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.     

The effect of gravitational and pressure torques on Titan's length-of-day variations

Cassini radar observations show that Titan's spin is slightly faster than synchronous spin. Angular momentum exchange between Titan's surface and the atmosphere over seasonal time scales corresponding to Saturn's orbital period of 29.5 year is the most likely cause of the observed non-synchronous rotation. We study the effect of Saturn's gravitational torque and torques between internal layers on the length-of-day (LOD) variations driven by the atmosphere. Because static tides deform Titan into an ellipsoid with the long axis approximately in the direction to Saturn, non-zero gravitational and pressure torques exist that can change the rotation rate of Titan. For the torque calculation, we estimate the flattening of Titan and its interior layers under the assumption of hydrostatic equilibrium. The gravitational forcing by Saturn, due to misalignment of the long axis of Titan with the line joining the mass centers of Titan and Saturn, reduces the LOD variations with respect to those for a spherical Titan by an order of magnitude. Internal gravitational and pressure coupling between the ice shell and the interior beneath a putative ocean tends to reduce any differential rotation between shell and interior and reduces further the LOD variations by a few times. For the current estimate of the atmospheric torque, we obtain LOD variations of a hydrostatic Titan that are more than 100 times smaller than the observations indicate when Titan has no ocean as well as when a subsurface ocean exists. Moreover, Saturn's torque causes the rotation to be slower than synchronous in contrast to the Cassini observations. The calculated LOD variations could be increased if the atmospheric torque is larger than predicted and or if fast viscous relaxation of the ice shell could reduce the gravitational coupling, but it remains to be studied if a two order of magnitude increase is possible and if these effects can explain the phase difference of the predicted rotation variations. Alternatively, the large differences with the observations may suggest that non-hydrostatic effects in Titan are important. In particular, we show that the amplitude and phase of the calculated rotation variations are similar to the observed values if non-hydrostatic effects could strongly reduce the equatorial flattening of the ice shell above an internal ocean.     

Elastic ice shells of synchronous moons: Implications for cracks on Europa and non-synchronous rotation of Titan

A number of synchronous moons are thought to harbor water oceans beneath their outer ice shells. A subsurface ocean frictionally decouples the shell from the interior. This has led to proposals that a weak tidal or atmospheric torque might cause the shell to rotate differentially with respect to the synchronously rotating interior. Applications along these lines have been made to Europa and Titan. However, the shell is coupled to the ocean by an elastic torque. As a result of centrifugal and tidal forces, the ocean would assume an ellipsoidal shape with its long axis aligned toward the parent planet. Any displacement of the shell away from its equilibrium position would induce strains thereby increasing its elastic energy and giving rise to an elastic restoring torque. In the investigation reported on here, the elastic torque is compared with the tidal torque acting on Europa and the atmospheric torque acting on Titan.Regarding Europa, it is shown that the tidal torque is far too weak to produce stresses that could fracture the ice shell, thus refuting an idea that has been widely advocated. Instead, it is suggested that the cracks arise from time-dependent stresses due to non-hydrostatic gravity anomalies from tidally driven, episodic convection in the satellite’s interior.Two years of Cassini RADAR observations of Titan’s surface have been interpreted as implying an angular displacement of ∼0.24° relative to synchronous rotation. Compatibility of the amplitude and phase of the observed non-synchronous rotation with estimates of the atmospheric torque requires that Titan’s shell be decoupled from its interior. We find that the elastic torque balances the seasonal atmospheric torque at an angular displacement ?0.05°, effectively coupling the shell to the interior. Moreover, if Titan’s surface were spinning faster than synchronous, the tidal torque tending to restore synchronous rotation would almost certainly be larger than the atmospheric torque. There must either be a problem with the interpretation of the radar observations, or with our basic understanding of Titan’s atmosphere and/or interior.     

Non-Newtonian topographic relaxation on Europa

Models of topographic support on Europa by lateral shell thickness variations have previously assumed a Newtonian ice viscosity. Here I show that using a more realistic stress-dependent viscosity gives relaxation times which can be significantly different. Topography of wavelength 100 km cannot be supported by lateral shell thickness variations for ∼50 Myr, unless the shell thickness is <10 km or the ice grain size >10 mm. Shorter wavelength topography would require even thinner shells, but may be supported elastically. Global-scale variations in shell thickness, however, can be supported for geological timescales if the shell thickness is O(10 km).     

