Permanently shaded areas near the poles of the Moon and Mercury may harbor water ice. We develop a physical model for migration of water molecules in the regolith and discover two pathways that can lead to accumulation of H2O in the subsurface. A small fraction of water molecules delivered, either continuously or abruptly, to permanently cold areas diffuses into the regolith and can remain there longer than on the surface. Higher temperatures lead to deeper burial. At constant temperature, this diffusive migration produces less than one molecular layer of volatile H2O on grains, because it is driven by differences in surface concentrations. The water is therefore expected to be in adsorbed form, and the amount stored in this fashion could be at most a few hundred ppm of H2O. A second pathway is pumping by diurnal temperature oscillations from a transient ice cover that may have formed during a large comet impact. It can lead to high ground ice densities, but the ground ice layer lasts not long beyond the disappearance of the ice cover. Both types of subsurface charging mechanism work best for temperatures typical of permanently shaded areas with sunlit surfaces in their field of view.
 Water molecules delivered to the Moon or Mercury from space transiently move along ballistic trajectories and can get trapped at permanently shaded areas near the poles, where sublimation rates are so low that ice would survive over billions of years [Watson et al., 1961a, 1961b; Arnold, 1979, 1987; Svitek and Murray, 1988; Vondrak and Crider, 2003]. After H2O molecules have arrived on the surface of cold traps, they can continue to hop into the regolith. Transport and deposition of water molecules has been studied previously. To transport and trapping, we add subsurface migration caused by molecular diffusion in the porous regolith, taking a different approach than that used by Cocks et al. .
 Permanently shaded areas near the lunar poles have not seen the sun for at least 2 billion years [Ward, 1975]. Volatiles are delivered to the Moon and Mercury by comets, meteoroids, interstellar dust particles, and solar wind. Estimates reach 1014 kg over 2 × 109 years for the Moon [Arnold, 1979], but are uncertain. Furthermore, many of the molecules are destroyed by photodissociation on their way to the cold traps. Moses et al.  estimate up to 6 × 1014 kg of water was delivered to Mercury over 3.5 billion years by micrometeorites. The ice that arrives in cold areas may last billions of years. As calculated below, the sublimation loss at 100 K is E ≈ 1 kg m−2Ga−1, the equivalent of a 1 mm thick layer in 1 billion years.
 Ice is lost not only by sublimation, but also by UV-radiation from the sun or the local interstellar medium [Morgan and Shemansky, 1991] and by sputtering from solar wind directly or from the tail of Earth's magnetosphere [Lanzerotti et al., 1981]. The destruction rates may exceed accumulation rates [Lanzerotti et al., 1981], which questions whether ice really exists in the cold traps.
 Ground ice would be better protected from destruction by space weathering than exposed ice. Crider and Vondrak [2003a, 2003b] have modeled burial by impact ejecta. We take a different approach. Section 2 describes the physical concepts that govern migration within extremely cold regolith in a vacuum environment. Previous studies [Watson et al., 1961b; Arnold, 1979; Butler et al., 1993; Butler, 1997] describe lateral migration to cold traps. The ratio of time of flight to residence time is large for ballistic hops on the dayside, but tiny in the pores of permanently shaded areas. Migration of inert gases is limited by diffusion through grains, as studied by Killen , in contrast to transport in intergrain pores, considered here. In section 3 quantitative equations for subsurface migration are developed. Section 4 contains analytical solutions to these equations for constant temperature; we consider the survival time of buried ice and deposition of water by large comet impacts and by continuous delivery to the cold trap. Section 5 discusses the role of impact gardening and adsorption. Sections 6 and 7 deal with multilayer coverage and time varying temperature. The last section summarizes the results.
 This work is mainly relevant to the Moon. The likely pure ice deposits on Mercury do not call for a mechanism that can at most fill pore spaces with ice. Yet the results apply equally to Mercury, and the results on outward migration furthermore to comets and minor bodies when ice is lost through a porous layer.
