Temperature dependent thermal inertia of homogeneous Martian regolith

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 116,, doi: /2011je003805, 2011 Temperature dependent thermal inertia of homogeneous Martian regolith Sylvain Piqueux 1 and Philip R. Christensen 2 Received 24 January 2011; revised 27 April 2011; accepted 2 May 2011; published 21 July [1] Past studies of the thermophysical properties of the Martian surface layer have assumed temperature independent thermal inertia, which is a function of the material density, specific heat, and bulk conductivity. In this paper, we evaluate the temperature driven variations of these quantities for particulated and cemented material under Martian conditions of atmospheric pressure and temperature. Temperature driven density variations are negligible. The specific heat of a basaltic material is strongly influenced by the temperature ( 75% increase from 150 to 315 K), inducing significant variations of the thermal inertia. The thermal conductivity of uncemented Martian regolith is weakly controlled by the solid phase conductivity and strongly controlled by the gaseous phase conductivity. As a result, the conductivity of the solid phase (i.e., composition, temperature) is unimportant, whereas medium to large variations (30 50%) of the bulk conductivity are associated with temperature induced fluctuations of the pore filling gas conductivity. Overall, the thermal inertia of uncemented Martian soils is predicted to vary significantly ( 80%) throughout the range of the observed surface temperatures. In the case of cemented soils, the contribution of the gas conductivity is generally small, and the solid phase (i.e., grains and cement) conductivity (i.e., composition, temperature) becomes more important. Consequently, the magnitude of the thermal inertia change for cemented soils is variable, and smaller than that predicted for uncemented materials (10 50%). Large diurnal and seasonal temperature variations only occur within the top material, and most of the near surface regolith does not experience large thermal inertia variations. The shapes of modeled diurnal temperature curves are not significantly modified (e.g., the 0200 LT (Martian local time) apparent thermal inertia of uncemented regolith is up to 15% lower than the average daily inertia of the top material). Thermally derived grain sizes are usually based on conductivity measurements operated at room temperature or above, where the thermal inertia and specific heat are higher than on Mars, implying that grain size predictions for Mars are currently underestimated. However, in situ observations by the Mars Exploration Rovers and thermal conductivity modeling suggest that this underestimation is not significant. Citation: Piqueux, S., and P. R. Christensen (2011), Temperature dependent thermal inertia of homogeneous Martian regolith, J. Geophys. Res., 116,, doi: /2011je Introduction [2] The thermal inertia I (J m 2 K 1 s 0.5 ) describes the ability of a material to resist a temperature change when subjected to a deviation from thermal equilibrium, after forcing a new boundary temperature or heat flux. The thermal inertia is defined by p I ¼ ffiffiffiffiffiffiffi kc ð1þ 1 Department of Earth Sciences, University of California, Riverside, California, USA. 2 Mars Space Flight Facility, Arizona State University, Tempe, Arizona, USA. Copyright 2011 by the American Geophysical Union /11/2011JE with k the thermal conductivity of the bulk material (J s 1 m 1 K 1 ), r the density (kg m 3 ), and c the specific heat (J kg 1 K 1 ). Previous studies of the Martian regolith have assumed that density and specific heat vary by small factors among most geological materials, an assumption that we discuss in this paper. The thermal conductivity varies by up to 4 orders of magnitude depending mainly on the physical characteristics of the material (e.g., grain size, from 10 4 Js 1 m 1 K 1 for dust size particles near the top of Olympus Mons to 2 J s 1 m 1 K 1 for objects larger than a diurnal skin depth, e.g., 20 cm), therefore being the primary controlling factor that determines the thermal inertia [Neugebauer et al., 1971]. For this reason, the thermal inertia is a valuable quantity to determine because it leads to the thermal conductivity of the sub- 1of18

