Lagrange versus 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,” which 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 _out , which is significantly longer than previous studies (with P _out being the initial orbital period of the outer body). We obtain the resulting disruption timescale T _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 _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° 160° become strongly stabilized with much longer disruption timescales. We show the sensitivity of the normalized disruption timescale T _d / P _out to the orbital parameters of triple system. The resulting T _d / P _out distribution is practically more useful in a broad range of astronomical applications than the stability criterion based on the Lyapunov divergence.

Keywords

• Celestial mechanics
• Three-body problem
• Multiple stars
• Compact binary stars
• Orbital evolution