2. Physics of Vapor Migration
 Here we develop a physical model for molecular diffusion at low temperature in an exosphere. To begin with, we rederive the expression for the sublimation rate of ice in vacuum. The rate of incident particles from a saturated atmosphere is [Langmuir, 1913; Watson et al., 1961b]
with Pv the equilibrium vapor pressure, k the Boltzmann constant, T temperature, and μ the mass of an H2O molecule. This is a maximum condensation rate, because not all incident molecules stick to the surface. A fraction (1 − α) of incident molecules is reflected from the surface, where α is the condensation coefficient. For ice, the coefficient α in the temperature range 40–120 K is in the range 1–0.7 [Haynes et al., 1992].
 The saturation vapor pressure Pv is defined by an equilibrium between particles that hit the surface and stick to the surface and the number of molecules that escape from a solid ice surface. When vapor is in equilibrium with ice, the evaporation rate equals the condensation rate,
The evaporation or sublimation rate is also applicable without atmosphere. It may appear counterintuitive that the sublimation loss into vacuum depends on the saturation vapor pressure Pv even when no vapor is present. The equilibrium pressure is defined by a balance between incoming and outgoing flux, and it encodes the energy barrier a molecule has to overcome to escape. This energy barrier is the same with or without the presence of vapor. Throughout this paper we abbreviate Eevaporation with E.
 At any nonzero temperature, molecules have a probability to leave the surface they are bound to. The mean residence time τ of a molecule is τ = θ/E, where θ is the number of molecules per area. The areal density of H2O molecules for solid ice is θ = (ρ/μ)2/3, where ρ is the density of solid ice. The numerical value is θ ≈ 1019 molecules/m2. In summary, the residence time of a molecule, in the absence of other loss mechanisms, is
Figure 1 shows the residence time of water molecules as a function of temperature. It assumes a simple form for Pv,
where pt and Tt are the triple point pressure and temperature and Q = 51.058 kJ/mol is the sublimation enthalpy. Below roughly 150 K, the molecules spend most of their time residing on the surface and a comparatively short time on ballistic jumps. This intuitively matches earlier concepts of molecules that hop at high temperatures on the dayside of the Moon, but do not hop at low temperatures on the nightside or in permanently shaded areas [Arnold, 1979; Butler, 1997]. Our physical model provides finite residence times at all temperatures and can describe the migration of H2O molecules within cold areas.
 Experimentally measured sublimation rates for H2O ice are shown in Figure 2. Sack and Baragiola  measured sublimation rates between 135 and 170 K, and rates near 135 K for films deposited and grown at lower temperatures. Below 140 K, fresh films have enhanced sublimation rates attributed to sublimation from regions containing amorphous ice. Precise rates depend not only on temperature, but also on history. Bryson et al.  determined sublimation rates between 132 and 187 K, noticing a transition around 150 K. Data from the International Critical Tables [Washburn et al., 2003] above −90°C are based on earlier experiments.
 The binding energy of molecules adsorbed to the regolith or on amorphous ice differs from that in crystalline ice [Speedy et al., 1996]. H2O deposited at very cold temperatures is expected to form amorphous ice. In the light of other uncertainties, we use equation (4), which closely follows that from the International Critical Tables, as a description of the saturation pressure in section 4. Physically adsorbed H2O is discussed in section 5.
3. Subsurface Migration
3.1. Statistical Model
 Water molecules on the surface can migrate into the porous, loose regolith by random jumps. The mean grain size in lunar soil samples is typically 45–100 μm [Heiken et al., 1991]. Our model uses discrete jumps of length = 75 μm, which represent a typical grain size diameter, assumed to approximate pore space size. The model involves one spatial dimension, the vertical. The temperature is assumed to be constant with time and increases linearly with depth, T(z) = T(0) + gz, where g is the geothermal gradient.
 The first type of model is a statistical simulation of individual molecules. At every time step of duration Δt, there is a probability Δt/τ for the molecule to leave the surface. Once released we assume molecules move with equal probability upward or downward. Molecules that move upward from the surface are assumed lost. Also, there is a probability (1 − α) that molecules reflect immediately rather than stick.
 Model calculations are carried out with a continuous supply of water molecules to the surface of a cold trap. Figure 3 shows the number of H2O molecules in the ground as a function of time. Only a fraction of the water supplied accumulates in the subsurface, most is lost from the surface, but the amount of H2O steadily increases with time. Lower temperatures favor retention of water, but higher temperatures lead to deeper burial (not shown). (The dependence of the model results on various parameters will become clear later, when we solve the model analytically.)
