广义相对论和四极矩扰动下共面非共振行星系统的长期动力学
Secular dynamics of coplanar, non-resonant planetary system under the general relativity and quadrupole moment perturbations
摘要：
我们构建了一个长期的理论，共面系统的N-行星不参与强平均运动共振，这是远离碰撞区。除了点到点的牛顿相互作用，我们考虑的广义相对论的恒星和最里面的行星的引力势的修正，也修改了这个潜在的四极矩和潮汐扭曲的恒星。我们关注的是等级行星系统。关于模型参数（质量，半长轴，恒星的自旋速率）的调查揭示了丰富的和非平凡的长期系统的动力学。我们的研究集中在它的均衡。经典久期理论预测的这种解对应于对准（模式I）或反对准（模式II）的顶点，可能会受到引力修正的强烈影响。所谓的真长期共振，这是一个新的特点，经典的两个行星的问题发现的Michtchenko和Malhotra（2004），可能会出现在其他地区，不同的相空间的广义模型。我们发现分叉模式II出现新的，但未知的文献中，secularly不稳定的平衡和相空间的复杂结构。这些平衡可能意味着长期不稳定的轨道配置，即使是最初的中等偏心。质点重力改正可以影响长期时间尺度的长期稳定性，而长期时间尺度的长期稳定性可能通过恒星的自转速率直接依赖于恒星的年龄。我们还分析了长期动力学的上升仙女座系统的广义模型的领域。同样在这个三行星系统的例子中，可能会出现新的长期平衡。
英文摘要：
We construct a secular theory of a coplanar system of N-planets not involved in strong mean motion resonances, and which are far from collision zones. Besides the point-to-point Newtonian mutual interactions, we consider the general relativity corrections to the gravitational potential of the star and the innermost planet, and also a modification of this potential by the quadrupole moment and tidal distortion of the star. We focus on hierarchical planetary systems. A survey regarding model parameters (the masses, semi-major axes, spin rate of the star) reveals a rich and non-trivial dynamics of the secular system. Our study is focused on its equilibria. Such solutions predicted by the classic secular theory, which correspond to aligned (mode I) or anti-aligned (mode II) apsides, may be strongly affected by the gravitational corrections. The so called true secular resonance, which is a new feature of the classic two-planet problem discovered by Michtchenko & Malhotra (2004), may appear in other, different regions of the phase space of the generalized model. We found bifurcations of mode II which emerge new, yet unknown in the literature, secularly unstable equilibria and a complex structure of the phase space. These equilibria may imply secularly unstable orbital configurations even for nitially moderate eccentricities. The point mass gravity corrections can affect the long term-stability in the secular time scale, which may directly depend on the age of the host star through its spin rate. We also analyze the secular dynamics of the upsilon Andromede system in the realm of the generalized model. Also in this case of the three-planet system, new secular equilibria may appear.
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