The flattenings of the layers of rotating planets and satellites deformed by a tidal potential

Abstract

We consider the Clairaut theory of the equilibrium ellipsoidal figures for differentiated nonhomogeneous bodies in nonsynchronous rotation (Tisserand, Mécanique Céleste, t.II, Chaps. 13 and 14) adding to it a tidal deformation due to the presence of an external gravitational force. We assume that the body is a fluid formed by $n$ homogeneous layers of ellipsoidal shape and we calculate the external polar flattenings $\epsilon _k,\, \mu _k$ and the mean radius $R_k$ of each layer or, equivalently, their semiaxes $a_k$, $b_k$, and $c_k$. To first order in the flattenings, the general solution can be written as $\epsilon _k=\mathcal {H}_k\epsilon _{h}$ and $\mu _k=\mathcal {H}_k\mu _{h}$, where $\mathcal {H}_k$ is a characteristic coefficient for each layer that depends only on the internal structure of the body and $\epsilon _{h}$ and $\mu _{h}$ are the flattenings of the equivalent homogeneous problem. For the continuous case, we study the Clairaut differential equation for the flattening profile using the Radau transformation to find the boundary conditions when the tidal potential is added. Finally, the theory is applied to several examples: (i) a body composed of two homogeneous layers, (ii) bodies with simple polynomial density distribution laws, and (iii) bodies following a polytropic pressure-density law.