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.     

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.     

Titan's 5-micron lightcurve

We report on repeated mid-resolution (R∼2000) spectroscopic observations of Titan, acquired between November 2002 and January 2003 with ISAAC at the ESO/VLT and covering the 4.84-5.05 μm range. These observations, which sample four different positions of Titan around Saturn, clearly indicate a variability of the 5-μm continuum albedo, with Titan's geometric albedo decreasing by ∼40% from Titan's leading side to the trailing side. This Titan 5-μm “lightcurve” appears to be in phase with the other near-infrared lightcurves. This can be understood in terms of a surface model in which water ice coexists in minor and variable proportions (13-25%, if pure) with a second, dark, component.     

Formation of the regular satellites of giant planets in an extended gaseous nebula I: subnebula model and accretion of satellites

We model the subnebulae of Jupiter and Saturn wherein satellite accretion took place. We expect each giant planet subnebula to be composed of an optically thick (given gaseous opacity) inner region inside of the planet’s centrifugal radius (where the specific angular momentum of the collapsing giant planet gaseous envelope achieves centrifugal balance, located at rCJ ∼ 15R_J for Jupiter and rCS ∼ 22R_S for Saturn) and an optically thin, extended outer disk out to a fraction of the planet’s Roche-lobe (R_H), which we choose to be ∼R_H/5 (located at ∼150 R_J near the inner irregular satellites for Jupiter, and ∼200R_S near Phoebe for Saturn). This places Titan and Ganymede in the inner disk, Callisto and Iapetus in the outer disk, and Hyperion in the transition region. The inner disk is the leftover of the gas accreted by the protoplanet. The outer disk may result from the nebula gas flowing into the protoplanet during the time of giant planet gap-opening (or cessation of gas accretion). For the sake of specificity, we use a solar composition “minimum mass” model to constrain the gas densities of the inner and outer disks of Jupiter and Saturn (and also Uranus). Our model has Ganymede at a subnebula temperature of ∼250 K and Titan at ∼100 K. The outer disks of Jupiter and Saturn have constant temperatures of 130 and 90 K, respectively.Our model has Callisto forming in a time scale ∼10⁶ years, Iapetus in 10⁶-10⁷ years, Ganymede in 10³-10⁴ years, and Titan in 10⁴-10⁵ years. Callisto takes much longer to form than Ganymede because it draws materials from the extended, low density portion of the disk; its accretion time scale is set by the inward drift times of satellitesimals with sizes 300-500 km from distances ∼100R_J. This accretion history may be consistent with a partially differentiated Callisto with a ∼300-km clean ice outer shell overlying a mixed ice and rock-metal interior as suggested by Anderson et al. (2001), which may explain the Ganymede-Callisto dichotomy without resorting to fine-tuning poorly known model parameters. It is also possible that particulate matter coupled to the high specific angular momentum gas flowing through the gap after giant planet gap-opening, capture of heliocentric planetesimals by the extended gas disk, or ablation of planetesimals passing through the disk contributes to the solid content of the disk and lengthens the time scale for Callisto’s formation. Furthermore, this model has Hyperion forming just outside Saturn’s centrifugal radius, captured into resonance by proto-Titan in the presence of a strong gas density gradient as proposed by Lee and Peale (2000). While Titan may have taken significantly longer to form than Ganymede, it still formed fast enough that we would expect it to be fully differentiated. In this sense, it is more like Ganymede than like Callisto (Saturn’s analog of Callisto, we expect, is Iapetus). An alternative starved disk model whose satellite accretion time scale for all the regular satellites is set by the feeding of planetesimals or gas from the planet’s Roche-lobe after gap-opening is likely to imply a long accretion time scale for Titan with small quantities of NH₃ present, leading to a partially differentiated (Callisto-like) Titan. The Cassini mission may resolve this issue conclusively. We briefly discuss the retention of elements more volatile than H₂O as well as other issues that may help to test our model.     

