Existence of a lower bound for the distance between point masses of relative equilibria for generalised quasi-homogeneous $n$-body problems and the curved $n$-body problem

TL;DR

This paper establishes a universal lower bound on the minimum distance between point masses in relative equilibria for generalized quasi-homogeneous and curved $n$-body problems, ensuring these configurations are well-separated.

Contribution

It proves the existence of a non-zero lower bound for distances in relative equilibria and demonstrates the compactness of the set of such solutions in curved spaces.

Findings

Universal lower bound for inter-mass distances in relative equilibria

Compactness of the set of relative equilibria solutions

Results extend to $n$-body problems in spaces of constant Gaussian curvature

Abstract

We prove that if for relative equilibrium solutions of a generalisation of quasi-homogeneous $n$-body problems the masses and rotation are given, then the minimum distance between the point masses of such a relative equilibrium has a universal lower bound that is not equal to zero. We furthermore prove that the set of such relative equilibria is compact and prove related results for $n$-body problems in spaces of constant Gaussian curvature.