Compressional stress effect on thermal conductivity of powdered materials: Measurements and their implication to lunar regolith

Icarus 267 (2016) 1–11

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Compressional stress effect on thermal conductivity of powdered materials: Measurements and their implication to lunar regolith Naoya Sakatani a,⇑, Kazunori Ogawa b, Yu-ichi Iijima a, Masahiko Arakawa b, Satoshi Tanaka a a b

Institute of Space and Astronautical Science, Japan Aerospace Exploration Agency, 3-l-l Yoshinodai, Chuo-ku, Sagamihara, Kanagawa 252-5210, Japan Department of Planetology, Graduate School of Science, Kobe University, 1-1, Rokkodai-cho, Nada-ku, Kobe, Hyogo 657-8501, Japan

a r t i c l e

i n f o

Article history: Received 26 August 2015 Revised 2 December 2015 Accepted 9 December 2015 Available online 17 December 2015 Keywords: Moon, surface Regoliths Experimental techniques

a b s t r a c t Thermal conductivity of powdered materials under vacuum conditions is a valuable physical parameter in the context of planetary sciences. We report results of thermal conductivity measurements of 90–106 lm and 710–1000 lm glass beads, and lunar regolith simulant using two different experimental setups for varying the compressional stress and the temperature, respectively. We found the thermal conductivity increase with the compressional stress, for example, from 0.003 to 0.008 W m
1 K
1 for the glass beads of 90–106 lm in diameter at the compressional stress less than 20 kPa. This increase of the thermal conductivity is attributed the areal enlargement of the contacts between particles due to their elastic deformation. The thermal conductivity increased also with temperature, which primarily represented enhancement of the radiative heat conduction between particles. Reduction of the estimated radiative conductivity from the effective thermal conductivity obtained in the first experiment yields the relation between the solid conductivity (conductive contribution through inter-particle contacts) and the compressional stress. We found that the solid conductivity is proportional to approximately 1/3 power of the compressional stress for the glass beads samples, while the regolith simulant showed a weaker exponent than that of the glass beads. We developed a semi-empirical expression of the thermal conductivity of the lunar regolith using our data on the lunar regolith simulant. This model enabled us to estimate a vertical distribution of the lunar subsurface thermal conductivity. Our model provides an examination for the density and compressional stress relationships to thermal conductivity observed in the in-situ measurements in Apollo 15 and 17 Heat Flow Experiments. Ó 2015 Elsevier Inc. All rights reserved.

1. Introduction From the past to the present in the Solar System, powdered materials have existed ubiquitously. For example, the planetesimals formed in the early solar nebula were composed of micronsized dust particles. Many planetary bodies including moons and asteroids are covered by fine grained material called ‘‘regolith”. Under vacuum conditions, the powdered materials have much lower thermal conductivity than intact rocks, so that the powdered materials act as thermal blanket on the bodies. Determination of the thermal conductivity of the surface regolith and its distribution in depth direction is one of the key issues for understanding thermal processes within planetary bodies. Concerning the Moon, the thermal conductivity of the surface regolith layer was measured as a part of Heat Flow Experiments in Apollo 15 and 17 missions (Langseth et al., 1972, 1973). It serves ⇑ Corresponding author. E-mail address: [email protected] (N. Sakatani). http://dx.doi.org/10.1016/j.icarus.2015.12.012 0019-1035/Ó 2015 Elsevier Inc. All rights reserved.

as an informative reference of thermal conductivity structure of lunar and other planetary regolith layers. In these experiments, two heat flow probes were emplaced in drilled boreholes at each site. Direct measurements of the thermal conductivity were carried out by activating electrical heaters on the probes. The resultant thermal conductivity of subsurface regolith ranged from 0.0141 W m
1 K
1 at the depth of 35 cm up to 0.0295 W m
1 K
1 at 233 cm depth. On the other hand, Langseth et al. (1976) estimated thermal diffusivity of the regolith using data on attenuation of probe’s periodic temperature induced by the annual temperature wave from the surface. Assuming a constant density and a constant specific heat of the regolith with depth, they revised the thermal conductivity values downward between 0.0091 and 0.0132 W m
1 K
1. The higher thermal conductivity deduced from the heater-activated measurements (Langseth et al., 1972, 1973) was considered to be affected by compression of the regolith (which means increase in bulk density of the regolith and compressional stress stored in the regolith media) due to the drilling process in the probe installation (Langseth et al., 1976; Grott et al.,

