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arXiv 1003.3308astro-ph.EP
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三行星系统的动力学:动力系统的一种方法

The Dynamics of Three-Planet Systems: an Approach from Dynamical System

Bungo Shikita, Hiroko Koyama, Shoichi Yamada

摘要:

我们详细研究了三颗行星在一颗中心恒星的影响下相互作用的运动。众所周知,有两个以上行星的系统在长时间保持准稳定后变得不稳定,导致一些行星的高度偏心轨道运动或喷射。在本文中,我们关注的是这种准稳定性以及随后的不稳定性的物理基础,并提倡所谓的“停滞运动”在相空间中,这已经在动力系统领域进行了探索。我们采用李雅普诺夫指数、轨道根数功率谱和准稳定运动持续时间分布来分析三行星系统的相空间结构,这是最简单的,也是最有代表性的一个表明不稳定性的相空间结构。从李雅普诺夫指数可以看出,系统在初始准稳定状态下几乎是非混沌的,而在此之后则变为间歇性混沌。非混沌运动在作用角图中产生水平密集带,而空洞对应于混沌运动。我们得到的轨道偏心率的功率谱的幂律。幂律分布也被发现的准稳定状态的持续时间。将这些结果综合起来,我们可以得到这样一幅图:相空间由所谓的KAM环面和卫星环面组成,它们嵌入混沌海中。卫星环面具有自相似分布,并且负责持续时间的无标度幂律分布。该系统在一个KAM环和卫星环之间长期处于停滞状态,并不时地向另一个KAM环移动,表现出间歇性的混沌行为.

英文摘要:

We study in detail the motions of three planets interacting with each other under the influence of a central star. It is known that the system with more than two planets becomes unstable after remaining quasi-stable for long times, leading to highly eccentric orbital motions or ejections of some of the planets. In this paper, we are concerned with the underlying physics for this quasi-stability as well as the subsequent instability and advocate the so-called "stagnant motion" in the phase space, which has been explored in the field of dynamical system. We employ the Lyapunov exponent, the power spectra of orbital elements and the distribution of the durations of quasi-stable motions to analyze the phase space structure of the three-planet system, the simplest and hopefully representative one that shows the instability. We find from the Lyapunov exponent that the system is almost non-chaotic in the initial quasi-stable state whereas it becomes intermittently chaotic thereafter. The non-chaotic motions produce the horizontal dense band in the action-angle plot whereas the voids correspond to the chaotic motions. We obtain power laws for the power spectra of orbital eccentricities. Power-law distributions are also found for the durations of quasi-stable states. All these results combined together, we may reach the following picture: the phase space consists of the so-called KAM tori surrounded by satellite tori and imbedded in the chaotic sea. The satellite tori have a self-similar distribution and are responsible for the scale-free power-law distributions of the duration times. The system is trapped around one of the KAM torus and the satellites for a long time (the stagnant motion) and moves to another KAM torus with its own satellites from time to time, corresponding to the intermittent chaotic behaviors.

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