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Dynamical Chaos in Planetary Systems
ISBN-13: 9783030521462 / Angielski / Miękka / 2021 / 390 str.
Dynamical Chaos in Planetary Systems
ISBN-13: 9783030521462 / Angielski / Miękka / 2021 / 390 str.
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"Shevchenko's book is a remarkably thorough and timely overview of the subject, suitable for any researcher in the field but ideally one with an appropriate background in Hamiltonian dynamics. ... This is an area where chaos as a tool becomes less descriptive and more diagnostic - the true sign of a subject reaching maturity; this book is a chronicle of that journey." (Carl Murray, The Observatory, Vol. 141 (1282), June, 2021)
Preface
I Origins and manifestations of dynamical chaos 11
1 Chaotic behaviour 13
1.1 Pendulum, resonances and chaos 13
1.2 Models of resonance ...... . 15
1.3 Interaction and overlap of resonances . 15
1.4 Symplectic maps in general 16
1.5 The standard map ....... . 18
1.6 The separatrix map ...... . 19
1.7 The separatrix algorithmic map 23
1.8 Geometry of chaotic layers . . . . 26
2 Numerical tools for studies of dynamical chaos 41
2.1 The Lyapunov exponents ....... . 41
2.2 The Poincare sections ......... . 50
2.3 Stability diagrams and dynamical charts 51
2.4 Statistics of Poincare recurrences 51
3 Adiabatic and non-adiabatic chaos: the Lyapunov timescales 53
3.1 Non-adiabatic chaos ... . 54
3.1.1 Chirikov's constant .... . 54
3.2 Adiabatic chaos .......... . 62
3.3 The Lyapunov timescales in resonance doublets and triplets 71
3.4 The Lyapunov exponents in resonance multiplets 74
4 Chaotic diffusion 79
4.1 Diffusion rates 79
4.1.1 Diffusion rates in resonance multiplets ..... . 79
4.1.2 Diffusion rates in resonance triplets and doublets 815 Lyapunov and diffusion timescales: relationships 85
5.1 Finite-time Lyapunov exponents 87
5.2 The generic relationship .... 87
5.3 Conditions for the relationship 90
5.4 Numerical examples . . . . . . 91
6 Widths of chaotic layers 99
6.1 Extents of chaotic domains . . . . . . . . . . . . 99
6.1.1 The separatrix split . . . . . . . . . . . . 102
6.1.2 Early estimates of the chaotic layer width 105
6.2 "Generic" width of the chaotic layer . . . . . . . 107
6.2.1 The layer width in the case of non-adiabatic chaos 109
6.2.2 The layer width in the case of adiabatic chaos . . . 109
6.3 Marginal resonances . . . . . . . . . . . . . . . . . . . . . 122
6.3.1 Marginal resonances in the case of non-adiabatic chaos . 123
6.3.2 Marginal resonances in the case of adiabatic chaos 124
6.3.3 Marginal resonances: theory versus simulations 124
6.3.4 Marginal resonances: phase space sections . . . . . 130
7 Orbital dynamics with encounters: the encounter and Kepler maps 133
7.1 The encounter map . . . . . . . . . . . . . . . . 134
7.1.1 Derivation of the encounter map . . . . 134
7.1.2 Width of the chaotic layer: the µ217 law 136
7.1.3 The Wisdom gaps . . . . . . 139
7.2 The Kepler map . . . . . . . . . . . 141
7.2.1 Prehistory of the Kepler map 143
7.2.2 Derivation of the Kepler map 146
7.2.3 Width of the chaotic layer: the µ215 law 153
7.2.4 The Kepler map as a generalized separatrix map 154
7.2.5 The Lyapunov and diffusion timescales of cometary motion158
8 Hamiltonian intermittency and Levy fl.ights in the three-body problem 161
8.1 Two kinds of Hamiltonian intermittency 162
8.2 Overview of generalized separatrix maps 163
8.3 Levy flights at the edge of escape: the distribution . . . . . . . . . 165
8.4 Levy flights at the edge of escape:the "TL - Tr" relation . . . . . . . 175
8.5 Ways of disruption of three-body systems 181
II Resonances and chaos in the Solar system 185
9 Order and chaos in the Solar system: historical background 189
10 Chaotic rotation 193
10.1 Chaotic rotation of satellites . . . . . . . . . . . 193
10.1.1 Spin-orbit resonances . . . . . . . . . . 196
10.1.2 Lyapunov timescales of chaotic rotation 200
10.1.3 Widths of chaotic layers . . . . . . . . . 202
10.1.4 Chaotic planar rotation and chaotic tumbling 203
10.1.5 Stability with respect to tilting the axis of rotation . 211
10.2 Chaotic obliquities of planets . . . . . . . . . . . . . . . . . 217
11 Chaotic orbital dynamics of minor bodies 221
11.1 Chaotic dynamics of satellite systems. . . . . . . . . . . 221
11.1.1 Generalization of the separatrix algorithmic map 224
11.1.2 The Miranda-Umbriel system . . . 228
11.1.3 The Mimas-Tethys system . . . . 231
11.1.4 The Prometheus-Pandora system. 234
11.2 Chaos in orbital dynamics of asteroids . . 245
