Regular Motions in Extra-Solar Planetary Systems

TL;DR

This review discusses the dynamics of exoplanetary systems, focusing on averaged equations, resonances, Hamiltonian formulation, and the evolution of systems into stationary solutions with apsidal corotation.

Contribution

It provides a comprehensive overview of the mathematical modeling and dynamical behaviors of multi-planet systems, including resonant interactions and Hamiltonian reduction techniques.

Findings

Oscillations of the angle between semi-major axes around a constant value.

Introduction of Hamiltonian equations for the N-planet problem.

Discussion of systems captured into resonance and their evolution.

Abstract

This paper is a review of the dynamics of a system of planets. It includes the study of averaged equations in both non-resonant and resonant systems and shows the great deal of situations in which the angle between the two semi-major axes oscillates around a constant value. It introduces the Hamiltonian equations of the $N$-planet problem and Poincar\'e's reduction of them to $3N$ degrees of freedom with a detailed discussion of the non-osculating ``canonical'' heliocentric Keplerian elements that should be used with Poincar\'e relative canonical variables. It also includes Beaug\'e's approximation to expand the disturbing function in the exoplanetary case where masses and eccentricities are large. The paper is concluded with a discussion of systems captured into resonance and their evolution to symmetric and asymmetric stationary solutions with apsidal corotation.