TY - JOUR

T1 - The dynamics of three-planet systems

T2 - An approach from a dynamical system

AU - Shikita, Bungo

AU - Koyama, Hiroko

AU - Yamada, Shoichi

PY - 2010

Y1 - 2010

N2 - 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 a 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. With 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.

AB - 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 a 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. With 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.

KW - Celestial mechanics

KW - Chaos

KW - Planets and satellites: dynamical evolution and stability

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U2 - 10.1088/0004-637X/712/2/819

DO - 10.1088/0004-637X/712/2/819

M3 - Article

AN - SCOPUS:77950277243

VL - 712

SP - 819

EP - 832

JO - Astrophysical Journal

JF - Astrophysical Journal

SN - 0004-637X

IS - 2

ER -