The rotation of Io with a liquid core

In a previous paper (Henrard, Celest. Mech. Dyn. Astron. 178, 144-153, 2005c) we have developed an analytical theory of the rotation of the Galilean satellite Io, considered as a rigid body and based on a synthetic theory of its orbital motion due to Lainey (Theorie Dynamique des Satellites Galileens. PhD dissertation, Observatoire de Paris, 2002) (see also Lainey et al., A&A, 420, 1171-1183 2004a; A&A, 427, 371-376, 2004b). One of the most important causes of departure of the actual rotation from the rigid theory is thought to be the existence of a liquid core, the size of which is unknown but would be an important piece of information concerning the structure of the interior of the satellite. In this contribution we develop the analytical theory of a liquid core contained in a cavity filled by an inviscid fluid of constant uniform density and vorticity. The theory is based on Poincare (Bull. Astron. 27, 321-356, 1910) model and is developed by a Lie transform perturbation method, very much like in our previous contribution. Our main conclusion is that the addition of a degree of freedom (the spin of the core) with a frequency close to the orbital frequency multiplies the possibility of resonances and that for some particular size of the core one may expect a (possibly small) region of chaotic behaviour in the vicinity of the Cassini state.