Tides on Europa: the membrane paradigm

TL;DR

This paper develops a membrane theory for Europa's icy crust to predict tidal responses and dissipation, providing analytical formulas and benchmarking tools that improve understanding of tidal tectonics and heat flow.

Contribution

It introduces a new membrane theory with depth-dependent rheology, deriving analytical formulas for tidal Love numbers and correcting previous computational errors.

Findings

Membrane formulas predict Love numbers with high accuracy for thin crusts.

Tidal heating in the crust is proportional to the imaginary part of the membrane spring constant.

Benchmarking revealed a significant error in the original SatStress code.

Abstract

Jupiter's moon Europa has a thin icy crust which is decoupled from the mantle by a subsurface ocean. The crust thus responds to tidal forcing as a deformed membrane, cold at the top and near melting point at the bottom. In this paper I develop the membrane theory of viscoelastic shells with depth-dependent rheology with the dual goal of predicting tidal tectonics and computing tidal dissipation. Two parameters characterize the tidal response of the membrane: the effective Poisson's ratio $\bar\nu$ and the membrane spring constant $\Lambda$, the latter being proportional to the crust thickness and effective shear modulus. I solve membrane theory in terms of tidal Love numbers, for which I derive analytical formulas depending on $\Lambda$, $\bar\nu$, the ocean-to-bulk density ratio and the number $k_2^o$ representing the influence of the deep interior. Membrane formulas predict $h_2$ and $k_2$ with an accuracy of a few tenths of percent if the crust thickness is less than one hundred kilometers, whereas the error on $l_2$ is a few percents. Benchmarking with the thick-shell software SatStress leads to the discovery of an error in the original, uncorrected version of the code that changes stress components by up to 40%. Regarding tectonics, I show that different stress-free states account for the conflicting predictions of thin and thick shell models about the magnitude of tensile stresses due to nonsynchronous rotation. Regarding dissipation, I prove that tidal heating in the crust is proportional to $Im \Lambda$ and that it is equal to the global heat flow (proportional to $Im k_2$) minus the core-mantle heat flow (proportional to $Im k_2^o$). As an illustration, I compute the equilibrium thickness of a convecting crust. More generally, membrane formulas are useful in any application involving tidal Love numbers such as crust thickness estimates, despinning tectonics or true polar wander.