ISBN: 3-540-67280-X
TITLE: Satellite Orbits
AUTHOR: Montenbruck, Oliver; Gill, Eberhard
TOC:

1 Around the World in a Hundred Minutes 1
1.1 A Portfolio of Satellite Orbits 1
1.1.1 Low-Earth Orbits 2
1.1.2 Orbits of Remote Sensing Satellites 3
1.1.3 Geostationary Orbits 4
1.1.4 Highly Elliptical Orbits 6
1.1.5 Constellations 7
1.2 Navigating in Space 8
1.2.1 Tracking Systems 8
1.2.2 A Matter of Effort 10
2 Introductory Astrodynamics 15
2.1 General Properties of the Two-Body Problem 16
2.1.1 Plane Motion and the Law of Areas 16
2.1.2 The Form of the Orbit 17
2.1.3 The Energy Integral 19
2.2 Prediction of Unperturbed Satellite Orbits 22
2.2.1 Kepler's Equation and the Time Dependence of Motion 22
2.2.2 Solving Kepler's Equation 23
2.2.3 The Orbit in Space 24
2.2.4 Orbital Elements from Position and Velocity 28
2.2.5 Non-Singular Elements 29
2.3 Ground-Based Satellite Observations 32
2.3.1 Satellite Ground Tracks 32
2.3.2 Satellite Motion in the Local Tangent Coordinate System 36
2.4 Preliminary Orbit Determination 39
2.4.1 Orbit Determination from Two Position Vectors 40
2.4.2 Orbit Determination from Three Sets of Angles 43
Exercises 47
3 Force Model 53
3.1 Introduction 53
3.2 Geopotential 56
3.2.1 Expansion in Spherical Harmonics 56
3.2.2 Some Special Geopotential Coefficients 59
3.2.3 Gravity Models 61
3.2.4 Recursions 66
3.2.5 Acceleration 68
3.3 Sun and Moon 69
3.3.1 Perturbing Acceleration 69
3.3.2 Low-Precision Solar and Lunar Coordinates 70
3.3.3 Chebyshev Approximation 73
3.3.4 JPL Ephemerides 75
3.4 Solar Radiation Pressure 77
3.4.1 Eclipse Conditions 80
3.4.2 Shadow Function 81
3.5 Atmospheric Drag 83
3.5.1 The Upper Atmosphere 86
3.5.2 The Harris­Priester Density Model 89
3.5.3 The Jacchia 1971 Density Model 91
3.5.4 A Comparison of Upper Atmosphere Density Models 98
3.5.5 Prediction of Solar and Geomagnetic Indices 102
3.6 Thrust Forces 104
3.7 Precision Modeling 107
3.7.1 Earth Radiation Pressure 107
3.7.2 Earth Tides 108
3.7.3 Relativistic Effects 110
3.7.4 Empirical Forces 112
Exercises 113
4 Numerical Integration 117
4.1 Runge­Kutta Methods 118
4.1.1 Introduction 118
4.1.2 General Runge­Kutta Formulas 120
4.1.3 Stepsize Control 121
4.1.4 Runge­Kutta­Nyström Methods 123
4.1.5 Continuous Methods 127
4.1.6 Comparison of Runge­Kutta Methods 129
4.2 MultistepMethods 132
4.2.1 Introduction 132
4.2.2 Adams­Bashforth Methods 134
4.2.3 Adams­Moulton and Predictor­Corrector Methods 136
4.2.4 Interpolation 140
4.2.5 Variable Order and Stepsize Methods 141
4.2.6 Stoermer and Cowell Methods 143
4.2.7 Gauss­Jackson or Second Sum Methods 145
4.2.8 Comparison of Multistep Methods 146
4.3 Extrapolation Methods 147
4.3.1 The Mid-Point Rule 147
4.3.2 Extrapolation 148
4.3.3 Comparison of Extrapolation Methods 150
4.4 Comparison 151
Exercises 154
5 Time and Reference Systems 157
5.1 Time 157
5.1.1 Ephemeris Time 160
5.1.2 Atomic Time 161
5.1.3 Relativistic Time Scales 162
5.1.4 Sidereal Time and Universal Time 165
5.2 Celestial and Terrestrial Reference Systems 169
5.3 Precession and Nutation 172
5.3.1 Lunisolar Torques and the Motion of the Earth's Rotation Axis 172
5.3.2 Coordinate Changes due to Precession 174
5.3.3 Nutation 178
5.4 Earth Rotation and Polar Motion 181
5.4.1 Rotation About the Celestial Ephemeris Pole 181