2 surface, hence providing fundamental physical properties of Martian surficial deposits. [3] Our understanding of the relationship between k and the nature of the soil relies primarily on laboratory measurements, where the effects of the atmospheric pressure [Smoluchowski, 1910; Deissler and Boegli, 1958; Wechsler and Glaser, 1965; Wechsler et al., 1972; Presley and Christensen, 1997], grain size [Smoluchowski, 1910; Wechsler and Glaser, 1965; Wechsler et al., 1972; Presley and Christensen, 1997], particle sorting [Presley and Craddock, 2006], composition [Deissler and Boegli, 1958; Wechsler et al., 1972], porosity [Fountain and West, 1970; Presley and Christensen, 2010a, 2010b], and grain cementation [Presley et al., 2009] have been studied and sometimes parameterized for direct use in the analysis of Martian surface temperature data. However, little work has been performed to quantify the effect of the temperature on the thermal inertia of materials under Martian conditions. [4] Thermophysical studies of airless bodies (e.g., Mercury, the Moon, asteroids) have long included temperaturedependent properties [Linsky, 1966] of the regolith for several reasons: [5] 1. These bodies usually experience large diurnal temperature variations (e.g., over 600 K for Mercury [Vasavada et al., 1999]), leading to large changes of the top material thermophysical properties. [6] 2. In the absence of atmosphere, the conduction of heat in particulate media is dominated by radiation [Watson, 1964]. Heat transfer by radiation is strongly temperature dependent. [7] 3. Several experimental and theoretical studies have precisely constrained and parameterized the specific heat [Robie et al., 1970] and vacuum bulk conductivity of particulate geologic material as a function of the temperature [Fountain and West, 1970]. [8] Linsky [1966] shows that in the case of the Moon, temperature dependent properties of regolith materials are fundamental parameters to consider. On Mars however, the range of surface temperatures is typically much smaller (e.g., K), and the bulk conductivity of the regolith is dominated by the presence of CO 2 gas in the pores, limiting the heat transfer by radiation to a marginal contribution [Fountain and West, 1970]. In addition, the complex geometrical relationship between the discontinuous solid phase (i.e., the grains) and the continuous gaseous phase (i.e., CO 2 ) prevents a simple prediction of the bulk conductivity of the Martian regolith and its temperature dependence. For these reasons, previous thermophysical studies of the Martian regolith have ignored the temperature dependence of the thermal inertia. [9] Measurements of the thermal conductivity of particulate samples under Martian conditions of pressure and temperature exist but are rare and have not been parameterized [Fountain and West, 1970]. They suggest a modest increase of the bulk conductivity of particulate material (e.g., 20% from 200 to 320 K). In situ measurements show a clear dependence of the Martian regolith thermal inertia with the temperature [Zent et al., 2010]. In this paper, we propose to quantify the temperature driven variations of the density, specific heat, and bulk thermal conductivity of the Martian regolith and evaluate their effects on modeled surface temperatures. 2. Material Properties [10] In this section, the properties of materials contributing to the thermal inertia of the Martian regolith (i.e., density, specific heat, and thermal conductivity) are described, as well as their dependence on the temperature Density [11] The pore filling gaseous phase is assumed to be composed entirely of CO 2. In this study, we consider a gas pressure of 510 Pa and a temperature range of K. Under those conditions, which we refer to as typical Martian conditions in the rest of this paper, the contribution of the CO 2 gas on the bulk density of a particulate regolith is always insignificant when compared to the solid phase and is ignored. [12] The density of solid geological materials varies by extremely small factors within the range of Martian temperatures and (less than 2% between 150 and 270 K for water ice) [Heuze, 1983]. The density of the regolith is considered to be independent of the temperature in this paper Specific Heat [13] Published specific heat values for geological solids at Mars temperatures are rare, but measurements using particulated lunar basaltic samples show a strong temperature dependence [Robie et al., 1970; Fujii and Osako, 1973]. Data acquired in the laboratory by Robie et al. [1970] and Fujii and Osako [1973] suggest a 75% increase of the specific heat in the K range, from 430 to 770 J kg 1 K 1. These measurements are in good agreement with previous measurements performed by Winter and Saari [1969] using other geological materials. Such variations indicate that the specific heat is the most temperaturesensitive thermophysical parameter of the Martian regolith and that thermally driven variations of c should be taken into account. [14] The contribution of the gas phase to the bulk specific heat of a soil is negligible when compared to the solid phase contribution; therefore, it is neither modeled nor further considered Thermal Conductivity [15] In this section, we aim to characterize the variations of the regolith bulk conductivity as a function of the temperature, which depend on the individual behavior of the gas and solid phases. To do so, we integrate these parameters individually in a model describing the bulk conductivity of the uncemented regolith to assess their respective contributions to the temperature dependence of the regolith bulk conductivity Gas Conductivity [16] The gas conductivity controls the bulk conductivity of loose Martian surficial deposits [Smoluchowski, 1910; Deissler and Boegli, 1958; Wechsler and Glaser, 1965; Wechsler et al., 1972; Presley and Christensen, 1997] despite being significantly smaller than the solid phase conductivity. The flow of heat from one solid grain to the 2of18

3 Figure 1. Grain size and temperature driven variations of the pore filling gas conductivity for a surface pressure of 510 Pa (average Martian pressure). Changing the pressure to extreme Martian pressures (i.e., Pa) does not significantly change the trend presented here. The variations are expressed as the percent change in the gas conductivity when compared to the value at 150 K for the same grain sizes. Contour lines indicate every 4% change of the gas thermal conductivity. Absolute values of the gas conductivity are from Piqueux and Christensen [2009a]. The pore filling gas conductivity is shown to increase by up to 60% over the range of Martian temperatures. next is impeded by the reduced contact area between the spherical solids [Jakosky, 1986]. Instead, heat has to flow from one grain to the next through the pore filling gas, whose thermal conductivity k gas is small and influenced by a series of intertwined parameters described by Piqueux and Christensen [2009a]. For this reason, the bulk conductivity of a particulated sample is controlled by the low conductivity of the gas and not by the high conductivity of the solid. [17] The gas conductivity k gas is strongly controlled by the ratio between the mean free path l of the CO 2 molecules and the typical size L of the pores (directly dependent on the grain size). This ratio is usually called the Knudsen number: Kn ¼ L ¼ kt p L ffiffi 2 d2 P with k the Boltzmann constant (J K 1 ), T the temperature (K), d the molecular collisional diameter of CO 2 molecules (m), and P the atmospheric pressure (Pa). When Kn is small (e.g., large grains, high pressure, or low temperature), the gas conductivity is large and the bulk conductivity is also large. When Kn is large (e.g., small grains, low pressure, or high temperature), the gas conductivity is small and the bulk conductivity is also small. The Martian surface temperature ð2þ varies by a factor of 2 as a function of the latitude, season, local time, and soil thermophysical properties. Equation (2) indicates that temperatures oscillating by a factor 2 ( K, extreme Martian temperatures) lead to large variations of Kn (factor 2), with higher Kn associated with higher temperatures. As Kn controls the pore filling gas conductivity, seasonal or diurnal oscillations of Kn driven by a temperature change (equation (2)) are associated with variations of the pore filling gas conductivity, with lower values at high temperatures [Piqueux and Christensen, 2009a]. This effect is offset by another temperature dependent transport property of gases: in the slip flow regime (i.e., when the mean free path of the pore filling gas molecules is much smaller than the size of the pores), the conductivity of CO 2 gas k 0 increases with increasing temperature, from to J s 1 m 1 K 1 between 150 and 315 K (from Vesovic et al. [1990] extrapolated for the lowest temperatures). This behavior is relatively well described by the kinetic theory of gases [Knudsen, 1934]. [18] Figure 1 integrates both effects and quantifies the variations of the pore filling gas conductivity as a function of the temperature and grain size. The pore filling gas conductivity always increases with increasing temperature and grain size [Piqueux and Christensen, 2009a], with values ranging from 0 to J s 1 m 1 K 1. For a given 3of18

