Andrade rheology in time-domain. Application to Enceladus' dissipation of energy due to forced libration

Abstract

The main purpose of this work is to present a time-domain implementation of the Andrade rheology, instead of the traditional expansion in terms of a Fourier series of the tidal potential. This approach can be used in any fully three dimensional numerical simulation of the dynamics of a system of many deformable bodies. In particular, it allows large eccentricities, large mutual inclinations, and it is not limited to quasi-periodic perturbations. It can take into account an extended class of perturbations, such as chaotic motions, transient events, and resonant librations.The results are presented by means of a concrete application: the analysis of the libration of Enceladus. This is done by means of both analytic formulas in the frequency domain and direct numerical simulations. We do not a priori assume that Enceladus has a triaxial shape, the eventual triaxiality is a consequence of the satellite motion and its rheology. As a result we obtain an analytic formula for the amplitude of libration that incorporates a new correction due to the rheology.Our results provide an estimation of the amplitude of libration of the core of Enceladus as 0.6% of that of the shell. They also reproduce the observed 10 GW of tidal heat generated by Enceladus with a value of 0.17 × 10 14 Pa·s for the global effective viscosity under both Maxwell and Andrade rheology.

Librations of a body composed of a deformable mantle and a fluid core

“…Section 5 contains our main contribution: an explicit system of ordinary differential equations for the rotational motion of a body made of a mantle and a fluid core. The rheology of the mantle can be given by any linear viscoelastic model, including accurate approximations to models with infinite memory as the Andrade Gevorgyan et al (2020) or the Sundberg-Cooper rheologies Gevorgyan (2021). The influence of oceans and other fluids bounded to the mantle can be incorporated into the rheology, as far as these effects can be considered in a spherically average sense.…”

From the internal structure to the librations of satellites and backs

“…In Figure 12 one can see the comparison of the dissipation rate between the two rheologies, in the second case the dissipation happens in the mantle and in the boundary between the two layers, dissipation at the boundary is much smaller that the one in the mantle, Figure 12 bottom. As in Gevorgyan et al (2019) the increase in the libration amplitude brings to an increase in the dissipation rate and this can be observed from Figure 12 by comparing the two top figures. The Tisserand frame angular velocity in Figure 11 is essentially the same for both rheologies.…”

“…For the second rheology dissipation occurs in the mantle and in the boundary between the two layers, dissipation at the boundary is much smaller that the one in the mantle. As in Gevorgyan et al (2019) the increase in the libration amplitude brings to an increase in the dissipation rate. In this case the force matrix in the guiding frame Jg, as defined in Section 2, is given by 3Gmp times The time dependent coefficients δJ g12 , δJ g13 , δJ g23 that appear in equation (3.21) are given by .127) where i = j and u and v are defined in (A.121).…”

“…In the second research paper, the models proposed in [14,15] and further developed in [17,39] were compared to the models used for IMPOP and JPL ephemerides, which are the source of the real data used to fit the parameters and compare the results. The influence of the permanent deformations, represented by the prestress, on the librations of celestial bodies was studied.…”

“…We intend to add multilayered rheologies to the modeling framework that we have been developing [14,15,39,40] envisaging applications to the icy satellites of our solar system. One reason to study the icy satellites is that they are less well understood than the terrestrial planets, displaying a strong coupling between thermal and orbital evolution that is almost entirely absent from terrestrial planets.…”