Theory of orbit determination

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
Part I Problem Statement and Requirements......Page 13
1.1 Orbits and observations......Page 15
1.2 The minimum principle......Page 17
1.3 Two interpretations......Page 18
1.4 Classification of the problem......Page 19
Tracking......Page 20
Catalogs......Page 22
Planetary systems......Page 24
1.5 How to read this book......Page 25
2.1 The equation of motion......Page 27
Existence and uniqueness of the solutions......Page 28
Lipschitz-continuity of the flow\n......Page 29
2.3 The variational equation......Page 30
Variational equation for second-order equations......Page 31
2.4 Lyapounov exponents......Page 32
2.5 Model problem dynamics......Page 33
3.1 Continuous random variables......Page 35
Jointly distributed random variables......Page 36
Independence, marginal and conditional probability......Page 38
Rotational invariance......Page 39
Two-dimensional Gaussian variables......Page 40
Multidimensional Gaussian variables......Page 41
3.3 Expected values and transformations......Page 42
Linear transformations of Gaussians......Page 43
Conditional probability density on a linear subspace......Page 44
4.1 Equation of motion and integrals......Page 45
Symmetries and integrals......Page 46
4.2 Coordinate changes......Page 48
Reduction of the two-body problem......Page 49
Solution of the two-body problem......Page 50
Barycentric coordinates......Page 52
Heliocentric coordinates......Page 53
4.4 Jacobian coordinates......Page 55
Existence and conditional uniqueness of Jacobian coordinates......Page 56
Planetary and binary type hierarchies......Page 58
The perturbing function......Page 59
Expansions in spherical harmonics......Page 60
The four-body case......Page 62
Perturbations in heliocentric coordinates......Page 64
How many bodies?......Page 66
Non-gravitational perturbations......Page 67
Part II Basic Theory......Page 69
5.1 Linear least squares......Page 71
Model problem......Page 72
5.2 Nonlinear least squares......Page 74
Differential corrections......Page 75
Convergence and comparison with the linear case......Page 76
5.3 Weighting of the residuals......Page 78
5.4 Confidence ellipsoids......Page 80
Conditional ellipsoids for nominal values......Page 81
Marginal ellipsoids......Page 82
Conditional ellipsoids for non-nominal values......Page 83
5.5 Propagation of covariance......Page 84
5.6 Model problem......Page 86
Observations of angle variables......Page 88
5.7 Probabilistic interpretation......Page 89
Normalization of the probability density......Page 91
5.8 Gaussian error models and outlier rejection......Page 92
Outlier removal for weak fit......Page 94
Binning and local Gaussian models......Page 96
Correlations......Page 97
6.1 Complete rank deficiency......Page 99
Curing rank deficiency......Page 100
Model problem with degeneracy......Page 102
6.2 Exact symmetries......Page 103
6.3 Approximate rank deficiency and symmetries......Page 105
Approximate symmetries......Page 106
A posteriori scaling......Page 108
A priori scaling......Page 109
6.5 Planetary systems: extrasolar planets......Page 110
One planet......Page 111
Circular approximation......Page 112
Rank deficiency of the exoplanet problem......Page 114
Exoplanetary systems......Page 115
6.6 Planetary systems: the Solar System......Page 116
Symmetries......Page 117
Relativistic effects......Page 118
Part III Population Orbit Determination......Page 123
7.1 Classification of the problem......Page 125
Attribution......Page 126
Linkage......Page 127
Linear theory......Page 128
Probabilistic interpretation......Page 130
Nonlinearity......Page 132
Restricted orbit identification......Page 133
Multistage identification procedure......Page 134
7.4 Nonlinear orbit identification......Page 136
Model identification problem......Page 137
Cycle slip estimation......Page 139
Constrained orbit identification......Page 141
The confidence ellipse......Page 142
Semilinear predictions......Page 144
Attributables......Page 145
Attribution penalty......Page 147
Attribution procedure......Page 148
Excluding interstellar orbits......Page 149
The inner boundary......Page 152
Excluding satellites of the Earth......Page 153
The tiny object boundary......Page 154
Delaunay triangulations......Page 156
Distance-dependent corrections......Page 160
Structure of the confidence regions......Page 161
Quasi-product structure......Page 163
8.4 Predictions from an attributable......Page 164
Triangulated ephemerides......Page 166
8.5 Linkage by sampling the admissible region......Page 168
Angular momentum and energy......Page 170
Equating the integrals......Page 171
Intersections between the curves......Page 172
Admissible regions for Earth satellites......Page 175
Sampling......Page 177
Radar attributable and admissible regions......Page 178
The two-body integrals method for radar observations......Page 181
9.1 Attributables and curvature......Page 183
Computation of curvature......Page 184
9.2 The method of Laplace......Page 186
9.3 The method of Gauss......Page 187
Gauss–Laplace equivalence......Page 189
Topocentric Laplace method......Page 191
Topocentric, Gauss–Laplace equivalence......Page 192
Problems in the topocentric Laplace method......Page 193
The intersection problem......Page 195
9.6 Charlier theory......Page 197
9.7 Generalization of the Charlier theory......Page 200
The singular curve......Page 201
An even or an odd number of solutions......Page 203
