Quasi-periodic Almost-collision Orbits in the Spatial Three-Body Problem

TL;DR

This paper proves the existence of a set of positive measure of quasi-periodic almost-collision orbits in the spatial three-body problem, extending previous results from planar cases.

Contribution

It provides a rigorous proof for the existence of quasi-periodic almost-collision orbits in the spatial three-body problem, confirming long-standing predictions.

Findings

Existence of positive measure set of such orbits in the spatial case

Extension of planar results to three-dimensional spatial problem

Rigorous mathematical proof of these orbits' existence

Abstract

In a system of particles, quasi-periodic almost-collision orbits are collisionless orbits along which two bodies become arbitrarily close to each other -- the lower limit of their distance is zero but the upper limit is strictly positive -- and which are quasi-periodic in a regularized system up to a change of time. The existence of such orbits was shown in the restricted planar circular three-body problem by A. Chenciner and J. Llibre, and later, in the planar three-body problem by J. F\'ejoz. In the spatial three-body problem, the existence of a set of positive measure of such orbits was predicted by C. Marchal. In this article, we present a proof of this fact.