Foundations of Space Dynamics

An introduction to orbital mechanics and spacecraft attitude dynamics

Foundations of Space Dynamics offers an authoritative text that combines a comprehensive review of both orbital mechanics and dynamics. The author a noted expert in the field covers up-to-date topics including: orbital perturbations, Lambert's transfer, formation flying, and gravity-gradient stabilization. The text provides an introduction to space dynamics in its entirety, including important analytical derivations and practical space flight examples.

Written in an accessible and concise style, Foundations of Space Dynamics highlights analytical development and rigor, rather than numerical solutions via ready-made computer codes. To enhance learning, the book is filled with helpful tables, figures, exercises, and solved examples.

This important book:

• Covers space dynamics with a systematic and comprehensive approach
• Is designed to be a practical text filled with real-world examples
• Contains information on the most current applications
• Includes up-to-date topics from orbital perturbations to gravity- gradient stabilization
• Offers a deep understanding of space dynamics often lacking in other textbooks

Written for undergraduate and graduate students and professionals in aerospace engineering, Foundations of Space Dynamics offers an introduction to the most current information on orbital mechanics and dynamics.

Preface xiii

1 Introduction 1

1.1 Space Flight 1

1.1.1 Atmosphere as Perturbing Environment 1

1.1.2 Gravity as the Governing Force 4

1.1.3 Topics in Space Dynamics 5

1.2 Reference Frames and Time Scales 5

1.2.1 Sidereal Frame 5

1.2.2 Celestial Frame 8

1.2.3 Synodic Frame 8

1.2.4 Julian Date 8

1.3 Classification of Space Missions 10

Exercises 10

References 11

2 Dynamics 13

2.1 Notation and Basics 13

2.2 Plane Kinematics 14

2.3 Newton’s Laws 16

2.4 Particle Dynamics 17

2.5 The n-Body Problem 20

2.6 Dynamics of a Body 24

2.7 Gravity Field of a Body 27

2.7.1 Legendre Polynomials 29

2.7.2 Spherical Coordinates 31

2.7.3 Axisymmetric Body 34

2.7.4 Spherical Body with Radially Symmetric Mass Distribution 37

Exercises 37

References 40

3 Keplerian Motion 41

3.1 The Two-Body Problem 41

3.2 Orbital Angular Momentum 43

3.3 Orbital Energy Integral 45

3.4 Orbital Eccentricity 46

3.5 Orbit Equation 49

3.5.1 Elliptic Orbit 53

3.5.2 Parabolic Orbit 56

3.5.3 Hyperbolic Orbit 56

3.5.4 Rectilinear Motion 58

3.6 Orbital Velocity and Flight Path Angle 60

3.7 Perifocal Frame and Lagrange’s Coefficients 63

Exercises 65

4 Time in Orbit 69

4.1 Position and Velocity in an Elliptic Orbit 70

4.2 Solution to Kepler’s Equation 75

4.2.1 Newton’s Method 76

4.2.2 Solution by Bessel Functions 78

4.3 Position and Velocity in a Hyperbolic Orbit 80

4.4 Position and Velocity in a Parabolic Orbit 84

4.5 Universal Variable for Keplerian Motion 86

Exercises 88

References 89

5 Orbital Plane 91

5.1 Rotation Matrix 91

5.2 Euler Axis and Principal Angle 94

5.3 Elementary Rotations and Euler Angles 97

5.4 Euler-Angle Representation of the Orbital Plane 101

5.4.1 Celestial Reference Frame 103

5.4.2 Local-Horizon Frame 104

5.4.3 Classical Euler Angles 106

5.5 Planet-Fixed Coordinate System 111

Exercises 114

6 Orbital Manoeuvres 117

6.1 Single-Impulse Orbital Manoeuvres 119

6.2 Multi-impulse Orbital Transfer 123

6.2.1 Hohmann Transfer 124

6.2.2 Rendezvous in Circular Orbit 127

6.2.3 Outer Bi-elliptic Transfer 130

6.3 Continuous Thrust Manoeuvres 133

6.3.1 Planar Manoeuvres 134

6.3.2 Constant Radial Acceleration from Circular Orbit 135

6.3.3 Constant Circumferential Acceleration from Circular Orbit 136

6.3.4 Constant Tangential Acceleration from Circular Orbit 139

