Calculation of single-scattering albedos: Comparison of Mie results with Hapke approximations

The grain size of water ice can be determined from its near-infrared spectrum, which has numerous diagnostic absorption bands of different opacities. Models that have been used to determine water ice grain size from infrared spectra of icy outer Solar System objects have shown discrepancies in modeled grain size of a factor of two or more. Here the single-scattering albedo calculated using the commonly used Hapke model given by Roush [Roush, T.L., 1994. Icarus 108, 243-254] is compared with the exact calculation for spheres from a Mie series. An earlier approximation of single-scattering albedo called the Hapke “slab” model is also used in the comparison. All three models are implemented using the same optical constants for water ice at ∼110 K. Results are displayed for a large range of grain sizes from 1 μm to 1 mm. In general neither Hapke model can mimic the Rayleigh effects from particles sized near the wavelength of light that the Mie model predicts. For 10 μm particles, the slab model matches the Mie calculation quite well, but larger sizes are more discrepant. The Hapke/Roush model grain size needs to be ∼2.5 times larger to mimic the Mie results, and there are additional discrepancies in the continuum levels and band strengths. The Mie calculation for spheres is recommended for analysis of unknown remote sensing measurements, as it can mimic the spectra of oblate, prolate, and hollow particles given by equivalent sphere theories.     

Heat flow and thickness of a convective ice shell on Europa for grain size-dependent rheologies

Although it is mostly accepted that the lower part of the ice shell of Europa is actively convective, there is still much uncertainty about the flow mechanism dominating the rheology of this convective layer, which largely depends on the grain size of the ice. In this work, we examined thermal equilibrium states in a tidally heated and strained convective shell, for two rheologies sensitive to grain size, grain boundary sliding and diffusion creep. If we take a lower limit of 70 mW m⁻² for the surface heat flow, according to some geological features observed, the ice grain size should be less than 2 or 0.2 mm for grain boundary sliding or diffusion creep respectively. If in addition the thickness of the ice shell is constrained to a few tens of kilometers and it is assumed that the thickness of the convective layer is related to lenticulae spacing, then grain sizes between 0.2 and 2 mm for grain boundary sliding, and between 0.1 and 0.2 mm for diffusion creep are obtained. Also, local convective layer thicknesses deduced from lenticulae spacing are more similar to those here derived for grain boundary sliding. Our results thus favor grain boundary sliding as the dominant rheology for the water ice in Europa's convective layer, since this flow mechanism is able to satisfy the imposed constraints for a wider range of grain sizes.     

On convection in ice I shells of outer Solar System bodies, with detailed application to Callisto

It has been argued that the dominant non-Newtonian creep mechanisms of water ice make the ice shell above Callisto's ocean, and by inference all radiogenically heated ice I shells in the outer Solar System, stable against solid-state convective overturn. Conductive heat transport and internal melting (oceans) are therefore predicted to be, or have been, widespread among midsize and larger icy satellites and Kuiper Belt objects. Alternatively, at low stresses (where non-Newtonian viscosities can be arbitrarily large), convective instabilities may arise in the diffusional creep regime for arbitrarily small temperature perturbations. For Callisto, ice viscosities are low enough that convection is expected over most of geologic time above the internal liquid layer for plausible ice grain sizes (?a few mm); the alternative for early Callisto, a conducting shell over a very deep ocean (>450 km), is not compatible with Callisto's present partially differentiated state. Moreover, if convection is occurring today, the stagnant lid would be quite thick (∼100 km) and compatible with the lack of active geology. Nevertheless, Callisto's steady-state heat flow may have fallen below the convective minimum for its ice I shell late in Solar System history. In this case convection ends, the ice shell melts back at its base, and the internal ocean widens considerably. The presence of such an ocean, of order 200 km thick, is compatible with Callisto's moment-of-inertia, but its formation would have caused an ∼0.25% radial expansion. The tectonic effects of such a late, slow expansion are not observed, so convection likely persists in Callisto, possibly subcritically. Ganymede, due to its greater size, rock fraction and full differentiation, has a substantially higher heat flow than Callisto and has not reached this tectonic end state. Titan, if differentiated, and Triton should be more similar to Ganymede in this regard. Pluto, like Callisto, may be near the tipping point for convective shutdown, but uncertainties in its size and rock fraction prevent a more definitive assessment.     

