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Icarus

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Glaciation on Mercury: Accumulation and flow of ice in permanentlyshadowed circum-polar crater interiors

James L. Fastooka,⁎, James W. Headb, Ariel N. Deutschb

a School of Computing and Information Science, University of Maine, Orono, ME 04469 USAbDepartment of Earth, Environmental and Planetary Sciences, Brown University, Providence, RI 02912 USA

A R T I C L E I N F O

Keywords:MercuryGlaciationPermanently shadowed regions, iceCold spots

A B S T R A C T

Radar-bright deposits that coincide with regions in permanent shadow typically found within impact craters atand near the poles of Mercury have been interpreted as being composed of water ice. We investigate the dynamicproperties of these solid water-ice deposits (glaciers) with the goal of constraining their movement, flow rates,possible deformation, and related structures and deposits. As an end-member, we treat the extreme case ofmaximum ice accumulation where these deposits fill the crater to the shadow line, an event we would onlyexpect following maximum accumulation conditions such as a large cometary impact or a comet shower. We findthat, given the extremely cold conditions and the limited thickness of glaciers, even under the most favorableaccumulation conditions, glaciation is cold-based, ice flow velocities are very low (4×10−8 m/yr) and, with theexception of a sublimation lag deposit, glaciation is unlikely to leave any significant impact on the terrain. Animportant accelerator of ice flow is found to be the contribution of lateral heat conduction from the surroundingextremely hot surface terrain (∼225 K average with up to 450 K maximums), which for 55 km craters yieldsvelocities of 10−3 m/yr while for smaller 10 km craters velocities may exceed 1m/yr, enough to leave a de-tectable ice deformation imprint such as drop moraines. However given that the observed deposits are sub-stantially below their maximum fill capacities these estimates are unlikely to be attained.

1. Introduction

Earth-based radar observations have revealed the presence of radar-bright materials near the poles of Mercury shown in Fig. 1 (Slade et al.,1992; Harmon and Slade, 1992; Butler et al., 1993; Harmon, 2007;Harmon et al., 1994; Harmon et al., 2011). Analysis of the radar datasuggests that these highly reflective materials are composed of nearlypure water ice, with less than ∼5% volume fraction of silicates(Butler et al., 1993). Furthermore, enhanced concentrations of hy-drogen in the north polar region of Mercury are also consistent with awater-ice composition for the ice, as suggested from measurements bythe Neutron Spectrometer on board the MErcury Surface, Space EN-vironment, GEochemistry, and Ranging (MESSENGER) spacecraft(Lawrence et al., 2013). Extensive data from MESSENGER providesevidence that these water-ice deposits are distributed within the per-manently shadowed terrains near the poles of Mercury (Chabot et al.,2012; Chabot 2013; Deutsch et al., 2016). Both reflectance(Neumann et al., 2013) and visible imaging (Chabot et al., 2014, 2016)of the permanently shadowed surfaces reveal that ice on Mercury todayis present in two configurations: (1) as stable surface ice, where surface

reflectance is anomalously high (Neumann et al., 2013) or (2) as stablesubsurface ice, buried beneath insulating materials that have anom-alously low reflectance (Neumann et al., 2013). This insulating layer isestimated to be relatively thin (∼10–30 cm thick) on the basis ofcomparative variation of the flux of epithermal and fast neutrons withlatitude (Lawrence et al., 2013). The low-reflectance materials are in-terpreted to have formed as a sublimation lag, and are suggested to becomposed of volatiles species other than water, including organic-richcompounds (Paige et al., 2013).

Thermal models suggest that water ice is stable within the perma-nently shadowed terrains on Mercury on geologic timescales(Paige et al., 1992; Paige 2013). Mercury provides a unique environ-ment for ice accumulation due to lack of an appreciable atmosphere totrap heat and the planet's low obliquity of 0.034° (Margot et al., 2012).Insolation on Mercury varies not only with latitude, but also withlongitude due to the planet's eccentric orbit and 3:2 spin-orbit re-sonance: the average insolation is higher at longitudes 0°E and 180°Ethan the average insolation at longitudes 90°E and 270°E(Vasavada et al., 1999). A detailed energy-balance model of the surfacetemperatures around and within polar craters (Vasavada et al., 1999)

https://doi.org/10.1016/j.icarus.2018.07.004Received 27 August 2017; Received in revised form 6 April 2018; Accepted 5 July 2018

⁎ Corresponding author.E-mail address: fastook@maine.edu (J.L. Fastook).

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Available online 30 July 20180019-1035/ © 2018 Elsevier Inc. All rights reserved.

T

suggests an annual mean temperature near 110 K in permanent shadow,and surface temperatures close to 400 K in sunlit regions, creating asubstantial thermal gradient across permanently shadowed terrains inthe polar regions.

Here, we describe possible dynamical properties of the ice accu-mulations in the unique thermal environment of Mercury. Whilethickness estimates for the current state of the ice deposits on Mercuryrange from several m (Harmon, 2007; Black et al., 2010) to ∼50m(Eke et al., 2017; Susorney et al., 2017; Deutsch et al., 2018), there islittle understanding of how the state of the ice accumulations may havechanged through time. Here, we explore dynamics of ice under mer-curian conditions in order to delineate the effects of ice deformationand the potential for flow that could modify the observed ice depositsover geologic time. In order to evaluate the potential for flow, we treatthe extreme end-member case in which ice accumulates in craters to theshadow line (permanent shadow terminator, or PST), and assess glacialmovement styles (wet-based or cold-based) and glacial flow ratescompared to glaciers on the Earth and Mars. We then analyze the role ofheat conduction from surrounding terrain into the margins and base ofthe glacier and its influence on flow rates. We conclude with an as-sessment of the range of processes involved in glaciation on Mercury(from initial ice accumulation to the post-glacial period), comparingthese to the Earth and Mars, and listing predictions for recognition ofperiods of earlier and more extensive glaciation on Mercury.

