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 non-synchronous rotation (Tisserand, Mécanique Céleste, t.II, Chap. 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 ǫ k , µ 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 ǫ k = H k ǫ h and µ k = H k µ h , where H k is a characteristic coefficient for each layer which only depends on the internal structure of the body and ǫ h , µ 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.

The Concentric Maclaurin Spheroid method with tides and a rotational enhancement of Saturn’s tidal response

“…Within these constraints, we show below that our extended CMS method yields results that are in excellent agreement with results from Folonier et al (2015).…”

“…Here we demonstrate that, for a rapidly-rotating giant planet, the latter terms make a significant contribution to the love numbers k nm , as well as (unobservably small) tidal contributions to the gravitational moments J n . Folonier et al (2015) presented a method for approximating the love numbers 75 of a non-homogeneous body using Clairaut theory for the equilibrium ellipsoidal figures. This results in an expression for the love number k 2 for a body composed of concentric ellipsoids, parameterized by their flattening parameters.…”

“…However, in bodies with more 80 complicated density distributions, the equipotential surfaces will have a more general spheroidal shape. Because of the small magnitude of tidal perturbations, the method of Folonier et al (2015) works in the limit of slow rotation despite this limitation. However, the method does not account for the coupled effect of tides and rotation, and does not predict love numbers of order higher than k 2 .…”

Tidal synchronization of an anelastic multi-layered body: Titan’s synchronous rotation

“…In this article we extended the static equilibrium figure of a multi-layered body, presented in Folonier et al (2015), to the viscous case, adapting it to allow the differential rotation of the layers. For this sake, we used the Newtonian creep tide theory, presented in Ferraz-Mello (2013) and Ferraz-Mello (2015a).…”

“…The static equilibrium figure of one body composed by N homogeneous layers, under the action of the tidal potential and the non-synchronous rotation, when all layers rotate with the same angular velocity, was calculated by Folonier et al (2015). 1 In this work, we need, beforehand, to extend these results to the case in which each layer has one different angular velocity.…”