# Stability of Broucke's Isosceles Orbit

## TL;DR

This paper analyzes the linear stability of Broucke's isosceles orbit with specific mass configurations, extending previous results and showing stability conditions through analytical and numerical methods.

## Contribution

It extends Yan's results to a new mass configuration, introduces a regularization method for binary collisions, and applies an analytical stability analysis using Roberts' method.

## Key findings

- Four-degrees-of-freedom orbit is unstable except for specific mass intervals.
- Two-degrees-of-freedom subset exhibits wider stability interval.
- Stability analysis reduces to computing three entries of a 4x4 matrix.

## Abstract

We extend the result of Yan to Broucke's isosceles orbit with masses $m_1$, $m_1$, and $m_2$ with $2m_1 + m_2 = 3$. Under suitable changes of variables, isolated binary collisions between the two mass $m_1$ particles are regularizable. We analytically extend a method of Roberts to perform linear stability analysis in this setting. Linear stability is reduced to computing three entries of a $4 \times 4$ matrix related to the monodromy matrix. Additionally, it is shown that the four-degrees-of-freedom setting has a two-degrees-of-freedom invariant set, and linear stability results in the subset comes `for free' from the calculation in the full space. The final numerical analysis shows that the four-degrees-of-freedom orbit is linearly unstable except for the interval $0.555 < m_1 < 0.730$, whereas the two-degrees-of-freedom orbit is linearly stable for a much wider interval.