 Molecular random walk leads to diffusive migration and we now obtain the corresponding diffusion coefficient. The probability to end up at a distance j after n hops is given by a binomial distribution,
where −n ≤ j ≤ n (either all odd or all even). This leads to a mean-square displacement of
One hop requires τ time, and n hops require nτ time. Hence the mean-square displacement is 〈z2〉 = 2t/τ. For a point source that spreads according to the diffusion equation 〈z2〉 = 2Dt. Hence the diffusion coefficient is
In this derivation, we have neglected the probability that a molecule immediately jumps a second time, that is, the sticking coefficient α is assumed to be 1.
3.2. Continuum Description
 The migration process can also be described in terms of a continuous mass density ρ(z, t). The outward flux from any site is σ/τ, where σ is the areal density. The net flux of molecules between two neighboring subsurface sites at depth zn and zn+1 = zn + is
The factor of 2 in the denominator appears, because a molecule can leave in one or the other direction. Mass density and surface density are related by ρ = μσ/. The flux at any depth is therefore
Local mass conservation dictates ∂ρ/∂t +∂J/∂z = 0. Hence the mass density evolves according to
 If the temperature is constant with depth, then the residence time can be pulled out of the derivative, and the migration is described by the diffusion equation
with a diffusion coefficient of D = 2/(2τ). This is the same coefficient we derived above based on the random walk model, equation (5). Note that the diffusion equation arises not because one gas diffuses through another, but from random migration of a single molecule species.
 When geothermal heating is not neglected, expansion of equation (6) yields
The first term on the right-hand side is a diffusive flux; the second term is an advective flux with an outward velocity w, due to a temperature increase with depth. The thermal conductivity of the most surficial layer on the Moon can be extremely low, and the temperature increase with depth correspondingly large [Heiken et al., 1991].
 The derivative of τ with respect to depth z depends on the geothermal temperature gradient g = ∂T/∂z. With the expression from equation (4),
The ratio of advective to diffusive flux at any time is
where Z is the length-scale over which ρ changes significantly. The geothermally induced flux becomes important when the relative change in temperature due to geothermal heating, Zg/T, becomes comparable to 1/(Q/kT − 1/2) ≈ 0.02. For a gradient of g = 1 K/m, this occurs at a depth of about 2 m.
4. Solutions to the Subsurface Migration Model
 The general continuum equation (7) can now be solved. In fact, we only solve the simpler equation (8) and are subject to the limitation from the geothermal temperature gradient described by equation (10). Throughout this section, we take the temperature to be constant in time.
4.1. Loss Rate of Buried Ice
 The first of several applications is the determination of the survival time of ice buried by a layer of regolith of thickness Δz. Obviously, buried ice lasts substantially longer than ice on the surface, because a molecule has to undergo many hops to escape. Our physical model allows us to determine its life time.
 The flux at the upper boundary of the ice layer is given by J = −D∂ρ/∂z. Assuming the amount of ice at question is much larger than could be buffered by partially “wetted” grains in the overlying dry layer, a linear density profile will be established after a transient period, that is, equation (8) evolves toward ∂2ρ/∂z2 = 0. Thereafter, ∂ρ/∂z is constant and the density gradient is determined by the difference between the surface density (essentially zero) and the density at the ice-regolith boundary (μ/). The loss rate is given by
Figure 4 shows this loss rate for ice buried beneath a 10 cm thick layer, compared to the loss rate of exposed ice. The total loss rate for exposed ice consists of a temperature independent space weathering rate, δ, and a temperature dependent sublimation loss rate, E.
 A subtle but important point in the derivation of this formula is that ρ in the immediate vicinity of the ice is μ/, which is much smaller than the bulk density of solid ice, because is much larger than the size of an H2O molecule. The first layer of grains on top of the ice layer receives a certain flux from the ice and must send back less than it receives to maintain its transport capability. The grains maintain less than a complete layer of volatile H2O molecules, and the interstitial voids are not filled at a higher density of H2O.