4 grain size, the gas conductivity typically increases by up to 60% over the range of the Martian temperatures. Overall, the gas conductivity may vary by 25% over the course of a day (assuming a 80 K diurnal amplitude) and 35% seasonally (assuming a 100 K seasonal amplitude, most likely to occur at high latitudes) Solid Phase Conductivity [19] Orbital and in situ data have revealed the compositional diversity of the Martian surficial deposits, with a high compositional diversity of particulate materials, (e.g., basaltic and glassy sands) [Christensen et al., 2001a, 2004a, 2004b; Morris et al., 2006], amorphous silica particulates [Squyres et al., 2008; Ehlmann et al., 2009], and hematite spherules [Christensen et al., 2000, 2001b; Lane et al., 2002]. A wide range of potentially cementing materials has also been detected (e.g., sulfates [Gellert et al., 2004; Gendrin et al., 2005; Langevin et al., 2005; Morris et al., 2006], carbonates [Bandfield et al., 2003; Ehlmann et al., 2008; Boynton et al., 2009], oxides (see a compilation of observations by Chevrier and Mathé [2007], chloridebearing phases [Osterloo et al., 2008], and water ice [Feldman et al., 2004; Bandfield, 2007; Kuzmin et al., 2009]) that may be involved with the observed consolidation of the regolith [Shorthill et al., 1976a, 1976b; Moore et al., 1982; Bell et al., 2004; Herkenhoff et al., 2004a, 2004b; Sullivan et al., 2008]. The presence of cement in the regolith may explain the moderate values of thermal inertia observed over a significant fraction of the surface [Jakosky and Christensen, 1986; Piqueux and Christensen, 2009b; Presley et al., 2009]. [20] Thermal conductivity measurements for geologic materials below room temperature are rare, and strongly temperature dependent [Touloukian et al., 1970]. A variety of thermal conductivity measurements for glassy material at and above room temperature suggests a continuous increase of the conductivity with increasing temperature. For glassy material, we have linearly extrapolated the measurements tabulated by Clauser and Huenges [1995] from Kanamori et al. [1968] from 300 to 1000 K to estimate the conductivity of fused silica from 150 to 315 K (e.g., Js 1 m 1 K 1, e.g., 35% increase). We find: k glass ¼ 0:6924 þ 0:0015 T A plot provided by Johnson [1960] indicates values for k glass twice as large as those provided by equation (3) but confirms the rise in conductivity with increasing temperature. We have used equations (3) (5) to generate Figures 3, 4, and For crystallized material, we have chosen to extrapolate the relationship provided by Clauser and Huenges [1995] for basaltic material from 273 to 1200 K giving J s 1 m 1 K 1 in the K range (i.e., 35% decrease): 474 k basalt ¼ 1:18 þ 350 þ ðt 273:15Þ In section 4, we consider halite as a cementing phase for the Martian regolith. Halite is a relevant material because (1) chloride bearing minerals may have been identified on Mars [Osterloo et al., 2008]; (2) the thermal conductivity of NaCl has been measured at Martian temperatures [Touloukian ð3þ ð4þ et al., 1970]; and (3) the thermal conductivity of halite is similar to other potential cementing phases (e.g., sulfates, other chloride bearing minerals, carbonates etc.) at room temperature or above [Diment and Pratt, 1988]. In this paper, the conductivity of halite is expressed following the formulation given by Clauser and Huenges [1995] for higher temperature samples but is consistent with measurements compiled by Touloukian et al. [1970] at Martian temperatures: 2960 k halite ¼ 2:11 þ 350 þ ðt 273:15Þ The values for k halite range from 10.9 J s 1 m 1 K 1 to 5.4 J s 1 m 1 K 1 in the K range (e.g., 100% decrease) Regolith Bulk Conductivity [21] To translate the variations of the gas and solid phase conductivities described above into bulk conductivity changes, we propose to use the model of heat conduction for planetary soils presented by Piqueux and Christensen [2009a] whose outputs have been shown to be consistent with a wide range of experimental results. In this model, a soil is assumed to be composed of perfect uniformly sized spherical grains whose geometrical arrangement is isotropic in all directions of space, and controls the packing density (e.g., soil porosity) of the system. Taking advantage of the symmetries resulting from this assumption, a representative unit of soil (i.e., a cell) can be described by a fraction of grains, cement, and gas whose individual thermophysical properties (e.g., density, specific heat, conductivity) are set by the modeler (Figure 2). Heat is forced to flow through these cells by setting a temperature gradient until the thermal steady state is reached. Each cell is treated as a small homogeneous regolith unit, where Fourier s law of heat conduction applies: the bulk conductivity k (J s 1 m 1 K 1 ) is linked to the size DZ (m), the temperature gradient DT = T bot T top (K), and the heat flux through the cell s boundaries Q (J s 1 m 2 ), following Q ¼ k DT DZ DT and DZ are set by the modeler. Q at the cell s upper (or lower) surface is integrated by the numerical code once the steady state reached, and k is then derived, providing a bulk thermal conductivity for each soil configuration. A radiative input (depending on the temperature and grain size) and a solid to solid contact contribution (due to the rough nonideal nature of the surface of the grains) are added independently following the expression provided by Watson [1964] and are usually small. The bulk conductivity of the cell can then be expressed as a function of any free parameter of the system. This numerical model of heat conduction through particulate material may incorporate any solid or gas phase properties (e.g., at any pressure, temperature, solid composition, presence of cement forming menisci in the intergrain regions, amount of cement, porosity). A lookup table generated by this model and used to calculate the bulk conductivity of the regolith is incorporated in the auxiliary material. 1 1 Auxiliary materials are available at ftp://ftp.agu.org/apend/je/ 2011je ð5þ ð6þ 4of18

