The Kepler Problem with Anisotropic Perturbations

TL;DR

This paper investigates the dynamics of a two-body gravitational problem with an added anisotropic perturbation, revealing conditions for chaos, collision, and escape behaviors depending on the perturbation degree.

Contribution

It introduces a detailed analysis of the anisotropic perturbation effects on the Kepler problem, including chaos and integrability conditions for specific perturbation degrees.

Findings

Positive measure of initial conditions lead to collisions and ejections for eta>2.

Flow on the zero-energy manifold is chaotic for eta>2 and eta.

The case eta=2 is integrable with heteroclinic connections between manifolds.

Abstract

We study a 2-body problem given by the sum of the Newtonian potential and an anisotropic perturbation that is a homogeneous function of degree $-\beta$, $\beta\ge 2$. For $\beta>2$, the sets of initial conditions leading to collisions/ejections and the one leading to escapes/captures have positive measure. For $\beta>2$ and $\beta\ne 3$, the flow on the zero-energy manifold is chaotic. For $\beta=2$, a case we prove integrable, the infinity manifold of the zero-energy level is a disconnected set, which has heteroclinic connections with the collision manifold.