Constraints on martian lobate debris apron evolution and rheology from numerical modeling of ice flow

Radar observations in the Deuteronilus Mensae region by Mars Reconnaissance Orbiter have constrained the thickness and dust concentration found within mid-latitude ice deposits, providing an opportunity to more accurately estimate the rheology of ice responsible for the formation of lobate debris aprons based on their apparent age of ∼100 Myr. We developed a numerical model simulating ice flow under martian conditions using results from ice deformation experiments, theory of ice grain growth based on terrestrial ice cores, and observational constraints from radar profiles and laser altimetry. By varying the ice grain size, the ice temperature, the subsurface slope, and the initial ice volume we determine the combination of parameters that best reproduce the observed LDA lengths and thicknesses over a period of time comparable to the apparent ages of LDA surfaces (90-300 Myr). We find that an ice temperature of 205 K, an ice grain size of 5 mm, and a flat subsurface slope give reasonable ages for many LDAs in the northern mid-latitudes of Mars. Assuming that the ice grain size is limited by the grain boundary pinning effect of incorporated dust, these results limit the dust volume concentration to less than 4%. However, assuming all LDAs were emplaced by a single event, we find that there is no single combination of grain size, temperature, and subsurface slope which can give realistic ages for all LDAs, suggesting that some or all of these variables are spatially heterogeneous. Based on our model we conclude that the majority of northern mid-latitude LDAs are composed of clean (?4 vol%), coarse (?1 mm) grained ice, but regional differences in either the amount of dust mixed in with the ice, or in the presence of a basal slope below the LDA ice must be invoked. Alternatively, the ice temperature and/or timing of ice deposition may vary significantly between different mid-latitude regions. Either eventuality can be tested with future observations.     

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.     

Europa’s ridged plains and smooth low albedo plains: Distinctive compositions and compositional gradients at the leading side-trailing side boundary

This investigation uses linear mixture modeling employing cryogenic laboratory reference spectra to estimate surface compositions and water ice grain sizes of Europa’s ridged plains and smooth low albedo plains. Near-infrared spectra for 23 exposures of ridged plains materials are analyzed along with 11 spectra representing low albedo plains. Modeling indicates that these geologic units differ both in the relative abundance of non-ice hydrated species and in the abundance and grain sizes of water ice. The background ridged plains in our study area appear to consist predominantly of water ice (∼46%) with approximately equal amounts (on average) of hydrated sulfuric acid (∼27%) and hydrated salts (∼27%). The solutions for the smooth low albedo plains are dominated by hydrated salts (∼62%), with a relatively low mean abundance of water ice (∼10%), and an abundance of hydrated sulfuric acid similar to that found in ridged plains (∼27%). The model yields larger water ice grain sizes (100 μm versus 50-75 μm) in the ridged plains. The 1.5-μm water ice absorption band minimum is found at shorter wavelengths in the low albedo plains deposits than in the ridged plains (1.498 ± .003 μm versus 1.504 ± .001 μm). The 2.0-μm band minimum in the low albedo plains exhibits a somewhat larger blueshift (1.964 ± .006 μm versus 1.983 ± .006 μm for the ridged plains).The study area spans longitudes from 168° to 185°W, which includes Europa’s leading side-trailing side boundary. A well-defined spatial gradient of sulfuric acid hydrate abundance is found for both geologic units, with concentrations increasing in the direction of the trailing side apex. We associate this distribution with the exogenic effects of magnetospheric charged particle bombardment and associated chemical processing of surface materials (the radiolytic sulfur cycle). However, one family of low albedo plains exposures exhibits sulfuric acid hydrate abundances up to 33% lower than found for adjacent exposures, suggesting that these materials have undergone less processing, thus implying that these deposits may have been emplaced more recently.Modeling identifies high abundances (to 30%) of magnesium sulfate brines in the low albedo plains exposures. Our investigation marks the first spectroscopic identification of MgSO₄ brine on Europa. We also find significantly higher abundances of sodium-bearing species (bloedite and mirabilite) in the low albedo plains. The results illuminate the role of radiolytic processes in modifying the surface composition of Europa, and may provide new constraints for models of the composition of Europa’s putative subsurface ocean.     