5 Figure 2. Basic organization of the numerical model of heat transfer for Martian soil. The geometry defines the porosity of the cell (here a simple cubic arrangement, but other geometries are considered [see Piqueux and Christensen, 2009a]), which is meshed to reach the numerical stability and an acceptable level of residuals. The user sets the boundary conditions and the material properties. Once a steady state is reached, the temperature and heat flux fields are determined numerically. 5of18

6 Figure 3. Effect of the solid phase conductivity on the bulk conductivity of Martian surficial deposits, as a function of the pore filling gas conductivity (i.e., pressure, temperature, grain size, and porosity) using the model described in Figure 2. Contour lines indicate bulk conductivities at J s 1 m 1 K 1 intervals. The solid phase conductivity is shown to have little influence on the bulk conductivity of uncemented regolith Effect of Solid Conductivity Changes on the Bulk Conductivity of Uncemented Regolith [22] In this section, we derive the bulk conductivity of uncemented regolith whose solid conductivity ranges from 0.25 to 20 J s 1 m 1 K 1 to cover a wide range of solid phase compositions and large solid phase conductivity variations due to temperature changes. Figure 3 shows that the solid phase conductivity has little influence on the bulk conductivity of a soil. The bulk conductivity is only noticeably influenced by the solid phase conductivity with low solid conductivities (e.g., less than 1 Js 1 m 1 K 1 ), which may be observed with glassy materials near 150 K. In addition, equation (3) suggests at most a variation of 0.2Js 1 m 1 K 1 of the conductivity of the solid phase over the range of the Martian temperatures, which might not result in significant variations of the bulk conductivity (Figure 3). This result confirms that the composition of the solid and its conductivity variations due to temperature changes have little impact on the thermal inertia of uncemented soils. Only solids characterized by extremely small thermal conductivities (i.e., <0.5 J s 1 m 1 K 1 ) seem to be partially controlling the bulk thermal conductivity. No common geologic solid with such low thermal conductivity from 150 to 315 K has been identified in the literature. We conclude that for uncemented regolith, variations of the solid phase conductivity due to temperature or composition changes have no significant influence on the bulk conductivity of the regolith Effect of Gas Conductivity Changes on the Bulk Conductivity of Uncemented Regolith [23] Piqueux and Christensen [2009a] show that small gas conductivity changes translate into large variations in bulk conductivity. In this section, we characterize the variations of the bulk thermal conductivity of a regolith due to temperature driven changes of the gas conductivity using the model presented in Figure 2 assuming fixed solid phase conductivity. The variations of the bulk conductivity of a soil due to the temperature driven changes of the porefilling gas conductivity vary with the pore size, with larger variations associated with larger particles (Figure 3). The largest grains modeled (i.e., 1 cm) display an extreme bulk conductivity range ( 120%) but such variations are not expected to occur naturally because high thermal inertia materials usually do not experience large temperature variations (Figure 4). Realistic variations of the bulk conductivity of an uncemented regolith due to temperature changes are 20 40% for 1 mm to 1 mm particles. [24] Figure 5 compares thermal conductivity measurements of particulate basaltic material by Fountain and West [1970] from 200 to 320 K under Martian pressure (i.e., 700 Pa of CO 2 ) and the conductivity prediction of the model discussed previously. The measurements show a thermal conductivity increase of 20% for mm grains in this temperature range, whereas our model predicts a conductivity increase of 30% (or 45% from 150 to 315 K, i.e., the full range of Martian surface temperatures). However, 6of18

7 Figure 4. Variations of the bulk conductivity due to temperature induced changes of the pore filling gas conductivity for a surface pressure of 510 Pa. The variations are expressed as the percent change of the bulk conductivity when compared to the value at 150 K for the same grain sizes. Contour lines indicate 5% change of the bulk thermal conductivity. Absolute values of the pore filling gas conductivity used to generate this figure are from Piqueux and Christensen [2009a], and the relationship between the gas conductivity and the bulk conductivity is derived from the model presented in Figure 2. The bulk conductivity of uncemented regolith is shown to possibly double over the range of surface Martian temperatures due to temperature changes of the pore filling CO 2 gas. our model also predicts larger bulk conductivity variations due to porosity changes than observed by these authors. This difference may originate from the modeling assumption that the pore sizes are proportional to the mean grain size, which is unlikely to be true for a sample of mm particles. In addition, the pores of polydispersed samples are expected to be smaller than pores of homogeneous samples because smaller particles may occupy some of the space Figure 5. Comparison between thermal conductivity measurements of mm particles of basalt under 700 Pa of CO 2 gas as a function of the temperature by Fountain and West [1970] (dots) and thermal conductivity trend modeled following [Piqueux and Christensen, 2009a] (lines). Red, blue, green, and black dots indicate sample bulk densities of 1500, 1300, 1130, and 790 kg m 3, respectively. The modeled conductivities predicted by the model of Piqueux and Christensen [2009a] overestimate the effect of the porosity, but the temperature dependencies are in good agreement with the measurements. 7of18

