Space Trajectories

Auteur

Max Cerf, Ph.D. is an emeritus expert in mission analysis and optimization at ArianeGroup, where he has been involved in space mission analysis and developing and deploying the company's Ariane launchers for 30 years. He is also serving as an Associate Professor at Sorbonne-Université, where his research focuses on control, optimization, and applied mathematics.

Texte du rabat

Authoritative reference covering essential concepts of orbital mechanics and explaining how they relate to advanced space trajectory applications Space Trajectories is the first book to offer a comprehensive exploration of orbital mechanics and trajectory optimization in a single volume. Beginning with a review of essential concepts, the book progresses to advanced space applications, highlighting methods used in today's space missions. The contents are organized into three parts. The first part delves into free orbital motion, covering topics such as Keplerian motion, perturbed motion, the three-body problem, orbit determination, and collision risks in orbit. The second part focuses on controlled orbital motion, discussing impulsive transfer, orbital rendezvous, thrust level optimization, low-thrust transfer, and space debris cleaning. The third part examines ascent and reentry, including launch into orbit, launcher staging, analytical solutions in flat Earth, interplanetary missions, and atmospheric reentry. Each chapter is written in a modular way, featuring conclusion summaries, key points, and suggestions for further investigation. Examples are included with detailed solutions methods that readers can apply to solve their own trajectory problems. Written by an expert of the topic who has performed guidance of Ariane launchers for 30 years, Space Trajectories includes information on:

• Keplerian motion, motion time law, universal formulation, equinoctial parameters, and Lagrange coefficients
• Osculating orbit, Gauss equations, gravitational and third body perturbations, Lissajous and Halo orbits, and invariant manifolds
• Astrometry measurements, Kalman filtering, orbit uncertainties, and collision probability
• Transfer in one, two, or three impulses, minimum-energy transfer, Lambert's problem, high- and low-thrust transfer, and interplanetary path
• Launch and reentry trajectories, propulsion systems, optimized thrust profiles, and launcher staging Space Trajectories is an essential reference for students and researchers aiming to quickly understand the main issues in astrodynamics and the way to design trajectories, as well as space engineers seeking to consolidate their knowledge in the field of optimization and optimal control applied to aerospace and space missions.

