Ice Shell Thickness Research Articles

Implications of nonsynchronous rotation on the deformational history and ice shell properties in the south polar terrain of Enceladus

Numerous studies of the south polar region of Saturn's moon Enceladus have focused on the four main fractures (the tiger stripes), their associated water plume, and the effects of diurnal tidal stresses on these fractures. However, due to their small magnitude (10–100 kPa), diurnal stresses alone are unable to completely explain the initial formation of the tiger stripes and the many additional fractures in the region. We suggest nonsynchronous rotation (NSR) stresses, induced by a freely rotating ice shell over a global ocean, are large enough to overcome the tensile strength of the ice shell and initiate fracturing on Enceladus. Using the tidal stress calculation program SatStressGUI, we demonstrate the dependence of the magnitude of the NSR stress on the thickness of the ice shell, particularly the brittle outer layer of the ice shell. We suggest the brittle layer in the south polar region is 2–4 km thick whereas in the more northern regions, it is 6–8 km. The difference is most likely due to the elevated energy flux at the south pole, compared to the regions to the north, resulting in a warmer and thinner ice shell in the south polar region. The thinner ice around the south pole permits NSR tensile stresses of ∼5 MPa, enough to fracture the ice. However, the thicker ice to the north reduces the NSR stress to less than 2 MPa making fracturing less likely. The difference in ice shell thickness, and resultant NSR stress, can help explain the activity in the south, but relative quiescence in the north. Our modeled NSR stress magnitudes and orientations suggests a NSR period on the order of 1 Myr and advocate that the tiger stripes have rotated ∼45° clockwise about the south pole since their initial formation in response to NSR of the ice shell. If the ice shell is rotating at a constant rate, the tiger stripes would thus be up to ∼100,000 years old. An apparent decreasing angular separation between consecutive fracture sets that formed in the south polar region suggests an ice shell that has weakened in strength over the course of its recent geologic history; this could be a result of increased heating, thinning of the ice shell, or a combination of the two. As the ice shell thinned, the NSR stresses were enhanced, making fracturing more likely after a smaller amount of NSR compared to the previous fracturing episode.

Powering prolonged hydrothermal activity inside Enceladus

Geophysical data from the Cassini spacecraft imply the presence of a global ocean underneath the ice shell of Enceladus 1 , only a few kilometres below the surface in the South Polar Terrain 2–4 . Chemical analyses indicate that the ocean is salty 5 and is fed by ongoing hydrothermal activity 6–8 . In order to explain these observations, an abnormally high heat power (>20 billion watts) is required, as well as a mechanism to focus endogenic activity at the south pole 9,10 . Here, we show that more than 10 GW of heat can be generated by tidal friction inside the unconsolidated rocky core. Water transport in the tidally heated permeable core results in hot narrow upwellings with temperatures exceeding 363 K, characterized by powerful (1–5 GW) hotspots at the seafloor, particularly at the south pole. The release of heat in narrow regions favours intense interaction between water and rock, and the transport of hydrothermal products from the core to the plume sources. We are thus able to explain the main global characteristics of Enceladus: global ocean, strong dissipation, reduced ice-shell thickness at the south pole and seafloor activity. We predict that this endogenic activity can be sustained for tens of millions to billions of years. Tidal forcing within a very porous (unconsolidated) core can generate enough energy to drive all the observed global features of Enceladus. This activity can be sustained up to several billion years.

Plume Activity and Tidal Deformation on Enceladus Influenced by Faults and Variable Ice Shell Thickness.

We investigated the effect of variations in ice shell thickness and of the tiger stripe fractures crossing Enceladus' south polar terrain on the moon's tidal deformation by performing finite element calculations in three-dimensional geometry. The combination of thinning in the polar region and the presence of faults has a synergistic effect that leads to an increase of both the displacement and stress in the south polar terrain by an order of magnitude compared to that of the traditional model with a uniform shell thickness and without faults. Assuming a simplified conductive heat transfer and neglecting the heat sources below the ice shell, we computed the global heat budget of the ice shell. For the inelastic properties of the shell described by a Maxwell viscoelastic model, we show that unrealistically low average viscosity of the order of 1013 Pa s is necessary for preserving the volume of the ocean, suggesting the important role of the heat sources in the deep interior. Similarly, low viscosity is required to predict the observed delay of the plume activity, which hints at other delaying mechanisms than just the viscoelasticity of the ice shell. The presence of faults results in large spatial and temporal heterogeneity of geysering activity compared to the traditional models without faults. Our model contributes to understanding the physical mechanisms that control the fault activity, and it provides potentially useful information for future missions that will sample the plume for evidence of life. Key Words: Enceladus—Tidal deformation—Faults—Variable ice shell thickness—Tidal heating—Plume activity and timing. Astrobiology 17, 941–954.

