The Poincaré-Hough model applied to the synchronous rotation at low eccentricity

This paper presents a study of the Poincaré-Hough model of rotation of the synchronous natural satellites, in which these bodies are assumed to be composed of a rigid mantle and a triaxial cavity filled with inviscid fluid of constant uniform density and vorticity. In considering an Io-like on a low eccentric orbit, we describe the different possible behaviors of the system, depending on the size, polar flattening and shape of the core. We use for that the numerical tool. Starting from a Hamiltonian formulation of the system, we derive the Hamilton equations before integrating them numerically. Then, we use a frequency analysis algorithm to give a quasi-periodic representation, allowing us to split the different contributions and to characterise the equilibrium of the system. We show that, even without orbital eccentricity, the system can have complex behaviors, in particular when the core is highly flattened. In such a case, the polar motion is forced by several degrees and longitudinal librations appear. We also get a shift of the obliquity when the polar flattening of the core is small.