Exoplanetary systems: The role of an equilibrium at high mutual inclination in shaping the global behavior of the 3-D secular planetary three-body problem
Exoplanetary systems: The role of an equilibrium at high mutual inclination in shaping the global behavior of the 3-D secular planetary three-body problem
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DOI：
10.1016/j.icarus.2007.05.007
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发表时间：
2007-11
期刊：
影响因子：
3.2
通讯作者：
Anne-Sophie Libert;J. Henrard
中科院分区：
文献类型：
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作者：
Anne-Sophie Libert;J. Henrard
基于高阶（12阶）微扰势的偏心率和倾角的幂展开，我们分析了两个非共面行星的长期相互作用。该模型基于行星三体问题，其可以通过众所周知的节点消除而减少到两个自由度[Jacobi，C. G. J.，1842.纳赫宇航员。XX，81-102]。我们引入了非奇异的标准变量，提出了问题的对称性。主要的动力学特征取决于平衡点的位置和稳定性，这些平衡点很容易用我们的分析模型找到。我们发现，存在一个平衡时，两个偏心率为零。当相互倾斜很小时，这种平衡是稳定的，但对于较大的相互倾斜，它变得不稳定，产生一个大的混沌区，并通过分叉，两个规则区域，所谓的Kozai共振。这种分析研究，这取决于只有两个参数（半长轴的比例和行星的质量比），使可能的问题进行大调查，使我们能够确定和量化其主要的动力学特征，周期轨道，定期和混乱的区域，等我们的分析模型的结果进行了说明，并证实了数值积分。
On the basis of a high-order (order 12) expansion of the perturbative potential in powers of the eccentricities and the inclinations, we analyze the secular interactions of two non-coplanar planets which are not in mean-motion resonance. The model is based on the planetary three-body problem which can be reduced to two degrees of freedom by the well-known elimination of the nodes [Jacobi, C.G.J., 1842. Astron. Nachr. XX, 81–102]. We introduce non-singular canonical variables which bring forward the symmetries of the problem. The main dynamical features depend on the location and stability of the equilibria which are easily found with our analytical model. We find that there exists an equilibrium when both eccentricities are zero. When the mutual inclination is small, this equilibrium is stable, but for larger mutual inclination it becomes unstable, generating a large chaotic zone and, by bifurcation, two regular regions, the so-called Kozai resonances. This analytical study which depends on only two parameters (the ratio of the semi-major axes and the mass ratio of the planets) makes possible a large survey of the problem and enables us to identify and quantify its main dynamical features, periodic orbits, regular and chaotic zones, etc. The results of our analytical model are illustrated and confirmed by numerical integrations.