 A dry layer has two important effects on the survival time of buried ice, in addition to the protection from space weathering it provides. It lowers the peak temperature experienced by the ice, which can be very important [Vasavada et al., 1999; Harmon et al., 2001], but this effect is not included in the constant temperature calculations here. Figure 4 shows the effect of the diffusive barrier the water has to migrate through. For time varying temperature, equation (11) generalizes to,
where angular brackets indicate the time average. The damping of temperature cycles by the overlying layer can result in a dramatic reduction of 〈E〉 at the ice-regolith boundary relative to 〈E〉 on the very surface.
Equation (11) is equivalent to the Knudsen flow through a porous medium. The Knudsen diffusion coefficient is on the order of DK = K, where is the mean thermal velocity of molecules = and K a geometry dependent constant of order one. The flux is JK = DKΔρ′/Δz, where ρ′ is not the total density of H2O, but the density of H2O in the gaseous phase. From the ideal gas law ρ′ = μP/(kT), where P is pressure. The Knudsen flux is, using equation (1),
This agrees with equation (11) within a prefactor. The flux is reduced relative to sublimation from an exposed ice surface by a factor of order /Δz. Knudsen flux calculations have been used for comets, the Moon, and Mercury by, e.g., Fanale and Salvail  and Salvail and Fanale . The ideal gas vapor density at low temperatures is so small that a pore space usually does not contain even a single molecule, yet the Knudsen formula turns out to be applicable.
4.2. Slow Continuous Water Delivery
 Here we consider the accumulation of subsurface H2O due to slow but steady supply of water molecules to the cold trap. Slow means the supply rate is less than the loss rate, so that the surface is never fully covered with H2O molecules. If σ0 denotes the areal density of H2O molecules on the surface, then σ0 ≤ θ. From the surface, molecules are lost to sublimation at rate σ0/(2τ). The sublimation rate is taken to be half of that of bulk ice, because the other half migrates at least one step into the subsurface and about as much comes back. This factor of 2 is, in any case, unimportant for order of magnitude estimates. The space weathering rate is assumed to be proportional to the fraction of surface area covered with H2O molecules σ0/θ, because only a fraction of hostile particles hit H2O molecules.
 The balance of H2O molecules on the surface is described by
where σ0 is the areal density of H2O molecules on the surface (number of molecules per area), s the rate of continuous H2O supply, θ the areal number density of H2O molecules for solid ice, and δ the destruction rate from space weathering. As will become apparent below, the net flux of molecules into the subsurface is relatively small and can be neglected in (13). Even when the space weathering rate exceeds the supply rate, H2O molecules are present on the surface. After a transitional period, σ0 approaches an equilibrium value
For short times, σ0(t) = st.
 The solution to the diffusion equation with the boundary condition ρ(0, t) = μσ0(∞)/ can be found as follows [e.g., Strauss, 1992]. Take ρ(z, t) = (z, t) + μσ0(∞)/, then also satisfies the diffusion equation and has the boundary condition (0, t) = 0 and the initial condition (z, 0) = −μσ0(∞)/ for z > 0. Using the reflection method, the solution can be obtained as
The total H2O mass is
For comparison, the total amount supplied is μst. The mean depth is
For testing purposes, the solution is compared to statistical model calculations (described in section 3.1) in Figure 5. Indeed, the statistical simulations agree with the continuum solutions for t ≫ τ.
Figure 6 shows the column integrated ground ice mass as a function of temperature, compared to the amount of H2O supplied over 1 billion years. According to equation (16), m is independent of . There is a temperature for which the amount of subsurface H2O is a maximum. At low temperature little H2O accumulates, because the migration is so slow. At temperatures higher than the optimum the loss from the surface is so fast that few source molecules reside on the surface at any time. From differentiation of equation (16) with respect to τ, the maximum with respect to temperature is achieved for 1/(2τ) = δ/, that is, when the sublimation rate σ0/τ is close to the space weathering rate δσ0/θ. The amount of subsurface H2O depends on s and δ, but never exceeds m = μθ, which it reaches when σ0(∞) = θ, as can be seen from equation (16).
 The volume density of H2O can be estimated by
or by the density at average depth,
Once σ0 has assumed its equilibrium value, the H2O molecules migrate deeper and deeper with time, but the overall density of H2O remains unchanged. This density is on the order of μσ0/, and it reaches at most monolayer coverage, because σ0 ≤ θ.
