Modeling the Cassini States of large icy satellites with an angular momentum approach

It is generally assumed that the large icy satellites of Jupiter and of Saturn are, like our Moon, in an equilibrium rotation state called a Cassini State. In this state, the rotation of the satellite is synchronous with the orbital motion and the precession rate of the rotation axis is equal to that of the normal to the orbit. Moreover, the spin axis of the satellite, the normal to its orbit and the normal to the inertial plane remain coplanar and the obliquity (the angle between the normal to the orbit and the spin axis) is constant over time. For satellites with a slow orbital precession rate like the large icy satellites, up to four Cassini states are possible, characterized by a (theoretically) constant obliquity close to 0 (CSI), ± π/2 (CSII and IV) and π (CSIII). From these four states, only two are stable: CSI and CSIII.

We here model these two stable Cassini States of triaxial satellites using an angular momentum approach. In our model, the motion of the spin motion in space is coupled with the polar motion of the satellite and, contrary to what is usually done in the classical Cassini States studies, we do not average the external gravitational torque over short period terms. We can therefore compute the mean obliquity value of the different satellites but also the nutations (small periodic variations) both in obliquity and in longitude, which are due to the periodic variations of the gravitational torque acting on the satellites. We identify and study two causes for the polar motion and their effects on the obliquity: the semi-diurnal polar motion due to the inclination and node precession and the long period polar motion due to the eccentricity and pericenter precession.