A second-order angular momentum theory for the Cassini states of large satellites with presence of a subsurface ocean

We here develop an angular momentum approach to study the Cassini states (CS) of large satellites such as the Galilean satellites and Titan. Contrary to classical approaches where obliquity is the solution of a trigonometric equation and is constant-over-time, our dynamical approach allows us to identify not only the mean obliquity of satellites, but also their nutation in space as well as their polar motion (PM) with respect to the solid surface.

Large icy satellites are assumed to be in a CSI configuration with an obliquity close to zero. We investigate this equilibrium state by solving the dynamical equations for a fully rigid satellite and without averaging the external torque over the mean anomaly. We obtain mean obliquities close to those obtained from the classical studies. In addition to that mean value, our second-order dynamical method allows us to identify CSI nutations at quasi semi-diurnal and diurnal periods (and with amplitudes ranging from tens to thousands of mas) as well as one diurnal and one long-period PM. For Io, Europa and Ganymede, the diurnal and long-period PMs are of the same order of magnitude, while the long-period PM dominates the solution for Callisto and Titan.

We next extend our dynamical model to include the presence of a subsurface ocean and predict the orientation of the spin axes of the outer shell, internal ocean and solid interior. We find five eigenmodes, including a Free Ocean Nutation (FON) and an Inner Chandler Wobble (ICW). Far from resonance with one of these eigenmode, the obliquity asymptotically tends towards values close to or slightly higher than that of a fully solid satellite. Depending on the shell and ocean thickness, PM is dominated by its diurnal or long-period term, with amplitudes that can reach tens or even hundreds of meters. Large amplitudes of nutation and PM can result from a resonance between the node precession and the FON or between the pericenter precession and the ICW.