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On the Stability of Quasi-Satellite Orbits in the Elliptic Restricted Three-Body Problem Dissertação Para a Obtenção De Grau

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On the Stability of Quasi-Satellite Orbits in the Elliptic Restricted Three-Body Problem Dissertação Para a Obtenção De Grau On the Stability of Quasi-Satellite Orbits in the Elliptic Restricted Three-Body Problem Application to the Mars-Phobos System Francisco da Silva Pais Cabral Dissertação para a obtenção de Grau de Mestre em Engenharia Aeroespacial Júri Presidente: Prof. Doutor Fernando José Parracho Lau Orientador: Prof. Doutor Paulo Jorge Soares Gil Vogal: Prof. Doutor João Manuel Gonçalves de Sousa Oliveira Dezembro de 2011 Acknowledgments The author wishes to acknowledge His thesis coordinator, Prof. Paulo Gil, for, one, presenting this thesis opportunity in the author’s field of interest and, second, for the essential orientation provided to the author that made this very same thesis possible, His professors, both in IST and TU Delft, for the acquired knowledge and transmitted passion in the most diverse fields, His university, IST, for providing the means to pursue the author’s academic and professional goals, His colleagues for their support and availability to discuss each others’ works, His friends for keeping the author sane. iii Abstract In this thesis, the design of quasi-satellites orbits in the elliptic restricted three-body problem is ad- dressed from a preliminary space mission design perspective. The stability of these orbits is studied by an analytical and a numerical approach and findings are applied in the study of the Mars-Phobos system. In the analytical approach, perturbation theories are applied to the solution of the unperturbed Hill’s equations, obtaining the first-order approximate averaged differential equations on the osculating elements. The stability of quasi-satellite orbits is analyzed from these equations and withdrawn conclu- sions are confirmed numerically. We also use the fast Lyapunov indicator, a chaos detection technique, to analyze the stability of the system. The study of fast Lyapunov indicator maps for scenarios of par- ticular interest provides a better understanding on the characteristics of quasi-satellite orbits and their stability. Both approaches are proven to be powerful tools for space mission design. Keywords: Quasi-Satellite Orbits, Elliptic Restricted Three-Body Problem, Stability, Perturbation The- ory, Fast Lyapunov Indicator, Mars-Phobos System. v Resumo Nesta tese, o planeamento de quase-orbitas´ no contexto do problema restrito dos tresˆ corpos el´ıptico e´ estudado de uma perspectiva do planeamento preliminar de missoes˜ espaciais. A estabilidade destas orbitas´ e´ estudada atraves´ de uma aboradagem anal´ıtica e de uma abordagem numerica´ e as descober- tas sao˜ aplicadas ao caso do sistema Marte-Fobos. Na abordagem anal´ıtica, teorias de perturbac¸ao˜ sao˜ aplicadas a` soluc¸ao˜ das equac¸oes˜ de Hill nao˜ perturbadas, obtendo-se as equac¸oes˜ diferenciais medias´ aproximadas de primeira ordem nos elementos osculadores. A estabilidade de quase-orbitas´ e´ analisada atraves´ destas equac¸oes˜ e as conclusoes˜ retiradas sao˜ confirmadas numericamente. Us- amos tambem´ o indicador rapido´ de Lyapunov, uma tecnica´ de detecc¸ao˜ de caos, para analisar a estabilidade do sistema. O estudo dos mapas do indicador rapido´ de Lyapunov para cenarios´ de par- ticular interesse vai-nos providenciar uma melhor compreensao˜ das caracter´ısticas de quase-orbitas´ e da sua estabilidade. Ambas as abordagens demonstram ser ferramentas poderosas no planeamento de missoes˜ espaciais. Palavras Chave: Quase-Orbitas,´ Problema Restrito dos Tresˆ Corpos El´ıptico, Estabilidade, Teoria de Perturbac¸ao,˜ Indicador Rapido´ de Lyapunov, Sistema Marte-Fobos. vii Contents Acknowledgments........................................... iii Abstract.................................................v Resumo................................................. vii List of Tables.............................................. xi List of Figures............................................. xiii List of Acronyms............................................ xv List of Symbols............................................. xix 1. Introduction 1 1.1. Motivation.............................................1 1.2. Problem Statement........................................2 1.3. Quasi-Satellite Orbits......................................3 1.4. Stability..............................................4 1.5. Bibliographic Review.......................................4 1.5.1. Orbits In The Three-Body Problem...........................5 1.5.2. Chaos Indicators.....................................6 1.6. Thesis Overview.........................................7 2. Dynamics 9 2.1. Classical Mechanics.......................................9 2.1.1. Lagrangian Mechanics.................................. 