The global shape of Europa: Constraints on lateral shell thickness variations

The global shape of Europa is controlled by tidal and rotational potentials and possibly by lateral variations in ice shell thickness. We use limb profiles from four Galileo images to determine the best-fit hydrostatic shape, yielding a mean radius of 1560.8±0.3 km and a radius difference a−c of 3.0±0.9 km, consistent with previous determinations and inferences from gravity observations. Adding long-wavelength topography due to proposed lateral variations in shell thickness results in poorer fits to the limb profiles. We conclude that lateral shell thickness variations and long-wavelength isostatically supported topographic variations do not exceed 7 and 0.7 km, respectively. For the range of rheologies investigated (basal viscosities from 10¹⁴ to ) the maximum permissible (conductive) shell thickness is 35 km. The relative uniformity of Europa's shell thickness is due to either a heat flux from the silicate interior, lateral ice flow at the base of the shell, or convection within the shell.     

On the origins of band topography, Europa

We use stereo-derived topography of extensional bands on Europa to show that these features can be elevated by 100-150 m with respect to the surroundings, and that the positive topography sometimes extends beyond the band margins. Lateral variations in shell thickness cannot maintain the observed topography for timescales greater than ∼0.1 Myr. Lateral density variations can maintain the observed topography indefinitely; mean density contrasts of 5 and 50 kg m⁻³ are required for shell thicknesses of 20 and 2 km, respectively. Density variations caused by temperature contrasts require either present-day heating or that bands are young features (<1 Myr old). Stratigraphic analyses suggest that these mechanisms are unlikely. The observation that bands form from ridges may be explained by an episode of shear-heating on ridges weakening the ridge area, and leading to strain localization during extension. Fracture porosity is likely to persist over Myr timescales in the top one-third to one-quarter of the conductive part of the ice shell. Lateral variations in this porosity (of order 20%) are the most likely mechanism for producing band topography if the ice shell is thin (≈2 km); porosity variations of 2% or less are required if the shell is thicker (≈20 km). If the ice shell is thick, lateral variations in salt content are a more likely mechanism. Warm ice will tend to lose dense, low-melting temperature phases and be buoyant relative to colder, salt-rich ice. Thus, lateral density variations will arise naturally if bands have been the sites of either localized heating or upwelling of warm ice during extension.     

Convective-conductive transitions and sensitivity of a convecting ice shell to perturbations in heat flux and tidal-heating rate: Implications for Europa

We investigate the response of conductive and convective ice shells on Europa to variations of heat flux and interior tidal-heating rate. We present numerical simulations of convection in Europa's ice shell with Newtonian, temperature-dependent viscosity and tidal heating. Modest variations in the heat flux supplied to the base of a convective ice shell, ΔF, can cause large variations of the ice-shell thickness Δδ. In contrast, for a conductive ice shell, large ΔF involves relatively small Δδ. We demonstrate that, for a fluid with temperature-dependent viscosity, the heat flux undergoes a finite-amplitude jump at the critical Rayleigh number Racr. This jump implies that, for a range of heat fluxes relevant to Europa, two equilibrium states—corresponding to a thin, conductive shell and a thick, convective shell—exist for a given heat flux. We show that, as a result, modest variations in heat flux near the critical Rayleigh number can force the ice shell to switch between the thin, conductive and thick, convective configurations over a ∼10⁷-year interval, with thickness changes of up to ∼10-30 km. Depending on the orbital and thermal history, such switches might occur repeatedly. However, existing evolution models based on parameterized-convection schemes have to date not allowed these transitions to occur. Rapid thickening of the ice shell would cause radial expansion of Europa, which could produce extensional tectonic features such as fractures or bands. Furthermore, based on interpretations for how features such as chaos and ridges are formed, several authors have suggested that Europa's ice shell has recently undergone changes in thickness. Our model provides a mechanism for such changes to occur.     

Shell thickness variations and the long-wavelength topography of Titan

The long-wavelength topography of Titan has an amplitude larger than that expected from tidal and rotational distortions at its current distance from Saturn. This topography is associated with small gravity anomalies, indicating a high degree of compensation. Both observations can be explained if Titan has a floating, isostatically-compensated ice shell with a spatially-varying thickness. The spatial variations arise because of laterally-variable tidal heating within the ice shell. Models incorporating shell thickness variations result in an improved fit to the observations and a degree-two tidal Love number h_2t consistent with expectations, without requiring Titan to have moved away from Saturn. Our preferred models have a mean shell thickness of ≈100 km in agreement with the observed gravity anomalies, and a heat flux appropriate to a chondritic Titan. Shell thickness variations are eliminated by convection; we therefore conclude that Titan’s ice shell is not convecting at the present day.     