2. Modeling the glacial flow process

2.1. Crater morphology and bed topography

Modeling glacial flow requires topography on which to reconstructthe glacier or ice sheet in question. The depth of a crater can be ap-proximated from the ratio of depth, d, to diameter, D, (d/D, (Nagel andFechtig, 1980)), which Vasavada et al. (1999) modeled for mercuriancraters from idealized crater geometries as 1:5 (depth is 1/5th thediameter) for craters less than 10 km in diameter and 1:25 for craterswith diameters of 100 km. We found from looking at topographic

profiles (transects) across a mercurian crater that the d/D ratio could goas low as 1:40 and that crater-wall slopes were typically ∼7.5° (seeBarnouin et al., 2012, his Fig. 1, for a 120 km crater). More recently,various researchers used topographic data acquired by the MercuryLaser Altimeter (MLA) to derive various power-law fits for depth-dia-meter relationships for both simple craters (crater less than 10 km indiameter and generally bowl-shaped) and complex craters (those largerthan 10 km and generally with a flat crater floor) (Barnouin et al., 2012;Talpe et al., 2012; Susorney et al., 2016). These power-law fits alsoprovide estimates of crater-wall slope ranging from 15–45° for small10 km craters to 5–30° for larger 100 km craters.

In our modeling, we utilize an idealized crater geometry that isdefined by various power-law fits, is a function only of crater diameter,and was derived from topographic measurements of more than 300mercurian craters (Susorney et al., 2016). Given only crater diameter,these power laws provide estimates of the depth, the rim-crest height,and the wall width from which we define our idealized crater geometry.Depth as a function of diameter is shown in Fig. 2(a). The red dot-dashline is depth from Talpe et al. (2012), and the blue dashed line is depthfrom Susorney et al. (2016), both with indicated uncertainties and goodagreement. In calculating the effective shadowing depth, we include therim-crest height, also from Susorney et al. (2016), indicated by the solidblack line. Also shown in the top gray box are the power-law fits for thewall widths, defined as the lateral distance between the rim crest andthe crater floor, (Susorney et al., 2016) that define the crater-wallslopes.

A profile generated from these power laws is shown in Fig. 2(b) for a20 km crater (black solid line). The blue dashed line indicates thespecified diameter, from which depth (red dashed line), rim-crestheight (green dashed line) and wall width (magenta dashed line) arederived using the power-law fits in Susorney et al. (2016). Also shownare permanent shadow terminator elevations for various latitudes(85–89°, red, green, blue, magenta, and cyan solid lines, respectively).The PST elevations increase with latitude to the increasingly grazingangle of the Sun coupled with planet's low obliquity. In the top gray boxthe profile is shown with a 1:1 scaling.

Fig. 1. Red circles indicate craters labeled onFigs. 8, 10, and 11 with location and size in-dicated in Table 1. Radar-bright deposits(Harmon et al., 2011) are shown in yellow,permanently shadowed terrain (Deutsch et al.,2016) is shown in blue, and regions with bothradar-bright deposits and permanent shadoware shown in pink (Deutsch et al., 2016). (Forinterpretation of the references to color in thisfigure legend, the reader is referred to the webversion of this article).

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For input to the ice sheet model, a 2-dimension rectangular array ofgrid points representing the bed topography is generated by rotatingthe bed profile obtained from the power laws about the center of thecrater. Performing an orthographic rotation of these grid coordinatessuch that the sun angle, defined by the specified latitude of the crater, isin a horizontal plane produces the shadow surface. Moving along eachgrid line we select the first highest point encountered, setting eachsubsequent point to be in shadow if it is below that first highest point.The original un-rotated height of that shadow surface defines the per-manent shadow terminator elevation.

An example of this simplified geometry based on Crater C (a dia-meter of 50 km at latitude 87.7° from Vasavada et al. (1999) (theirTable 1) is shown in Fig. 3. Fig. 3(a) shows bed topography, Fig. 3(b)the ice surface elevation, Fig. 3(c) the ice thickness, and Fig. 3(d) a 3Dperspective view of the crater with the potential permanently shadowedvolume shown as completely full of ice. While this end-member ex-ample greatly exceeds the realistic volumes for water-deposits on

Mercury, given that ice thickness are estimated between several m(Harmon, 2007; Black et al., 2010) and ∼50m thick (Eke et al., 2017;Susorney et al., 2017; Deutsch et al., 2018), we model here the extremecase of maximum ice thickness allowable, the case in which these de-posits fill the crater to the PST.

2.2. Ice loss and supply

This maximum ice configuration is then passed to our ice dynamicsmodel (UMISM, the University of Maine Ice Sheet Model), a finite-element shallow-ice approximation model that has been used ex-tensively both on Earth (Johnson and Fastook 2002; Kleman et al.,2002; Hooke and Fastook, 2007) and Mars (Fastook et al., 2008;Fastook et al., 2011; Fastook et al., 2012; Fastook et al., 2013;Fastook and Head, 2014; Fastook and Head, 2015), here adapted forMercury gravity and thermal environment. UMISM is a thermo-mechanically-coupled ice sheet model, where ice properties that controldeformation are functions of temperature, which itself is derived from atime-dependent solution of the energy equation within the ice sheetvolume, accounting for advection of heat by moving ice, internal shearheating, and geothermal heat flux.

An ablation rate of −1.0m/yr is assumed in the sunlit areas, suf-ficient to quickly remove any ice that moves out of the PST volume.Minor accumulation (10−10 m/yr) is assumed for the ice surface withinthe PST, consistent with estimates of water-ice delivery via impacts toMercury (Moses et al., 1999). With such a low accumulation rate the icesheet surface will basically attempt to “relax” into a configurationwhere the driving stress (the product of density, gravitational accel-eration, thickness, and surface slope) is minimized and any observedvelocity is a result of internal deformation and not of mass-balancedriven flux to accommodate mass input from surface accumulation.

