Secular perturbations of asteroids with high inclination and eccentricity

Secular perturbations of asteroids with high inclination and eccentricity moving under the attraction of the sun and Jupiter are studied on the assumption that Jupiter's orbit is circular. After short-periodic terms in the Hamiltonian are eliminated, the degree of freedom for the canonical equations of motion can be reduced to 1. Since there is an energy integral, the equations can be solved by quadrature. When the ratio of the semi-major axes of the asteroid and Jupiter takes a very small value, the solutions are expressed by elliptic functions. When the z component of the angular momentum (that is, Delaunay's H) of the asteroid is smaller than a certain limiting value, there are both a stationary solution and solutions corresponding to libration cases. The limiting value of H increases as the ratio of the semimajor axes increases, i.e., the corresponding limiting inclination drops from 39.2° to 1.8° as the ratio of the axes increases from 0.0 to 0.95.