Equilibrium Convection on a Tidally Heated and Stressed Icy Shell of Europa for a Composite Water Ice Rheology

Water ice I rheology is a key factor for understanding the thermal and mechanical state of the outer shell of the icy satellites. Ice flow involves several deformation mechanisms (both Newtonian and non-Newtonian), which contribute to different extents depending on the temperature, grain size, and applied stress. In this work I analyze tidally heated and stressed equilibrium convection in the ice shell of Europa by considering a composite viscosity law which includes diffusion creep, basal slip, grain boundary sliding and dislocation creep, and. The calculations take into account the effect of tidal stresses on ice flow and use grain sizes between 0.1 and 100 mm. An Arrhenius-type relation (useful for parameterized convective models) is found then by fitting the calculated viscosity between 170 and 273 K to an exponential regression, which can be expressed in terms of pre-exponential constant and effective activation energy. I obtain convective heat flows between ~40 and ~60 mW m^?2, values lower than those usually deduced (~100 mW m^?2) from geological indicators of lithospheric thermal state, probably indicating heterogeneous tidal heating. On the other hand, for grain sizes larger than ~0.3 mm the thicknesses of the ice shell and convective sublayer are ~20–30 km and ~5–20 km respectively, values in good agreement with the available information for Europa. So, some fundamental geophysical characteristics of the ice shell of Europa could be arising from the properties of the composite water ice rheology.     

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).     

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.     

Louth crater: Evolution of a layered water ice mound

We report on observations made of the ∼36 km diameter crater, Louth, in the north polar region of Mars (at 70° N, 103.2° E). High-resolution imagery from the instruments on the Mars Reconnaissance Orbiter (MRO) spacecraft has been used to map a 15 km diameter water ice deposit in the center of the crater. The water ice mound has surface features that include roughened ice textures and layering similar to that found in the North Polar Layered Deposits. Features we interpret as sastrugi and sand dunes show consistent wind patterns within Louth over recent time. CRISM spectra of the ice mound were modeled to derive quantitative estimates of water ice and contaminant abundance, and associated ice grain size information. These morphologic and spectral results are used to propose a stratigraphy for this deposit and adjoining sand dunes. Our results suggest the edge of the water ice mound is currently in retreat.     

Ultraviolet observations of Phoebe from the Cassini UVIS

Observations of Saturn's distant moon Phoebe were made at far-ultraviolet (FUV) (1100-1900 Å) and extreme-ultraviolet (EUV) (600-1100 Å) wavelengths by the Cassini Ultraviolet Imaging Spectrograph (UVIS) during the Cassini spacecraft flyby on June 11, 2004. These are the first UV spectra of Phoebe and the first detection of water ice on a Solar System surface using FUV wavelengths. The characteristics of water ice in the FUV are presented, and Hapke models are used to interpret the spectra in terms of composition and grain size; the use of both areal and intimate mixing models is explored. Non-ice species used in these models include carbon, ice tholin, Triton tholin, poly-HCN and kerogen. Satisfactory disk-integrated fits are obtained for intimate mixtures of ∼10% H₂O plus a non-ice species. Spatially resolved regions of higher (∼20%) and lower (∼5%) H₂O ice concentrations are also detected. Phoebe does not display any evidence of volatile activity. Upper limits on atomic oxygen and carbon are 5×¹⁰¹¹ and 2×¹⁰¹² atoms/cm², respectively, for solar photon scattering. The UVIS detection of water ice on Phoebe, and the ice amounts detected, are consistent with IR measurements and contribute to the evidence for a Phoebe origin in the outer Solar System rather than in the main asteroid belt.     