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2010). At present, the thermal conductivity around 0.01 W m
1 K
1 as estimated by Langseth et al. (1976) has been believed to be the typical thermal conductivity of the lunar subsurface regolith. The thermal conductivity of the lunar regolith can be also estimated from cooling history of surface nighttime temperature. Keihm and Langseth (1973) reported that the thermal conductivity of upper a few centimeters of the lunar regolith layer is in the order of 0.001 W m
1 K
1, and suggested that the there is rapid conductivity increase by an order of magnitude from the surface to a few tens cm depth. The subsurface thermal conductivity seems consistent with the conductivity inferred from the annual temperature wave analysis by Langseth et al. (1976). Vasavada et al. (2012) also found the similar abrupt change in the thermal conductivity using surface temperature data of lunar equatorial region obtained from Diviner Radiometer on-board Lunar Reconnaissance Orbiter. These previous works considered that the rapid change in the conductivity is related to the density increase with depth, and they derived empirical relationships between the density and conductivity to fit the surface temperature data. However, the derived density dependence and absolute values of the conductivity is not consistent with the thermal conductivity of regolith samples returned by the Apollo missions and measured in laboratories, which was lower than the subsurface conductivity by several factors even at a possible maximum density of 1950 kg m
3 (Cremers, 1971). A possible explanation of the depth dependent thermal conductivity is that the thermal conductivity of the deeper regolith is enhanced by self-weighted compressional stress in addition to the density (Horai, 1981). However, the quantitative dependence of thermal conductivity on compressional stress has not been investigated experimentally. This would give an important information for the determination of the thermal conductivity distribution in vertical direction of the lunar and other planetary surface regolith layers. In this work, we measured thermal conductivity of powdered materials under vacuum conditions with controlling compressional stress within the samples. The derived compressional stress dependence of the thermal conductivity was utilized to estimate the vertical thermal conductivity structure of the regolith layers on the Moon. 2. Thermal conductivity of powdered materials Heat transfer within powdered materials under vacuum conditions contains two contributions. One is thermal conduction within particles and across inter-particle contacts, and the other is thermal radiation through void spaces between the particles. Effective thermal conductivity k is expressed as,

k ¼ ksolid þ krad ;

ð1Þ

where ksolid is the solid conductivity, and krad is the radiative conductivity (Wechsler et al., 1972). The solid conductivity is controlled by thermal conductance at the contacts between the particles and the array structure of the particles (or networks of the contacts). Significantly low contact conductance of the narrow contact area between the particles reduces the effective thermal conductivity compared to that of the solid material by several orders of magnitude. Some theoretical formulae for the solid conductivity were proposed concerning equal-sized spheres (Halajian and Reichman, 1969; Chan and Tien, 1973). However, they can be applied only to certain selected powders and packing structures, and have not been verified experimentally. In our previous study (Sakatani et al., 2012), thermal conductivity of glass beads was measured under vacuum and a variation of the thermal conductivity in depth direction was detected in a

vertically-extended sample container. The thermal conductivity increased by several factors within a depth range from 1 to 30 cm without effective density variations. A similar trend was identified from a laboratory experiment by Langseth et al. (1974) in a deeper depth range from 39 to 181 cm. Sakatani et al. (2012) concluded that the self-weighted compressional stress increases the inter-particle contact area (or solid conductivity) by elastic deformation of the particles. The experiments by Sakatani et al. (2012) and Langseth et al. (1974) suggested that the solid conductivity is dependent of the compressional stress with an exponent of 0.37–0.50. However, these exponent values were somewhat inaccurate, because the self-weighted stress distribution within the powdered materials packed in a container with finite horizontal dimensions was disturbed by frictional force between the particles and the inner wall of the container, and because evaluation of the radiative conductivity contributing to the measured effective thermal conductivity was insufficient. These issues are addressed in this work. The thermal conductivity measured by experiments contains both solid and radiative contributions, as indicated by Eq. (1). Determination of each conductivity term is essential for understanding the heat transfer mechanism in the powdered materials. One of the methods for this is measuring temperature dependence of the thermal conductivity. The radiative conductivity depends strongly on temperature. Watson (1964) modeled radiative heat transfer between the particle surfaces as the radiation between multiple parallel plates. According to his model, the radiative conductivity is proportional to the third powder of temperature,

krad ¼ BT 3 ;