 Quantitatively, a monolayer cover corresponds to μ/ ≈ 4 g/m3, on the order of a few ppm of the regolith density. Measurements of grain surface areas in lunar soil samples are on the order of 500 m2/kg [Heiken et al., 1991]. Multiplied with the areal density of a monolayer of H2O molecules, μθ, this provides an H2O fraction of a few hundred ppm [Hodges, 2002]. This is substantially more than estimated from our one-dimensional model, due to actual surface to volume ratios of grains and the presence of fines in the lunar soils. We conclude that up to a few hundred ppm of H2O could accumulate by this pathway. The maximum amount of H2O that could be stored in this fashion is less than the minimum amount inferred by [Lawrence et al., 2006]. Moreover, excess water is likely concentrated in areas smaller than the footprint of the Lunar Prospector neutron spectrometer, and the highest subsurface mixing ratios form only for optimal temperature and benign space weathering rates.
4.3. Initial Ice Cover
 Here we consider an initial, thick ice layer that may have formed during a giant comet impact. The exposed ice layer disappears after a time t0, due to losses, but some of the subsurface H2O that accumulates during this time will last beyond t0.
 The upper boundary condition is ρ(0, t) = μ/ as long as the ice layer is present, and ρ(0, t) = 0 afterward. For constant surface density, we can reuse result (15) from the previous section, but replace σ0(∞) with θ,
Integrated over depth, this yields the total ground ice mass for as long as an ice cover is present,
 After the ice cover is lost, we obtain the solution by using as initial condition the final density distribution ρ(z, t0) from (18) and again employ the method of reflected sources,
Although this integral cannot be carried out explicitly, it is possible to obtain the total mass by swapping the integration variables y and z.
For tt0, when the cover lasted a relatively short period, m → μθt0/. In this case, the total remaining H2O mass is proportional to the duration t0 of the ice cover.
 In the same way, by swapping integrals, we obtain the mean depth
Figure 7 shows again a comparison of statistical model calculations for verification purposes. In this comparison θ equals 40 molecules.
Figure 8 shows the column integrated H2O mass as a function of temperature, equation (20), compared to the amount of H2O in the initial ice layer. The assumed initial ice cover is M = 100 kg/m2, approximately a 10 cm thick layer, and this mass is equivalent to what is delivered to the cold trap in Figure 6 over the same total time period of 1 billion years. The duration of the ice cover is calculated from δ, τ′, and M, according to M = μ(θ/τ′ + δ) t0. Here, τ′ corresponds to the molecular bond in the ice layer, which can differ from the bond between the migrating H2O molecules and the regolith grain surfaces. But we do eventually assume τ′ = τ. The maximum amount of subsurface H2O for any combination of weathering rate and initial ice thickness is obtained when the ice cover lasts until the present, m = μθ from equation (19), and is indicated by a dash line in Figure 8.
 The volume density of H2O can be estimated by
The expression in parenthesis is always ≤ 1. Hence the maximum H2O density corresponds to monolayer coated grains, the same as described at the end of section 4.2.
 Meteorite impacts lead to turnover and burial of regolith. This does not necessarily destroy the ice and may in fact protect it [Arnold, 1979], but quantitative estimates of diffusive migration are only relevant if it proceeds faster than the turnover or burial. Figure 9 shows the mean ice depth from equation (17) for several temperatures in comparison with estimates of the gardening depth [Heiken et al., 1991; Arnold, 1975]. For temperatures above ∼110 K (using residence times for crystalline ice), gardening never significantly affects the subsurface migration of ice.
 Radiation destroys molecules on the surface of the cold trap before they can migrate downward. The existence of a condensation coefficient α < 1 may be important, because some molecules that land on cold areas will jump to protected pores without spending time on the surface. In hiding places not directly exposed to space weathering, molecules would experience substantially lower destruction rates. We have not considered this effect in our model calculations, but it may enhance the amount of H2O in the subsurface under the scenario discussed in section 4.2.