Measurements of the elastic thickness under ancient lunar terrain

The part of a planet's gravity that is coherent with its topography provides information about the deflection of its crust after loading, and hence the resistance of its lithosphere to bending at the time the load was emplaced. We used observed line of sight accelerations from Lunar Prospector, together with the accelerations we would have expected if anomalies in the gravity field were only caused by topography, to estimate the admittance and coherence between topography and gravity over several regions of the lunar nearside. We then compared our estimates to what we would expect if the lithosphere behaved as a floating elastic plate or shell, assuming a linear relationship between topography and gravity. We found in the region surrounding the southern highland crater Clavius that the data can be modeled using a thin plate with Te=12±5 km and uncorrelated loads at the surface and base of the upper crust. A spherical shell model with surface loading is less satisfactory: to fit the admittance adequately requires topography with wavelengths over 400 km to be formed when Te≈ 1 km and the remainder when Te≈7.5 km. By contrast, the apparent lack of compensation around the youngest giant impact basins requires a plate with Te>80 km or a shell with Te>25 km. Our results indicate the thickness of the lunar lithosphere increased from ≈12 km in the pre-Nectarian to >25 km in the Nectarian.     

Forced obliquity and moments of inertia of Titan

The obliquity of Titan is small, but certainly non-zero, and may be used to place constraints on Titan's internal structure. The measured gravity coefficients of Titan imply that it is non-hydrostatic and thus the normal Darwin-Radau approach to determining internal structure cannot be applied. However, if the obliquity is assumed to be tidally damped (that is, in a Cassini state) then combining the obliquity with the measured gravity coefficients allows Titan's moment of inertia to be determined without invoking hydrostatic equilibrium. For polar moment values in the range (0.3<C/MR²<0.4), tidally-damped obliquity values of (0.115°<|ε|<0.177°) result. If the inferred moment value exceeds 0.4, this strongly suggests the presence of a near-surface ice shell decoupled from the interior, probably by a subsurface ocean.     

The three-dimensional structure of Saturn’s E ring

Saturn’s diffuse E ring consists of many tiny (micron and sub-micron) grains of water ice distributed between the orbits of Mimas and Titan. Various gravitational and non-gravitational forces perturb these particles’ orbits, causing the ring’s local particle density to vary noticeably with distance from the planet, height above the ring-plane, hour angle and time. Using remote-sensing data obtained by the Cassini spacecraft in 2005 and 2006, we investigate the E-ring’s three-dimensional structure during a time when the Sun illuminated the rings from the south at high elevation angles (>15°). These observations show that the ring’s vertical thickness grows with distance from Enceladus’ orbit and its peak brightness density shifts from south to north of Saturn’s equator plane with increasing distance from the planet. These data also reveal a localized depletion in particle density near Saturn’s equatorial plane around Enceladus’ semi-major axis. Finally, variations are detected in the radial brightness profile and the vertical thickness of the ring as a function of longitude relative to the Sun. Possible physical mechanisms and processes that may be responsible for some of these structures include solar radiation pressure, variations in the ambient plasma, and electromagnetic perturbations associated with Saturn’s shadow.     

Nonhydrostatic effects and the determination of icy satellites’ moment of inertia

