We consider the Clairaut theory of the equilibrium ellipsoidal figures fordifferentiated non-homogeneous bodies in non-synchronous rotation adding to ita tidal deformation due to the presence of an external gravitational force. Weassume that the body is a fluid formed by $n$ homogeneous layers of ellipsoidalshape and we calculate the external polar flattenings and the mean radius ofeach layer, or, equivalently, their semiaxes. To first order in theflattenings, the general solution can be written as $\epsilon_k={\calH}_k*\epsilon_h$ and $\mu_k={\cal H}_k*\mu_h$, where $\cal{H}_k$ is acharacteristic coefficient for each layer which only depends on the internalstructure of the body and $\epsilon_h, \mu_h$ are the flattenings of theequivalent homogeneous problem. For the continuous case, we study the Clairautdifferential equation for the flattening profile, using the Radautransformation to find the boundary conditions when the tidal potential isadded. Finally, the theory is applied to several examples: i) a body composedof two homogeneous layers; ii) bodies with simple polynomial densitydistribution laws and iii) bodies following a polytropic pressure-density law.
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