论文标题
分层三重系统动力学中的高阶效应。四极方术语
Higher-order effects in the dynamics of hierarchical triple systems. Quadrupole-squared terms
论文作者
论文摘要
我们分析了层次三重系统的世俗演化到遥远的第三体对内二进制诱导的四极扰动中的二阶进化。 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.在瞬时轨道元件演化的Lagrange行星方程中，二阶效应是由获得每个元素的一阶解，由常数（或缓慢变化）的零件和振动扰动片组成，并将其重新插入后，以获得二阶溶液。在两个轨道时标的平均值以获得长期演变之后，这些二阶四杆（$ q^2 $）条款有望产生订单$（a/a）^6 $的效果。但是，我们发现轨道平均值实际上通过外部轨道周期的比例（$ \ sim（a/a）^{3/2} $的一个因子增强了二阶项。对于具有低质量第三身体的系统，$ q^2 $效果很小，但是对于具有可比性或非常巨大的第三身体的系统（例如，绕太阳能质量星星绕的阳光jupiter系统）或$ 100 \，m_ \ odot $ binary System System在一阶溶液中发生的逆行和背部。这些结果与Luo，Katz和Dong的结果完全一致，该结果使用“校正的双重平均”方法得出。
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.
