Oscillatory collision approach in the Earth-Moon restricted three body problem

TL;DR

This paper proves the existence of oscillatory orbits in the Earth-Moon restricted three-body problem, where orbits repeatedly approach and recede from the primary mass, using topological and computational methods.

Contribution

It introduces a novel combination of topological tools, interval computations, and Levi-Civita regularization to rigorously establish oscillatory orbits in the three-body problem.

Findings

Existence of orbits oscillating near the primary masses.

Validation of dynamics through rigorous interval computations.

Application of shadowing arguments to original coordinates.

Abstract

We consider the Earth-Moon planar circular restricted three body problem and present a proof of the existence orbits, which approach arbitrarily close to one of the primary masses, and at the same time after each approach they move away from the mass to a prescribed distance. In other words the orbits oscillate between being arbitrarily close to collision and away from it. We achieve our goal with the use of topological tools combined with rigorous interval computations. We use the Levi-Civita regularization and validate that the dynamics in the regularized coordinates leads to a good topological alignment between various sets. We then perform shadowing arguments that this leads to the required dynamics in the original coordinates of the system.