Creep tide theory: equations for differentiated bodies with aligned layers

Abstract

The creep tide theory is used to establish the basic equations of the tidal evolution of differentiated bodies formed by aligned homogeneous layers in co-rotation. The mass concentration of the body is given by the fluid Love number $k_f$. The formulas are given by series expansions valid for high eccentricity systems. They are equivalent to Darwin’s equations, but formally more compact. An application to the case of Enceladus, with $k_f=0.942$, is discussed.

Appendix: Equivalence of Darwin’s tidal theory for viscous bodies and the creep tide theory

In our previous papers, the equivalence of the variational equations derived from the creep tide theory and those of Darwin’s constant time lag (or CTL) theories was several times stressed. This equivalence is reinforced in this paper by the extension of the creep tide theory to a differentiated body with aligned co-rotating layers and the introduction of the actual fluid Love numbers.³

The restriction to the CTL theories stems from the fact that all versions of Darwin’s theory published in the XX$^{th}$ century (revisited in Ferraz-Mello et al. 2008) followed what was dubbed “Fall schwacher Reibung” by Gerstenkorn (1955), or “weak friction approximation” (Alexander 1973), in which the phase shifts, or lags, $\sigma _k$ are assumed to be small quantities. This postulate introduces in the theories one stringent approximation: Darwin’s “height" (also called “fraction of equilibrium tide”) $\cos \sigma _k$ becomes, in the first order of approximation, equal to 1 and disappears from the equations. When the factors $\cos \sigma _k$ missing in the CTL theories are reintroduced, we have total equivalence of the creep tide theory and Darwin’s theory for homogeneous bodies.

The approach resulting from the introduction of the weak friction hypotheses was discussed by Efroimsky and Williams (2009) in a section of their paper, with the title “The stone rejected by the builders”. They showed that the weak friction approximation was the culprit for some apparent singularities appearing in the equations, near the synchronism of rotational and orbital motions, when the ad hoc lags were taken proportional to a negative power of the frequency.

In Darwin paper (1880a), the phase shifts are inserted by hand, both in the case where they are kept undetermined and in the cases where they are fixed in agreement with his 1879 paper. However, under no circumstances did he assume that the phase shifts are small. On the contrary, there are in his paper (Darwin 1880a) examples with phase shifts close to 45$^{\circ }$ (the angles $f, g, \dots$ adopted by him correspond to $\frac{1}{2}\sigma _k$ in the creep tide theory). High phase shifts appear in the tidal deformation of bodies with very high viscosity (see examples in Ferraz-Mello et al. 2020).

Differences, however, exist between the two theories. The most obvious one is that the quantities playing the role of relaxation factor in these theories, ${\mathfrak {p}}$ and $\gamma$, respectively, are related to the viscosity according to different laws (such that $\gamma = 4.75 {\mathfrak {p}}$). In addition, even in the case of viscous bodies, Darwin prefers to introduce the phase shifts by hand, while in the creep tide theory they are introduced through the (approximated) solution of the creep differential equation. Major differences, however, appear when we consider the parametric version of the creep tide theory (Folonier et al. 2018; Ferraz-Mello et al. 2020). These equations allow us to obtain a system of differential equations for the parameters defining the shape, orientation, and rotation of the body. The simultaneous integration of these equations can be done without the need for any hypotheses on the rotation of the deformed body.