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Table of Contents
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