8 Figure 6. Thermal inertia variations for uncemented regolith as a function of the temperature and grain size for a surface pressure of 510 Pa and assuming a constant porosity of 36%. This model includes the specific heat and bulk conductivity variations with temperature. Contour lines are every 50 J m 2 K 1 s 0.5. The thermal inertia of uncemented regolith may double over the range of Martian temperatures. available between the larger grains. While the difference between the measurements and the model may impact grain size predictions from thermal inertia measurements, the modeled and observed temperature dependence trends are similar and in good agreement. 3. Thermal Inertia Variations of Uncemented Regolith 3.1. Maximum Thermal Inertia Variations [25] Temperature dependent specific heats and bulk thermal conductivities (resulting from the temperaturedependent gas and solid conductivities) are incorporated in equation (1) to derive the thermal inertia of particulated material under typical Martian conditions as a function of the temperature. Results are presented in Figure 6. To generate Figure 6, the porosity is assumed to be constant and equal to 36% for convenience and because this value has been observed in natural particulated deposits [Denekamp and Tsur Lavie, 1981]. Small grain samples are often characterized by higher porosities, i.e., larger than 80%. Laboratory [Presley and Christensen, 2010a] and modeling work [Piqueux and Christensen, 2009a] show that higher porosities are associated with lower conductivities, suggesting that the thermal inertia values reported in Figure 6 for grains smaller than 100 mm are overestimated. However, the purpose of Figure 6 is to illustrate the temperature dependence of the thermal inertia, which is mostly independent of the porosity effect (Figure 5) and mainly controlled by the variations of the specific heat of the solid phase. [26] As expected, under typical Martian conditions, the thermal inertia always increases with increasing temperature, following the combined effects of the specific heat and bulk conductivities (Figure 4). For most grain sizes, the thermal inertia nearly doubles between 150 and 315 K, with a slightly more pronounced increase for larger grains, due to the larger radiative conduction contribution Application to Mars Surface Temperature Modeling [27] We have shown that the thermal inertia of the regolith may double over the range of the Martian temperatures. However, only the near surface material experiences large surface temperature variations, and most of the regolith does not. We define the apparent thermal inertia as the thermal inertia value that is derived from a surface temperature measurement assuming a homogeneous regolith. The apparent thermal inertia of the regolith integrates the thermophysical properties of the first few diurnal skin depths. As a result, the apparent thermal inertia is not expected to vary as much as the top material inertia in the case of the Martian regolith. [28] Surface temperature modeling at the equator including temperature dependent inertias of particulate material do not show significant modification of the shape and amplitude of 8of18

9 Figure 7. Modeled diurnal temperature curves using temperature dependent (solid line) and fixed thermal inertias (dotted lines). Thermal inertia values are given in J m 2 K 1 s 0.5. Temperature dependent thermal inertias are given at 230 K. Conditions are Ls 0, latitude 0 N, atmospheric opacity of 0.2, surface albedo of 0.2. Local time is expressed as 1/24 fraction of the day. Small modifications of the modeled diurnal curves result from the inclusion of a temperature dependent thermal inertia. diurnal curves when compared to results from temperatureindependent models (Figure 7). Near noon, when the surface material is the warmest, the thermal inertia is the highest and the predicted temperature is lower by up to 5 K when using a temperature dependent model. Conversely, before dawn, a temperature dependent model predicts temperatures up to 4 K warmer than a temperature independent model (Figure 8). High inertia material displays the largest temperature difference. For comparison, diurnal temperature modeling of fine dust (i.e., thermal inertia of 50 J m 2 K 1 s 0.5 )is barely impacted by the use of a temperature dependent thermal inertia model, with a daytime modeling difference of <2 K and nighttime modeling difference of 3 K. [29] Figure 9 shows the relationship between the apparent thermal inertia and the average (i.e., at 230 K) thermal inertia of a temperature dependent regolith at 0200 LT (Martian local time), when thermal inertia using Thermal Emission Spectrometer onboard Mars Global Surveyor are usually determined. Because nighttime surface temperatures are lower than average, the apparent thermal inertia at night is lower than the average thermal inertia of the top material. The magnitude of this difference depends on the overall thermal inertia of the material, and increases with increasing thermal inertia. For low thermal inertia material (i.e., 50 J m 2 K 1 s 0.5 ) this difference is small, as suggested by Figures 7 and 8, whereas it becomes larger with higher inertia material. For a regolith characterized by a thermal inertia of 300 J m 2 K 1 s 0.5 at 230 K, the nighttime apparent thermal inertia is 250 J m 2 K 1 s 0.5,or 15% lower than average. Near 0400 LT, the thermal inertia is poorly constrained by the surface temperature and little thermophysical information about the regolith properties can be derived. When the surface temperature is the warmest (near noon, the exact local time of highest surface temperature depends on the thermal inertia, see Figure 7), the apparent thermal inertia is predicted to be higher than average Thermally Derived Grain Sizes [30] Presently, Martian grain sizes are derived from thermal inertia data using thermal conductivity measurements by Presley and Christensen [1997] at room temperature or above and assuming a fixed specific heat. Most often, Martian nighttime temperatures are utilized (i.e., the coolest temperatures), because they are most sensitive to the thermal inertia, and least sensitive to the topography and albedo. Consequently, the temperature difference between the laboratory and the Martian surface is as much as 150 K, inducing a large drop of (1) the pore filling thermal conductivity (Figures 1 and 4) and (2) of the specific heat value of the soil. Figure 6 suggests that the drop in thermal inertia for typical particulate material due to lower temperatures may reach up to a factor two. Thus, thermally derived grain sizes must take the soil temperature into account. [31] Figure 10 compiles the modeling results for uncemented soil discussed in this paper with previous published work. The relationship between the thermal inertia and various grain sizes is displayed using (1) the conductivity derived by Presley and Christensen [1997] in the laboratory assuming rc =10 6 (assumption proposed by Neugebauer et al. [1971] followed by numerous subsequent thermal inertia studies deriving grain sizes) and (2) the temperaturedependent conductivity [Piqueux and Christensen, 2009a] and specific heat [Robie et al., 1970]. In both cases, the atmospheric pressure is equal to 510 Pa. 9of18

