# Linear Stability of Elliptic Relative Equilibria of Restricted Four-body Problem

## TL;DR

This paper analyzes the linear stability of elliptic relative equilibria in a restricted four-body problem with three primaries forming a Lagrangian triangle, revealing stability regions based on mass parameters and eccentricity.

## Contribution

It provides a comprehensive bifurcation diagram for stability regions, employing $ ext{om}$-Maslov index and differential operator theories, and identifies two stable sub-regions.

## Key findings

- Full bifurcation diagram of stability regions
- Identification of two stable sub-regions
- Dependence of stability on mass parameters and eccentricity

## Abstract

In this paper, we consider the linear stability of the elliptic relative equilibria of the restricted 4-body problems where the three primaries form a Lagrangian triangle. By reduction, the linearized Poincar\'e map is decomposed to the essential part, the Keplerian part and the elliptic Lagrangian part where the last two parts have been studied in literature. The linear stability of the essential part depends on the masses parameters $\alpha$, $\beta$ with $\alpha\geq \beta >0$ and the eccentricity $e\in[0,1)$. Via $\om$-Maslov index theory and linear differential operator theory, we obtain the full bifurcation diagram of linearly stable and unstable regions with respect to $\alpha$, $\beta$ and $e$. Especially, two linearly stable sub-regions are found.