ð2Þ
1


4

where B is a temperature-independent constant in W m K unit. An increase in the thermal conductivity with temperature can be primarily interpreted as increase of the radiative conductivity. Most investigators treated the solid conductivity as a constant independent of temperature (Merrill, 1969; Fountain and West, 1970; Cremers et al., 1970). Fitting of the equation of k ¼ ksolid þ BT 3 to the temperature-dependent thermal conductivity data with ksolid and B being the free parameters yields the solid conductivity and the radiative conductivity (or coefficient B) for a given sample. In practice, the solid conductivity is also dependent on temperature, because the thermal conductivity of the solid material and elastic coefficients (Young’s modulus and Poisson’s ratio) also vary with temperature. According to the model by Halajian and Reichman (1969), the solid conductivity is directly proportional to the thermal conductivity of the solid material, while it varies with 1/3 power of the elastic coefficients. The former effect would be effective on the temperature dependence of the solid conductivity, so that we improve the solid conductivity expression by including the temperature dependence as ksolid ¼ Akm ðTÞ, where A is a non-dimensional constant independent of the temperature and km is the temperature-dependent thermal conductivity of the solid material. The temperature dependence of the effective thermal conductivity can be written as,

k ¼ Akm ðTÞ þ BT 3 :

ð3Þ

The values of A and B, representing the solid and radiative contributions, respectively, are estimated by the fitting of Eq. (3) with a known function km ðTÞ. 3. Experimental method In order to investigate the effect of the compressional stress on the solid conductivity, two types of experiments were carried out. One is the thermal conductivity measurements under vacuum as a function of the compressional stress. This experiment is called

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N. Sakatani et al. / Icarus 267 (2016) 1–11

ð4Þ

Because a sintered block of the lunar regolith simulant could not be prepared and measured due to the small amount that was available, we refer to the thermal conductivity data for several basalt rocks summarized by Desai et al. (1974). The thermal conductivity of the basalt decreases with rising temperature below 1400 K (Fig. 2), and linear fitting to all data shown in Fig. 2 yields the following expression on km ðTÞ for the basaltic material,
4

km ðTÞ ¼
9:53  10 T þ 2:40:

ð5Þ

Eqs. (4) and (5) are used to determine the radiative contributions in Section 4.2.

Cumulative Weighted Fraction (% )

40

20

0 10

100

1000

Particle Size ( Fig. 1. Particle size distribution of two glass beads and lunar regolith simulant. The curves for the glass beads were obtained by our microscopic observation, and that for the regolith simulant was from Horai (1981). A shaded area represents the range of particle size distribution of Apollo lunar soil samples shown by Horai (1981).

Table 2 Density and porosity of the samples. The porosity is defined as u ¼ 1
q=qs , where q is bulk density and qs is true density of the materials. qs was measured by water pycnometer method, qs ¼ 2480 kg m
3 for the glass beads and qs ¼ 2990 kg m
3 for the regolith simulant. Material

Particle size

Bulk density (kg m
3)

Porosity

Glass beads Glass beads Regolith simulant

90–106 lm 710–1000 lm Mean size of 74 lm

1530 1500 1700

0.39 0.40 0.43

3.5

Sintered glass Basalt from Desai et al. (1974)

)

km ðTÞ ¼ 8:50  10 T þ 0:855:

60

-1


4

Regolith Simulant

80

K

We used three powdered samples; two kinds of soda-lime glass beads (particle diameters ranged from 90 to 106 lm and from 710 to 1000 lm), and a lunar regolith simulant. The glass beads are almost spherical in shape and uniform particle size, so that they serve as simple model materials. Elemental compositions of these glass beads are given in Table 1. The lunar regolith simulant is from the same material as used by Horai (1981); it has a particle size distribution comparable with Apollo 12 lunar soils as shown in Fig. 1, and consists of crushed terrestrial basalt powders. The larger and smaller particles than 74 lm are Knippa and Berkeley basalt powders, respectively. Bulk density and porosity of the samples are summarized in Table 2. In order to investigate the thermal conductivity of the solid materials km ðTÞ in Eq. (3), we prepared a consolidated glass block by sintering the 90–106 lm glass beads at the temperature of 1120 °C for 24 h. The porosity of this glass block is less than 1%, so that the radiative heat transfer through the voids is negligible. A line heat sensor (see Section 3.2) was sandwiched by the two glass plates prepared from the sintered glass block, and its thermal conductivity was measured at the temperature from 240 to 355 K (Fig. 2). The thermal conductivity of the glass increased with the temperature, and a linear regression yielded the following equation of km for the glass material,

