Higher-order effects in the dynamics of hierarchical triple systems: Quadrupole-squared terms
Higher-order effects in the dynamics of hierarchical triple systems: Quadrupole-squared terms
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DOI：
10.1103/physrevd.103.063003
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发表时间：
2020-11
影响因子：
5
通讯作者：
C. Will
中科院分区：
文献类型：
--
作者：
C. Will
我们分析了由遥远的第三天体在内部双星上引起的四极扰动中分层三重系统向二阶的长期演化。牛顿三体运动方程以半长轴 $a/A$ 之比的幂展开，成为一对有效的单体开普勒运动方程，受到一系列多极扰动的扰动，表示为四极、$O[(a/A{)}^{3}]$、八极、$O[(a/A{)}^{4}]$ 等等。在瞬时轨道元素演化的拉格朗日行星方程中，二阶效应是通过获得每个元素的一阶解而产生的，该解由一个常数（或缓慢变化）部分和一个振荡微扰部分组成，并将其重新插入方程以获得二阶解。在对两个轨道时间尺度进行平均以获得长期演化后，这些二阶四极 (${Q}^{2}$) 项预计会产生 $(a/A{)}^{6}$ 阶的效应。然而，我们发现轨道平均值实际上通过外轨道周期与内轨道周期之比$\ensuremath{\sim}(A/a{)}^{3/2}$增强了二阶项。对于具有低质量第三天体的系统，${Q}^{2}$ 影响很小，但对于具有相当质量或非常大质量的第三天体的系统，例如绕太阳质量恒星运行的太阳-木星系统，或绕 ${10}^{6}\text{ }\text{ 运行的 $100\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ 双星系统}{M}_{\ensuremath{\bigodot}}$大质量黑洞，${Q}^{2}$效应可以完全抑制一阶解中发生的内轨道从顺行到逆行以及返回的翻转。这些结果与 Luo、Katz 和 Dong 使用“校正双平均”方法得出的结果完全一致。
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, $\ensuremath{\sim}(A/a{)}^{3/2}$. 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 $100\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ binary system orbiting a ${10}^{6}\text{ }\text{ }{M}_{\ensuremath{\bigodot}}$ 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.