The limiting curve......Page 205
Ellipsoid long axis and weak direction......Page 209
The weak direction vector field......Page 210
Constrained differential corrections......Page 211
Parameterizing and sampling the LOV......Page 212
Orbit determination......Page 214
Multiple ephemerides and recovery......Page 215
Multiple orbit identification......Page 216
Recursive attribution......Page 217
Qualitative analysis......Page 219
10.3 Selection of a metric......Page 220
Coordinates to express initial conditions......Page 221
Comparison of different LOVs......Page 222
Uncertainty of curvature......Page 224
The infinite distance limit......Page 225
10.4 Surface of variations......Page 226
10.5 The definition of discovery......Page 227
Test of possible definitions......Page 229
11.1 Operational constraints of Solar System surveys......Page 231
From detections to tracklets......Page 233
From tracks to identifications......Page 234
Binary tree method......Page 235
Recursive attribution method......Page 236
Normalization procedure......Page 238
Discordant identifications......Page 239
An example......Page 240
Merging discordant identifications......Page 241
Orbit identification......Page 242
Tracklet management......Page 243
Quality control metrics......Page 244
Simulation results......Page 245
11.6 Recovery of low confidence detections......Page 247
Recovery simulations......Page 248
12 IMPACT MONITORING......Page 249
12.1 Target planes......Page 251
Linear predictions on target planes......Page 252
Stationary points of the Keplerian distance function......Page 254
Uncertainty of the MOID......Page 255
The minimal distance maps and their singularities......Page 256
Geometric definition of the regularization......Page 257
Potentially hazardous asteroids......Page 259
12.3 Virtual asteroids......Page 260
The line of variations as geometric sampling......Page 261
The LOV trace on the target planes......Page 262
12.4 Target plane trails......Page 263
The principle of the simplest geometry......Page 264
Returns to close approach......Page 265
Algorithms for the minimum close approach distance......Page 267
12.5 Reliability and completion of impact monitoring......Page 268
Generic completion......Page 269
12.6 The current monitoring systems......Page 270
Part IV Collaborative Orbit Determination......Page 271
Gravity of point masses......Page 273
Mass and gravity of an extended body......Page 275
Harmonic functions......Page 276
Spherical symmetry......Page 277
13.2 Spherical harmonics......Page 278
Zonal spherical harmonics......Page 279
Tesseral spherical harmonics......Page 281
Expansion in spherical harmonics......Page 283
Total mass and center of mass......Page 285
Moments of inertia......Page 286
Recursion formulae......Page 287
Orthogonality......Page 288
Normalization......Page 289
Completeness......Page 291
Equatorial orbit......Page 292
Kaula expansion......Page 293
13.5 Frequency analysis, ground track, and resonance......Page 296
Resonance......Page 297
14 NON-GRAVITATIONAL PERTURBATIONS......Page 299
Interaction of radiation with the surface......Page 300
Secular perturbations......Page 302
14.2 Thermal emission......Page 306
The Yarkovsky effect......Page 309
Diurnal Yarkovsky effect......Page 310
Reflected radiation pressure......Page 311
Visible light......Page 312
Infrared radiation......Page 313
14.4 Drag......Page 314
14.5 Active spacecraft effects......Page 315
Possible solutions......Page 316
Maneuvers and leakages......Page 317
14.6 Case study: asteroid orbiter......Page 318
Photo-gravitational symmetry......Page 319
Deflection by impact and its measure......Page 321
15.1 Local-global decomposition......Page 323
15.2 Case study: satellite laser ranging......Page 326
15.3 Perturbation model......Page 327
15.4 Local geodesy......Page 329
Selection of the passes and data preparation......Page 330
15.5 Symmetries and rank deficiencies......Page 331
Constraints and rigidity of the network......Page 333
Stability test......Page 334
16 SATELLITE GRAVIMETRY......Page 335
Navigation systems......Page 336
Accelerometers......Page 337
Apparent accelerations......Page 338
Calibration......Page 340
Gradiometers......Page 341
Apparent accelerations......Page 342
Local–global decomposition......Page 343
16.3 Gradiometer missions......Page 345
Gradiometer error models......Page 347
A posteriori gradiometer calibration......Page 348
Local–global correlations......Page 349
16.4 Resonant decomposition......Page 350
16.5 Polar gaps......Page 351
Principal components analysis......Page 353
Symmetry and degeneration......Page 354
Outside the polar caps......Page 355
Limitations of gravimetry missions......Page 356
16.6 Satellite-to-satellite tracking......Page 357
Laser Doppler interferometry for gravimetry......Page 358
The numerical precision requirements......Page 360
17.1 Science goals for an orbiter around Mercury......Page 361
Geophysics of Mercury......Page 362
17.2 Interplanetary tracking......Page 363
Time-scales and science goals......Page 367
Accelerometer observables......Page 369
Relativity in Mercury-centric orbit......Page 370
17.3 The gravimetry experiment......Page 368
17.4 The rotation experiment......Page 372
The observing conditions......Page 374
17.5 The relativity experiment......Page 376
17.6 Global data processing......Page 379
The local–global decomposition......Page 380
Line of sight symmetry......Page 381
Conclusions......Page 382
References......Page 383
Index......Page 391