Free Translational Oscillations of Icy Bodies With a Subsurface Ocean Using a Variational Approach

Abstract

We analyze the influence of the interior structure of an icy body with an internal ocean on the relative translational motions of its solid constituents. We consider an isolated body differentiated into three homogeneous layers with spherical symmetry: an external ice-I layer, a subsurface ammonia-water ocean, and a rocky inner core. This composition represents icy bodies such as Europa, Titania, Oberon, and Triton, as well as Pluto, Eris, Sedna, and 2004 DW. We construct the equations of motion by assuming that the solid constituents are rigid and that the ocean is an ideal fluid, the internal motion being characterized by the relative translations of the solids and the induced flow in the fluid. Then we determine the dynamics of the icy body using the methods of analytical mechanics, that is, we compute the kinetic energy and the gravitational potential energy, and obtain the Lagrangian function. The resulting solution of the Lagrange equations shows that the solid layers perform translational oscillations of different amplitudes with respect to the barycenter of the body. We derive the dependence of the frequency of the free oscillations of the system on the characteristics of each layer, expressing the period of the oscillations as a function of the densities and masses of the ocean and the rocky inner core, and the mass of the icy body. We apply these results to previously developed subsurface models and obtain numerical values for the period and the ratio between the amplitudes of the translational oscillations of the solid components. The features obtained are quite different from the cases of Earth and Mercury. Our analytical formulas satisfactorily explain the source of these differences. When models of the same icy body, compatible with the existence of an internal ocean, differ in the thickness of the ice-I layer, their associated periods experience a relative variation of at least 10%. In particular, the different models for Titania and Oberon exhibit a larger variation of about 37% and 30%. This indicates an absolute difference of the order of three and two hours, respectively. This suggests that the free period of the internal oscillations might provide a new procedure to constrain the internal structure of icy bodies with a subsurface ocean.

Period of the Slichter mode of Mercury and its possible observation

“…6). This surprising result is probably due to the fact that the contribution of the mantle motion to the Slichter mode period is reduced by the low density difference between the inner and the outer cores (Escapa & Fukushima 2011). When the inner core is very small compared to the outer core, the frequency of the Slichter mode can be approximated as…”

Revisiting the Slichter mode of Mercury in the context of differentiated models

“…Although the characteristics of the rigid layers are different, the dynamics can be described following the variational method employed by Escapa & Fukushima (2011). Hence, we only briefly describe the main steps of the development.…”

“…These equations are similar to Eqs. (9) in Escapa & Fukushima (2011), substituting the former density of the inner core by its mean densityρ s in the definition of α (Eq. (1)).…”

“…Hence, we aim at obtaining results by means of an alternative and independent analytical model of the Slichter mode based on the variational approach constructed in Escapa & Fukushima (2011).…”

“…In Sect. 2 we model the internal translational dynamics of the considered body by means of the variational formalism by Escapa & Fukushima (2011). The theory, although being general, is applied to differentiated models of Mercury (Van Hoolst & Jacobs 2003).…”