General Relativistic Effects on Hill Stability of Multi-Body Systems I: Stability of Three-Body Systems Containing a Massive Black Hole

TL;DR

This paper extends the analysis of Hill stability in three-body systems to include general relativistic effects, deriving an analytical criterion and confirming that relativity tends to destabilize such systems compared to Newtonian predictions.

Contribution

It provides the first analytical criterion for relativistic Hill stability in three-body systems, validated through numerical simulations, enhancing understanding of stability near massive black holes.

Findings

Relativity decreases the Hill stability of three-body systems.

An analytical sufficient condition for relativistic Hill stability is derived.

Numerical simulations confirm the criterion's validity and usefulness.

Abstract

We study the effects of general relativistic gravity on the Hill stability, that is, the stability of a multi-body system against a close approach of one orbit to another, which has been hitherto studied mainly in Newtonian mechanics and applied to planetary systems. We focus in this paper on the three-body problem and extend the Newtonian analyses to the general relativistic regime in the post-Newtonian approximation. The approximate sufficient condition for the relativistic Hill stability of three-body systems is derived analytically and its validity and usefulness are confirmed numerically. In fact, relativity makes the system more unstable than Newtonian mechanics in the sense of the Hill stability as expected by our theoretical prediction. The criterion will be useful to analyze the results of large-scale N-body simulations of dense environments, in which the stability of three-body sub-systems is important.