标题： 高阶效应在等级三体系统动力学中的影响。 四极矩平方项 显示英文标题

标题： Higher-order effects in the dynamics of hierarchical triple systems. Quadrupole-squared terms

摘要： 我们分析了等级三体系统的长期演化，考虑由远处第三体引起的内双星的四极摄动的二阶效应。 牛顿三体运动方程在半长轴比值 $a/A$ 的幂展开下，变成一对有效的单体开普勒运动方程，受到一系列多极摄动（记为四极、 $O[(a/A)^3]$、八极、 $O[(a/A)^4]$ 等）的扰动。 在瞬时轨道元素演化的拉格朗日行星方程中，二阶效应来源于获得每个元素的一阶解，包括一个常数（或缓慢变化）部分和一个振荡的扰动部分，再将其重新代入方程以得到二阶解。 经过两个轨道时间尺度的平均以获得长期演化后，这些二阶四极（$Q^2$）项预计会产生 $(a/A)^6$ 阶效应。 然而我们发现，轨道平均实际上将二阶项增强了外轨道周期与内轨道周期之比的倍数，即 $ \sim (A/a)^{3/2}$。 对于具有低质量第三天体的系统，$Q^2$效应很小，但对于具有相当质量和非常高质量的第三天体的系统（例如围绕太阳质量恒星运行的太阳-木星系统，或者围绕$10^6 \, M_\odot$大质量黑洞运行的$100 \, M_\odot$双星系统），$Q^2$效应可以完全抑制在一阶解中发生的从顺行到逆行再回到顺行的内轨道翻转现象。这些结果与Luo、Katz和Dong使用“修正双重平均法”得出的结果完全一致。 显示英文摘要

摘要： We analyze the secular evolution of hierarchical triple systems to second-order in the quadrupolar perturbation induced on the inner binary by the distant third body. The Newtonian three-body equations of motion, expanded in powers of the ratio of semimajor axes $a/A$, become a pair of effective one-body Keplerian equations of motion, perturbed by a sequence of multipolar perturbations, denoted quadrupole, $O[(a/A)^3]$, octupole, $O[(a/A)^4]$, and so on. In the Lagrange planetary equations for the evolution of the instantaneous orbital elements, second-order effects arise from obtaining the first-order solution for each element, consisting of a constant (or slowly varying) piece and an oscillatory perturbative piece, and reinserting it back into the equations to obtain a second-order solution. After an average over the two orbital timescales to obtain long-term evolutions, these second-order quadrupole ($Q^2$) terms would be expected to produce effects of order $(a/A)^6$. However we find that the orbital average actually enhances the second-order terms by a factor of the ratio of the outer to the inner orbital periods, $ \sim (A/a)^{3/2}$. For systems with a low-mass third body, the $Q^2$ effects are small, but for systems with a comparable-mass or very massive third body, such as a Sun-Jupiter system orbiting a solar-mass star, or a $100 \, M_\odot$ binary system orbiting a $10^6 \, M_\odot$ massive black hole, the $Q^2$ effects can completely suppress flips of the inner orbit from prograde to retrograde and back that occur in the first-order solutions. These results are in complete agreement with those of Luo, Katz and Dong, derived using a "Corrected Double-Averaging" method.