10 Figure 8. Temperature differences as a function of the local time for the curves presented in Figure 7. Variable temperature differences are predicted as a function of the local and thermal inertia. Before dawn, when thermal inertias are usually derived, the surface temperature of a given regolith may be a few kelvin cooler than predicted when using a fixed thermal inertia model. Figure 9. Apparent thermal inertia (i.e., thermal inertia derived using a fixed thermal inertia) versus temperature dependent thermal inertia (given at 230 K) for the regolith described in Figures 7 and 8. Because the temperature dependent thermal inertia model predicts cooler predawn temperatures, the 0200 LT apparent thermal inertia is lower than the average material regolith thermal inertia calculated at 230 K. 10 of 18

11 Figure 10. Relationship between thermal inertia and typical grain size at 510 Pa, using (1) the thermal conductivity relationship provided by Presley and Christensen [1997] and assuming a fixed specific heat (dotted line) and (2) the temperature dependent conductivity from Piqueux and Christensen [2009a] using a porosity of 36% in conjunction with the temperature dependent specific heat provided by Robie et al. [1970] (solid lines). The gray shading indicates the 10% uncertainty reported by Presley and Christensen [1997]. Theoretical error bars for the temperature dependent model are similar. [32] Figure 10 illustrates the potential effect of the temperature on the apparent grain sizes; for grains smaller than 100 mm, our model predicts grain sizes that are always larger than those derived using [Presley and Christensen, 1997]. Figure 10 assumes a fixed porosity of 36%, whereas small grains are usually associated with much higher porosities [see, e.g., Presley and Christensen, 1997, Table 1], indicating that the thermal inertias calculated for the finest material may be somewhat overestimated. For mm grains, only warm soils (i.e., K) are associated with conductivities similar to those predicted by Presley and Christensen [1997], but typical nighttime temperatures of particulated materials on Mars are usually lower. Finally, grains larger than 400 mm are predicted to display similar thermal inertias regardless of the model. Overall, Figure 10 shows that in general, the high temperatures of the laboratory compared to typical nighttime Martian temperatures result in underestimated grain sizes, especially for the finest materials (i.e., <400 mm) Coherence With in Situ Observations [33] Fergason et al. [2006] compare Mini TES thermally derived grain sizes using Presley and Christensen s [1997] laboratory work with Microscope Imager (MI) data onboard both Mars Exploration Rovers. They did not find any significant disagreement between both instruments derived grain sizes, but significant limitations prevented a detailed comparison. These limitations are the MI resolution, which only clearly identifies individual grains larger than 210 mm (I 200 J m 2 K 1 s 0.5 )[Herkenhoff et al., 2004b] and the thermal inertia determination error (i.e., 10 20%). As a consequence of these combined uncertainties, the relationship between grain size and thermal inertia synthesized on Figure 10 is not incompatible with MI data acquired on the Martian surface. Two examples of consistent results between MI observations and the model are presented. [34] 1. In the model from Fergason et al. [2006] the particulated material in Gussev Middle Ground Hollow is shown to have a thermal inertia of 150 J m 2 K 1 s 0.5. Our numerical model suggests a grain size of 80 mm assuming a highly porous material (e.g., 50%, consistent with 80 mm dust [from Presley and Christensen, 1997, Table 1]) at 280 K (Fergason et al. [2006] provided 45 mm), and MI suggests a typical grain size smaller than 100 mm. [35] 2. In the model from Fergason et al. [2006] and Herkenhoff et al. [2004a], the thermal inertia of aeolian bed forms near the rim of Bonneville crater is characterized by a thermal inertia of 250 J m 2 K 1 s 0.5, associated by Fegason et al. and Herkenhoff et al. with a typical grain size of 415 mm (our model provides 550 mm). MI found a trimodal distribution dominated by mm grains overlaid by a layer of 1 2 mm grains. [36] While providing a loose validation due to the uncertainties stemming from MI and Mini TES limitations, these observations confirm that the grain sizes derived with 11 of 18