The heat flow of Europa

The heat flow from Europa has profound implications for ice shell thickness and structure, as well as for the existence of an internal ocean, which is strongly suggested by magnetic data. The brittle-ductile transition depth and the effective elastic thickness of the lithosphere are here used to perform heat flow estimations for Europa. Results give preferred heat flow values (for a typical geological strain rate of ¹⁰⁻¹⁵ s⁻¹) of 70-110 mW m⁻² for a brittle-ductile transition 2 km deep (the usually accepted upper limit for the brittle-ductile transition depth in the ice shell of Europa), 24-35 mW m⁻² for an effective elastic thickness of 2.9 km supporting a plateau near the Cilix impact crater, and >130 mW m⁻² for effective elastic thicknesses of ?0.4 km proposed for the lithosphere loaded by ridges and domes. These values are clearly higher than those produced by radiogenic heating, thus implying an important role for tidal heating. The ?19-25 km thick ice shell proposed from the analysis of size and depth of impact structures suggests a heat flow of ?30-45 mW m⁻² reaching the ice shell base, which in turn would imply an important contribution to the heat flow from tidal heating within the ice shell. Tidally heated convection in the ice shell could be capable to supply ∼100 mW m⁻² for superplastic flow, and, at the Cilix crater region, ∼35-50 mW m⁻² for dislocation creep, which suggests local variations in the dominant flow mechanism for convection. The very high heat flows maybe related to ridges and domes could be originated by preferential heating at special settings.     

Strength of the lithosphere of the Galilean satellites

Several approaches have been used to estimate the ice shell thickness on Callisto, Ganymede, and Europa. Here we develop a method for placing a strict lower bound on the thickness of the strong part of the shell (lithosphere) using measurements of topography. The minimal assumptions are that the strength of faults in the brittle lithosphere is controlled by lithostatic pressure according to Byerlee's law and the shell has relatively uniform density and thickness. Under these conditions, the topography of the ice provides a direct measure of the bending moment in the lithosphere. This topographic bending moment must be less than the saturation bending moment of the yield strength envelope derived from Byerlee's law. The model predicts that the topographic amplitude spectrum decreases as the square of the topographic wavelength. This explains why Europa is rugged at shorter wavelengths (∼10 km) but extremely smooth, and perhaps conforming to an equipotential surface, at longer wavelengths (>100 km). Previously compiled data on impact crater depth and diameter [Schenk, P.M., 2002. Nature 417, 419-421] on Europa show good agreement with the spectral decrease predicted by the model and require a lithosphere thicker than 2.5 km. A more realistic model, including a ductile lower lithosphere, requires a thickness greater than 3.5 km. Future measurements of topography in the 10-100 km wavelength band will provide tight constraints on lithospheric strength.     

Thermal convection in ice-I shells of Titan and Enceladus

Cassini-Huygens observations have shown that Titan and Enceladus are geologically active icy satellites. Mitri and Showman [Mitri, G., Showman, A.P., 2005. Icarus 177, 447-460] and McKinnon [McKinnon, W.B., 2006. Icarus 183, 435-450] investigated the dynamics of an ice shell overlying a pure liquid-water ocean and showed that transitions from a conductive state to a convective state have major implications for the surface tectonics. We extend this analysis to the case of ice shells overlying ammonia-water oceans. We explore the thermal state of Titan and Enceladus ice-I shells, and also we investigate the consequences of the ice-I shell conductive-convective switch for the geology. We show that thermal convection can occur, under a range of conditions, in the ice-I shells of Titan and Enceladus. Because the Rayleigh number Ra scales with δ³/ηb, where δ is the thickness of the ice shell and ηb is the viscosity at the base of the ice-I shell, and because ammonia in the liquid layer (if any) strongly depresses the melting temperature of the water ice, Ra equals its critical value for two ice-I shell thicknesses: for relatively thin ice shell with warm, low-viscosity base (Onset I) and for thick ice shell with cold, high-viscosity base (Onset II). At Onset I, for a range of heat fluxes, two equilibrium states—corresponding to a thin, conductive shell and a thick, convective shell—exist for a given heat flux. Switches between these states can cause large, rapid changes in the ice-shell thickness. For Enceladus, we demonstrate that an Onset I transition can produce tectonic stress of ∼500 bars and fractures of several tens of km depth. At Onset II, in contrast, we demonstrate that zero equilibrium states exist for a range of heat fluxes. For a mean heat flux within this range, the satellite experiences oscillations in surface heat flux and satellite volume with periods of ∼50-800 Myr even when the interior heat production is constant or monotonically declining in time; these oscillations in the thermal state of the ice-I shell would cause repeated episodes of extensional and compressional tectonism.     

