Modern Celestial Mechanics: Dynamics in the Solar System

✍ Scribed by Alessandro Morbidelli

• Publisher: CRC Press
• Year: 2002
• Tongue: English
• Leaves: 370
• Series: Advances in Astronomy and Astrophysics
• Edition: 1st
• Category: Library

✦ Synopsis

Celestial Mechanics has achieved spectacular results on the structure and evolution of the Solar System in the last 20 years. This book describes recent results on Solar System dynamics, with a solid theoretical basis and is strongly focused on the dynamics of planets and of small bodies. Modern Celestial Mechanics will be of great interest to graduate students and researchers of astronomy and astrophysics.

✦ Table of Contents

PREFACE......Page 6
INTRODUCTION: OBJECTS OF THE SOLAR SYSTEM, PROBLEMS OF CELESTIAL MECHANICS......Page 14
1.1 Equations of motion......Page 24
1.2 Orbital elements......Page 25
1.3 Perturbations of the two-body problem......Page 29
1.4 Hamiltonian systems and the two-body problem......Page 31
1.5 Perturbations in Hamiltonian form......Page 33
1.6 Canonical transformations......Page 35
1.7 Properties of Hamiltonian flow......Page 38
1.8 Integrable Hamiltonians......Page 39
1.9 Action-angle variables......Page 41
1.9.1 Delaunay variables......Page 42
1.9.2 Hamilton equations in Delaunay variables for the restricted and the planetary problem......Page 46
1.9.3 D'Alembert rules......Page 48
1.10 Integrable dynamics......Page 49
2.1 Introduction to perturbation theory......Page 52
2.2 Lie series approach......Page 53
2.3 The small divisors problem......Page 55
2.3.1 Normal forms......Page 58
2.4 Beyond the first order......Page 59
2.4.1 Example of computation of the optimal order of the normal form......Page 61
2.4.2 Generation of high-order harmonics by the normalization process......Page 62
2.5 Averaging over the mean motions......Page 64
2.5.1 Secular normal form......Page 66
2.5.2 Mean motion resonant normal form......Page 70
3.1 Kolmogorov's theorem......Page 72
3.1.1 Sketch of proof of Kolmogorov's theorem......Page 73
3.2 Properties of KAM tori......Page 76
3.3 Numerical examples......Page 80
4.1 The integrable approximation......Page 84
4.2 Resonant action-angle variables......Page 87
4.3.1 Size of the remainder......Page 91
4.3.2 Resonant invariant tori......Page 92
4.3.3 Splitting of separatrices......Page 94
4.3.4 Size of the chaotic region......Page 97
5.1 Monitoring the time evolution in phase space......Page 102
5.2 Lyapunov exponents......Page 105
5.2.1 Numerical computation of the MLE......Page 107
5.3 Frequency analysis......Page 111
5.3.1 Numerical determination of the frequencies......Page 115
5.4.1 Fast Lyapunov indicator......Page 116
5.4.2 Helicity and twist angles......Page 117
5.4.3 Mean exponential growth factor of nearby orbits......Page 118
5.4.4 Mean, maximal and minimal values of the actions......Page 119
6.1 Two degrees of freedom......Page 120
6.1.1 Heteroclinic intersections......Page 124
6.2.1 The Nekhoroshev theorem......Page 127
6.2.2 Nekhoroshev structure......Page 132
6.3 Exploring the dynamical structure of a given system......Page 135
7.1 Lagrange-Laplace solution......Page 140
7.2 Higher-order solutions......Page 144
7.3 Chaotic secular motion of the planets......Page 148
7.4 Spin axes dynamics......Page 153
8.1 The linear integrable approximation......Page 162
8.2 The Kozai integrable approximation......Page 167
8.2.1 Kozai dynamics inside the orbit of the main perturber......Page 169
8.2.2 Kozai dynamics outside the orbit of the main perturber......Page 173
8.2.3 Action-angle variables for the Kozai Hamiltonian......Page 179
8.3 Proper elements......Page 180
8.3.1 Asteroid families......Page 185
8.4 Secular resonances......Page 187
8.4.1 Secular resonant dynamics......Page 192
8.4.2 The anomalous case of the ν₆ resonance......Page 200
9.1 A simple integrable approximation......Page 208
9.1.1 Phase protection from planetary collisions......Page 219
9.1.2 The case of the 1/1 resonance......Page 221
9.2 Mean motion resonance overlapping......Page 223
9.2.1 Threshold for overlapping in the vicinity of the planet......Page 229
9.2.2 Overlapping of resonances with different planets......Page 230
9.3 Resonant multiplets......Page 233
9.4 The modulated pendulum approximation......Page 237
10.1 Origin of the resonant perturbation terms......Page 244
10.1.1 The direct effect......Page 245
10.1.2 The indirect effect......Page 247
10.1.3 Inclusion of both direct and indirect effects in the asteroid problem......Page 252
10.2 Three-body resonant multiplets......Page 253
10.3 The asteroid and Kuiper belts......Page 260
10.4 Chaotic dynamics of the giant planets......Page 267
11.1 Successive elimination of harmonics......Page 270
11.2 The mean motion resonant dynamical system......Page 273
11.2.1 Secondary resonances......Page 275
11.2.2 The Kozai dynamics......Page 279
11.2.3 Perihelion secular resonances......Page 284
11.2.4 Nodal secular resonances......Page 289
11.2.5 Three-body resonances......Page 290
11.3.1 The 3/1 resonance......Page 293
11.3.2 The 2/1 resonance......Page 299
11.3.3 The 3/2 resonance......Page 305
11.4 The major resonances in the Kuiper belt......Page 307
11.4.1 The 2/3 resonance......Page 308
11.4.2 The 1/2 resonance......Page 311
11.5 The 1/1 resonances......Page 313
12.1 Detection of the chaotic zones......Page 316
12.2 Chaotic diffusion and macroscopic instability......Page 325
12.3 Analytic estimates of Lyapunov time and instability time......Page 333
12.4 Do the KAM and Nekhoroshev theorems apply for small body dynamics?......Page 340
REFERENCES......Page 344
INDEX......Page 362