12 Figure 11. Bulk conductivity variations of a cemented regolith as a function of the amount of cement and gas conductivity (i.e., pressure, temperature, grain size, porosity). At this resolution, the gas conductivity (i.e., temperature driven variations of the gas conductivity) has little to no influence on the bulk conductivity of cemented soils. this model are not inconsistent with in situ observations performed at the surface of Mars. They also show that a temperature independent model provides consistent grain sizes with MI observations, suggesting that the temperature effect on the derived grain size is not a significant parameter. [37] We conclude that the low Martian surface temperatures relative to typical laboratory temperatures result in an underestimation of Martian fines but that this underestimation is not significant and close to the measurement uncertainties. 4. Thermal Inertia Variations of Cemented Regolith [38] Piqueux and Christensen [2009b] predict that for a wide range of planetary surface conditions, the gas conductivity has a reduced influence on the bulk conductivity of cemented regolith, except for loosely consolidated soils. Their results also suggest that the solid phase conductivity (grains and cement) becomes important and may partially control the bulk conductivity of the material. In this section, we characterize the variations of the bulk conductivity of cemented regolith due to temperature driven conductivity changes of the gaseous and solid phases. For clarity, we arbitrarily define a cemented regolith as containing over 0.1% in volume of the original porosity of cementing phase Thermophysical Properties of Cemented Regolith Effect of Gas Conductivity Changes on the Bulk Conductivity of Cemented Regolith [39] In a cemented regolith, the solid phase constitutes a continuous high conductivity medium where heat may flow much more efficiently than in an uncemented regolith in which the flow of heat is impeded in the intergrain region and must flow through the low conductivity gaseous phase [Jakosky and Christensen, 1986]. As a result, the thermal conductivity of the gaseous phase weakly controls the bulk conductivity of cemented regolith. Figure 11 shows the bulk conductivity of soils characterized by various bond fractions (from 0. to 1., these values represent the fraction of original porosity occupied by the cement) and for a wide range of pore filling gas conductivities covering the domain of micrometer to centimeter size grains from 150 to 315 K. Figure 11 confirms that the pore filling gas conductivity, which is controlled in part by the temperature, has very little influence on the bulk conductivity and inertia of the cemented soil. However, for very weakly cemented material (i.e., below the resolution of Figure 11) a domain exists where the gaseous contribution to the bulk conductivity is similar to the cement phase contribution [Piqueux and Christensen, 2009b]. Using the resolution of Figure 11, we determine that this domain is reached for cement volume significantly lower than 0.1% of the total soil volume (assuming that the cement concentrates a menisci in the intergrain regions). 12 of 18

13 Figure 12. Bulk conductivity variations of a cemented regolith as a function of the amount of cement and conductivity (i.e., temperature) of the cementing phase. In this case, the bonding phase is halite, and the pore filling gas conductivity is constant (0.002 J s 1 m 1 K 1 ). Where more than 35% of the void space is occupied by halite, all the cement menisci are interconnected, forming a continuous medium for heat to flow (cement supported sample). Contour lines are every 0.2 J s 1 m 1 K 1. [40] For weakly to highly cemented materials and assuming a pendular geometry for the cement (i.e., cement volume >0.1% of the original pore volume), the temperature driven variations of the gas conductivity have no significant influence on the bulk conductivity and inertia of the cemented soil. When the amount of cement occupies 35% of the original porosity or more, the cementing menisci become large enough to merge together and form a continuous medium supporting the grains ( cement supported ). When this threshold is reached, the conductivity increases significantly as the heat flows freely through the high conductivity cementing phase of the regolith Effect of Cement Conductivity Changes on the Bulk Conductivity of Cemented Regolith [41] In this section, we estimate the effects of temperaturedriven variations of the cementing phase conductivity on the bulk conductivity of the regolith, assuming that the grains and the gas phases have fixed conductivities. Potential cementing phases display a wide range of thermal conductivities, highly dependent on the temperature (e.g., equation (5), also see tables of Touloukian et al. [1970], Clauser and Huenges [1995], and Diment and Pratt [1988]). By increasing the contact area between grains, the control of the gas phase is reduced as shown on Figure 11, but simultaneously, the dependence of the bulk conductivity on the cementing phase conductivity increases. As opposed to the conductivity of CO 2 gas, the thermal conductivity of cements decreases with increasing temperature. As a result, the cementing phase tends to decrease the bulk conductivity of a soil as the temperature increases. Figure 12 illustrates this trend, with a soil characterized by a fixed grain conductivity (2.9 J s 1 m 1 K 1, basalt conductivity at 200 K) and a cementing phase whose conductivity ranges from 10.9 to 5.4 J s 1 m 1 K 1 (halite conductivity from 150 to 315 K). For all degrees of cementation, the bulk conductivity decreases from 40% (highly cemented material) to 0% (amount of cement approaching 0) between 150 and 315 K Effect of Grain Conductivity Changes on the Bulk Conductivity of Cemented Regolith [42] In this section, we quantify the effects of temperaturedriven variations of the thermal conductivity of the spherical grain of a cemented regolith assuming a fixed conductivity for the cementing phase Glassy Grains [43] Unlike the conductivity of crystallized materials at Martian temperatures, the conductivity of glassy material increases with increasing temperature. A modest increase of the bulk conductivity of a regolith is modeled from 150 to 315 K, in accordance with equation (3) (typically 10%), and this increase is only slightly dependent on the grains intrinsic conductivity (e.g., temperature and composition) as shown on Figure 13a. Figure 13a confirms that temperaturedriven variations of a cemented regolith made of glassy grains are not due to the temperature dependence of glass Basaltic Particles [44] In this case, the grains are assigned a basaltic composition, and their thermal conductivity decreases with increasing temperature. We note that basaltic material usually contains a fraction of glassy material, but the influence of the 13 of 18

14 Figure 13. Variations of the bulk conductivity of a cemented regolith assuming a fixed cement conductivity (i.e., 8.5 J s 1 m 1 K 1, halite at 200 K) and fixed pore filling gas conductivity (0.002 J s 1 m 1 K 1 ) but a variable grain conductivity as a function of the temperature. (a) The case of glassy grains, whose conductivity increases with increasing temperature (contour lines every 0.4 J s 1 m 1 K 1 )and(b)the case of basaltic grains whose conductivity decreases with increasing temperature (contour lines every 0.4 J s 1 m 1 K 1 ). The temperature driven variations of the conductivity of the grains translate into large bulk conductivity change in the case of cemented regolith. 14 of 18