10 2.1.2. Hamiltonian Mechanics................................. 11 2.1.3. The Variational Equations................................ 12 2.2. The N-Body Problem....................................... 13 2.3. The Restricted Three-Body Problem.............................. 14 2.4. Modifications of the Restricted Three-Body Problem..................... 17 2.4.1. The Spatial Restricted Three-Body Problem...................... 17 2.4.2. The Elliptic Restricted Three-Body Problem...................... 17 2.4.3. Change Of Origin.................................... 19 2.5. Equations of Motion....................................... 20 2.5.1. Hamilton’s Equations of Motion............................. 20 2.5.2. An Invariant Relation................................... 20 ix 2.6. Chaos Indicators......................................... 22 2.6.1. Lyapunov Characteristic Exponents.......................... 23 2.6.2. Fast Lyapunov Indicator................................. 24 3. QSO Solutions and Stability 27 3.1. Linearization of the Equations of Motion............................ 27 3.2. Unperturbed Hill’s Equations.................................. 29 3.2.1. Homogeneous Solution................................. 30 3.2.2. General Solution..................................... 32 3.2.3. Stability Considerations................................. 33 3.2.4. Constants of Integration................................. 35 3.2.5. Constant Transformation................................. 35 3.3. Influence of the Second Primary................................ 37 3.3.1. Region of Stability.................................... 37 3.3.2. Approximate Solutions in the Osculating Elements.................. 41 3.4. Application to the Mars-Phobos System............................ 47 4. Numerical Exploration of QSOs 49 4.1. Numerical Integration...................................... 49 4.2. Implementation Validation.................................... 49 4.2.1. Validation Test...................................... 50 4.2.2. Implementation Language................................ 50 4.3. Computational Parameters & Methodology.......................... 53 4.3.1. Integration Method and Time-Step........................... 53 4.3.2. Orbit Escape....................................... 56 4.4. FLI Maps............................................. 56 4.4.1. Planar QSOs....................................... 57 4.4.2. Three-Dimensional QSOs................................ 65 4.4.3. Velocity Maps....................................... 73 5. Conclusions 77 A. Programming Code 79 Bibliography 92 x List of Tables 1.1. Mars & Phobos Parameters...................................3 3.1. Amplitudes for Values of the Mean Motion Ratio........................ 48 4.1. Validation Test Results...................................... 51 4.2. MatLab & C Performance Comparison I............................ 51 4.3. MatLab & C Performance Comparison II............................ 52 4.4. Reference Values......................................... 53 4.5. Parameter Computation With Chosen Time-Step....................... 55 4.6. Orbits - Mean Orbital Motion Ratio............................... 60 xi List of Figures 1.1. Hill Sphere and Region of Influence of Phobos........................2 3.1. Analysis of the behavior of xnp(f) and ynp(f) ......................... 33 3.2. Analysis of the behavior of xnp(f) and ynp(f) with C3 = −e C2 ............... 33 3.3. Parametric plot of the stable solutions............................. 34 3.4. Osculating Elements....................................... 37 3.5. Orbital Resonance........................................ 48 4.1. FLI Behavior........................................... 52 4.2. Performance Comparison.................................... 54 4.3. Position and Velocity Error Analysis.............................. 55 4.4. FLI Map - x0 Vs. y_0 ........................................ 58 4.5. FLI Map - y0 Vs. x_ 0 ........................................ 59 4.6. Planar QSOs - Tangential Entry................................. 59 4.7. 2:1 Mean Motion Orbit...................................... 60 4.8. FLI Map - y0 Vs. y_0 ........................................ 62 4.9. FLI Map & QSO - y0 Vs. y_0 ................................... 63 4.10.FLI Map - Initial True Anomaly.................................. 64 4.11.FLI Map - Vertical Velocity.................................... 66 4.12.3D QSO.............................................. 67 4.13.FLI Map - z0 Vs. y_0 ........................................ 69 4.14.FLI Map - z0 Vs. z_0 ........................................ 71 4.15.3D QSOs - Amplitude Ratio................................... 72 4.16.3D QSOs - Large Amplitude Orbits............................... 72 4.17.FLI Map - Velocities I....................................... 74 4.18.FLI Map - Velocities II...................................... 75 xiii List of Acronyms 3BP Three-Body
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