Tidal heating and the long-term stability of a subsurface ocean on Enceladus

Abstract

Tidal dissipation has been suggested as the heat source for the south polar thermal anomaly on Enceladus. We find that under present-day conditions and assuming Maxwellian behavior, tidal dissipation is negligible in the silicate core. Dissipation may be significant in the ice shell if the shell is decoupled from the silicate core by a subsurface ocean. We have run a series of self-consistent convection and conduction models in 2D axisymmetric and 3D spherical geometry in which we include the spatially-variable tidal heat production. We find that in all cases, the shell removes more heat from the interior than can be produced in the core by radioactive decay, resulting in cooling of the interior and the freezing of any ocean. Under likely conditions, a 40-km thick ocean made of pure water would freeze solid on a ∼30 Ma timescale. An ocean containing other chemical components will have a lower freezing point, but even a water–ammonia eutectic composition will only prolong the freezing, not prevent it. If the eccentricity of Enceladus were higher ($e⩾0.015$) in the past, the increased dissipation in the ice shell may have been sufficient to maintain a liquid layer. We cannot therefore rule out the presence of a transient ocean, as a relic of an earlier era of greater heating. If the eccentricity is periodically pumped up, the ocean may have thickened and thinned on a similar timescale as the orbital evolution, provided the ocean never froze completely. We conclude that the current heat flux of Enceladus and any possible subsurface ocean is not in steady-state, and is the remnant of an epoch of higher eccentricity and tidal dissipation.

Introduction

The discovery of the thermal anomaly in the south polar region of Enceladus (Spencer et al., 2006), has launched a great deal of interest in potential activity in the ice shell. $5.8\pm 1.9\text{GW}$ of heat have been observed pouring out of this region (Spencer et al., 2006), and the global heat flux may be an order of magnitude higher. Such a heat flux was unexpected from a body as small as Enceladus (252 km radius), which under normal circumstances would have cooled very quickly. A likely heat source for the observed thermal anomaly is ongoing internal heating due to tidal dissipation (Matson et al., 2007), an idea first examined by Squyres et al. (1983) and Ross and Schubert (1989) well before this observation. The tidally dissipated energy, ${\dot{E}}_t$ within a satellite is a strong function of its orbital frequency ω, eccentricity e, and radius $R_s$ (Segatz et al., 1988)

$${\dot{E}}_t=-\frac{21}{2}\mathrm{Im}(k_2)\frac{{(\omega R_s)}^5}{G}{e}^2,$$

where G is the gravitational constant, and $k_2$ is the degree-2 tidal Love number. The dissipation is also a function of the material properties inside the satellite, which are globally described by $k_2$, and which can be found analytically for a homogeneous body (Ross and Schubert, 1989). For a homogeneous body,

$$k_2=\frac{3/2}{1+19\tilde{\mu }/(2\rho g{R}_s)},$$

where ρ is the density, g is the gravitational acceleration, and $\tilde{\mu }$ is the complex rigidity which depends on the rheology.

Enceladus, however, is unlikely to be homogeneous. The tidal dissipation and $k_2$ depend not only on the material properties of the planet, but the distribution of material as well. The thermal anomaly suggests a warm, differentiated interior (Schubert et al., 2007), consisting of a silicate core, overlain by an icy mantle, with a possible ocean separating the two layers (Porco et al., 2006, Nimmo et al., 2007).

The goals of this paper are to determine to what extent tidal dissipation may occur in Enceladus, and how that heat is distributed; to explore how that heat is removed through convection and/or conduction; and to place constraints on the resulting long-term stability and depth of the putative ocean. This paper is organized as follows. We first describe our models for tidal dissipation in multilayered bodies, and present the results of those models applied to Enceladus. In the following sections we present the results of convection and conduction modeling of Enceladus' ice shell including viscoelastic tidal heating. We find that under present-day conditions, tidal heating in the interior of the ice shell and the silicate core is insufficient to prevent an ocean from freezing. Finally, we discuss the implications of those results for the thermal evolution of Enceladus. We suggest that if there is a subsurface ocean on Enceladus today, it cannot be in thermal equilibrium. The current thermal activity is likely a remnant from an epoch of greater heating and higher eccentricity (Meyer and Wisdom, 2007), and the south polar thermal anomaly may be due to a shallow heat source, such as shear heating (Nimmo et al., 2007), rather than volumetric Maxwellian tidal dissipation.