Linear stability of the n-gon relative equilibria of the (1+n)-body problem

TL;DR

This paper analyzes the linear stability of regular n-gon configurations in the (1+n)-body problem, revealing stability conditions depending on the number of bodies and mass arrangements, with stability for larger even n.

Contribution

It provides a detailed stability analysis of n-gon relative equilibria, identifying conditions for stability based on the parity of n and mass ratios, extending previous results.

Findings

Linearly stable for even n ≥ 14

Stability depends on mass ratio intervals for n=8,10,12

Odd n cases have different stability conclusions

Abstract

We consider the linear stabilities of the regular n-gon relative equilibria of the (1+n)-body problem. It is shown that there exist at most two kinds of infinitesimal bodies arranged alternatively at the vertices of a regular n-gon when n is even, and only one set of identical infinitesimal bodies when n is odd. In the case of n even, the regular n-gon relative equilibrium is shown to be linearly stable when n>=14. In each case of n=8,10,12, linear stability can also be preserved if the ratio of two kinds of masses belongs to an open interval. When n is odd, the related conclusion on the linear stability is recalled.