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 shares another large domain in the rotation 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 oscillations around the synchronism is similar to the orbital period. Being that the current rotational states of this sort of satellite should be synchronous (Thomas and Helfenstein 2020), our results can be taken into account in evolutionary studies of their rotation.