Brittle ice shell thickness of Enceladus from fracture distribution analysis

We determine the depth of fracture penetration in multiple regions of Enceladus by performing self-similar clustering and length distribution analysis of fractures. The statistical characterization of fault-population attribute, such as length and clustering, provide a productive avenue for exploring deformation rate, stress transmission mode, rheology of the medium, and mechanical stratification of the ice satellite. Through this analysis, we estimate the depth of the mechanical discontinuity of Enceladus’ ice shell that is the depth to which fractures penetrate the brittle ice layer above the ductile one.In this work, we find that for the South Polar Terrain (SPT), the brittle ice shell interested by fracture penetration is about 30 km and corresponds to the total depth of the ice shell because the SPT has a very high thermal gradient and, hence, fractures likely reach the ocean-ice interface. In the other regions analyzed, the depth of fracture penetration increases from 31 to 70 km from the South Pole to northern regions up to 75°.

Effect of the tiger stripes on the deformation of Saturn's moon Enceladus

AbstractEnceladus is a small icy moon of Saturn with active jets of water emanating from fractures around the south pole, informally called tiger stripes, which might be connected to a subsurface water ocean. The effect of these features on periodic tidal deformation of the moon has so far been neglected because of the difficulties associated with implementation of faults in continuum mechanics models. Here we estimate the maximum possible impact of the tiger stripes on tidal deformation and heat production within Enceladus's ice shell by representing them as narrow zones with negligible frictional and bulk resistance passing vertically through the whole ice shell. Assuming a uniform ice shell thickness of 25 km, consistent with the recent estimate of libration, we demonstrate that the faults can dramatically change the distribution of stress and strain in Enceladus's south polar region, leading to a significant increase of the heat production in this area.

Mechanics of evenly spaced strike-slip faults and its implications for the formation of tiger-stripe fractures on Saturn’s moon Enceladus

We present the first mechanical analysis based on realistic rheology and boundary conditions on the formation of evenly spaced strike-slip faults. Two quantitative models employing the stress-shadow concept, widely used for explaining extensional-joint spacing, are proposed in this study: (1) an empirically based stress-rise-function model that simulates the brittle-deformation process during the formation of evenly spaced parallel strike-slip faults, and (2) an elastic plate model that relates fault spacing to the thickness of the fault-hosting elastic medium. When applying the models for the initiation and development of the tiger-stripe fractures (TSF) in the South Polar Terrain (SPT) of Enceladus, the mutually consistent solutions of the two models, as constrained by the mean spacing of the TSF at ∼35km, requires that the brittle ice-shell thickness be ∼30km, the elastic thickness be ∼0.7km, and the cohesive strength of the SPT ice shell be ∼30kPa. However, if the brittle and elastic models are decoupled and if the ice-shell cohesive strength is on the order of ∼1MPa, the brittle ice shell would be on the order of ∼10km.

