Dynamics and Mission Design Near Libration Points, Volume III: Advanced Methods for Collinear Points

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Table of Contents

Preface v

Chapter 1 Quasi-periodic Solutions Near the Equilateral Points of the Earth-Moon System 1

1.1 Introduction 1

1.1.1 Expansion of the Equations 2

1.2 Idea of the Resolution Method 4

1.3 The Algebraic Manipulator 4

1.3.1 Storing Fourier Series 5

1.3.2 Basic Subroutines 6

1.3.3 Making Operations 7

1.4 The Newton Method 9

1.4.1 Some Remarks about the Jacobian Matrix 9

1.4.2 Adding Terms to the Newton Process 10

1.4.3 High Level Routines 10

1.5 The Program 13

1.5.1 Test of the Program 15

1.6 Results of the Algebraic Manipulator 15

1.7 Numerical Refinement 21

1.7.1 Final Results 22

1.8 The Neighbourhood of the Computed Nearly Quasi-periodic Solution 25

1.9 Problems and Extensions 31

Chapter 2 Global Description of the Orbits Near the L1 Point of the Earth-Sun System in the RTBP 33

2.1 Introduction 33

2.2 The Equations of Motion 33

2.3 Formal Series Solutions 36

2.3.1 A Program to Obtain the Formal Series Solutions 39

2.3.2 Results and Formal Tests 41

2.3.3 Numerical Tests 47

2.4 On the Convergence of the Series 51

2.5 Towards a Description of the Neighbourhood of L1 53

2.6 Discussion on the Use of Lissajous Orbits 54

2.7 Effective Reduction to the Central Manifold 56

2.7.1 The Method 56

2.7.2 The Program and the Results 60

2.7.3 Numerical Simulations and Tests 63

2.8 Conclusions 66

Chapter 3 Quasi-periodic Halo Orbits 69

3.1 Numerical Refinement 69

3.1.1 Computation of the Initial Values 70

3.1.2 The Differential Matrix DF(Q) 72

3.1.3 Computing the Poincare Map and its Differential .72

3.2 Main Program and Basic Routines 77

3.2.1 Numerical Results 78

3.3 The Equations of Motion for the Simulations of the Control 83

3.4 The Effect of Errors 83

3.5 When a Control is Applied 84

3.6 Magnitudes Related to the Control 85

3.7 Description of the Program 86

3.8 Numerical Results 88

3.8.1 Discussion on the Effect of the Different Errors and Magnitudes 91

3.8.2 Conclusions 91

Chapter 4 Transfer From the Earth to a Halo Orbit 93

4.1 Introduction 93

4.2 Local Approximation of the Stable Manifold 94

4.2.1 Main Program and Basic Routines 96

4.2.2 Output and Sample of Results 97

4.3 Globalization of the Manifold 100

4.3.1 Main Program and Basic Routines 101

4.3.2 Output and Sample of Results 101

4.4 Selecting Passages Near the Earth 104

4.4.1 Main Program and Basic Routines 105

4.4.2 Output and Sample of Results 106

4.5 Ranges in the Manifold Suited for the Transfer 108

4.5.1 Main Programs and Basic Routines 108

4.5.2 Output and Sample of Results 109

4.6 Characteristics of the Orbits Near the Earth 114

4.6.1 Main Program and Basic Routines 114

4.6.2 Output and Sample of Results 114

4.6.3 Some Numerical Explorations 117

4.7 Conclusions 126

Appendix A The JPL Model 127

A.1 Implementation and Use 127

A.2 Basic Subroutines 130

Appendix B Reference Systems and Equations of Motion 133

B.1 The Restricted Three Body Problem 133

B.2 The Real Problem 137

B.2.1 Reference Systems 137

B.2.2 Equations of Motion 139

Appendix C The Model Equations Near the Equilateral Points in the Earth-Moon System 147

C.1 Introduction 147

C.2 Systems of Reference 147

C.3 The Lagrangian 149

C.4 The Hamiltonian and the Related Expansions 150

C.5 Some Useful Expansions 151

C.6 Fourier Analysis: The Relevant Frequencies and the Related Coefficients 154

C.7 Simplified Normalized Equations 164

C.8 Numerical Tests 170

Appendix D Transfer Between Halo Orbits of the RTBP 177

D.1 Introduction 177

D.2 Strategy of the Method 177

Bibliography 187

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