 The binding energy of molecules adsorbed to the regolith differs from that in ice. The sublimation enthalpy of crystalline ice is 51 kJ/mol, that of amorphous ice formed by deposition at low temperature is 0.45eV or 43 kJ/mol [Sack and Baragiola, 1993]. de Leeuw et al.  report adsorption energies for H2O on forsterite surfaces of 100–172 kJ/mol. The adsorption energy does not determine the residence time by itself. The average time of stay τ of the molecule on the surface of a substrate is
where ν approximately corresponds to the vibrational frequency of the bond between the H2O molecule and the substrate surface, typically 1012–1014 Hz, and Q is the energy of adsorption [Adamson, 1982; Atkins, 1986]. We have not found values for ν for H2O on olivine at cold temperatures in the literature.
 We do not have the appropriate data to quantitatively describe the residence times of adsorbed molecules. Hence we emphasized general results and equations. The residence time for crystalline ice is used in example cases.
 A chemisorbed or physically strongly adsorbed layer of H2O molecules may exist on lunar grains that is stable indefinitely [Hodges, 2002; Cocks et al., 2002]. Since the stability of this layer is so strong, it may preexist, and our calculations only consider the volatile water.
 The appropriate residence time for the type of migration process considered in sections 3.1 and 4.2 is the adsorption of H2O on a substrate. It is possible to imagine situations where changes in binding energies with layer are important, but when the first layer is extremely strongly adsorbed and preexistent, our results remain valid after they are adjusted for more accurate values of τ.
 The upper bound on H2O accumulation is independent of the residence times, because it corresponds to the mass of a monolayer, which in turn is determined by the available surface area. This maximum mass should equal the amount that is chemisorbed, essentially up to doubling the possible hydrogen content.
 A further complication arises when an ice layer, with its binding energy, faces regolith grains with a different finite binding energy, as in sections 4.1 and 4.3, because surface concentrations can compensate for differences in residence time to achieve a flux balance. The loss rate of buried ice is still governed by the residence time of molecules in ice.
6. Multilayer Coverage
 When a layer of H2O covers up an earlier layer, not every molecule is free to escape. We now develop equations for multilayer coverage, which are especially relevant for time varying temperatures. We imagine again a one-dimensional model with discrete sites spaced by a distance = 75 μm, which represent a typical grain size diameter.
 The fraction of a site's area coated with H2O is denoted with cn, which ranges from 0 to 1. For a site at depth zn = n, the outward flux is cnEn, while the received flux from both sides is (cn−1En−1 + cn+1En+1)/2. The areal density σn is described by
 In a continuum formulation, we can write (23) and (24) in terms of a volume mass density ρ = μσ/,
Here, c(z, t) is a spatially averaged cn. The associated mass flux is J = −μ(/2)∂(cE)/∂z.
 When the temperature is constant with depth and c ≤ 1, the evaporation rate E = /τ can be pulled out of the derivative in (25), and what remains is a diffusion equation for the coverage:
The right-hand side vanishes when the temperature is constant with depth.
 The flux may be decomposed as
Transport can be caused by differences in surface concentrations c or by differences in sublimation rates E. At constant temperature, only the concentration driven migration occurs, while geothermal heat and temperature oscillations are causes of the second form of transport. With these equations at hand, we can now proceed to study migration when the sublimation rate E changes with time and depth.
7. Pumping Effect
 Transport of water can result from differences in sublimation rates with depth. This does not require a change of mean temperature with depth. Time integration of (27) for temperature oscillations around a constant mean yields ρ = tμ (/2) (∂2 〈E〉/∂z2), where 〈E〉 is the sublimation rate averaged over time. Since the sublimation rate depends nonlinearly on temperature, a time varying temperature, even when periodic, causes pumping of molecules to depth. Due to the convex shape of Pv, 〈E〉 is larger for larger temperature amplitudes. Diurnal temperature amplitudes decay with depth and 〈E(T)〉 therefore decreases with depth. The ground ice mass increases proportionally with time as long as there is an ice source on the surface.
 The temperature is modeled by a sinusoidally varying surface temperature: T = Tm + Ta exp(−z/λ) sin(z/λ − ωt) + gz, which represents a solution to the heat conduction equation. Here, Tm is the mean temperature, Ta the temperature amplitude, λ the thermal skin depth (assumed 0.1 m), ω the diurnal period (of the Moon), and g the geothermal gradient (assumed 1 K/m). Reality will have a more complex temperature history and depth dependence [Salvail and Fanale, 1994; Vasavada et al., 1999], but our model temperature contains the essential ingredients.
