Theoretical Analysis of Asteroid Belt Based on the Liouvillian Dynamics

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Other Title
  • リウビル力学に基づいた小惑星分布の縞構造の分析
  • 交流 リウビル力学に基づいた小惑星分布の縞構造の分析
  • コウリュウ リウビル リキガク ニ モトズイタ ショウワクセイ ブンプ ノ シマ コウゾウ ノ ブンセキ

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Abstract

<p>The distribution of asteroids in solar system is not uniform due to the perturbation of the gravity of Jupiter. There are Kirkwood's gaps in the distribution at some resonance points, while there are also peaks in the distribution at other resonance points. We investigate the distribution of asteroids in terms of the eigenvalue problem of the Liouvillian defined by the Poisson bracket with the Hamiltonian instead of solving trajectory dynamics. In the three-body problem of an asteroid, Sun and Jupiter, unperturbed eigenstates of the Liouvillian for the two-body problem are degenerate due to the resonance. We demonstrate that the perturbation of Jupiter on the distribution of asteroids at resonance is described by the degenerate perturbation theory that is a technique developed in quantum mechanics. At the Kirkwood's gap, the perturbation of Jupiter removes the degeneracy and results in the level repulsion in the eigenvalues of the Liouvillian. For 3 : 2 resonance, Hilda triangle in the asteroid distribution is successfully reproduced by the spatial pattern of the steady state with zero eigenvalue of the Liouvillian. These results show that the Liouvillian dynamics based on the concept of the classical states is useful in analyzing classical systems at resonance.</p>

Journal

  • Butsuri

    Butsuri 72 (2), 121-126, 2017

    The Physical Society of Japan

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