Fig. 2. (a) Crater depths are estimated for 10–55 km diameter craters from the power-law fits of Susorney et al. (2016) in the blue dashed line, and of Talpe et al.(2012) in the red dot-dash line. The effective shadow depth (calculated as the sum of the crater depth and rim-crest height) is shown by the solid black line. The wallwidth, defined as the distance between the crater rim crest and the crater floor, is shown by the solid black line in the top gray box. (b) Transect across a 20 kmidealized crater defined by diameter (blue dashed line) from which depth (red dashed line), rim-crest height (green dashed line), and wall width (magenta dashedline) are derived through Susorney et al. (2016) power-law fits. Also shown are the permanent shadow terminator elevations for various latitudes. The gray box at thetop contains the same transect at a 1:1 scale. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

Table 1Location and size of labeled craters in Figs. 1, 8, 10, and 11.

Crater locations

Name Diameter (km) Latitude (N) Longitude (E)

C 52.0 87.8 188.8Chesterton 37.2 88.5 233.1Tryggvadóttir 31.0 89.6 188.4L 20.9 85.2 292.5Petronius 36.0 86.1 319.5Remarque 25.9 84.9 353.5Tolkien 50.0 88.8 148.9Y 18.2 87.6 210.2

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This is the first fundamental difference between glaciation on Mercuryand on planetary bodies such as Earth and Mars: the lack of an atmo-sphere on Mercury drives relatively high ablation rates, and thereforeprecludes net ice accretion in the accumulation zone, resulting in mass-balance driven flux to accommodate mass input from seasonal, annual,or longer-term surface ice accumulation.

2.3. Mechanical properties of the ice

Boundary conditions for the thermodynamic component of UMISM,used to calculate the mechanical properties of the ice (i.e., its hardness),include (1) mean-annual surface temperature, taken to be 110 K forexposed water ice (Vasavada et al., 1999], and (2) basal geothermalheat flux. Because the initial modeled ice filling the permanently sha-dowed volume is constrained in thickness (basically the crater depth)and the slope is shallow (basically the sun angle), as well as the fact thatthe ice is cold (∼110 K), modeled velocities are extremely low, on theorder of 10−8 m/yr for a uniform geothermal flux of 50mW/m2. This is7 to 8 orders of magnitude slower than ice flow velocities of meters todecameters per year for typical glaciers on Earth (Paterson 1994]. Ice

flow velocity, shown in Fig. 4, is highest at the base of the crater wallwhere the maximum case thickness explored here is greatest at just over2400m, which we again acknowledge is unlikely to have ever beenattained on Mercury, even during an episode of heavy bombardment bywater-delivering comets. Long-term deformation with this velocitydistribution would result in a thinning upstream toward the crater rimand a thickening downstream toward the crater center as the profilerelaxes to a minimum driving stress configuration. This downstreamthickening would tend to drive the ice in the crater center past the PSTand into the illuminated portion of the crater, causing rapid ablationthere. However, with velocities this low, little deformation would takeplace even over billion-year time spans. Maximum forward velocitieswould result in less than a meter glacial advance into sunlight over a100 million years.

Warmer ice (assumed to be ordinary ice Ih) is exponentially softerthan colder ice due to the temperature dependence of the rate para-meter, A(T), in the non-linear Glen's Flow Law (Glen, 1958;Paterson and Budd, 1982) relating strain rate ( ∈ ) to stress (τ). Theexponent, n, is usually taken to be 3.

= A T τɛ̇ ( ) n (1)

Fig. 3. Idealized geometry for Crater C from Vasavada et al. (1999), Table 1, with diameter of 50 km at 87.7°N, 188.8°E. (a) Bed elevation (m), (b) surface elevation(m), (c) ice thickness (m) that in our extreme-case simulation fills the permanently-shadowed volume, and (d) a 3D view of the maximum end-member case treatedhere in which ice is filled to the shadow line (permanent shadow terminator PST). (For interpretation of the references to color in this figure legend, the reader isreferred to the web version of this article).

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The Arrhenius relationship is widely used to describe the tempera-ture dependence of many physical properties, including chemical re-action rates, diffusion coefficients, as well as creep phenomena such asis described by Glen's Flow Law.

=

−A T A e( ) OQ

RT (2)

Here R is the Boltzman constant, T is temperature in Kelvins, and Ao

(the pre-exponent factor) and Q (the activation energy) are constantsobtained by fitting to data. Since Arrhenius is considered to be rea-sonably accurate down to a homologous temperature (the temperaturedivided by the melting point) of 0.4 (Cadek, 1988), we expect thisreasonably extends the measured temperature dependence of ice(Paterson, 1994) down to the cold temperatures in Mercury's perma-nently shadowed craters (∼110 K). At these temperatures, a one-degreeincrease in temperature doubles the rate parameter in Glen's Flow Law,effectively doubling the velocity obtained by integrating vertical strainrates from the bed to the ice surface to find the column-averaged ve-locities necessary for the conservation of mass calculation in UMISM.

Ice at depth is warmer than ice at the surface due to the insulatingpower of the ice thickness. As such, the heat flux delivered to the baseof the ice is a key parameter controlling ice deformation, and hence, theice velocity. Vasavada et al. (1999) uses an estimate for the heat flux of20mW/m2 (Schubert et al., 1988). An estimate based on the height of ascarp and its implications for the depth of the brittle-ductile transition,as well as the depth of melting of the crustal-mantle transition yields asurface heat flux that ranges from 30 to 60mW/m2 (Nimmo andWatters, 2004), but their method is based on the assumption of30–40mW/m2 originating from the mantle (which they allow could beas much as 10mW/m2 higher) with the rest provided by the radiogeniccontribution of the crust.