15 temperature on the conductivity of basalt shows that the crystallized phase dominates the thermophysical properties. For this reason, basaltic material is considered here as a crystallized phase. This modeling configuration provides results similar to the previous one, but the bulk conductivity decreases with increasing temperature, as expected from equation (4) (Figure 13b). In this case, the change in absolute bulk conductivity is larger than that obtained from the model with glassy material (Figure 13a) because the temperature dependence of the conductivity of crystallized materials is higher than that of glass (typically 25%, up to 40%). [45] Overall, our model shows that the influence of the temperature and solid phase composition on the bulk conductivity of cemented materials is variable, often important, and should be taken into account (Figure 12). We note that the thermal conductivity of highly cemented materials is less dependent on the composition of the grains and temperature of the soil than poorly cemented materials (Figure 13) Temperature Dependent I for Cemented Soils [46] In this section, the full dependence of the thermal inertia of cemented soils with temperature is presented under typical Martian conditions (Figure 14) and discussed for a wide range of degrees of cementation. The thermal inertia values given here for cemented soils integrate the temperature effect on the gas, grains, and cementing phase conductivities, as well as on the specific heat of the material as described in previous sections of this paper. For completeness, Figure 14 illustrates the case of a glassy (Figure 14a) and a basaltic (Figure 14b) regolith. [47] In both cases, the thermal inertia increases with increasing temperature, mainly due to the temperature effect on the specific heat of the solid phases (spheres and cement). However, the largest thermal inertia increase is obtained with glassy materials because the conductivity of the grains also increases with increasing temperature (equation (3) and Figure 13a) whereas the decreasing conductivity of basaltic grains with increasing temperature partially offsets the increase of the specific heat (equation (4) and Figure 13b). As a result, the thermal inertia of a glassy cemented soil typically increases by 20% from 150 to 315 K, whereas this increase is just 10% with cemented basaltic grains. In the former case, the increase of the thermal inertia with increasing temperature is larger when medium amounts of cement are present (i.e., less than 30% of the void space), because the control of the grain conductivity is best expressed in this domain (Figure 13a). Figure 14a shows that for low to medium amounts of cement in a glassy particulated soil, the thermal inertia may increase by up to 50% over the range of the Martian surface temperatures. [48] Overall, temperature driven variations of the thermal inertia of cemented materials are smaller than those for loose samples due to the competing trends imposed by the increasing value of the specific heat with increasing temperature, and the decreasing cement conductivity value with increasing temperature Application to Mars Surface Temperature Modeling [49] Cemented materials usually display higher thermal inertia values than uncemented samples [Piqueux and Christensen, 2009b; Presley et al., 2009] and therefore display smaller diurnal temperature variations. In addition, we have shown that the thermal inertia of cemented material is generally less temperature dependent than uncemented material due to the reduced contribution of the gas conductivity, and the opposite temperature dependence trends of the specific heat and cement phase conductivities. Modeled diurnal temperature curves including a temperaturedependent and a fixed thermal inertia show small differences at any time of the day (i.e., less than 0.5 K), even with low cement content (0.1% in volume of the original porosity). These results are independent of the grain size because the gas conductivity (controlled by the pore size) is unimportant in a cemented regolith. Such small temperature differences are below the detection threshold of surface and orbiting instruments. 5. Conclusions [50] Large variations of the thermal inertia for geologic materials are predicted under Martian conditions of pressure and temperature. For uncemented material, the following pertain. [51] 1. The increase of the specific heat with increasing temperature ( 75% from 150 K to 315 K for basaltic material) is the primary source of the temperature dependence of the thermal inertia. [52] 2. The pore filling gas conductivity is also strongly associated with the temperature and increases with increasing temperature, leading to bulk conductivity increases of 30 50% depending on the grain size and temperature. [53] 3. The variations of the solid phase conductivity (due to temperature changes or to compositional heterogeneities) have little influence on the bulk conductivity and inertia of Martian soils. [54] 4. The thermal inertia of loose material varies strongly with the temperature, typically 75% in the K range. Soils characterized by low thermal inertias are less affected by temperature changes than larger grains. [55] 5. Diurnal temperature curves generated with a model including temperature dependent thermal inertia show small distortions compared to temperatures generated by a model with fixed thermal inertia. The apparent thermal inertia is the lowest at night when the temperature is the coldest and is <15% lower than the average daily apparent inertia. [56] 6. Thermally derived grain sizes on Mars usually rely on nighttime (i.e., cold) temperature measurements, and are based on laboratory experiments performed at room temperature or above. As a result, Martian grain sizes inferred from thermal inertia data are underestimated by factors related to the grain size and temperature. More accurate grain size determination from thermal inertia data may incorporate the soil temperature at the time of the measurement, but is not necessary. [57] In the case of cemented material, the following pertain. [58] 1. The thermal conductivity of the cement and particle phases partially controls the bulk conductivity of cemented regolith. Except for glassy material, the conductivity of solids decreases with increasing temperature, resulting in a decrease of the bulk conductivity of cemented regolith with increasing temperature. 15 of 18

16 Figure 14. Variations of the thermal inertia of soil a cemented by halite, with (a) glassy or (b) basaltic grains, a constant pore filling gas conductivity of J s 1 m 1 K 1, and an increasing solid phase specific heat with temperature following Fujii and Osako [1973]. Contour lines are every 100 J m 2 K 1 s 0.5. The thermal inertia of cemented material is shown to always increase with increasing temperature, regardless of the grain composition. The increase in inertia is typically smaller than that modeled with uncemented material. 16 of 18