The orbital-thermal evolution and global expansion of Ganymede

The tectonically and cryovolcanically resurfaced terrains of Ganymede attest to the satellite's turbulent geologic history. Yet, the ultimate cause of its geologic violence remains unknown. One plausible scenario suggests that the Galilean satellites passed through one or more Laplace-like resonances before evolving into the current Laplace resonance. Passage through such a resonance can excite Ganymede's eccentricity, leading to tidal dissipation within the ice shell. To evaluate the effects of resonance passage on Ganymede's thermal history we model the coupled orbital-thermal evolution of Ganymede both with and without passage through a Laplace-like resonance. In the absence of tidal dissipation, radiogenic heating alone is capable of creating large internal oceans within Ganymede if the ice grain size is 1 mm or greater. For larger grain sizes, oceans will exist into the present epoch. The inclusion of tidal dissipation significantly alters Ganymede's thermal history, and for some parameters (e.g. ice grain size, tidal Q of Jupiter) a thin ice shell (5 to 20 km) can be maintained throughout the period of resonance passage. The pulse of tidal heating that accompanies Laplace-like resonance capture can cause up to 2.5% volumetric expansion of the satellite and contemporaneous formation of near surface partial melt. The presence of a thin ice shell and high satellite orbital eccentricity would generate moderate diurnal tidal stresses in Ganymede's ice shell. Larger stresses result if the ice shell rotates non-synchronously. The combined effects of satellite expansion, its associated tensile stress, rapid formation of near surface partial melt, and tidal stress due to an eccentric orbit may be responsible for creating Ganymede's unique surface features.     

A ∼25 K temperature of formation for the submicron ice grains which formed comets

Following the observations of ice grains in cometary comae and their size distributions, we reexamined experimentally our previous conclusion that the ice grains which agglomerated to form comet nuclei were formed at ∼25 K. The suggestion of a ∼25 K formation temperature was confirmed experimentally. Moreover, we suggest that these ice grains had to be of submicron size.     

Numerical modeling of endogenic thermal anomalies on Europa

A variety of recent resurfacing features have been observed on Europa, which may produce thermal anomalies detectable by a future mission. However, the likelihood of such a detection depends on their size and lifetimes. The results of this numerical study suggest that the lifetime of a thermal anomaly associated with the emplacement of 100 m of water onto the surface of Europa is several hundred years, and ∼10 years for 10 m of water. If warm ice is emplaced on the surface instead of liquid water, these lifetimes decrease by up to a factor of two. Exploration of model parameters indicates that a thin insulating surface layer can double thermal anomaly lifetimes, anomalies emplaced at a latitude of 80° can remain detectable nearly a factor of two longer than those at equatorial latitudes, and anomalies on the night side can remain detectable for up to ∼20% longer than those on the day side. High temperatures are very short-lived as the surface ice cools very rapidly to below 200 K due to sublimation cooling. Assuming steady-state resurfacing, the number of detectable thermal anomalies associated with the emplacement of 100 m of water would be on the order of 10 if the typical resurfacing area is 15 km². If recent resurfacing is dominated by chaos regions with typical areas of 100 to 1000 km² and lifetimes of 1000 to 4000 years, the number of detectable thermal anomalies would be on the order of 1 to 10.     

Stratospheric superrotation in the TitanWRF model

TitanWRF general circulation model simulations performed without sub-grid-scale horizontal diffusion of momentum produce roughly the observed amount of superrotation in Titan’s stratosphere. We compare these results to Cassini-Huygens measurements of Titan’s winds and temperatures, and predict temperature and winds at future seasons. We use angular momentum and transformed Eulerian mean diagnostics to show that equatorial superrotation is generated during episodic angular momentum ‘transfer events’ during model spin-up, and maintained by similar (yet shorter) events once the model has reached steady state. We then use wave and barotropic instability analysis to suggest that these transfer events are produced by barotropic waves, generated at low latitudes then propagating poleward through a critical layer, thus accelerating low latitudes while decelerating the mid-to-high latitude jet in the late fall through early spring hemisphere. Finally, we identify the dominant waves responsible for the transfers of angular momentum close to northern winter solstice during spin-up and at steady state. Problems with our simulations include peak latitudinal temperature gradients and zonal winds occurring ∼60 km lower than observed by Cassini CIRS, and no reduction in zonal wind speed around 80 km, as was observed by Huygens. While the latter may have been due to transient effects (e.g. gravity waves), the former suggests that our low (∼420 km) model top is adversely affecting the circulation near the jet peak, and/or that we require active haze transport in order to correctly model heating rates and thus the circulation. Future work will include running the model with a higher top, and including advection of a haze particle size distribution.     