An additional consideration is the lateral transport of heat from thewarm sunlit surroundings of the cold permanently shaded crater in-terior. This lateral transport into a crater is illustrated with a simplifiedgeometry consisting of a rectangular region of water ice with a coldsurface and a depth/diameter relationship appropriate for mercuriancraters embedded in rock with a hot surface that is heated from belowby a geothermal heat flux. A 2D solution of the steady-state heat flowequation (Becker, 1981) for such a rectangular domain with a “coldspot” on the surface results in a depression of the isotherms below the

cold spot. This depression and warping of the isotherms tends to directadditional heat from the surroundings beneath the hot surface terraininto the center of the cooler region beneath the cold spot. This is de-monstrated for a 50 km cold spot in Fig. 4 for a nominal heat flux of50mW/m2 entering the base of the domain at a depth of 20 km.Fig. 5(a) illustrates the conductivity, showing higher conductivity rock(Eppelbaum et al., 2014) surrounding the low conductivity ice-filledregion. The temperatures within the modeled region are shown inFig. 5(b), and Fig. 5(c) shows the enhanced interior heat fluxes beneaththe simplified crater exceeding 150mW/m2. Heat delivered to thebottom of the ice-filled region is shown in Fig. 5(d); most of the heat isconcentrated near the edges of the crater, although even at the centerthe delivered flux exceeds the geothermal flux by a factor of 1.5(75mW/m2).

The effect of lateral heat flux is influenced by the diameter of such asimplified crater, with the effect being most pronounced near the edgesat the surface-temperature discontinuity. For this reason smaller cratershave much higher heat fluxes at their centers, decreasing as the craterdiameter increases. This effect is illustrated in Fig. 6, which shows theheat flux delivered to the bottom of the simplified crater as a function ofdiameter for a nominal heat flux of 50mW/m2 at the base of the 20 kmdeep domain. This effect is further demonstrated in Fig. 7, which showsthe temperature (top row) and heat flux fields (bottom row) beneaththe cold spot for diameters 10, 20, 30, 40, and 50 km (a–e, respec-tively). Temperature isotherms are bowed down beneath the cold spot,and since heat flows perpendicularly to the isotherms, heat is pre-ferentially focused onto the base of the cold spot. The region of en-hanced lateral flux is most intense beneath the edge of the crater, wherethe temperature discontinuity is largest (400 to 110 K). With thesmallest 10 km crater modeled here, the two contributions from theright and the left overlap in the center, resulting in the largest flux atthe center of the crater. Larger craters result in a deeper penetration ofthe temperature anomaly, and hence steeper, temperature gradientscarrying more heat flux, but since they are farther apart, the synergisticcontribution is less at the center of the crater and less enhanced flux isdelivered. Accounting for the lateral transport of heat conductedthrough the bedrock to small craters suggests the most potential forwarming from enhanced geothermal heat flux, and the least likelihoodthat deposits would survive for significant periods of time.

3. Results and discussion

3.1. The effects of enhanced heat fluxes

Having derived the significantly increased heat flux that can bedelivered to the base of the crater from lateral conduction (e.g., Figs. 6and 7), we now can apply the ice sheet model with these enhancedfluxes instead of the uniform 50mW/m2, an example of which wasshown in Figs. 3 and 4. To investigate the effect of the enhanced lateralheat flux, we run the ice sheet model with both uniform and enhancedheat fluxes for various size craters and various latitudes (10 to 55 km in5 km steps and 85 to 89.5° in 0.5° steps, a total of 100 model runs).

Fig. 8(a) shows the base-10 logarithm of the maximum velocityobserved in each uniform flux model run (a value of −10 correspondsto 10−10 m/yr). Labeled black stars indicate the size and latitude ofcraters with known deposits from Vasavada et al. (1999) Table 1. Fig. 1and Table 1 show locations and size of these craters. Crater C, which isas yet unnamed, was shown as an example in Figs. 3 and 4. Since depthis determined from the depth/diameter relationship, larger craters ingeneral display higher velocities, primarily due to the thicker ice thatcan exist in the deeper craters. A secondary effect is the slope, which issteeper at lower latitudes due to the sun being higher above the hor-izon. This effect is negated by the thinner ice available with thesesteeper slopes, resulting in an optimal latitude at 87.0o where velocity ishighest (4.1× 10−8 m/yr). Lowest velocities (6.7× 10−12 m/yr) areobserved in the smallest craters at the highest latitudes where the sun

Fig. 4. Velocity field for Crater C from Vasavada et al. (1999), Table 1, for auniform geothermal heat flux of 50mW/m2. Crater rim crest region is outlinedin white. (For interpretation of the references to color in this figure legend, thereader is referred to the web version of this article).

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angle is lowest.We acknowledge that the model is not accounting for longitudinal

changes in insolation, since temperature shows both latitudinal andlongitudinal symmetries on the surface of Mercury due to the 3:2 spin-orbit resonance of the planet. The insolation at longitudes 90°E and270° is less than the insolation at 0°E and 180°E (Vasavada et al., 1999)and at this point, we are accounting only for latitudinal variations.Future modeling should investigate the longitudinal effects, given thatmapping of radar-bright materials within permanently shadowed cra-ters suggest that cold pole longitudes provide a more favorable thermalenvironment for water ice (Chabot et al., 2013).

With a specified surface temperature and a specified heat flux at thebase of the ice, the ice column is warmer toward the bed, resulting insofter, more easily deformable ice. Fig. 8(b) shows the maximumtemperature (with the surface uniformly at 110 K) observed at the bed,generally at the thickest point at the base of the crater wall on thesunward side. As would be expected, the thickest ice in large, high-latitude craters with low sun angles experience the greatest warming ofthe bed (34.5 K).

Re-running the same set of crater sizes and latitudes, but with theenhanced heat fluxes that includes lateral heat transport of Fig. 6, weobtain a distinctly different appearing pattern of temperatures at thebed, shown in Fig. 8(d). Immediately obvious is that in all cases the bed

of the ice is much warmer. The warmest ice (253 K) is now in the smallcraters that experience the greatest enhancement of the heat flux at thebase of the ice (Fig. 6). Since higher latitude leads to lower sun angles,and hence thicker ice, basal ice is still warmest at high latitudes.

