论文标题
在高相互倾斜的轨道中，具有两个行星的外极系统中的相空间结构
The phase-space architecture in extrasolar systems with two planets in orbits of high mutual inclination
论文作者
论文摘要
我们重新审视了世俗的3D行星三体问题，目的是提供统一的形式主义，以研究两个行星轨道之间相互倾斜度$ i_ {mut} $逐渐更高值的相空间的结构。我们提出了一种“簿记”技术（减少雅各比之后），将世俗的哈密顿量的明确分解为$ h_ {sec} = h_ {planar} +h_ {space} $，其中$ h_ {space} $包含所有条款，这些条款取决于$ i_ {mut} $。我们从数值上比较了通过轨道偏心率或多极扩展中通过膨胀获得的几种模型。我们找到最小所需的截断订单以准确表示动力学。我们探索过渡，因为$ i_ {mut} $增加，从类似平面的'到``lidov-kozai''制度。使用典型的（非层次）示例，我们展示了在3D情况下，在很大程度上也重现了平面案例的相结合世俗动力学的相相肖像结构。我们半分析$ i_ {mut} $的水平几乎可以整合。在此制度中，我们提出了一种正常形式的方法，可以通过半分析来计算几乎可综合状态的基本周期轨道（apsidal colotation共振）。另一方面，随着能量的增加，系统逐渐移动到“利多夫 - 科扎伊”制度。后者由两个倾斜的周期性轨道（$ C_1 $和$ C_2 $）的两个不同家族主导，其中$ C_2 $通过通常的Lidov-Kozai机制变得不稳定。我们讨论上述周期轨道家族之间的联系。最后，我们从数值上研究了两个行星的不同质量和半轴轴比的相位肖像形式，旨在确定上述现象是如何在一个或多个层次结构范围内选择的。
We revisit the secular 3D planetary three-body problem aiming to provide a unified formalism for studying the structure of the phase space for progressively higher values of the mutual inclination $i_{mut}$ between the two planets' orbits. We propose a `book-keeping' technique yielding (after Jacobi reduction) a clear decomposition of the secular Hamiltonian as $H_{sec}=H_{planar} +H_{space}$, where $H_{space}$ contains all terms depending on $i_{mut}$. We numerically compare several models obtained via expansion in the orbital eccentricities or via multipole expansion. We find the mimimum required truncation orders to accurately represent the dynamics. We explore the transition, as $i_{mut}$ increases, from a `planar-like' to a `Lidov-Kozai' regime. Using a typical (non-hierarchical) example, we show how the structure of the phase portraits of the integrable secular dynamics of the planar case is reproduced to a large extent also in the 3D case. We estimate semi-analytically the level of $i_{mut}$ up to which the dynamics remains nearly-integrable. In this regime, we propose a normal form method by which the basic periodic orbits of the nearly-integrable regime (apsidal corotation resonances) can be computed semi-analytically. On the other hand, as the energy increases the system gradually moves to the `Lidov-Kozai' regime. The latter is dominated by two different families of inclined periodic orbits ($C_1$ and $C_2$), of which $C_2$ becomes unstable via the usual Lidov-Kozai mechanism. We discuss the connection between the above families of periodic orbits. Finally, we study numerically the form of the phase portraits for different mass and semi-major axis ratios of the two planets, aiming to establish how generic are the phenomena reported above as the systems parameters are chosen close to one or more hierarchical limits.
