Specific Values of rp0 ...................................................... 93 4.2.4. Out-of-plane Long-Term Trajectory Evolution ............................................................ 94 vi Page 5. DESIGNING TRAJECTORIES IN A TIDALLY-INFLUENCED ENVIRONMENT ......... 97 5.1. Mission Design for Cassini End-of-Life: Point Solutions .................................................. 97 5.1.1. Saturn Impact Example ................................................................................................ 97 5.1.2. Long-Term Orbit Beyond Phoebe ................................................................................ 99 5.1.3. Cassini Escape Trajectories ........................................................................................ 101 5.2. Long-Term Maps: Identifying Particular Trajectories ...................................................... 102 5.3. Transitioning CR3BP trajectories to the Ephemeris Model .............................................. 107 5.4. Mission Design for Cassini End-of-Life Using Periapsis Poincaré Maps ........................ 110 5.4.1. Example 1: Impact and Escape Trajectories ............................................................... 111 5.4.2. Example 2: Long-term Orbit Beyond Phoebe ............................................................ 112 5.5. Mission Design for Titan Capture ..................................................................................... 116 5.6. Mission Design for Earth-Moon Transfer ......................................................................... 121 6. INCLUDING THE INFLUENCE OF AN ADDITIONAL SMALLER PRIMARY ........... 125 6.1. The Flyby Kick Function .................................................................................................. 126 6.2. Inbound versus Outbound Flybys in the CR3BP .............................................................. 137 6.3. Flyby Periapsis Maps .......................................................................................................
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