Exercises 141

References 143

7 Relative Motion in Orbit 145

7.1 Hill-Clohessy-Wiltshire Equations 148

7.2 Linear State-Space Model 151

7.3 Impulsive Manoeuvres About a Circular Orbit 153

7.3.1 Orbital Rendezvous 153

7.4 Keplerian Relative Motion 155

Exercises 158

8 Lambert’s Problem 161

8.1 Two-Point Orbital Transfer 161

8.1.1 Transfer Triangle and Terminal Velocity Vectors 162

8.2 Elliptic Transfer 164

8.2.1 Locus of the Vacant Focii 165

8.2.2 Minimum-Energy and Minimum-Eccentricity Transfers 166

8.3 Lambert’s Theorem 168

8.3.1 Time in Elliptic Transfer 169

8.3.2 Time in Hyperbolic Transfer 173

8.3.3 Time in Parabolic Transfer 175

8.4 Solution to Lambert’s Problem 177

8.4.1 Parameter of Transfer Orbit 178

8.4.2 Stumpff Function Method 179

8.4.3 Hypergeometric Function Method 185

Exercises 188

References 190

9 Orbital Perturbations 191

9.1 Perturbing Acceleration 191

9.2 Osculating Orbit 192

9.3 Variation of Parameters 194

9.3.1 Lagrange Brackets 197

9.4 Lagrange Planetary Equations 199

9.5 Gauss Variational Model 209

9.6 Variation of Vectors 214

9.7 Mean Orbital Perturbation 219

9.8 Orbital Perturbation Due to Oblateness 220

9.8.1 Sun-Synchronous Orbits 225

9.8.2 Molniya Orbits 226

9.9 Effects of Atmospheric Drag 227

9.9.1 Life of a Satellite in a Low Circular Orbit 228

9.9.2 Effect on Orbital Angular Momentum 229

9.9.3 Effect on Orbital Eccentricity and Periapsis 231

9.10 Third-Body Perturbation 235

9.10.1 Lunar and Solar Perturbations on an Earth Satellite 238

9.10.2 Sphere of Influence and Conic Patching 243

9.11 Numerical Methods for Perturbed Keplerian Motion 246

9.11.1 Cowell’s Method 246

9.11.2 Encke’s Method 246

Exercises 250

References 254

10 Three-Body Problem 255

10.1 Equations of Motion 256

10.2 Particular Solutions by Lagrange 257

Equilibrium Solutions in a Rotating Frame 257

Conic Section Solutions 259

10.3 Circular Restricted Three-Body Problem 261

10.3.1 Equations of Motion in the Inertial Frame 261

10.4 Non-dimensional Equations in the Synodic Frame 263

10.5 Lagrangian Points and Stability 267

10.5.1 Stability Analysis 268

10.6 Orbital Energy and Jacobi’s Integral 270

10.6.1 Zero-Relative-Speed Contours 272

10.6.2 Tisserand’s Criterion 275

10.7 Canonical Formulation 276

10.8 Special Three-Body Trajectories 278

10.8.1 Perturbed Orbits About a Primary 279

10.8.2 Free-Return Trajectories 279

Exercises 282

Reference 283

11 Attitude Dynamics 285

11.1 Euler’s Equations of Attitude Kinetics 286

11.2 Attitude Kinematics 288

11.3 Rotational Kinetic Energy 290

11.4 Principal Axes 292

11.5 Torque-Free Rotation of Spacecraft 294

11.5.1 Stability of Rotational States 295

11.6 Precession and Nutation 298

11.7 Semi-Rigid Spacecraft 299

11.7.1 Dual-Spin Stability 301

11.8 Solution to Torque-Free Euler’s Equations 303

11.8.1 Axisymmetric Spacecraft 304

11.8.2 Jacobian Elliptic Functions 307

11.8.3 Runge-Kutta Solution 308

11.9 Gravity-Gradient Stabilization 312

Exercises 321

12 Attitude Manoeuvres 323

12.1 Impulsive Manoeuvres with Attitude Thrusters 323

12.1.1 Single-Axis Rotation 324

12.1.2 Rigid Axisymmetric Spin-Stabilized Spacecraft 326

12.1.3 Spin-Stabilized Asymmetric Spacecraft 330

12.2 Attitude Manoeuvres with Rotors 330

12.2.1 Reaction Wheel 332

12.2.2 Control-Moment Gyro 333

12.2.3 Variable-Speed Control-Moment Gyro 334

Exercises 335

References 337

A Numerical Solution of Ordinary Differential Equations 339

A.1 Fixed-Step Runge-Kutta Algorithms 339

A.2 Variable-Step Runge-Kutta Algorithms 340

A.3 Runge-Kutta-Nyström Algorithms 342

References 343

B Jacobian Elliptic Functions 345

Reference 346

Index 347

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