Stability of the regular $n$-gon rotating equilibria with logarithm interaction

TL;DR

This paper analyzes the linear stability of regular n-gon rotating equilibria in the n-body problem with logarithmic interactions, providing explicit stability bounds with a central mass and identifying stability conditions without it.

Contribution

It offers explicit stability bounds for regular n-gons with a central mass and characterizes stability for n-gons without a central mass.

Findings

Stability depends on bounds of the central mass M.

Explicit equations for stability bounds are derived.

Regular n-gons are stable for n=2 to 6 without a central mass.

Abstract

We study the linear stability of regular $n$-gon rotating equilibria in the $n$-body problem with logarithm interaction. In the presence of a central mass $M$, linear stability is insured if $M$ is bounded below and above by constants depending on the number and mass of the (equal) outer $n$ bodies. Moreover, we provide explicit equations of these bounds. In the absence of a central mass we find that the regular $n$-gon is linearly stable for $n =2,3,\ldots 6$ only.