Linear stability of inner case of double averaged spatial restricted elliptic three body problem

Abstract

We study the secular effects in the motion of an asteroid with negligible mass in a spatial restricted elliptic three body problem with arbitrary inclination. Averaging over mean anomalies of the asteroid and the planet are applied to obtain the double averaged Hamiltonian system. It admits a two-parameter family of orbits corresponding to the motion of the third body in the plane of primaries' motion. The aim of our investigation is to analyze the stability of these orbits in inner case. We show that they are stable in the linear approximation and give descriptions of linear stability with respect to the eccentricity and argument of periapsis of asteroid. Numerical simulations of different types of orbits are performed as well.