Thermal evolution of Pluto and implications for surface tectonics and a subsurface ocean

Determining whether or not Pluto possesses, or once possessed, a subsurface ocean is crucial to understanding its astrobiological potential. In this study we use a 3D convection model to investigate Pluto’s thermal and spin evolution, and the present-day observational consequences of different evolutionary pathways. We test the sensitivity of our model results to different initial temperature profiles, initial spin periods, silicate potassium concentrations and ice reference viscosities. The ice reference viscosity is the primary factor controlling whether or not an ocean develops and whether that ocean survives to the present day. In most of our models present-day Pluto consists of a convective ice shell without an ocean. However if the reference viscosity is higher than 5 × 10¹⁵ Pa s, the shell will be conductive and an ocean should be present. For the nominal potassium concentration the present-day ocean and conductive shell thickness are both about 165 km; in conductive cases an ocean will be present unless the potassium content of the silicate mantle is less than 10% of its nominal value. If Pluto never developed an ocean, predominantly extensional surface tectonics should result, and a fossil rotational bulge will be present. For the cases which possess, or once possessed, an ocean, no fossil bulge should exist. A present-day ocean implies that compressional surface stresses should dominate, perhaps with minor recent extension. An ocean that formed and then re-froze should result in a roughly equal balance between (older) compressional and (younger) extensional features. These predictions may be tested by the New Horizons mission.     

Thermal-orbital evolution of Io and Europa

Coupled thermal-orbital evolution models of Europa and Io are presented. It is assumed that Io, Europa, and Ganymede evolve in the Laplace resonance and that tidal dissipation of orbital energy is an internal heat source for both Io and Europa. While dissipation in Io occurs in the mantle as in the mantle dissipation model of Segatz et al. (1988, Icarus 75, 187), two models for Europa are considered. In the first model dissipation occurs in the silicate mantle while in the second model dissipation occurs in the ice shell. In the latter model, ice shell melting and variations of the shell thickness above an ocean are explicitly included. The rheology of both the ice and the rock is cast in terms of a viscoelastic Maxwell rheology with viscosity and shear modulus depending on the average temperature of the dissipating layer. Heat transfer by convection is calculated using a parameterization for strongly temperature-dependent viscosity convection. Both models are consistent with the present orbital elements of Io, Europa, and Ganymede. It is shown that there may be phases of quasi-steady evolution with large or small dissipation rates (in comparison with radiogenic heating), phases with runaway heating or cooling and oscillatory phases during which the eccentricity and the tidal heating rate will oscillate. Europa's ice thickness varies between roughly 3 and 70 km (dissipation in the silicate layer) or 10 and 60 km (dissipation in the ice layer), suggesting that Europa's ocean existed for geological timescales. The variation in ice thickness, including both convective and purely conductive phases, may be reflected in the formation of different geological surface features on Europa. Both models suggest that at present Europa's ice thickness is several tens of km thick and is increasing, while the eccentricity decreases, implying that the satellites evolve out of resonance. Including lithospheric growth in the models makes it impossible to match the high heat flux constraint for Io. Other heat transfer processes than conduction through the lithosphere must be important for the present Io.     