Modeling the polar motion of Titan

The angular momentum of the atmosphere and of the hydrocarbon lakes of Titan have a large equatorial component that can excite polar motion, a variable orientation of the rotation axis of Titan with respect to its surface. We here use the angular momentum obtained from a General Circulation Model of the atmosphere of Titan and from an Ocean Circulation Model for Titan’s polar lakes to model the polar motion of Titan as a function of the interior structure. Besides the gravitational torque exerted by Saturn on Titan’s aspherical mass distribution, the rotational model also includes torques arising due to the presence of an ocean under a thin ice shell as well as the influence of the elasticity of the different layers.The Chandler wobble period of a solid and rigid Titan without its atmosphere is about 279years. The period of the Chandler wobble is mainly influenced by the atmosphere of Titan (−166years) and the presence of an internal global ocean (+135 to 295years depending on the internal model) and to a lesser extent by the elastic deformations (+3.7years).The forced polar motion of a solid and rigid Titan is elliptical with an amplitude of about 50m and a main period equal to the orbital period of Saturn. It is mainly forced by the atmosphere of Titan while the lakes of Titan are at the origin of a displacement of the mean polar motion, or polar offset. The subsurface ocean can largely increase the polar motion amplitude due to resonant amplification with a wobble free mode of Titan. The amplitudes as well as the main periods of the polar motion depend on whether and which forcing period is close to the period of a free mode. For a thick ice shell, the polar motion mainly has an annual period and an amplitude of about 1km. For thinner ice shells, the polar motion amplitude can reach several tens of km and shorter periods become dominant. We demonstrate that for thick ice shells, the ice shell rigidity weakly influences the amplitude of the polar motion. For thin ice shells, the level of the resonant amplification of the polar motion amplitude depends on the ice shell rigidity. Future observations of the polar motion of Titan could help constraining some properties of its interior structure as the ice shell thickness and ocean density.

A common origin for ridge-and-trough terrain on icy satellites by sluggish lid convection

Ridge-and-trough terrain is a common landform on outer Solar System icy satellites. Examples include Ganymede’s grooved terrain, Europa’s gray bands, Miranda’s coronae, and several terrains on Enceladus. The conditions associated with the formation of each of these terrains are similar: heat flows of order tens to a hundred milliwatts per meter squared, and deformation rates of order 10−16–10−12s−1. Our prior work shows that the conditions associated with the formation of these terrains on Ganymede and the south pole of Enceladus are consistent with vigorous solid-state ice convection in a shell with a weak surface. We show that sluggish lid convection, an intermediate regime between the isoviscous and stagnant lid regimes, can create the heat flow and deformation rates appropriate for ridge and trough formation on a number of satellites, regardless of the ice shell thickness. For convection to deform their surfaces, the ice shells must have yield stresses similar in magnitude to the daily tidal stresses. Tidal and convective stresses deform the surface, and the spatial pattern of tidal cracking controls the locations of ridge-and-trough terrain.

A passive probe for subsurface oceans and liquid water in Jupiter’s icy moons

We describe an interferometric reflectometer method for passive detection of subsurface oceans and liquid water in jovian icy moons using Jupiter’s decametric radio emission (DAM). The DAM flux density exceeds 3000 times the galactic background in the neighborhood of the jovian icy moons, providing a signal that could be used for passive radio sounding. An instrument located between the icy moon and Jupiter could sample the DAM emission along with its echoes reflected in the ice layer of the target moon. Cross-correlating the direct emission with the echoes would provide a measurement of the ice shell thickness along with its dielectric properties. The interferometric reflectometer provides a simple solution to sub-jovian radio sounding of ice shells that is complementary to ice penetrating radar measurements better suited to measurements in the anti-jovian hemisphere that shadows Jupiter’s strong decametric emission. The passive nature of this technique also serves as risk reduction in case of radar transmitter failure. The interferometric reflectometer could operate with electrically short antennas, thus extending ice depth measurements to lower frequencies, and potentially providing a deeper view into the ice shells of jovian moons.

Titan’s internal structure inferred from its gravity field, shape, and rotation state

