Dynamical chaos in planetary systems

Author(s)

    • Shevchenko, Ivan I.

Bibliographic Information

Dynamical chaos in planetary systems

Ivan I. Shevchenko

(Astrophysics and space science library, v. 463)

Springer, c2020

Available at  / 4 libraries

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Note

Includes bibliographical references (p. 345-368) and index

Description and Table of Contents

Description

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.

Table of Contents

PrefaceI 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 81 5 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 217 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 215 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

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Details

  • NCID
    BC0253627X
  • ISBN
    • 9783030521431
  • Country Code
    sz
  • Title Language Code
    eng
  • Text Language Code
    eng
  • Place of Publication
    Cham
  • Pages/Volumes
    xii, 376 p.
  • Size
    25 cm
  • Classification
  • Subject Headings
  • Parent Bibliography ID
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