A model for the interior structure, evolution, and differentiation of Callisto

The recently measured dimensionless moment of inertia (MoI) factor for Callisto of 0.3549±0.0042 (Anderson et al., 2001, Icarus, 153, 157-161) poses a problem: its value cannot be explained by a model in which Callisto is completely differentiated into an ice shell above a rock shell and an iron core such as its neighboring satellite Ganymede nor can it be explained by a model of a homogeneous, undifferentiated ice-rock satellite. We show that Callisto may be incompletely differentiated into an outer ice-rock shell in which the volumetric rock concentration is close to the primordial one at the surface and decreases approximately linearly with depth, an ice mantle mostly depleted of rock, and an about 1800 km rock-ice core in which the rock concentration is close to the close-packing limit. The ice-rock shell thickness depends on uncertain rheology parameters and the heat flow and can be roughly 50 to 150 km thick. We show that if Callisto accreted from a mix of metal bearing rock and ice and if the average size of the rocks was of the order of meters to tens of meters, then Callisto may have experienced a gradual, but still incomplete unmixing of the two components. An ocean in Callisto at a depth of 100-200 km is difficult to obtain if the ice is pure H₂O and if the ice-rock lithosphere is 100 km or more thick; a water ocean is more plausible for ice contaminated by ammonia, methane or salts; or for pure H₂O at a depth of 400-600 km.     

Coupled convection and tidal dissipation in Europa’s ice shell using non-Newtonian grain-size-sensitive (GSS) creep rheology

We present self-consistent, fully coupled two-dimensional (2D) numerical models of thermal evolution and tidal heating to investigate how convection interacts with tidal dissipation under the influence of non-Newtonian grain-size-sensitive creep rheology (plausibly resulting from grain boundary sliding) in Europa’s ice shell. To determine the thermal evolution, we solved the convection equations (using finite-element code ConMan) with the tidal dissipation as a heat source. For a given heterogeneous temperature field at a given time, we determined the tidal dissipation rate throughout the ice shell by solving for the tidal stresses and strains subject to Maxwell viscoelastic rheology (using finite-element code Tekton). In this way, the convection and tidal heating are fully coupled and evolve together. Our simulations show that the tidal dissipation rate can have a strong impact on the onset of thermal convection in Europa’s ice shell under non-Newtonian GSS rheology. By varying the ice grain size (1-10 mm), ice-shell thickness (20-120 km), and tidal-strain amplitude (0-4 × 10⁻⁵), we study the interrelationship of convection and conduction regimes in Europa’s ice shell. Under non-Newtonian grain-size-sensitive creep rheology and ice grain size larger than 1 mm, no thermal convection can initiate in Europa’s ice shell (for thicknesses <100 km) without tidal dissipation. However, thermal convection can start in thinner ice shells under the influence of tidal dissipation. The required tidal-strain amplitude for convection to occur decreases as the ice-shell thickness increases. For grain sizes of 1-10 mm, convection can occur in ice shells as thin as 20-40 km with the estimated tidal-strain amplitude of 2 × 10⁻⁵ on Europa.     

Comment on “Mechanics of tidally driven fractures in Europa's ice shell” by S. Lee, R.T. Pappalardo, and N.C. Makris [2005. Icarus 177, 367-379]

We show Lee, Pappalardo, and Makris' [2005. Icarus 177, 367-379] argument that surface cracks in Europa's icy shell penetrate 3-10 times deeper in the presence of subsurface ocean is not correct. We use numerical calculations to demonstrate that there is at most 50% increase in penetration depth for a crack opening in a shell of finite thickness compared to a half-space. We also propose a simple equation based on force balances to estimate the maximum thickness of an ice shell that can be opened under tensile stress. Our calculations show that a crack can only penetrate 330-m-thick ice shell under 200 kPa far-field tensile stress and half of that if the stress is 100 kPa. But the presence of water would allow crack penetrate ∼4.0 km into the ice shell with zero porosity.     

Thermal and topographic tests of Europa chaos formation models from Galileo E15 observations

Stereo topography of an area near Tyre impact crater, Europa, reveals chaos regions characterised by marginal cliffs and domical topography, rising to 100-200 m above the background plains. The regions contain blocks which have both rotated and tilted. We tested two models of chaos formation: a hybrid diapir model, in which chaos topography is caused by thermal or compositional buoyancy, and block motion occurs due to the presence of near-surface (1-3 km) melt; and a melt-through model, in which chaos regions are caused by melting and refreezing of the ice shell. None of the hybrid diapir models tested generate any melt within 1-3 km of the surface, owing to the low surface temperature. A model of ocean refreezing following melt-through gives effective elastic thicknesses and ice shell thicknesses of 0.1-0.3 and 0.5-2 km, respectively. However, for such low shell thicknesses the refreezing model requires implausibly large lateral density contrasts (50-100 kg m⁻³) to explain the elevation of the centres of the chaos regions. Although a global equilibrium ice shell thickness of ≈2 km is possible if Europa's mantle resembles that of Io, it is unclear whether local melt-through events are energetically possible. Thus, neither of the models tested here gives a completely satisfactory explanation for the formation of chaos regions. We suggest that surface extrusion of warm ice may be an important component of chaos terrain formation, and demonstrate that such extrusion is possible for likely ice parameters.     

