Geography of the rotational resonances and their stability in the ellipsoidal full two body problem

A fourth-order Hamiltonian describing the planar full two body problem is obtained, allowing for a mapping out of the geography of spin-spin-orbit resonances. The expansion of the mutual potential function up to the fourth-order results in the angles to come through one single harmonic and consequently the rotation of both bodies and mutual orbit are coupled. Having derived relative equilibria, stability analysis showed that the stability conditions are independent of physical and orbital characteristics. Simultaneously chaotic motion of bodies is investigated through the Chirikov diffusion utilizing geographic information of the complete resonances. The results show that simultaneous chaos among the binary asteroids is not expected to be prevalent due to the mass distribution of primary in compare with secondary. If mass distribution of bodies is of the same order, simultaneous chaos and global instability are achievable.