Orbital Motion in the Vicinity of the Non-collinear Equilibrium Points of a Contact Binary Asteroid

The orbital motion around the non-collinear equilibrium points (EPs) of a contact binary asteroid is investigated in this paper. A contact binary asteroid is an asteroid consisting of two lobes that are in physical contact. Here, 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 and _C40, and the sphere by a straightforward point mass model. The non-collinear EPs are linearly stable for asteroids with slow rotation rates, and become unstable as the rotation rate 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. For the unstable EPs, the motions around them are unstable as well. Therefore, the linear feedback control law based on low thrust is introduced to stabilize the motion and track the reference trajectory. In addition, more control force is required as any of the injection error, the amplitude of the analytical reference orbit or the rotation rate of the asteroid increases. For small orbits around the EPs, the third-order analytical solution can serve as a good reference trajectory. However, for large amplitude orbits, accurate numerical orbits are to be used as reference. This avoids an extra control force to track the less accurate third-order analytical solution.