Fig. 8(f) shows the difference between Fig. 8(b) and (d), which is theamount of warming due solely to the use of the enhanced heat fluxes.The greatest warming (∼121 K) occurs for the small 10 km, highestlatitude craters, and the least warming (34 K) occurs for the largest55 km, lowest latitude craters. This warmer ice, being softer and easierto deform, yields higher velocities, as shown in Fig. 8(c). The overallpattern is very different from the uniform 50mW/m2 heat flux case ofFig. 8(a), with the highest velocities now in the smallest craters (as highas 5m/yr at 87.0°). The order of magnitude speedup due to this warmersofter ice is shown in Fig. 8(e). Even with large craters at low latitudes,where the warming is least (34.5 K), a 4.3 orders of magnitude speedupis observed (from 2.6×10−8 m/yr to 5.3×10−4 m/yr). Large craterswhere uniform-flux velocities are a maximum, experience a 4.6 ordersof magnitude increase in velocity with the enhanced fluxes (from4.1×10−8 m/yr to 1.6×10−3 m/yr). Small craters at middle latitudes(87.5°) see the largest speedup (10.9 orders of magnitude faster, albeitwith very low velocities to begin with; from 6.7× 10−12 m/yr to5.2×10−1 m/yr).

Given the sensitivity to heat flux observed with the uniform versus

Fig. 5. Lateral heat transport from hot (∼400 K) sunlit terrain to cold (∼110 K) crater interior. (a) Conductivity, (b) temperature, and (c) heat flux are shown for a50 km diameter cold spot, and (d) the heat delivered to the bottom of a completely ice-filled crater is modeled. Boundary conditions consist of specified temperaturesacross the top (grading from a ∼400 K sunlit terrain to a ∼110 K permanently shadowed cold spot) and specified flux along the bottom at 20 km depth (50mW/m2).(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

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enhanced heat flux, an experiment was run for various base heat fluxes,both with (solid line) and without (dashed line) lateral heat transport.Using as an example, Crater C (displayed in Fig. 3 and 4), with a dia-meter of 50 km and at a latitude of 87.7°, we obtained the results shownin Fig. 9 for basal heat flux ranging from 0 to 100mW/m2. Whileanything over 60mW/m2 is probably unreasonable for anything excepta landscape experiencing volcanism or a recent projectile impact, weshow results to emphasize how strongly dependent velocity is on heatflux through the Arrheniusly-activated thermal softening of ice. Overthe range from 10 to 100mW/m2 we see more than eight orders ofmagnitude increase in velocity from 1.4×10−12 m/yr to4.0×10−4 m/yr for the crater with uniform heat flux, and more thanfive orders of magnitude increase in velocity from 3.6× 10−6 m/yr to

1.09m/yr when lateral heat transport is included.. Note that for theunreasonable 100mW/m2 heat flux value, the maximum observed ve-locity of 4× 10−4 m/yr for the uniform heat flux case is within anorder of magnitude of velocities observed on slow–moving terrestrialglaciers such as the debris-covered Mullins glacier in the Antarctic DryValleys (Marchant et al., 2007), whereas in the lateral heat transportcase as little as 50mW/m2 is sufficient to match the velocity of theslow-moving terrestrial glaciers.

Figs. 10 and 11 show results analogous to Fig. 8 for base geothermalheat fluxes of 20 and 30mW/m2. Patterns of maximum velocity andbasal temperatures with and without lateral heat transport are similarto those shown in Fig. 8(a–d) for the 50mW/m2 case, except of coursethat both temperatures and velocities are lower. Acceleration factorsand warming due to lateral heat transport show the same patterns asFig. 8(e) and (f).

Again, for the uniform flux cases the thickest ice in large, high-la-titude craters with low sun angles experience the greatest warming ofthe bed (13.8 K for 20mW/m2 and 20.5 K for 30mW/m2). The greatestwarming due to lateral heat transport (115 K for 20mW/m2 and 117 Kfor 30mW/m2) is still in the small 10 km, highest latitude craters, andthe least warming (33.6 K for 20mW/m2 and 33.8 K for 30mW/m2) isstill found in the largest 55 km, lowest latitude craters. For the uniformflux cases we still see an optimal latitude for 55 km craters (85.5°,4.0× 10−11 m/yr for 20mW/m2 and 86.0°, 4.4× 10−10 m/yr for30mW/m2) where velocity is highest. Lowest velocities(2.1× 10−14 m/yr for 20mW/ m2 and 1.6× 10−13 m/yr for 30mW/m2) are still observed in the smallest craters at the highest latitudeswhere the sun angle is lowest.

The highest velocities for the lateral heat transport case are still inthe smallest craters (0.88 m/yr for 20mW/m2 and 1.6 m/yr for 30mW/m2, both at 87.0°). Speedup due to the warmer softer ice for large low-latitude craters where the warming speedup spans 5.2 orders of mag-nitude increase (from 4.0× 10−11 m/yr to 6.5×10−6 m/yr) for20mW/ m2 and 4.9 orders of magnitude (from 3.9× 10−10 m/yr to2.9×10−5 m/yr) for 30mW/m2. Large craters where uniform-fluxvelocities are a maximum experience a 5.3 orders of magnitude increasein velocity with the enhanced fluxes (from 4.1×10−11 m/yr to8.5×10−6 m/yr) for 20mW/ m2 and a 5.1 orders of magnitude in-crease in velocity (from 4.3×10−10 m/yr to 5.0×10−5 m/yr) for30mW/ m2. Small craters at high latitudes (89.5°) experience the

Fig. 6. Heat flux delivered to the bottom of an idealized crater for nominal heatflux of 50mW/m2 at the base of the 20 km domain.

Fig. 7. Temperature fields (top row) and heat flux fields (bottom row) for (a) 10, (b) 20, (c) 30, (d) 40, and (e) 50 km-diameter (left to right) cold spots. In all of thefigures, the black and white outlined rectangle indicates the low conductivity ice. In the flux figures (bottom row), a green line outlines the region where the fluxexceeds the 50mW/m2

flux specified along the bottom of the domain at a depth of 20 km delineating regions of enhanced heat flux. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this article).

