Secular instability in the spatial three-body problem

TL;DR

This paper demonstrates the presence of chaotic dynamics, including horseshoe structures, in the secular evolution of the spatial three-body problem when one body is far away, revealing complex eccentricity variations.

Contribution

It provides an explicit computation of homoclinic solutions and Melnikov potential, establishing chaos in the secular regime of the spatial three-body problem.

Findings

Existence of horseshoe dynamics in secular evolution

Chaotic eccentricity variations in the three-body system

Explicit homoclinic solutions for the secular system

Abstract

Consider the spatial three-body problem, in the regime where one body revolves far away around the other two, in space, the masses of the bodies being arbitrary but fixed; in this regime, there are no resonances in mean motions. The so-called secular dynamics governs the slow evolution of the Keplerian ellipses. We show that it contains a horseshoe and all the chaotic dynamics which goes along with it, corresponding to motions along which the eccentricity of the inner ellipse undergoes large, random excursions. The proof goes through the surprisingly explicit computation of the homoclinic solution of the first order secular system, its complex singularities and the Melnikov potential.