A simple analytical model for the secular resonance ?6 in the asteroidal belt

Abstract

When asteroids are in the secular resonance ν6, the variation of the eccentricity becomes very large. In this paper, the dynamics of this secular resonance ν6 is investigated by a simple analytical model, in which the third degree terms of the eccentricity and inclination are taken into account. The eccentricity variations of asteroids located near this resonance are represented clearly by the diagrams of equi-Hamiltonian curves on the plane of $$\tilde \omega - \tilde \omega _S$$ versuse ( $$\tilde \omega ,\tilde \omega _S :$$ the longitude of perihelion of asteroids and Saturn,e: the eccentricity of asteroids). These diagrams predict that the eccentricity of these asteroids suffers a large increase or decrease, and that the secular resonance argument $$\tilde \omega - \tilde \omega _S$$ librates about 0° and 180°. In order to confirm these predictions, numerical integrations are carried out over one million years. By these integrations, it is found that the eccentricity of secular resonant asteroids becomes more than 0.8, and that the libration about 0° also exists, as well as the libration about 180°. The strongly depopulated region in the asteroidal belt, which corresponds to the position of the secular resonance ν6, is also explained well by this analytical model.