We compare the moment of inertia (MOI) of a simple hydrostatic, two layer body as determined by the Radau–Darwin Approximation (RDA) to its exact hydrostatic MOI calculated to first order in the parameter q = Ω²R³/GM, where Ω, R, and M are the spin angular velocity, radius, and mass of the body, and G is the gravitational constant. We show that the RDA is in error by less than 1% for many configurations of core sizes and layer densities congruent with those of solid bodies in the Solar System. We then determine the error in the MOI of icy satellites calculated with the RDA due to nonhydrostatic effects by using a simple model in which the core and outer shell have slight degree 2 distortions away from their expected hydrostatic shapes. Since the hydrostatic shape has an associated stress of order ρΩ²R² (where ρ is density) it follows that the importance of nonhydrostatic effects scales with the dimensionless number σ/ρΩ²R², where σ is the nonhydrostatic stress. This highlights the likely importance of this error for slowly rotating bodies (e.g., Titan and Callisto) and small bodies (e.g., Saturn moons other than Titan). We apply this model to Titan, Callisto, and Enceladus and find that the RDA-derived MOI can be 10% greater than the actual MOI for nonhydrostatic stresses as small as ∼0.1 bars at the surface or ∼1 bar at the core–mantle boundary, but the actual nonhydrostatic stresses for a given shape change depends on the specifics of the interior model. When we apply this model to Ganymede we find that the stresses necessary to produce the same MOI errors as on Titan, Callisto, and Enceladus are an order of magnitude greater due to its faster rotation, so Ganymede may be the only instance where RDA is reliable. We argue that if satellites can reorient to the lowest energy state then RDA will always give an overestimate of the true MOI. Observations have shown that small nonhydrostatic gravity anomalies exist on Ganymede and Titan, at least at degree 3 and presumably higher. If these anomalies are indicative of the nonhydrostatic anomalies at degree 2 then these imply only a small correction to the MOI, even for Titan, but it is possible that the physical origin of nonhydrostatic degree 2 effects is different from the higher order terms. We conclude that nonhydrostatic effects could be present to an extent that allows Callisto and Titan to be fully differentiated.     

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.     

Infrared spectroscopy of crystalline and amorphous diacetylene (C4H2) and implications for Titan's atmospheric composition

New laboratory spectra of crystalline and amorphous diacetylene ice have been recorded in the range of 7000-500 cm⁻¹ (1.4-20 μm) to aid in the identification of solid diacetylene on Saturn's moon Titan. We have established that amorphous diacetylene ice is stable only at temperatures less than 70±1 K. With respect to observations on Titan, the best approach would be to utilize future space-based telescopes to search for the ν₄ (3277/3271 cm⁻¹) in absorption against the reflected light from the sun and the slightly weaker ν₈ absorption bands (676/661 cm⁻¹) in absorption against the continuum emission.     

The D/H Ratio in Methane in Titan: Origin and History

We propose a new interpretation of the D/H ratio in CH₄ observed in the atmosphere of Titan. Using a turbulent evolutionary model of the subnebula of Saturn (O. Mousis et al. 2002, Icarus156, 162-175), we show that in contrast to the current scenario, the deuterium enrichment with respect to the solar value observed in Titan cannot have occurred in the subnebula. Instead, we argue that values of the D/H ratio measured in Titan were obtained in the cooling solar nebula by isotopic thermal exchange of hydrogen with CH₃D originating from interstellar methane D-enriched ices that vaporized in the nebula. The rate of the isotopic exchange decreased with temperature and became fully inhibited around 200 K. Methane was subsequently trapped in crystalline ices around 10 AU in the form of clathrate hydrates formed at 60 K, and incorporated into planetesimals that formed the core of Titan. The nitrogen-methane atmosphere was subsequently outgassed from the decomposition of the hydrates (Mousis et al. 2002). By use of a turbulent evolutionary model of the solar nebula (O. Mousis et al. 2000, Icarus148, 513-525), we have reconstructed the entire story of D/H in CH₄, from its high value in the early solar nebula (acquired in the presolar cloud) down to the value measured in Titan's atmosphere today. Considering the two last determinations of the D/H ratio in Titan—D/H=(7.75±2.25)×10⁻⁵ obtained from ground-based observations (Orton 1992, In: Symposium on Titan, ESA SP-338, pp. 81-85), and D/H=(8.75^+3.25_−2.25)×10⁻⁵, obtained from ISO observations (Coustenis et al. 2002, submitted for publication)—we inferred an upper limit of the D/H ratio in methane in the early outer solar nebula of about 3×10⁻⁴. Our approach is consistent with the scenario advocated by several authors in which the atmospheric methane of Titan is continuously replenished from a reservoir of clathrate hydrates of CH₄ at high pressures, located in the interior of Titan. If this scenario is correct, observations of the satellite to be performed by the radar, the imaging system, and other remote sensing instruments aboard the spacecraft of the Cassini-Huygens mission from 2004 to 2008 should reveal local disruptions of the surface and other signatures of the predicted outgassing.     

