递阶三重系统的Lagrange与Lyapunov稳定性：依赖于内外轨道间的相互倾斜
Lagrange vs. Lyapunov stability of hierarchical triple systems: dependence on the mutual inclination between inner and outer orbits
摘要：
虽然已经有许多研究检查的稳定性的分层三重系统，“稳定性”的含义是有点模糊，并已在以前的文献中有不同的解释。本文着重于“拉格朗日稳定性”，它大致是指对逃逸的一个机构从系统中，或“破坏”的三重系统的稳定性，相对于“类似李雅普诺夫稳定性”，这是有关系统动力学的混沌性质。我们使用直接的$N$体模拟计算三重系统的演化，直到$10^7 P_\mathrm{out}$，这比以前的研究要长得多（$P_\mathrm{out}$是外体的初始轨道周期）。我们得到的破坏时间尺度$T_\mathrm{d}$作为一个功能的三重轨道参数，特别注意的内部和外部轨道之间的相互倾斜，$i_\mathrm{mut}$的依赖。通过这样做，我们明确澄清了天文三元组的拉格朗日和李雅普诺夫稳定性之间的差异。此外，我们发现，冯齐佩尔-Kozai-Lidov振荡显着不稳定的倾斜三重态（大致与$60^\circ < i_\mathrm{mut} < 150^\circ$）相对于那些与$i_\mathrm{mut}=0^\circ$。另一方面，具有$i_\mathrm{mut}>160^\circ$的逆行三联体在长得多的破坏时间尺度下变得强烈稳定。给出了三重系统轨道参数对标准化破裂时标$T_\mathrm{d}/P_\mathrm{out}$的敏感性。由此产生的$T_\mathrm{d}/P_\mathrm{out}$分布实际上是更有用的，在广泛的天文应用比稳定性判据的基础上的李雅普诺夫散度。
英文摘要：
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.
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