Third-order analytical solutions around non-collinear equilibrium points of a contact binary asteroid

The third-order analytical solution for orbital motion around the non-collinear equilibrium points (EPs) of a contact binary asteroid is constructed in this paper. A contact binary asteroid is an asteroid consisting of two lobes that are in physical contact. It is represented by the combination of an ellipsoid and a sphere. The gravity field of the ellipsoid is approximated by a spherical harmonic expansion with terms C₂₀, C₂₂, C₄₀, and, and the sphere by a straightforward point mass model. The non-collinear EPs are linearly stable for the asteroids with slow rotation rates, and become unstable as the rotation rates goes up. To study the motion around the stable EPs, a third-order analytical solution is constructed, by the Lindstedt-Poincaré (LP) method. A good agreement is found between this analytical solution and numerical integrations for the motion in the vicinity of the stable EPs. Its accuracy decreases when the orbit goes further away from the EPs and the asteroid rotates faster.