5.4.2 Free Eulerian Precession 182
5.4.3 Observation and Extrapolation of Polar Motion 183
5.4.4 Transformation to the International Reference Pole 185
5.5 Geodetic Datums 185
Exercises 190
6 Satellite Tracking and Observation Models 193
6.1 Tracking Systems 193
6.1.1 Radar Tracking 193
6.1.2 Laser Tracking 202
6.1.3 The Global Positioning System 203
6.2 Tracking Data Models 208
6.2.1 Transmitter and Receiver Motion 208
6.2.2 Angle Measurements 209
6.2.3 Range Measurements 213
6.2.4 Doppler Measurements 215
6.2.5 GPS Measurements 217
6.3 Media Corrections 219
6.3.1 Interaction of Radiation and Atmosphere 219
6.3.2 Tropospheric Refraction 221
6.3.3 Ionospheric Refraction 225
Exercises 229
7 Linearization 233
7.1 Two-Body State Transition Matrix 235
7.1.1 Orbital-Elements Transition Matrix 235
7.1.2 Keplerian-to-Cartesian Partial Derivatives 236
7.1.3 Cartesian-to-Keplerian Partial Derivatives 238
7.1.4 The State Transition Matrix and Its Inverse 239
7.2 Variational Equations 240
7.2.1 The Differential Equation of the State Transition Matrix 240
7.2.2 The Differential Equation of the Sensitivity Matrix 241
7.2.3 Form and Solution of the Variational Equations 241
7.2.4 The Inverse of the State Transition Matrix 243
7.3 Partial Derivatives of the Acceleration 244
7.3.1 Geopotential 244
7.3.2 Point-Mass Perturbations 247
7.3.3 Solar Radiation Pressure 248
7.3.4 Drag 248
7.3.5 Thrust 249
7.4 Partials of the Measurements with Respect to the State Vector 250
7.5 Partials with Respect to Measurement Model Parameters 252
7.6 Difference Quotient Approximations 253
Exercises 255
8 Orbit Determination and Parameter Estimation 257
8.1 Weighted Least-Squares Estimation 258
8.1.1 Linearization and Normal Equations 260
8.1.2 Weighting 262
8.1.3 Statistical Interpretation 263
8.1.4 Consider Parameters 265
8.1.5 Estimation with A Priori Information 266
8.2 Numerical Solution of Least-Squares Problems 268
8.2.1 QR Factorization 268
8.2.2 Householder Transformations 270
8.2.3 Givens Rotations 272
8.2.4 Singular Value Decomposition 274
8.3 Kalman Filtering 276
8.3.1 Recursive Formulation of Least-Squares Estimation 277
8.3.2 Sequential Estimation 280
8.3.3 Extended Kalman Filter 282
8.3.4 Factorization Methods 283
8.3.5 Process Noise 284
8.4 Comparison of Batch and Sequential Estimation 286
Exercises 289
9 Applications 293
9.1 Orbit Determination Error Analysis 293
9.1.1 A Linearized Orbit Model 294
9.1.2 Consider Covariance Analysis 297
9.1.3 The GEODA Program 299
9.1.4 Case Studies 300
9.2 Real-Time Orbit Determination 303
9.2.1 Model and Filter Design 303
9.2.2 The RTOD Program 306
9.2.3 Case Studies 307
9.3 Relay Satellite Orbit Determination 312
9.3.1 Mathematical Models 312
9.3.2 The TDRSOD Program 313
9.3.3 Case Study 315
Appendix A 319
A.1 Calendrical Calculations 319
A.1.1 Modified Julian Date from the Calendar Date 321
A.1.2 Calendar Date from the Modified Julian Date 322
A.2 GPS Orbit Models 324
A.2.1 Almanac Model 325
A.2.2 Broadcast Ephemeris Model 326
Appendix B 329
B.1 Internet Resources 329
B.2 The Enclosed CD-ROM 330
B.2.1 Contents 330
B.2.2 System Requirements 331
B.2.3 Executing the Programs 331
B.2.4 Compilation and Linking 332
B.2.5 Index of Library Functions 335
List of Symbols 339
References 347
Index 361
END
