A Hamiltonian for 1/1 Rotational Secondary Resonances, and Application to Small Satellites of Saturn and Jupiter

In this work, we study the dynamics of rotation of the small satellites Methone and Aegaeon and revisit previous works on the rotation of Prometheus, Metis, and Amalthea. In all cases, the surfaces of section computed with the standard spin-orbit model reveal that the synchronous regime with small amplitude of libration shares another large domain in the phase space. We reproduce and apply the hamiltonian theory given in Wisdom (2004) to analytically characterize the detected structure as being a secondary resonance where the period of oscillation of the physical libration is similar to the orbital period of the satellite. We also show that the amplitude of libration around the secondary resonance is always larger than in the case of the other mode. Since the current rotational states of these sorts of satellites should be synchronous, our results can be considered in evolutionary studies of their rotation.