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.     

Channel morphometry, sediment transport, and implications for tectonic activity and surficial ages of Titan basins

Fluvial features on Titan and drainage basins on Earth are remarkably similar despite differences in gravity and surface composition. We determined network bifurcation (R_b) ratios for five Titan and three terrestrial analog basins. Tectonically-modified Earth basins have R_b values greater than the expected range (3.0-5.0) for dendritic networks; comparisons with R_b values determined for Titan basins, in conjunction with similarities in network patterns, suggest that portions of Titan’s north polar region are modified by tectonic forces. Sufficient elevation data existed to calculate bed slope and potential fluvial sediment transport rates in at least one Titan basin, indicating that 75 mm water ice grains (observed at the Huygens landing site) should be readily entrained given sufficient flow depths of liquid hydrocarbons. Volumetric sediment transport estimates suggest that ∼6700-10,000 Titan years (∼2.0-3.0 × 10⁵ Earth years) are required to erode this basin to its minimum relief (assuming constant 1 m and 1.5 m flows); these lowering rates increase to ∼27,000-41,000 Titan years (∼8.0-12.0 × 10⁵ Earth years) when flows in the north polar region are restricted to summer months.     

The dependence of surface tidal stress on the internal structure of Europa: The possibility of cracking of the icy shell

This study shows dependence of the surface diurnal tidal stress on the internal structure of Europa. Its purpose is to investigate possibility of cracking of the icy shell. The stress is evaluated under the plausible model of the internal structure constrained by the gravity field data. The possible effective stress at the sub-Jovian point decreases with thickening of the shell, while it does not depend on the core radius and the thickness of the H₂O layer. The range of this value is from 0.095 to 0.161 MPa, which does not exceed the tensile strength of ice. The stress required for the surface cracking would be mainly due to longer period deformations, especially non-synchronous rotation. And/or the actual strength of the ice at the surface would be smaller because of the preexisting cracks than that at a laboratory of the same temperature.     

Do explosive ice ejections occur on Jupiter’s and Saturn’s satellites?

The possibility of an explosive mechanical instability of ice (the Bridgman effect) in the thick icy shells of Jupiter’s and Saturn’s satellites is discussed in principle. The Bridgman effect is an explosive instability of dielectric solid bodies, which disintegrate into microscopic fragments under a quasistatic uniaxial loading in open compression systems at high pressures. The explosive instabilities of ice recently discovered in laboratory experiments with the Bridgman effect are also expected to occur in the extensive deep layers of the shells of icy planetary satellites (for example, in the case of episodical formation of major cracks in their lithospheres due to tidal forces, nonsynchronous rotation of the satellites, or extremely powerful impacts). The depths of occurrence of mechanically unstable ice in the thick crusts of Ganymede, Europa, and Titan, taken as examples, are crudely estimated using a pure-ice model without a possible ammonia admixture. The estimated thickness of the explosive-instability zone in the icy crust of Ganymede (under the assumption that the crust is ～75 km thick) ranges from ～7 to ～27 km at depths from ～40 to ～67 km, depending on the scaling parameter E = 0.2–1. This parameter relates the experimentally determined thicknesses of the ice samples in which the Bridgman effect occurs under laboratory conditions to the expected thicknesses of the explosively unstable layers in the envelopes of the icy satellites. Explosive effects are possible not throughout the entire thickness of the unstableice layer but only within some part of it, several centimeters to several tens of meters in thickness. According to the estimated location of the unstable layer in the crust of Europa (for an assumed crust thickness of ～30 km), such a layer can exist only at scaling factors E < 0.6 at depths ranging from ～21 to ～28 km. For Titan, if its crust is ～100 km thick, the thickness of the unstable layer is similarly estimated to range from ～15 to ～55 km at depths from ～37 to ～92 km for a scaling parameter E lying within the range 0.2–1. At E 0.2, which is quite possible, explosive instabilities of ice could also be expected on the Earth, in the icy shells of Antarctica and Greenland at depths from ～1 to ～1.5 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.