Frost grain size metamorphism: Implications for remote sensing of planetary surfaces

An understanding of the rates of frost grain growth is essential to the goal of relating spectral data on surface mineralogy to the physical history of a planetary surface. Models of grain growth kinetics have been constructed for various frosts based on their individual thermodynamic properties and on the difference in binding energy between molecules on plane vs curved faces. A steady state situation can occur on planetary surfaces in which thermal elimination of small grains competes with their creation, usually by meteorite impact. We utilize predicted grain growth rates to explain telescopic spectral data on condensate surfaces throughout the solar system. On Pluto, predicted CH₄ ice grain growth rates are very high despite the low temperature, resulting in a multicentimeter optical path. This explains the strong CH₄ absorption band depths, which otherwise would require large amounts of CH₄ gas. On the Uranian and Saturnian satellites, extremely slow grain growth rates are predicted because of the low vapor pressure of H₂O at the existing average surface temperatures. This may explain evidence for fine grain size and peculiar microstructure. On Io, ordinary thermal exchange is more effective than sputtering in promoting grain growth because of the properties of SO₂. Over much of Io's disk, submicron size grains of SO₂ could plausibly reconfigure into a surface glaze on a timescale comparable to the resurfacing rate. This may explain the relatively strong SO₂ signature in Io's infrared absorption spectrum as opposed to its weaker manifestation in the visible spectrum. In spite of lower sputtering fluxes, sputtering plays a more important role in grain growth for Europa, Ganymede, and Callisto than on Io. This is a result of high rates of thermally activated grain growth and resurfacing on Io. The sequence of H₂O-ice absorption band depths (related to the mean grain size) is J₂(T) ～ J₃(T) > J₂(L) > J₃(L) ～ J₄(T) ～ J₄(L), where L = leading and T = trailing. This is to be expected if sputtering were dominant. The calculations show, however, that neither thermalized exchange fluxes nor sputtering exchange fluxes can produce the implied grain growth or the ordering by ice absorption band depths of the six satellite hemispheres. Only sputtering control by simple ejection of H₂O from the satellites, as the dominant cause of shorter mean lifetimes for smaller exposed grains, can satisfactorily explain the data. Some observations, which suggest that there are vertical grain size gradients, may result from a steady state balance between intense near surface production of fine frost by comminution, coupled with ongoing ubiquitous grain growth in the vertical column. In certain cases, e.g., Europa and Enceladus, the possibility exists that endogenic activity as well as comminution could affect grain size—at least locally. It is concluded that not only ice identification and mapping, but ice grain size mapping is an important experiment to be conducted on future missions.     

Putative ice flows on Europa: Geometric patterns and relation to topography collectively constrain material properties and effusion rates

Europa's surface exhibits numerous small dome-like and lobate features, some of which have been attributed to fluid emplacement of ice or slush on the surface. We perform numerical simulations of non-Newtonian flows to assess the physical conditions required for these features to result from viscous flows. Our simulations indicate that the morphology of an ice flow on Europa will be, at least partially, influenced by pre-existing topography unless the thickness of the flow exceeds that of the underlying topography by at least an order of magnitude. Three classes of features can be identified on Europa. First, some (possibly most) putative flow-like features exhibit no influence from the pre-existing topography such as ridges, although their thicknesses are generally on the same order as those of ridges. Therefore, flow processes probably cannot explain the formation of these features. Second, some observed features show modest influence from the underlying topography. These might be explained by ice flows with wide ranges of parameters (ice temperatures >230 K, effusion rates >10⁷ m³ year⁻¹, and a wide range of grain sizes), although surface uplift (e.g., by diapirism) and in situ disaggregation provide an equally compelling explanation. Third, several observed features are completely confined by pre-existing topographic structures on at least one side; these are the best known candidates for flow features on Europa. If these features resulted from solid-ice flows, then temperatures >260 K and grain sizes <2 μm are required. Such small grain sizes seem unlikely; low-viscosity flows such as ice slurries or brines provide a better explanation for these features. Our results provide theoretical support for the view that many of Europa's lobate features have not resulted from solid-ice flows.     