11.2.1 The D'Alembert rules . . . . . . . 249
11.2.2 Resonant structure of the asteroid and Kuiper belts 251
11.2.3 Chaos in orbital dynamics of TNOs 254
11.2.4 Two-body resonances . . . . . . . . . . . . . . . . . 256
11.2.5 Three-body resonances. . . . . . . . . . . . . . . . . 259
11.2.6 Statistics of asteroids in two-body and three-body reso- nances . . . . . . .262
11.2.7 Lyapunov exponents in three-body resonances. . . 266
11.2.8 Statistics of mean motion resonances: an overview 268
11.2.9 Secular resonances . . . . . . . . . . . . . 270
11.2.lODiffusion timescales of asteroidal motion . . . . 271
11.3 Binary and multiple asteroids and TNOs. . . . . . . . 275
11.3.1 Chaotic zones around rotating contact binaries 276
11.3.2 Ida and Dactyl . . . . 276
11.4 Chaos in cometary dynamics 278
11.4.1 The Halley comet 278
12 Chaotic orbital dynamics of planets
12.1 Relevant three-body resonances ...
III Dynamics of exoplanets
13 Exoplanets: an overview
13.1 History and methods of discovery of exoplanets
13.2 Definition of a planet . . . . . . . . . .
13.3 Typology and properties of exoplanets
13.3.1 Types of exoplanets ....
13.3.2 Types of planetary systems
13.4 Planetary configurations . . . . . .
13.5 Dominant resonances . . . . . . . . . . . . . . . . . . . .
14 Secular dynamics of hierarchical planetary systems 309
15 Location and interaction of resonances 313
15.1 The circumprimary case (case of the outer perturber) 313
15.2 The circumbinary case (case of the inner perturber) 314
15.3 Apsidal precession of circumbinary orbits 315
15.4 The Mardling theory . . . . . . . . . . . . . . . . . . 316
16 Chaos as a clearing agent 323
16.1 Stability criteria and chaotic clearing effects 323
16.2 The Hill criterion and the Hill sphere . . . . 324
16.3 The Wisdom criterion and the Wisdom gap 325
16.4 The Mustill-Wyatt relation . . . . . . . . . 327
16.5 The Kepler map criterion and the circumbinary clearance effect 327
16.6 The Holman-Wiegert criteria for circumbinary and circumstellar chaos. . . . . . . . . 329
16.7 Chaotic clearing effects in planetary systems . . . . . . . . . . . 329
17 Chaotic zones around gravitating binaries 331
17.1 Radial extent of the circumbinary chaotic zone . . . . . . . . . 336
17.2 Stability diagrams for circumbinary exoplanets . . . . . . . . . 339
17.3 The mass parameter threshold and the diversity of observed ex- osystems ........... 339
18 Chaos in multiplanet systems 345
18.1 Multiplanet systems of single stars 345
18.2 Chaotic multiplanet systems. 346
18.3 Anomalous systems . . . . . . . . . 351
19 Chaos in planetary systems of binary stars 353
19.1 S-systems and P-systems. . . 353
19.2 The a Centauri A-B system . . . . 353
19.3 The 16 Cyg system . . . . . . . . . 354
19.4 The Kepler circumbinary systems . 355
19.5 The Moriwaki-Nakagawa criterion and formation of circumbinary planets . . . . . . . . 365
20 The Lidov-Kozai effect and chaos in exoplanetary systems 367
20.1 LKE in multiplanet systems . . . . . . . . . . . 368
20.2 LKE in planetary systems of binary stars . . . 370
20.3 Chaos in the planetary motion subject to LKE 372
21 Challenges and prospects 375
Appendix A
Appendix B
Appendix C
Bibliography
Ivan I. Shevchenko is Professor at Saint Petersburg State University and Head of the Department of Celestial Mechanics and Dynamical Astronomy at Pulkovo Observatory of the Russian Academy of Sciences, Saint Petersburg, Russia. He is author of the book "The Lidov-Kozai Effect - Applications in Exoplanet Research and Dynamical Astronomy", Springer (2017).
This is the first monograph dedicated entirely to problems of stability and chaotic behaviour in planetary systems and its subsystems. The author explores the three rapidly developing interplaying fields of resonant and chaotic dynamics of Hamiltonian systems, the dynamics of Solar system bodies, and the dynamics of exoplanetary systems. The necessary concepts, methods and tools used to study dynamical chaos (such as symplectic maps, Lyapunov exponents and timescales, chaotic diffusion rates, stability diagrams and charts) are described and then used to show in detail how the observed dynamical architectures arise in the Solar system (and its subsystems) and in exoplanetary systems. The book concentrates, in particular, on chaotic diffusion and clearing effects. The potential readership of this book includes scientists and students working in astrophysics, planetary science, celestial mechanics, and nonlinear dynamics.
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