Existence and Stability of Symmetric Periodic Simultaneous Binary Collision Orbits in the Planar Pairwise Symmetric Four-Body Problem

TL;DR

This paper proves the existence and analyzes the stability of symmetric periodic orbits with binary collisions in a symmetric four-body problem, extending previous work and providing numerical stability estimates across different mass ratios.

Contribution

It extends the analytic existence results of symmetric periodic orbits with binary collisions to a more general four-body problem and numerically investigates their stability across mass ratios.

Findings

Existence of symmetric periodic orbits with binary collisions in the four-body problem.

Stable orbits for mass ratios between 0.54 and 1.

Unstable orbits for mass ratios between 0 and 0.53.

Abstract

We extend our previous analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized fully symmetric equal mass four-body problem to the analytic existence of a symmetric periodic simultaneous binary collision orbit in a regularized planar pairwise symmetric equal mass four-body problem. We then use a continuation method to numerically find symmetric periodic simultaneous binary collision orbits in a regularized planar pairwise symmetric 1, m, 1, m four-body problem for $m$ between 0 and 1. Numerical estimates of the the characteristic multipliers show that these periodic orbits are linearly stability when $0.54\leq m\leq 1$, and are linearly unstable when $0<m\leq0.53$.