Level repulsion and threefold degeneracy of eigenstates of the Liouvillian in the Kirkwood gaps of asteroid belt

Abstract

The size of the Kirkwood gaps in the distribution of main belt asteroids in the solar system is estimated in terms of the eigenvalue problem of the Liouville operator (the Liouvillian). The problem is treated as an example of a restricted three-body problem that consists of an asteroid, Sun, and Jupiter. Jupiter is treated as a, perturbation on the asteroidSun two-body problem. It is well known that the physical dimension of the eigenvalue of the Liouvillian is a frequency. We found that the eigenstate of the Liouvillian of this twobody problem has a threefold degeneracy at the resonance point inside the Kirkwood gaps. Using the degenerate perturbation theory which has been extensively developed in quantum mechanics, we can analyze the resonance effect on the resonance point without divergence. The perturbation due to Jupiter removes the degeneracy and results in a level repulsion in the eigenvalues, just as the same mechanism of the level repulsion in the eigenvalues of the Hamiltonian for an electron in quantum solid-state physics. As a result, the spectrum of the Liouvillian for the three-body problem has a band structure in the frequency space. Due to the resonance effect, the sizes of the gaps in the band structure are linearly proportional to the intensity of the perturbation, instead of the second order obtained in off-resonance region for the usual perturbation theory. Hence, the resonance effect is much larger than the off-resonance effect, as expected. Since the degeneracy is threefold, there is a stable eigenstate inside the resonance region. The stability is independent of the intensity of the perturbation determined by the eccentricity of the asteroid. This explains the stability of some asteroids with moderately large eccentricity that are located on the resonance orbit inside the Kirkwood gaps.