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largest speedup (12.4 orders of magnitude faster from 2.1× 10−14 m/yr to 5.6× 10−2 m/yr for 20mW/m2 and 11.9 orders of magnitudefaster from 1.6×10−13 m/yr to 1.2×10−1 m/yr for 30mW/m2).

All of these results from Figs. 8, 10, and 11 are summarized inTable 2.

4. Summary and conclusions

On the basis of our analysis of the factors that are important in iceaccumulation and flow processes in permanently shadowed areas onMercury, we characterize Mercury glaciation as follows (Fig. 12):

1. Geometry of ice accumulation: In order for water ice to accu-mulate on Mercury, permanently shadowed regions are requiredand thus ice accumulation depends on location and on the size ofthe depression (Fig. 12(a)); impact crater interiors are clearly themost likely geologic environment for such accumulation, althoughvolcanic calderas and pit craters could create a similar environ-ment, as could some shallower topographic depressions and roughterrain at higher latitudes.

2. Ice accumulation environment: The ice accumulation environ-ment on Mercury is fundamentally different than on planetarybodies such as Earth and Mars. The lack of an atmosphere onMercury precludes atmospheric deposition as snow and net iceaccretion in the accumulation zone, resulting in mass-balancedriven flux to accommodate mass input from seasonal, annual, orlonger-term surface ice accumulation. Instead, water molecules aredelivered to the permanently shadowed cold traps by thermal mi-gration processes, and are derived from some combination of in-ternal sources (magmatic volatiles) and external sources (cometaryor other water-rich impact events).

Fig. 8. (a) Base-10 logarithm of the maximum observed velocity for various crater sizes and latitudes with a uniform heat flux of 50mW/m2. (b) Temperature of thebed warmed by a uniform heat flux of 50mW/m2. (c) Base-10 logarithm of the maximum observed velocity calculated with the enhanced heat fluxes. (d) Maximumobserved temperatures with lateral heat transport and non-uniform basal heat fluxes. (e) Ratio of enhanced flux velocities to those generated with a uniform of50mW/m2. (f) Warming of the bed due to the use of a non-uniform enhanced heat flux. Labeled black stars are a sample of north polar craters with ice deposits,which are also listed in Table 1 and shown in Fig. 1. All results are for the maximum end-member in which ice completely fills the craters to the PST. (Forinterpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

Fig. 9. Base-10 logarithm of the maximum observed velocity for shallow craterswith uniform heat flux (solid line) and for craters with basal flux enhanced bythe inclusion of lateral conduction of heat through the bedrock.

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3. Ice accumulation rates: Unknown are the actual ice accumulationrates and time of emplacement. Moses et al. (1999; their Table 1)assess a wide range of volatile sources (Interplanetary dust parti-cles, asteroids, Jupiter family comets, and Halley-type comets) and

calculate that they would deliver from 1 to 95m of ice to the coldtraps over 3.5×109 yr, an accumulation rate of 2.8× 10−10 to2.7×10−9 m/yr. They point out that delivery in their Monte Carlosimulation is dominated by a few large impactors, so the uniform

Fig. 10. Analogous to Fig. 8 but for a heat flux of 20mW/m2. (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article).

Fig. 11. Analogous to Fig. 8 but for a heat flux of 30mW/m2. (For interpretation of the references to color in this figure legend, the reader is referred to the webversion of this article).

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rate is a minimum and could have been larger episodically. Theoverall impact flux has declined through time, but individual co-metary impacts or comet showers are likely to be the most sig-nificant water source in recent geologic history. Water can also bedelivered to planetary surface through volcanic outgassing(Wilson and Head, 2008; Head and Wilson, 2015), however thelow-FeO content in surface silicates on Mercury suggests highlyreducing conditions (Robinson and Taylor, 2010). Even if water isoutgassed in mercurian eruptions, declining rates of volatile inputfrom effusive and explosive eruptions (Wilson and Head, 2008;Denevi et al., 2013; Ostrach et al., 2015; Prockter et al., 2010; Headand Wilson, 2015; Head et al., 2009) suggest that volcanism has notbeen a major source of water in the last third of the history ofMercury.Moses et al. (1999) point out that the observed purity of the radar-bright materials supports episodic large accumulations, because theice would be much dustier with slow continuous deposition. Epi-sodic deposition is also consistent with the observed sharp albedoboundaries of the deposits, which also suggests that the ice wasdelivered relatively recently (Chabot et al., 2014). Crider andKillen (2005) estimates the rate of burial of ice deposits in polarcraters on Mercury to be 0.43 cm/Myr. If the ice deposits in thepolar craters on Mercury consist of clean ice and the deposits areburied by 20 cm of regolith, they must be relatively recently em-placed, less that ∼50 Myrs ago (Crider and Killen, 2005).

4. Predicted ice stratigraphy: On the basis of stochastic episodicsupply events (individual impactors that vary in water content,velocity, impact angle, impact location and frequency), individualsupply events are predicted to produce a layered accumulationsequence representing the cumulative record of individual events.Current lack of knowledge of the specific role of scattered light onfavorable water molecule destinations within the cold trap, and anincomplete understanding of thermal migration dynamics in gen-eral, precludes a more detailed picture, but water molecules areunlikely to be emplaced in a layer of even thickness across the coldtrap footprint. Also uncertain is the time between events; the pre-sence of an organic-rich lag deposit on some current ice occur-rences (Neumann et al., 2013; Chabot et al., 2014) suggests thatablation between events could produce a distinctive stratigraphy ofpaired ice layers overlain by a darker lag.

