Generalized Hill-Stability Criteria for Hierarchical Three-Body Systems at Arbitrary Inclinations

TL;DR

This paper develops a generalized theoretical framework for assessing the stability of hierarchical three-body systems at any inclination, extending previous co-planar criteria and validating with extensive numerical simulations.

Contribution

It introduces a novel approach to derive Hill-stability criteria for arbitrarily inclined systems, incorporating secular evolution and Lidov-Kozai cycles, and combines analytic and numerical methods for a comprehensive stability formula.

Findings

Excellent agreement with theory up to 120° inclination

Stability radius increases at high inclinations

Polynomial fits improve stability predictions at extreme inclinations

Abstract

A fundamental aspect of the three-body problem is its stability. Most stability studies have focused on the co-planar three-body problem, deriving analytic criteria for the dynamical stability of such pro/retrograde systems. Numerical studies of inclined systems phenomenologically mapped their stability regions, but neither complement it by theoretical framework, nor provided satisfactory fit for their dependence on mutual inclinations. Here we present a novel approach to study the stability of hierarchical three-body systems at arbitrary inclinations, which accounts not only for the instantaneous stability of such systems, but also for the secular stability and evolution through Lidov-Kozai cycles and evection. We generalize the Hill-stability criteria to arbitrarily inclined triple systems, explain the existence of quasi-stable regimes and characterize the inclination dependence of their stability. We complement the analytic treatment with an extensive numerical study, to test our analytic results. We find excellent correspondence up to high inclinations $(\sim120^{\circ}$), beyond which the agreement is marginal. At such high inclinations the stability radius is larger, the ratio between the outer and inner periods becomes comparable, and our secular averaging approach is no longer strictly valid. We therefore combine our analytic results with polynomial fits to the numerical results to obtain a generalized stability formula for triple systems at arbitrary inclinations. Besides providing a generalized secular-based physical explanation for the stability of non co-planar systems, our results have direct implications for any triple systems, and in particular binary planets and moon/satellite systems; we briefly discuss the latter as a test case for our models.