Libration Heating and the Thermal State of Enceladus&#8217;s Ice Shell
Authors: Shao Wencheng , Nimmo Francis 
Published: July 2021

AbstractForced librations can enhance tidal dissipation in the ice shell of Enceladus, but how such librations might affect the thermal state of Enceladus has not been investigated. Here we use the elastic libration model established by Van Hoolst et al. (2013) to investigate the effect of the libration heating on Enceladus. We find that libration heating in the ice shell is sensitive to the shell thickness, ranging from less than 1 GW to 10 GW. However, the libration heating overall is insufficient to match the inferred conductive heat loss of Enceladus. Thus, either Enceladus is not in thermal equilibrium, or (more likely) alternative heating mechanisms below the shell are taking place. If such heat sources exist, our calculations show that Enceladus would reside in a stable thermal equilibrium resisting small perturbations to the shell thickness. This does not support the occurrence of a runaway melting proposed by Luan and Goldreich (2017). These results place limits on the possible thermal history of Enceladus and emphasize the need of quantifying the tidal dissipation inside the ocean or the core.1. IntroductionCassini observed that Enceladus is losing heat at a high rate (e.g., Spencer et al. 2006). Cassini&#8217;s observations also indicate the presence of a global subsurface ocean inside Enceladus (e.g., Iess et al. 2014). These two observations require Enceladus to possess high endogenic heat to keep in thermal balance.Tidal dissipation is a good candidate for the heat requirement. The shell tidal dissipation has been studied and seems insufficient to match the observed high heat flow (e.g., Behounkova et al. 2017). The ice shell experiences forced librations. These forced librations can enhance the shell tidal dissipation. Previous studies did not address the question how libration heating affects the thermal state of Enceladus. Luan and Goldreich (2017) proposed a thermal runaway scenario for Enceladus&#8217;s ice shell. Then the question emerges whether libration heating can lead to such a scenario. In this study, we will shed light on these questions.2. MethodologyWe use the elastic libration model established by Van Hoolst et al. (2013) to calculate the forced librations. This model can examine the forced librations of a tidally locked satellite with three homogeneous layers: an ice shell, a subsurface ocean and a rocky core. This model considers the finite elasticity of the shell in the calculations.We construct 41 interior models with shell thickness varying from 5-50 km. The core size is assumed as 190 km (Hemingway et al. 2018). Densities of ice and water are fixed as 900 and 1000 kg/m3 and the ice shell viscosity is depth-dependent. Required tidal displacements are calculated via the model of Roberts and Nimmo (2008). Orbital variation data are from JPL/Horizon. We estimate ice-shell libration heating by the formula in Wisdom (2004). The conductive heat loss rate is estimated through a formula in Hemingway et al. (2018).3. Results and discussionsLibration heating (Figure 1) is greatly dependent on the shell thickness. The libration heating reduces from 10 GW to less than 1 GW as the shell gets thicker from 5 to 50 km. However, this shell libration heating is less than the observed high heat flow on Enceladus. This suggests that additional heat sources in the ocean or in the silicate core are taking place if Enceladus is in equilibrium.Figure 1 (a) Shell&#8217;s tidal dissipation rate (including the effect of the diurnal forced libration), (b) Love number k2 and (c) dissipation factor Qs for different interior models. The blue region is the inferred shell thickness from the libration data (Van Hoolst et al. 2016).We add a constant heat source to study the thermal equilibrium of Enceladus. Comparing the total tidal heat to the conductive loss of Enceladus (Figure 2), we find a stable equilibrium state for Enceladus, which resists small perturbations to the shell thickness and does not favor the occurrence of a runaway melting on Enceladus.We also consider the influence of orbital eccentricity and shell bottom viscosity on our results (Figure 2). We find that only under some extreme situations (high eccentricity and thin shell), Enceladus&#8217;s thermal equilibrium could be unstable and vulnerable to small perturbations to the shell thickness. Our results indicate that episodic heating or runaway melting (if it occurred) is unlikely to originate from the librations of Enceladus&#8217;s ice shell.Figure 2 Total global heating rate (red) versus conductive cooling rate (blue) for interior models with the shell basal viscosity of (a) 1014 Pa s and (b) 1013 Pa s. The heating consists of dissipation in the ice shell and a heat source of 25 GW below the shell. Different line styles indicate different orbital eccentricities. Enceladus&#8217;s surface temperature is taken as 75 K. Stable and unstable equilibrium points are marked out.ReferencesBaland, R.-M., Yseboodt, M. &amp; Van Hoolst, T. The obliquity of Enceladus. Icarus 268, 12&#8211;31 (2016).B&#283;hounkov&#225;, M., Sou&#269;ek, O., Hron, J. &amp; &#268;adek, O. Plume Activity and Tidal Deformation on Enceladus Influenced by Faults and Variable Ice Shell Thickness. Astrobiology 17, 941&#8211;954 (2017).Hemingway, D., Iess, L., Tajeddine, R., &amp; Tobie, G. The interior of Enceladus. Enceladus and the icy moons of Saturn, 57-77 (2018).Iess, L. et al. The Gravity Field and Interior Structure of Enceladus. Science 344, 78&#8211;80 (2014).Luan, J. &amp; Goldreich, P. Enceladus: three-act play and current state. AGU Fall Meeting Abstracts 51, (2017).Roberts, J. H. &amp; Nimmo, F. Tidal heating and the long-term stability of a subsurface ocean on Enceladus. 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