# Linear stability of the elliptic relative equilibrium with $(1 +n)$-gon central configurations in planar $n$-body problem

## TL;DR

This paper analyzes the linear stability of a specific class of elliptic relative equilibria in the planar n-body problem, showing stability for large central mass and instability for small mass.

## Contribution

It provides new stability criteria for the $(1+n)$-gon elliptic relative equilibrium in the planar n-body problem, especially for large central mass.

## Key findings

- Stable for $n \\geq 8$ and large $m$ across all eccentricities.
- Instability results for small central mass.
- Stability depends on the size of the central mass.

## Abstract

We study the linear stability of $(1+n)$-gon elliptic relative equilibrium (ERE for short), that is the Kepler homographic solution with the $(1+n)$-gon central configurations. We show that for $n\geq 8$ and any eccentricity $e\in[0,1)$, the $(1+n)$-gon ERE is stable when the central mass $m$ is large enough. Some linear instability results are given when $m$ is small.