Several quantities measured by the Cassini-Huygens mission provide insight into the interior of Titan: the second-degree gravity field coefficients, the shape, the tidal Love number, the electric field, and the orientation of its rotation axis. The measured obliquity and tides, as well as the electric field, are evidence for the presence of an internal global ocean beneath the icy shell of Titan. Here we use these different observations together to constrain the density profile assuming a four-layer interior model (ice I shell, liquid water ocean, high pressure ice mantle, and rock core). Even though the observed second degree gravity field is consistent with the hydrostatic relation J2=10C22/3, which is a necessary but not sufficient condition for a synchronous satellite to be in hydrostatic equilibrium, the observed shape of the surface as well as the non-zero degree-three gravity signal indicate some departure from hydrostaticity. Therefore, we do not restrain our range of assumed density profiles to those corresponding to the hydrostatic value of the moment of inertia (0.34). From a range of density profiles consistent with the radius and mass of the satellite, we compute the obliquity of the Cassini state and the tidal Love number k2. The obliquity is computed from a Cassini state model for a satellite with an internal liquid layer, each layer having an ellipsoidal shape consistent with the measured surface shape and gravity field. The observed (nearly hydrostatic) gravity field is obtained by an additional deflection of the ocean–ice I shell interface, assuming that the layers have uniform densities. We show that the measured obliquity can be reproduced only for internal models with a dense ocean (between 1275 and 1350kgm−3) above a differentiated interior with a full separation of rock and ice. We obtain normalized moments of inertia between 0.31 and 0.33, significantly lower than the expected hydrostatic value (0.34). Evolutionary mechanisms leading to a significant departure from hydrostatic equilibrium while J2=10C22/3 remain an open issue. The tidal Love number is found to be mostly sensitive to the ocean density and to a lesser extent to the ice shell thickness. By combining obliquity and tidal Love number constraints, we show that the thickness of the outer ice shell is at least 40km and the ocean thickness is less than 100km, with an averaged density of 1300–1350kgm−3. The elevated density (>3400kgm−3) found for the rocky core further suggests that it might possess a significant fraction of iron.

Shape, topography, gravity anomalies and tidal deformation of Titan

Gravity measurements and elevation data from the Cassini mission have been used to create shape, global topography and gravity anomaly models of Titan that enable an improved understanding of its outer ice I shell structure. We provide constraints on the averaged ice shell thickness and its long-wavelength lateral variations, as well as the density of the subsurface ocean using gravity anomalies, the tidal Love number k2 measurement and long-wavelength topography. We found that Titan’s surface topography is consistent with an approximate isostatically compensated ice shell of variable thickness, likely in a thermally conductive or in a subcritical convective state, overlying a relatively dense subsurface ocean.

Active formation of ‘chaos terrain’ over shallow subsurface water on Europa

Europa, the innermost icy satellite of Jupiter, has a tortured young surface and sustains a liquid water ocean below an ice shell of highly debated thickness. Quasi-circular areas of ice disruption called chaos terrains are unique to Europa, and both their formation and the ice-shell thickness depend on Europa's thermal state. No model so far has been able to explain why features such as Conamara Chaos stand above surrounding terrain and contain matrix domes. Melt-through of a thin (few-kilometre) shell is thermodynamically improbable and cannot raise the ice. The buoyancy of material rising as either plumes of warm, pure ice called diapirs or convective cells in a thick (>10 kilometres) shell is insufficient to produce the observed chaos heights, and no single plume can create matrix domes. Here we report an analysis of archival data from Europa, guided by processes observed within Earth's subglacial volcanoes and ice shelves. The data suggest that chaos terrains form above liquid water lenses perched within the ice shell as shallow as 3 kilometres. Our results suggest that ice-water interactions and freeze-out give rise to the diverse morphologies and topography of chaos terrains. The sunken topography of Thera Macula indicates that Europa is actively resurfacing over a lens comparable in volume to the Great Lakes in North America.

Limits of Enceladus's ice shell thickness from tidally driven tiger stripe shear failure

[1] Enceladus's south polar thermal anomaly and water-rich plumes suggest the existence of a subsurface ocean, which is overlain by an ice shell of uncertain thickness. Our objective is to constrain Enceladus's ice shell thickness, through assessment of tidally driven Coulomb failure of Enceladus's tiger stripe faults. We find that thin to moderate ice shell thicknesses (<40 km) support shear failure along the tiger stripes, assuming low ice coefficients of friction (0.1–0.3) and shallow fault depths (<3 km). These results are marginally consistent with the minimum ice shell thickness which can permit convection within Enceladus's ice shell. A plausible scenario is one in which the heat loss and tectonic style of Enceladus has changed through time, with convection initiating in a thick ice shell, and tiger stripe activity commencing as the ice shell thinned.

