Lagrange vs. Lyapunov stability of hierarchical triple systems: dependence on the mutual inclination between inner and outer orbits

Abstract

While there have been many studies examining the stability of hierarchical triple systems, the meaning of ``stability'' is somewhat vague and has been interpreted differently in previous literatures. The present paper focuses on ``Lagrange stability'', which roughly refers to the stability against the escape of a body from the system, or ``disruption'' of the triple system, in contrast to ``Lyapunov-like stability'' that is related to the chaotic nature of the system dynamics. We compute the evolution of triple systems using direct $N$-body simulations up to $10^7{P}_{\mathrm{out}}$, which is significantly longer than previous studies (with $P_{\mathrm{out}}$ being the initial orbital period of the outer body). We obtain the resulting disruption timescale $T_{\mathrm{d}}$ as a function of the triple orbital parameters with particular attention to the dependence on the mutual inclination between the inner and outer orbits, $i_{\mathrm{mut}}$. By doing so, we have clarified explicitly the difference between Lagrange and Lyapunov stabilities in astronomical triples. Furthermore, we find that the von Zeipel-Kozai-Lidov oscillations significantly destabilize inclined triples (roughly with $60^{\circ }<i_{\mathrm{mut}}<150^{\circ }$) relative to those with $i_{\mathrm{mut}}=0^{\circ }$. On the other hand, retrograde triples with $i_{\mathrm{mut}}>160^{\circ }$ become strongly stabilized with much longer disruption timescales. We show the sensitivity of the normalized disruption timescale $T_{\mathrm{d}}/P_{\mathrm{out}}$ to the orbital parameters of triple system. The resulting $T_{\mathrm{d}}/P_{\mathrm{out}}$ distribution is practically more useful in a broad range of astronomical applications than the stability criterion based on the Lyapunov divergence.

arXiv comment13 pages, 3 figures, ApJ, in press

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