Fracture penetration in planetary ice shells

The proposed past eruption of liquid water on Europa and ongoing eruption of water vapor and ice on Enceladus have led to discussion about the feasibility of cracking a planetary ice shell. We use a boundary element method to model crack penetration in an ice shell subjected to tension and hydrostatic compression. We consider the presence of a region at the base of the ice shell in which the far-field extensional stresses vanish due to viscoelastic relaxation, impeding the penetration of fractures towards a subsurface ocean. The maximum extent of fracture penetration can be limited by hydrostatic pressure or by the presence of the unstressed basal layer, depending on its thickness. Our results indicate that Europa's ice shell is likely to be cracked under 1-3 MPa tension only if it is ?2.5 km thick. Enceladus' ice shell may be completely cracked if it is capable of supporting ∼1-3 MPa tension and is less than 25 km thick.     

A whole-moon thermal history model of Europa: Impact of hydrothermal circulation and salt transport

A whole-moon numerical model of Europa is developed to simulate its thermal history. The thermal evolution covers three phases: (i) an initial, roughly 0.5 Gyr-long period of radiogenic heating and differentiation, (ii) a long period from 0.5 Gyr to 4 Gyr with continuing radiogenic heating but no tidal dissipative heating (TDH), and (iii) a final period covering the last 0.5 Gyr until the present, during which TDH is active. Hydrothermal plumes develop after the initial period of heating and differentiation and transport heat and salt from Europa’s silicate mantle to its ice shell. We find that, even without TDH, vigorous hydrothermal convection in the rocky mantle can sustain flow in an ocean layer throughout Europa’s history. When TDH becomes active, the ice shell melts quickly to a thickness of about 20 km, leaving an ocean 80 km or more deep. Parameterized convection in the ice shell is non-uniform spatially, changes over time, and is tied to the deeper ocean–mantle dynamics. We also find that the dynamics are affected by salt concentrations. An initially non-uniform salt distribution retards plume penetration, but is homogenized over time by turbulent diffusion and time-dependent flow driven by initial thermal gradients. After homogenization, the uniformly distributed salt concentrations are no longer a major factor in controlling plume transport. Salt transport leads to the formation of a heterogeneous brine layer and salt inclusions at the bottom of the ice shell; the presence of salt in the ice shell could strongly influence convection in that layer.     

On the origin of south polar folds on Enceladus

Recent high-resolution Cassini images of the south polar terrain of Enceladus reveal regions of short-wavelength deformation, inferred to be compressional folds between the Baghdad and Damascus tiger stripes (Spencer, J.R., Barr, A.C., Esposito, L.W., Helfenstein, P., Ingersoll, A.P., Jaumann, R., McKay, C.P., Nimmo, F., Waite, J.H. [2009a]. Enceladus: An active cryovolcanic satellite. In: Saturn after Cassini-Huygens. Springer, New York, pp. 683-722). Here, we use Fourier analysis of the bright/dark variations to show that the folds have a dominant wavelength of 1.1 ± 0.4 km. We use the simple model of lava flow folding from Fink (Fink, J. [1980]. Geology 8, 250-254) to show that the folds could form in an ice shell with an upper high-viscosity boundary layer of thickness <400 m, with a driving stress of 40-80 kPa, and strain rate between 10⁻¹⁴ s⁻¹ and 10⁻¹² s⁻¹. Such deformation rates imply resurfacing of the SPT in 0.05-5 Myr, consistent with its estimated surface age. Measurements of fold topography and more sophisticated numerical modeling can narrow down the conditions of fold formation and provide valuable constraints on the thermal structure of the ice shell on Enceladus.     