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 (1–10mm), ice-shell thickness (20–120km), 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 1mm, no thermal convection can initiate in Europa’s ice shell (for thicknesses <100km) 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 1–10mm, convection can occur in ice shells as thin as 20–40km with the estimated tidal-strain amplitude of 2×10−5 on Europa.

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 δ 3 / η b , where δ is the thickness of the ice shell and η b 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.

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 10 14 to 10 15 Pa s ) the maximum permissible (conductive) shell thickness is 35 km. The relative uniformity of Europa's shell thickness is due to either a heat flux ⩾ 7 mW m −2 from the silicate interior, lateral ice flow at the base of the shell, or convection within the shell.

Shear heating as the origin of the plumes and heat flux on Enceladus

Enceladus, a small icy satellite of Saturn, has active plumes jetting from localized fractures ('tiger stripes') within an area of high heat flux near the south pole. The plume characteristics and local high heat flux have been ascribed either to the presence of liquid water within a few tens of metres of the surface, or the decomposition of clathrates. Neither model addresses how delivery of internal heat to the near-surface is sustained. Here we show that the most likely explanation for the heat and vapour production is shear heating by tidally driven lateral (strike-slip) fault motion with displacement of approximately 0.5 m over a tidal period. Vapour produced by this heating may escape as plumes through cracks reopened by the tidal stresses. The ice shell thickness needed to produce the observed heat flux is at least 5 km. The tidal displacements required imply a Love number of h2 > 0.01, suggesting that the ice shell is decoupled from the silicate interior by a subsurface ocean. We predict that the tiger-stripe regions with highest relative temperatures will be the lower-latitude branch of Damascus, Cairo around 60 degrees W longitude and Alexandria around 150 degrees W longitude.

Estimates of Europa’s ice shell thickness and strain rate from flanking cracks and bulge along Ridge R

Europa, the second Galilean satellite outward from Jupiter, has an outer layer of water of about 100 km thick and an outmost ice shell. The thickness of the ice shell is very important in understanding Europa’s habitability and thermal history, but estimates from different studies are very inconsistent, ranging from 0.2 to 30 km. Here we obtain an estimate of the ice shell thickness from locations of flanking crack and forebulge along Ridge R. Considering the water’s heating process to nearby ice shell in the crack, a flexure model is applied and it suggests the thickness of an ice shell to be 500–1500 m without a convective layer. Compared with previous studies using the same method but ignoring the water’s heating process, the rationality and accuracy have been improved dramatically in our results. We also get some constraints on the strain rate \(\dot \varepsilon \) and the characteristic temperature T c, which defines the base of the elastic layer.

Clathrate Hydrates of Oxidants in the Ice Shell of Europa

Europa's icy surface is radiolytically modified by high-energy electrons and ions, and photolytically modified by solar ultraviolet photons. Observations from the Galileo Near Infrared Mapping Spectrometer, ground-based telescopes, the International Ultraviolet Explorer, and the Hubble Space Telescope, along with laboratory experiment results, indicate that the production of oxidants, such as H2O2, O2, CO2, and SO2, is a consequence of the surface radiolytic chemistry. Once created, some of the products may be entrained deeper into the ice shell through impact gardening or other resurfacing processes. The temperature and pressure environments of regions within the europan hydrosphere are expected to permit the formation of mixed clathrate compounds. The formation of carbon dioxide and sulfur dioxide clathrates has been examined in some detail. Here we add to this analysis by considering oxidants produced radiolytically on the surface of Europa. Our results indicate that the bulk ice shell could have a approximately 1.7-7.6% by number contamination of oxidants resulting from radiolysis at the surface. Oxidant-hosting clathrates would consequently make up approximately 12-53% of the ice shell by number relative to ice, if oxidants were entrained throughout. We examine, in brief, the consequences of such contamination on bulk ice shell thickness and find that clathrate formation could lead to substantially thinner ice shells on Europa than otherwise expected. Finally, we propose that double occupancy of clathrate cages by O2 molecules could serve as an explanation for the observation of condensed-phase O2 on Europa. Clathrate-sealed, gas-filled bubbles in the near surface ice could also provide an effective trapping mechanism, though they cannot explain the 5771 A (O2)2 absorption.

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.