 Experimentally measured sublimation rates for H2O ice were shown in Figure 2. We use the rates of Sack and Baragiola , extrapolated to lower temperatures with constant enthalpy.
 Three levels of models are used in our simulations. The first simulates discrete sites and a time varying temperature and uses equations (23) and (24). This requires time steps much shorter than one month. The second level of model uses time-averaged evaporation rates in (23), leading to much faster simulations. The third and fastest type of model solves the continuum equations (25) and (26), which allows for a spatial resolution coarser than . We have compared results from each of these three models with each other for verification, and only present results obtained with the last of these models here.
Figure 10 shows the result of model calculations. Beginning with an ice cover and dry regolith, molecules migrate downward, and some of them remain in the subsurface after the ice cover has disappeared. There are four phases in the time evolution, as seen in Figure 10a. Initially, transport is dominated by diffusively migrating H2O molecules (proportional to the square root of time), followed by a period when most mass is transferred by pumping (proportional with time). After the ice cover has disappeared, the ice-rich layer disperses rapidly, followed by a slow loss of the diffusively accumulated H2O. Figure 10b shows snapshots of the depth distribution, with a clear bend seen between multilayer coverage and less than monolayer coverage. These simulations make clear that multiple layers of ice accumulate on grains through pumping. Figure 10c shows the maximum areal density, maxzσ/θ; it can be used to precisely separate the four phases of the time evolution. Since the ground ice mass increases proportionally with time during the pumping dominated phase, large volume densities can accumulate when the ice cover lasts sufficiently long.
 Higher temperatures enhance the pumping, because absolute sublimation rates are higher, but they also reduce the lifetime of the ice cover. Hence the ground ice mass is again expected to be highest for an intermediate temperature.
 This paper investigated the migration of H2O molecules in the lunar and Mercurian regolith by random hops. Among our general results are an expression for the effective diffusion coefficient of H2O molecules (5) and a transport equation for subsurface migration (7). Transport with partial or multiple molecular layers is described by equations (25) and (26).
 The survival time of ice buried beneath porous regolith can be estimated with equation (11). The loss rate turns out to be equivalent to the Knudsen flux through a porous medium. The diffusive barrier reduces the ice loss, compared to the sublimation loss of exposed ice at the same temperature, roughly by a factor of molecule hop length divided by layer thickness.
 We then discussed two scenarios for subsurface accumulation of externally derived water at constant temperature. One assumes a gradual supply of water to the surface and the other an initial ice layer. Our calculations indicate that in an appropriate temperature range water molecules can travel into the regolith before they are destroyed. Higher temperatures lead to deeper burial, and the deeper burial leads to preservation during impact gardening. We find there is an optimum temperature which maximizes the subsurface H2O content and an intrinsic limitation to the density of H2O that can accumulate by this type of migration. The grain coverage is limited to less than one layer of volatile H2O molecules, because differences in surface concentration are necessary for net transport of additional molecules. The process should lead to a small amount of H2O, depending on the grain size probably up to a few hundred ppm in excess of what is chemisorbed and extremely strongly adsorbed. The most favorable temperatures are in the range 110–130 K, if assumed residence times are also characteristic for adsorption. Both charging scenarios work most efficiently for temperatures typical of areas permanently shaded from direct sunlight but with sunlit surfaces in their field of view.
 Temperature oscillations provide a qualitatively different pathway for molecule transport than described above, driven by differences in (average) sublimation rates. Although this mechanism can produce a higher density of ice in the subsurface, the ice-rich layer disappears rapidly after the ice cover is gone.
 We conclude that neither of the two pathways would create abundant ice reservoirs, unless there is an even more massive ice cover on top. However, the expectation is that, as long as H2O is delivered to permanently shaded areas at all, a fraction remains, adsorbed in the uppermost regolith, not at the coldest temperatures but for temperatures that allow limited migration of water molecules.
 It is a pleasure to thank Paul Lucy, Marilena Stimpfl, and Ashwin Vasavada for insightful discussions. This material is based upon work supported by the National Aeronautics and Space Administration through the NASA Astrobiology Institute under Cooperative Agreement NNA04CC08A issued through the Office of Space Science.