5. The nature of glacial flow and ice flow velocities in the max-imum ice accumulation case (crater filled to the permanentlyshadowed terminator, as modeled here): As an end-member forthe maximum possible extent of glaciation in permanently sha-dowed regions, we treated the extreme case where ice deposits fillthe crater to the shadow line (PST) (Fig. 12(b)), a situation wewould only expect following maximum accumulation conditions

such as a large cometary impact or a comet shower. We found thattemperatures are sufficiently cold that even under these maximumaccumulation conditions, flow velocities were vanishingly small(10−8 to 10−11 m/yr without lateral conduction), compared totypical flow velocities for slow, cold debris-cover glaciers on Earth(∼10−2 m/yr (Marchant et al., 2007)), and closer to, but stillconsiderably below, typical flow velocities on Mars (∼10−5 m/yr(Fastook et al., 2011)). Including lateral conduction increases po-tential velocities for craters filled to the PST to 10−3 m/yr for the55 km craters and to potentially 1m/yr for 10 km craters, certainlyenough to produce observable deformation. However, these are forthe extreme case of thick ice (>1000m even in the 10 km craters).Given current estimates of the timing of ice deposits (<50 millionyears old (Crider and Killen, 2005)) and thickness of the ice (be-tween several m (Harmon, 2007; Black et al., 2010)) and ∼50m(Eke et al., 2017; Susorney et al., 2017; Deutsch et al., 2018), thissuggests that the ice on Mercury today was deposited and likely hasnot advanced at all.

6. The effects of lateral conduction on glacier thermal structureand flow rates: The extreme temperature difference (Paige et al.,2013) between permanently shadowed regions (75–100 K) andadjacent regions illuminated by the Sun (200–400 K) means thatheat will be conducted through the glacial substrate toward thedeposit base (Fig. 12(a) and (b)). We explored a range of para-meters and found that although maximum effects on ice thermalproperties varied depending on crater geometry and illumination,the net effect was insufficient to bring flow style into the realm ofwet-based conditions, even for a maximum end-member case whereice is filled to the PST (Fig. 12(c)). When assisted by lateral con-duction the maximum ice flow rates range from 10−3 to 1m/yr onMercury, a rate that would certainly produce observable deforma-tion over the lifetime of these deposits. For small ice patches(Deutsch et al., 2017), lateral conduction is likely to be a limitingfactor in ice accumulation and retention. In the smallest crater forwhich we modeled the lateral heat flow (10 km, Fig. 6), the heatflux delivered to the base of the crater was five times the basicgeothermal heat flux (50mW/m2), and was increasing ex-ponentially as crater size decreases. For a feature ∼1 km in size theflux would be over ten times the basic flux delivering considerableheat to the base of the deposit making its long-term existence un-likely.

7. Style of glaciation: wet-based or cold-based?: On the basis of ouranalysis, all glaciation on Mercury in the range of maximum ac-cumulation conditions that we treated would be frozen to the bed,and thus cold-based (Fig. 12(c)). Cold-based glaciation would se-verely limit the production of geomorphic features (such asdrumlins, eskers and other features typical of wet-based glaciation)

Table 2A summary of results shown in Figs. 8, 10, and 11 for the three different geothermal heat flux cases: 20, 30, and 50mW/m2.

Heat flux 20mW/m2 30mW/m2 50mW/m2

Uniform heat fluxMax warming from

surface55 km 89.5° 13.8 K 55 km 89.5° 20.5 K 55 km 89.5° 34.5 K

Max velocity 55 km 85.5° 4.1×10−11 m/yr 55 km 86.0° 4.4× 10−10 m/yr 55 km 87.0° 4.1× 10−8 m/yrMin velocity 10 km 89.5° 2.1×10−14 m/yr 10 km 89.5° 1.6× 10−13 m/yr 10 km 89.5° 6.7× 10−12 m/yrLateral heat transportHighest velocity 10 km 87.0° 0.88m/yr 10 km 87.0° 1.6m/yr 10 km 87.0° 5.3m/yrMax warming due to

transport10 km 89.5° 115 K 10 km 89.5° 117 K 10 km 89.5° 121 K

Min warming 55 km 85.0° 33.6 K 55 km 85.0° 33.8 K 55 km 85.0° 34.1 KSpeedup for min

warming5.21(orders)

4.0×10−11 to 6.5× 10−6 m/yr 4.87(orders)

3.9×10−10 to 2.9× 10−5 m/yr 4.31(orders)

2.6× 10−08 to 5.3× 10−4 m/yr

Speedup at uniform max 5.32(orders)

4.1×10−11 to 6.5× 10−6 m/yr 5.06(orders)

4.4×10−10 to 5.0× 10−5 m/yr 4.60(orders)

4.1× 10−08 to 1.6× 10−3 m/yr

Max speedup 10 km 89.5° 12.4 (orders) 10 km 89.5° 11.9 (orders) 10 km 89.5° 10.9 (orders)2.1× 10−14 to 5.6× 10−02 m/yr 1.6× 10−13 to 0.12m/yr 6.7× 10−12 to 0.52m/yr

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that might provide evidence of previous thick ice deposits in per-manently or formerly permanently shadowed regions (Fig. 12(d)).Wet-based conditions are predicted to occur only in special situa-tions such as: (1) a magmatic hot spot (likely to occur only earlierin the history of Mercury), (2) a magmatic intrusion into ice de-posits, as seen on Earth and Mars (Head and Wilson, 2002;

Head and Wilson, 2007; Wilson and Head, 2002, Wilson and Head,2007; Kadish et al., 2008; Scanlon et al., 2014, Scanlon, 2015), or(3) an area of elevated geothermal flux remaining from the waningstages of a newly formed impact at high enough latitudes to pro-duce permanent shadow (the initial thermal flux is likely to besufficiently high that water ice would not accumulate).

