标题： Brown Hamiltonian-II的扩展。 修改后的von Zeipel-Lidov-Kozai效应的分析研究 显示英文标题

标题： Extensions of Brown Hamiltonian-II. Analytical study on the modified von Zeipel-Lidov-Kozai effects

摘要： 在弱分层三体系统中，来自内双星和外双星周期振荡的非线性扰动在其长期动力学演化中起着关键作用。 在此背景下，我们在论文I中开发了一个扩展的布朗哈密顿量，它作为描述修正的冯·泽佩尔-利多夫-科扎伊（ZLK）振荡的基本模型。 本研究旨在分析该扩展框架内的ZLK振荡特征，重点关注相空间结构、ZLK中心的位置、达到的最大偏心率、摆动周期的边界以及触发ZLK共振所需的临界倾角。 在扩展哈密顿量下，我们引入了Lidov积分C_ZLK，它是哈密顿量和角动量z分量的组合，用于表征修正后的ZLK特性。 发现摆动和环绕周期由C_ZLK=0分隔，这与经典理论一致。 此外，我们使用摄动技术推导了这些ZLK特性的解析表达式。 解析预测与数值结果进行了比较，显示出两者之间良好的一致性。 值得注意的是，结果表明，在布朗修正的影响下，顺行和逆行区域中的ZLK特性不再对称。 最后，我们进行了$N$-体积分，涉及数百万个轨道以生成动力学图，其中数值结构被扩展模型导出的解析解很好地捕捉到了。 显示英文摘要

摘要： In triple systems of weak hierarchies, nonlinear perturbations arising from the periodic oscillations associated with the inner and outer binaries play a crucial role in shaping their long-term dynamical evolution. In this context, we have developed an extended Brown Hamiltonian in Paper I, which serves as a fundamental model for describing the modified von Zeipel-Lidov-Kozai (ZLK) oscillations. The present work aims to analyze the characteristics of ZLK oscillations within this extended framework, focusing on phase-space structures, the location of ZLK center, the maximum eccentricity reached, the boundaries of librating cycles, and the critical inclination required to trigger ZLK resonance. Under the extended Hamiltonian, we introduce the Lidov integral C_ZLK, which is a combination of the Hamiltonian and the z-component of angular momentum, to characterize the modified ZLK properties. It is found that the librating and circulating cycles are separated by C_ZLK=0, which is consistent with the classical theory. Furthermore, we derive analytical expressions of these ZLK properties using perturbation techniques. Analytical predictions are compared to numerical results, showing an excellent agreement between them. Notably, the results reveal that ZLK characteristics in prograde and retrograde regimes are no longer symmetric under the influence of Brown corrections. At last, we conduct $N$-body integrations about millions of orbits to generate dynamical maps, where the numerical structures are well captured by the analytical solutions derived from the extended model.