Computer assisted proofs for transverse collision and near collision orbits in the restricted three body problem

TL;DR

This paper introduces a flexible computer-assisted framework for proving the existence of collision and near-collision orbits in the restricted three body problem, confirming long-studied numerical results analytically.

Contribution

It develops an a-posteriori method applicable to multiple coordinate systems, enabling rigorous proofs of complex celestial orbits previously only numerically observed.

Findings

Proved existence of transverse ejection/collision orbits.

Confirmed Strömgren's asymptotic periodic orbits.

Established families of orbits passing through collision.

Abstract

This paper considers two point boundary value problems for conservative systems defined in multiple coordinate systems, and develops a flexible a-posteriori framework for computer assisted existence proofs. Our framework is applied to the study collision and near collision orbits in the circular restricted three body problem. In this case the coordinate systems are the standard rotating coordinates, and the two Levi-Civita coordinate systems regularizing collisions with each of the massive primaries. The proposed framework is used to prove the existence of a number of orbits which have long been studied numerically in the celestial mechanics literature, but for which there are no existing analytical proofs at the mass and energy values considered here. These include transverse ejection/collisions from one primary body to the other, Str\"{o}mgren's assymptotic periodic orbits (transverse homoclinics for $L_{4,5}$), families of periodic orbits passing through collision, and orbits connecting $L_4$ to ejection or collision.