8. Detecting the presence of previous ice deposits: Under currentobliquity conditions (0.034° (Margot et al., 2012)), ice deposits arestable in the present permanently shadowed regions and ablatewith time from scattered light. On the basis of our findings, fol-lowing complete ablation of the ice, little positive evidence shouldremain indicating that ice previously existed in these regions due tothe cold-based nature of the glaciation and the extremely low flowvelocities predicted for the ice deposits (Fig. 12(d)). Developmentof a low-albedo layer (Neumann et al., 2013; Chabot et al., 2014) ofpossible organic composition (Paige et al., 2013) could result in alag deposit remaining on the crater floor following complete loss ofice. However, low-albedo lag deposits on Mercury are found ex-clusively in permanently shadowed regions, suggesting that they,too, are volatile-rich (Paige et al., 2013), thus even this glaciation-associated remnant proxy for previous ice deposits would not sur-vive for an extended period of time beyond water ice loss. However,their existence in permanently shadowed terrains that are of highertemperatures than terrains where water ice is exposed suggests thatthese volatile species are stable at higher temperatures than waterice (Paige et al. 2013; Neumann et al., 2013; Chabot et al. 2016).Cold-based glacial deposits are often accompanied by drop mor-aines, which are ridges of supraglacial debris deposited at themargins of the ice deposit (Head and Marchant, 2003; Kadish et al.,2014; Mackay et al., 2014). Drop moraines form at a stationary icefront where the forward advance rate is equal to the marginal ab-lation rate: ice continues to move forward in the glacier but anysupraglacial debris is deposited at the stationary front as a moraine(Paterson, 1994). On the basis of the extremely slow moving ice inour simulations even for a maximum end-member case where ice isfilled to the PST, and the low likelihood of significant waxing andwaning of the ice deposit front, we would not expect abundant dropmoraines in the geologic record. Furthermore, the paucity of po-tential debris observed on current Mercury ice deposits, and theextremely slow flow rates that might serve to redistribute debris,also reduce the possibility of drop moraines. Future mission ima-ging observations (Benkhoff et al., 2010) into the permanentlyshadowed regions illuminated by scattered light might search for(1) marginal deposits of low-albedo sublimation lags (a type of dropmoraine), (2) boulder piles or ridge segments formed from debrisexcavated through the glacier by impacts, or (3) rockfalls from thecrater wall, transported laterally to the glacier margin, and subse-quently deposited (Fig. 12(d)). One of the strongest tell-tale signs ofthe previous presence of cold-based ice is the paucity of smallcraters in permanently shadowed portions of the crater floor(caused by “filtering” of craters formed solely in the ice), relative tothe number of craters in the sunlit portions of the floor. This, too,can be investigated with high-resolution images of permanentlyshadowed surface.

9. Rates of horizontal ice movement versus vertical ablation:Terrestrial and martian cold-based glaciers advance relativelyslowly, but still rapidly (six to ten orders of magnitude) comparedto cold-based glaciers on Mercury. Because of this, if ice on Mercuryever accumulated to a significant thickness, then downward arealablation rates may exceed flow velocity rates for forward move-ment ice, resulting in a different geometry of evolution than thattypically seen on terrestrial glaciers (Fig. 12(c)). If the ice depositsablate more quickly than they flow one would expect the low-al-bedo material would cover the entirety of the ice deposits. This mayaccount for the fact that in almost every example of ice-bearingcraters where there is a lag deposit, the low-reflectance material

Fig. 12. Schematic profiles illustrating ice deposits and potential glaciation inpermanently shadowed regions of Mercury. (a) Geometry and nomenclature;(b) maximum accumulation stage, (c) glacial flow stage, (d) predicted residualdeposits.

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covers the entirety of the ice deposit. Additional modeling of iceablation rates in permanently shadowed areas under various con-ditions on Mercury, as well as knowledge of the effects of a low-albedo lag, are needed to assess this further.Clearly, in the cold environment of the Mercury polar and circum-polar craters, flow velocities are extremely low (10−8 m/yr), andhence the potential for significant deformation of the ice deposits issmall. It is clear that accounting for the enhanced flux of heat fromthe surrounding hot sun-lit terrain is important, offering orders ofmagnitude speedup over a uniform flux case. As is expected, larger,thicker deposits with the greatest surface slope would flow thefastest, leading to an optimal thermal environment conducive to theaccumulation of thick ice due to the lower sun angle at high lati-tudes that in our model defines the surface slope of the deposit.However, given the estimate of ∼50m for the current deposits(with >1000m required for even the modeled low velocities) weexpect the deposits are supply limited, and that they are basicallystagnant unmoving deposits, reflecting the extreme efficiency of thecold-trapping mechanism.

10. Implications for lunar polar volatiles: In a manner similar toMercury, the lack of an atmosphere on the Moon prohibits atmo-spheric volatile deposition as snow and net ice accretion in theaccumulation zone. Therefore, lunar polar volatiles should alsomigrate to cold traps via thermal processes (e.g., Crider andVondrak, 2002), after possible delivery from water-rich impactors(e.g., Cintala, 1992) or volcanic outgassing (e.g., Needham andKring, 2017), or through in-situ reactions from solar wind im-plantation (e.g., Crider and Vondrak, 2000). Given that radarmeasurements of lunar polar deposits show that they are moretenuous than mercurian polar deposits (e.g., Lawrence et al., 2017),and that lunar polar deposits appear to be much more spatiallyheterogeneous on the surface (e.g., Hayne et al., 2015) in com-parison to the polar deposits on Mercury, it is plausible that polarice deposits on the Moon are relatively less thick and more de-graded than those on Mercury. Therefore, given the results of ourmodeling, we consider that the flow velocities of any lunar polardeposits to be negligible, even when considering the effects of lat-eral conduction. However, if ice were ever to accumulate in suchamounts as to approach the permanently shadowed terminatorregions on the Moon, as modeled here for Mercury, then we wouldexpect the style of glaciation of these deposits to be cold-based aswell. If the ice on the Moon is relatively ancient (e.g., Siegler et al.,2016), then the ice stratigraphy within lunar polar craters mayshow layering correlated with the accumulation sequences, re-presenting both the stochastic episodic supply events, as well as theintermittent deposition of ejecta from impact events (e.g.,Hurley et al., 2012).

Acknowledgments

We gratefully acknowledge financial support to JWH from the NASASolar System Exploration Research Virtual Institute (SSERVI) grant forEvolution and Environment of Exploration Destinations under co-op-erative agreement number NNA14AB01A at Brown University; thisanalysis was part of a precursor and perspective study for the natureand emplacement of volatile deposits on the Moon. This work wassupported by NASA under grant NNX16AT19H issued through theHarriett G. Jenkins Graduate Fellowship to A. N. Deutsch.

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