AI 中文总结
研究高偏心率下分层三体问题中摆动 Kozai-Lidov 循环的长期八极矩演化，通过解析得出二阶项剩余动力学为单摆，给出法向角动量振荡等结果，模型与数值积分相符，适用于多种分层三体情况。
AI 中文摘要
我们在双平均受限分层三体问题中，解析求解了具有负 Kozai 常数的摆动 Kozai-Lidov 循环（近日点幅角摆动的循环）的长期八极矩演化。摆动循环在轨道角动量的法向分量消失时达到极端偏心率，但其缓慢动力学此前未得到解决。我们表明，在八极矩强度的二阶项中，剩余动力学是一个具有显式系数的单摆。该单摆预测了法向角动量的缓慢振荡，其振幅与八极矩强度呈线性关系，在一个约为长期时间尺度除以八极矩强度的时间尺度上演化，同时给出轨道翻转的明确判据。在分层三体典型的八极矩强度下，这些振幅与旋转循环的振幅相当，在摆动窗口的部分区域甚至可能超过旋转循环的振幅，前提是等待相应更长的时间尺度。该解析模型与在摆动窗口内对双平均方程的数值积分结果一致，适用于研究偏心 Kozai-Lidov 效应的分层三体（从热木星形成到引力波源）的典型八极矩强度。
英文摘要
We solve analytically the long-term octupole evolution of \textit{librating} Kozai-Lidov cycles - those with a negative Kozai constant, in which the argument of pericenter librates - in the double-averaged restricted hierarchical three-body problem. Librating cycles reach extreme eccentricities when the normal component of the orbital angular momentum vanishes, just as rotating cycles do, but their slow dynamics was left unsolved: as noted by Katz, Dong \& Malhotra (2011), the azimuth of the eccentricity vector jumps by half a turn every cycle, so the leading-order octupole kick alternates in sign and cancels pairwise. We show that the surviving dynamics, at second order in the octupole strength, is a simple pendulum with explicit coefficients. The pendulum predicts slow oscillations of the normal angular momentum with amplitude linear in the octupole strength, evolving on a timescale of order the secular timescale divided by the octupole strength, together with an explicit criterion for orbital flips. At octupole strengths typical of hierarchical triples, these amplitudes are comparable to - and in part of the librating window can exceed - those of rotating cycles, provided one waits the correspondingly longer timescale. The analytic model agrees with numerical integrations of the double-averaged equations across the librating window, for octupole strengths typical of the hierarchical triples in which the eccentric Kozai-Lidov effect is studied - from hot-Jupiter formation to gravitational-wave sources.
Journal refMon Not R Astron Soc (2026)
