We study the stationary points of the hierarchical three body problem in the planetary limit (m_2, m_3 << m_1) at both the quadrupole and octupole orders. We demonstrate that the extension to octupole order preserves the principal stationary points of the quadrupole solution in the limit of small outer eccentricity e_2 but that new families of stable fixed points occur in both prograde and retrograde cases. The most important new equilibria are those that branch off from the quadrupolar solutions and extend to large e_2. The apsidal alignment of these families is a function of mass and inner planet eccentricity, and is determined by the relative directions of precession of omega_1 and omega_2 at the quadrupole level. These new equilibria are also the most resilient to the destabilizing effects of relativistic precession. We find additional equilibria that enable libration of the inner planet argument of pericentre in the limit of radial orbits and recover the non-linear analogue of the Laplace-Lagrange solutions in the coplanar limit. Finally, we show that the chaotic diffusion and orbital flips identified with the Eccentric Kozai Lidov mechanism and its variants can be understood in terms of the stationary points discussed here.

中文翻译：

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层次三体问题的静止点

我们在四极和八极阶研究行星极限 (m_2, m_3 << m_1) 中分层三体问题的驻点。我们证明了八极阶数的扩展在小外偏心率 e_2 的极限内保留了四极解的主要驻点，但是在顺行和逆行情况下都会出现新的稳定不动点族。最重要的新平衡是那些从四极解分支出来并扩展到大 e_2 的平衡。这些家族的顶点排列是质量和内行星偏心率的函数，由四极水平上 omega_1 和 omega_2 的相对进动方向决定。这些新的平衡也是相对论进动的不稳定影响最有弹性的。我们发现了额外的平衡，可以在径向轨道的极限内平衡内行星中心的自变量，并在共面极限内恢复拉普拉斯-拉格朗日解的非线性类似物。最后，我们表明可以根据此处讨论的驻点来理解与偏心 Kozai Lidov 机制及其变体相关的混沌扩散和轨道翻转。