Contenu

About the Author xv

Foreword xvii

Acknowledgments xix

Introduction xxi

Part I Free Orbital Motion 1

1 Two-Body Problem 3

1.1 Introduction 3

1.2 Keplerian Motion 3

1.2.1 Dynamic Model 4

1.2.2 Prime Integrals 5

1.2.3 Orbit Shape 7

1.3 Motion Time Law 12

1.3.1 Elliptical Orbit 12

1.3.2 Hyperbolic Orbit 14

1.3.3 Parabolic Orbit 15

1.3.4 Lagrange Coefficients 16

1.3.5 Universal Variable 18

1.4 Orbital Parameters 23

1.4.1 Classical Orbital Parameters 23

1.4.2 Relation to Position and Velocity 23

1.4.3 Equinoctial Orbital Parameters 25

1.4.4 Earth Orbits 25

1.5 Conclusion 31

1.5.1 The Key Points 31

1.5.2 To Go Further 31

2 Perturbed Motion 33

2.1 Introduction 33

2.2 Unperturbed Motion 34

2.2.1 Keplerian Model 34

2.2.2 Orbital Parameters 34

2.2.3 Useful Frames 36

2.3 Perturbed Motion 36

2.3.1 Osculating Orbit 37

2.3.2 Derivation Formula 37

2.3.3 Gauss Equations 38

2.3.4 Lagrange Equations 41

2.3.5 Equinoctial Parameters 43

2.3.6 Integration Methods 44

2.4 Gravitational Perturbations 48

2.4.1 Gravitational Potential 49

2.4.2 Spherical Body 49

2.4.3 Nonspherical Body 51

2.4.4 First Zonal Term 54

2.5 Other Perturbations 58

2.5.1 Third-Body Attraction 58

2.5.2 Atmospheric Friction 64

2.5.3 Radiation Pressure 65

2.6 Conclusion 65

2.6.1 The Key Points 65

2.6.2 To Go Further 66

3 Three-Body Problem 67

3.1 Introduction 67

3.2 Circular-Restricted Three-Body Problem 68

3.2.1 Motion Equations 68

3.2.2 Accessible Region 69

3.2.3 Lagrange Points 71

3.3 Periodic Orbits 74

3.3.1 Linearized Solution 74

3.3.2 Periodic Solution 80

3.3.3 Halo Orbits 83

3.4 Transfers 86

3.4.1 Linearization in the Orbit Vicinity 86

3.4.2 Invariant Manifolds 88

3.4.3 Transfer Strategies 89

3.5 Conclusion 93

3.5.1 The Key Points 93

3.5.2 To Go Further 93

4 Orbit Determination 95

4.1 Introduction 95

4.2 Measurements 96

4.2.1 Observation System 96

4.2.2 Measurements 100

4.2.3 Directions of Stars 102

4.3 Preliminary Orbit Estimation 104

4.3.1 Position Measurements 104

4.3.2 Direction Measurements 107

4.4 Continuous Orbit Estimation 110

4.4.1 Least Squares 111

4.4.2 Differential Correction 113

4.4.3 Kalman Filtering 116

4.5 Conclusion 119

4.5.1 The Key Points 119

4.5.2 To Go Further 119

5 Collision Risks 121

5.1 Introduction 121

5.2 Orbit Uncertainties 122

5.2.1 Orbital Motion 122

5.2.2 Ellipsoid of Uncertainty 124

5.2.3 Gaussian Model 126

5.3 Conjunction 129

5.3.1 Numerical Simulation 129

5.3.2 Orbit-to-Orbit Distance 129

5.3.3 Trajectory-to-Orbit Distance 133

5.3.4 Combined Covariance 134

5.4 Risk of Collision 136

5.4.1 Short Conjunction 137

5.4.2 Probability of Collision 138

5.4.3 Analytical Formula 140

5.4.4 Maximum Probability 141

5.4.5 Long Conjunction 143

5.5 Conclusion 145

5.5.1 The Key Points 145

5.5.2 To Go Further 145

Part II Controlled Orbital Motion 147

6 Impulsive Transfer 149

6.1 Introduction 149

6.2 Orbit Target 150

6.2.1 Problem Formulation 150

6.2.2 Transfer in One Impulse 151

6.2.3 Transfer in Two or Three Impulses 155

6.3 Point Target 160

6.3.1 Problem Formulation 160

6.3.2 Minimum-Energy Transfer 161

6.3.3 Minimum-Eccentricity Transfer 162

6.3.4 Noncollinear Transfer 163

6.3.5 Collinear Transfer 168

6.4 Point and Time Target 171

6.4.1 Lambert's Problem 171

6.4.2 Lambert's Theorem 171

6.4.3 Transfer Time Equation 175

6.4.4 Universal Variable 181

6.4.5 Solution Methods 183

6.5 Conclusion 184

6.5.1 The Key Points 184

6.5.2 To Go Further 184

7 Orbital Rendezvous 187

7.1 Introduction 187

7.2 Phasing and Transfer 188

7.2.1 Orbital Model 188

7.2.2 Phasing 189

7.2.3 Transfer 190

7.2.4 Visibility 197

7.3 Target in Circular Orbit 198

7.3.1 Hill-Clohessy-Wiltshire Equations 198

7.3.2 Free Motion 200

7.3.3 Maneuvers 204

7.3.4 Approach Scenario 206

7.4 Control Laws 206

7.4.1 Optimum Control 206

7.4.2 Specific Controls 210

7.5 Conclusion 214

7.5.1 The Key Points 214

7.5.2 To Go Further 214

8 Optimal Thrust Level 215

8.1 Introduction 215

8.2 Problem Formulation 216

8.2.1 Optimal Control Problem 216

8.2.2 Conditions for Optimality 217

8.2.3 Property of the Velocity Costate 218

8.3 Analytical Solution 221

8.3.1 Direction of Thrust 221

8.3.2 Costate Vector 225

8.3.3 Injection Point and Direction 226

8.3.4 Reduced Problem 228

8.3.5 Performance Estimate 229

8.4 Application 230

8.4.1 Optimized Thrust Profile 230

8.4.2 Fixed Thrust Level 232

8.5 Conclusion 236

8.5.1 The Key Points 236

8.5.2 To Go Further 237

9 Low-Thrust Transfer 239

9.1 Introduction 239

9.2 Problem Formulation 240

9.2.1 Dynamics 240

9.2.2 Optimal Control Problem 242

9.2.3 Local Control Laws 245

9.2.4 Edelbaum's Solution 249

9.3 Transfers to the Geostationary Orbit 251

9.3.1 Dynamic Model 251

9.3.2 Optimal Control Problem 253

9.3.3 Solution Method 255

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