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Inertial modes of a freely rotating ellipsoidal planet and their relation to nutations

We compute the inertial modes of a freely rotating two-layer planetary model with an ellipsoidal inviscid fluid core and a perfectly rigid mantle. We present a method to derive analytical formulae for the frequencies of the Free Core Nutation (FCN) and Chandler Wobble (CW) which are valid to all orders of the dynamical flattening of the core and mantle, and we show how the FCN and CW are the direct generalisation of the purely fluid Spin-Over mode (SO) and of the Eulerian Wobble (EW) to the case where the mantle can oscillate freely around a state of steady rotation. Through a numerical computation for an axisymmetric (oblate spheroidal) planet, we demonstrate that all other inertial modes of the steadily rotating fluid core are also free modes of the freely rotating two-layer planet.