Chaotic Dynamics in Planetary Systems , 1st ed. 2023
Astronomy and Planetary Sciences Series

The main theme of the book is the presentation of techniques used to identify chaotic behavior in the evolution of conservative mechanical systems and their application to astronomical systems. It results from graduate courses given by the author over the years both at university and at several international summer schools.

Along the book surfaces of section, Lyapunov characteristic exponents, frequency maps, MEGNO, dense grid maps, etc., are presented and discussed in connection with the applications. The initial chapter is devoted to the presentation of the main ideas of the chaotic dynamics of conservative systems in plain language so that they can be accessible to a wide range of professionals and students of physical sciences. The applications are mainly related to the motions in the solar system and extrasolar planetary systems.

Another chapter is devoted to the applications to asteroids showing how the asteroidal belt is sculpted by chaos and resonances. The contrasting existence of gaps in the distribution of the asteroids and groups of asteroids in resonances is thoroughly discussed. 

The interest in applications to planetary systems is growing since the discovery of systems of resonant planets around some stars of the solar neighborhood. Exoplanets added a lot of cases to a problem that was before restricted to the planets of our solar system. The book includes an account of results already existing about compact systems.

1.1 Planetary motion

1.2 Chaotic motion of the planets

1.3 Information loss due to chaos

1.4 Chaos in the rotation of Hyperion

1.6 Conservative Mechanics

1.7 Two-body problem

1.8 Flow incompressibility (Liouville)

1.9 Laws of conservation (First Integrals) and complete integration

1.11 Integrability

1.12 Surfaces of Section. Poincar´e maps

1.13 The motion of a star in a galaxy with axial symmetry

1.13.1 The H´enon-Heiles system

1.15 Two degrees of freedom

1.16 The homoclinic entanglement. Regime transitions

1.17 Phobos and Hyperion

1.18 Enceladus and Dione

1.19 Perturbed systems. Resonance and Libration

1.20 The KAM theory (Kolmogorov - Arnold - Moser)

Chapter 2 Resonant Asteroidal Dynamics

2.1 Resonant Asteroids

2.2 Asteroids in the restricted three-body model

2.2.1 The resonance 2:1

2.2.3 The resonance 3:1

2.2.4 The Pluto-Neptune resonance

2.3 Close encounters. Swing-by

2.4 Asteroids in the restricted three-body elliptic model

2.5.1 From “Himmelsmechanik” to “Atommechanik”

2.6 The Alinda gap

2.6.1 Regimes of motion in the resonance 3:1

2.6.2 The origin of the Alinda gap

2.6.4 Alinda, Quetzalcoatl, Seneca, Syrinx and Toutatis

2.7 Digital filtering

2.8 The Hecuba gap and the Zhongguo group

2.9 Lyapunov characteristic exponents (LCE)

2.11 The theory of LCEs. Variational equations

2.12 The maximum Lyapunov exponent (mLCE)

2.12.1 Calculation of the other LCEs

2.13 Exponential divergence and information loss

2.15 Events. Sudden orbital transitions

2.15.1 Stable chaos

2.16 Fast Lyapunov indicators (FLI)

2.17 The Hecuba gap asteroids

2.17.2 The Griquas

2.17.3 Resonant asteroids in cometary orbits

2.18 Hilda group

2.19 Gaps vs. Groups

Chapter 3 Planetary Systems. Exoplanets

3.1 Chaos in the Solar System

3.2 The use of the Fourier Transform to diagnose chaos

3.3 Chaos around the giant planets. Dynamical maps

3.4 Frequency analysis of weakly chaotic systems

3.6 The rotations of the Earth and Mars

3.7 Frequency analysis on dense grids. Arnold web

3.8 Other strategies in Frequency Analysis. Resonant asteroids

3.9 Dynamical maps on dense grids. Trojan asteroids

3.11 Exoplanets

3.11.1 Example: Upsilon Andromedae

3.12 MEGNO

3.12.1 Example: The super resonance of GJ 876

3.14 Resonant Chains

3.14.1 Example: TOI-178

3.14.2 Example: HR 8799

3.15 Apsidal Corotation Resonance (ACR)

3.16 Dynamical power spectrum (Frequency map)

Index

The author, Professor Emeritus at the Institute of Astronomy, Geophysics and Atmospheric Sciences of the University of Sao Paulo is a scientist working on Planetary Sciences. In the past 60 years, he studied several topics in planetary dynamics: asteroids, resonances, tides, chaos, and exoplanets. The International Astronomical Union named “Ferraz-Mello” the resonant asteroid 1983 XF (5201). Professor Ferraz-Mello is Doctor Honoris Causa of Paris Observatory and, in 2015, won the Brouwer Award of the American Astronomical Society. 

Uses accessible lange to present concepts of the chaotic dynamics of conservative mechanical systems

Shows step by step how occurrence of chaotic motions can be identified

Discusses resonances in the solar system and compact exoplanetry systems