Ice table depth variability near small rocks at the Phoenix landing site, Mars: A pre-landing assessment

We combine thermal simulations of ground ice stability near small rocks with extrapolations of the abundance of rocks at the Phoenix landing site based on HiRISE rock counts to estimate the degree of ice table depth variability within the 3.8 m² workspace that can be excavated during the mission. Detailed predictions of this kind are important both to test current ground-ice theory and to optimize soil investigations after landing. We find that Phoenix will very likely have access to at least one rock in the diameter range 5 cm to 1 m. Our simulations, which assume the ice to be in diffusive equilibrium with atmospheric water vapor, indicate that all rocks in this size range are associated with an annulus of deep ice-free soil. Ice table depth variability of 1-5 cm is very likely at the landing site due to the presence of small rocks. Further, there are scenarios in which Phoenix might exploit the presence of individual large rocks and/or the arrangement of small rocks to sample soils at depths >10 cm below the average depth predicted from orbit (∼4 cm). Scale analysis to constrain uncertainties in simulation results indicates that estimates of maximum depths may be somewhat conservative and that ice table depressions associated with individual rocks could be deeper and laterally more extended than indicated by formal predictions by mm to cm.     

Lunar soil grain size distribution

A comprehensive review has been made of the currently available data for lunar grain size distributions. It has been concluded that there is little or no statistical difference among the large majority of the soil samples from the Apollo 11, 12, 14, and 15 missions. The grain size distribution for these soils has reached a steady state in which the comminution processes are balanced by the aggregation processes. The median particle size for the steady-state soil is 40 to 130 µm. The predictions of lunar grain size distributions based on the Surveyor television photographs have been found to be quantitatively in error and qualitatively misleading.     

Subsurface heat transfer on Enceladus: Conditions under which melting occurs

Given the heat that is reaching the surface from the interior of Enceladus, we ask whether liquid water is likely and at what depth it might occur. The heat may be carried by thermal conduction through the solid ice, by the vapor as it diffuses through a porous matrix, or by the vapor flowing upward through open cracks. The vapor carries latent heat, which it acquires when ice or liquid evaporates. As the vapor nears the surface it may condense onto the cold ice, or it may exit the vent without condensing, carrying its latent heat with it. The ice at the surface loses its heat by infrared radiation. An important physical principle, which has been overlooked so far, is that the partial pressure of the vapor in the pores and in the open cracks is nearly equal to the saturation vapor pressure of the ice around it. This severely limits the ability of ice to deliver the observed heat to the surface without melting at depth. Another principle is that viscosity limits the speed of the flow, both the diffusive flow in the matrix and the hydrodynamic flow in open cracks. We present hydrodynamic models that take these effects into account. We find that there is no simple answer to the question of whether the ice melts or not. Vapor diffusion in a porous matrix can deliver the heat to the surface without melting if the particle size is greater than ∼1 cm and the porosity is greater than ∼0.1, in other words, if the matrix is a rubble pile. Whether such an open matrix can exist under its own hydrostatic load is unclear. Flow in open cracks can deliver the heat without melting if the width of the crack is greater than ∼10 cm, but the heat source must be in contact with the crack. Frictional heating on the walls due to tidal stresses is one such possibility. The lifetime of the crack is a puzzle, since condensation on the walls in the upper few meters could seal the crack off in a year, and it takes many years for the heat source to warm the walls if the crack extends down to km depths. The 10:1 ratio of radiated heat to latent heat carried with the vapor is another puzzle. The models tend to give a lower ratio. The resolution might be that each tiger stripe has multiple cracks that share the heat, which tends to lower the ratio. The main conclusion is that melting depends on the size of the pores and the width of the cracks, and these are unknown at present.