The great thickness debate: Ice shell thickness models for Europa and comparisons with estimates based on flexure at ridges

Estimates of the thickness of the ice shell of Europa range from <1 to >30 km. The higher values are generally assumed to be estimates of the entire ice shell thickness, which may include a lower ductile layer of ice, whereas many of the smaller thickness estimates are based on analyses that only consider that portion of the ice layer that behaves elastically at a particular strain rate. One example of the latter is flexure analysis, in which the elastic ice layer is modeled as a plate or sphere that is flexed under the weight of a surface load. We present calculations based on flexure analysis in which we model the elastic ice layer as flexing under a line-load caused by ridges. We use precisely located, parallel flanking cracks as indicators of the location of greatest tensile stress induced by flexure. Our elastic thickness results are spatially variable: ∼500-2200 m (two sites) and ∼200-1000 m (one site). Thorough analysis of Europan flexure studies performed by various researchers shows that the type of model selected causes the greatest variability in the thickness results, followed by the choice of Young's modulus, which is poorly constrained for the Europan ice shell. Comparing our results to those of previously published flexure analyses for Europa, we infer spatial variability in the elastic ice thickness (at the time of load emplacement), with smooth bands having the thinnest elastic ice thickness of all areas studied. Because analysis of flexure-induced fracturing can only reveal the elastic thickness at the time of load emplacement, calculated thickness variability between features having different ages may also reflect a temporal variability in the thickness of Europa's ice shell.     

Impact basin relaxation at Iapetus

We investigate impact basin relaxation on Iapetus by combining a 3D thermal evolution model (Robuchon, G., Choblet, G., Tobie, G., Cadek, O., Sotin, C., Grasset, O. [2010]. Icarus 207, 959-971) with a spherical axisymmetric viscoelastic relaxation code (Zhong, S., Paulson, A., Wahr, J. [2003]. Geophys. J. Int. 155, 679-695). Due to the progressive cooling of Iapetus, younger basins relax less than older basins. For an ice reference viscosity of 10¹⁴ Pa s, an 800 km diameter basin relaxes by 30% if it formed in the first 50 Myr but by 10% if it formed at 1.2 Gyr. Bigger basins relax more rapidly than smaller ones, because the inferred thickness of the ice shell exceeds the diameter of all but the largest basins considered. Stereo topography shows that all basins 600 km in diameter or smaller are relaxed by 25% or less. Our model can match the relaxation of all the basins considered, within error, by assuming a single basin formation age (4.36 Ga for our nominal viscosity). This result is consistent with crater counts, which show no detectable age variation between the basins examined.     

Heat flow, lenticulae spacing, and possibility of convection in the ice shell of europa

Two opposing models to explain the geological features observed on Europa’s surface have been proposed. The thin-shell model states that the ice shell is only a few kilometers thick, transfers heat by conduction only, and can become locally thinner until it exposes an underlying ocean on the satellite’s surface. According to the thick-shell model, the ice shell may be several tens of kilometers thick and have a lower convective layer, above which there is a cold stagnant lid that dissipates heat by conduction. Whichever the case, from magnetic data there is strong support for the presence of a layer of salty liquid water under the ice. The present study was performed to examine whether the possibility of convection is theoretically consistent with surface heat flows of ∼100-200 mW m⁻², deduced from a thin brittle lithosphere, and with the typical spacing of 15-23 km proposed for the features usually known as lenticulae. It was obtained that under Europa’s ice shell conditions convection could occur and also account for high heat flows due to tidal heating of the convective (nearly isothermal) interior, but only if the dominant water ice rheology is superplastic flow (with activation energy of 49 kJ mol⁻¹; this is the rheology thought dominant in the warm interior of the ice shell). In this case the ice shell would be ∼15-50 km thick. Furthermore, in this scenario explaining the origin of the lenticulae related to convective processes requires ice grain size close to 1 mm and ice thickness around 15-20 km.     

Thermal Equilibrium States of Europa's Ice Shell: Implications for Internal Ocean Thickness and Surface Heat Flow

Ice-shell thickness and ocean depth are calculated for steady state models of tidal dissipation in Europa's ice shell using the present-day values of the orbital elements. The tidal dissipation rate is obtained using a viscoelastic Maxwell rheology for the ice, the viscosity of which has been varied over a wide range, and is found to strongly increase if an (inviscid) internal ocean is present. To determine steady state values, the tidal dissipation rate is equated to the heat-transfer rate through the ice shell calculated from a parameterized model of convective heat transfer or from a thermal conduction model, if the ice layer is found to be stable against convection. Although high dissipation rates and heat fluxes of up to 300 mWm⁻² are, in principle, possible for Europa, these values are unrealistic because the states for which they are obtained are thermodynamically unstable. Equilibrium models have surface heat flows around 20 mWm⁻² and ice-layer thicknesses around 30 km, which is significantly less than the total thickness of the H₂O-layer. These results support models of Europa with ice shells a few tens of kilometers thick and around 100-km-thick subsurface oceans.