Generalized Hill-stability criteria for hierarchical three-body systems at arbitrary inclinations

Abstract

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