Glass Beads 710-1000

3

-1

3.1. Sample selection

100

Thermal Conductivity (W m

‘‘stress-controlled experiment” in the following part of this paper. The another is the thermal conductivity measurement as a function of the temperature, called ‘‘temperature-controlled experiment”, so that the radiative conductivity was evaluated using Eq. (3). Subtraction of the constant radiative conductivity from the effective thermal conductivity values obtained by the stress-controlled experiment produces a relation between the solid conductivity and the compressional stress.

2.5 2 1.5 1 0.5 200

400

600

800

1000

1200

1400

Temperature (K)

3.2. Method for thermal conductivity measurements The thermal conductivity was measured by the line heat source method (Carslaw and Jaeger, 1959) in both experiments. The thermal conductivity of the sample can be estimated from transient temperature change of a line heater emplaced in samples. After

Table 1 Composition of the glass beads. Composition

Fraction (%)

SiO2 Na2 O þ K2 O CaO MgO Al2 O3 Fe2 O3 Other

70–73 12–16 7–12 2–5 1.0–2.5 0.05–0.15 <1.0

Fig. 2. Thermal conductivity of sintered glass and basalt rocks as a function of the temperature. Results of linear regressions for each sample are also shown.

sufficiently-long heating time, a linear relation between the temperature and the natural logarithm of time appears, whose slope angle is inversely proportional to the thermal conductivity of the surrounding material, as expressed by the following equation.

T¼

q ln t þ b; 4pk

ð6Þ

where T is temperature of the heater at the heating time t, q is heat generation per unit length of the line heater, k is thermal conductivity of the surrounding material, and b is a constant. This method has often been applied to powdered materials in past studies (e.g., Presley and Christensen, 1997a).

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In our experiments, a nichrome wire with a diameter of 180 lm and electrical resistance per unit length of 42.35 X m
1 at about 298 K was used as a line heat source. An alumel–chromel thermocouple was bonded by glue at the mid-length point of the nichrome wire to measure its temperature. This set of the nichrome wire and the thermocouple works as a sensor of the thermal conductivity. The sensors were suspended in sample containers. Details for the sample containers with the line heat source sensors are described in Sections 3.3 and 3.4. Time range of the linear regime, in which a relation between the temperature and heating duration is represented by Eq. (6), was restricted using the same method in Presley and Christensen (1997a). Considering the diameter and length of our nichrome wire, and the sample dimensions, appropriate time range for the linear regime is from 400 to 1000 s after the onset of heating (Sakatani et al., 2012). The thermal conductivity is determined by least square fitting of Eq. (6) within this time range.

(10) (9)

(8)

(11) (7) (6) (5) (13) (2) (1) (14) (12)

(13)

(4) (1) Samples (2) Line Heat Source System (3) Sample Container (4) Stress Transducers (5) Weights (6) Guides for Weights (7) Strings (8) Ultrasonic Motor (9) Gears (10) Rotational Shaft (11) Bearing (12) Exhaust System (13) Vacuum Chamber (14) Electronic Devices

3.3. Experimental setup for compressional stress dependence

within void spaces is of the order of 10
7 W m
1 K
1 according to a model by Piqueux and Christensen (2009). Since this is lower than the thermal conductivity measured in this experiment by three orders of magnitude, and is below the measurement resolution, the effect of the gas conduction on the thermal conductivity is negligible. The stress values of the stress transducers started to decrease just after the evacuation, and gradually returned up to the original values before the evacuation. This means differential gas pressure between the front and back sides of the detector planes of the stress transducers; i.e., the gas evacuation from the inside of the transducers was less efficient than the void spaces within the samples. It took at least 3 days for the re-equilibrium

K-type thermocouple 100 µm

40 mm

In the stress-controlled experiment, we measured thermal conductivities of three samples listed in Table 2 as a function of the compressional stress. Fig. 3 shows the experimental configuration. The compression system consisted of six brass blocks and its driving mechanics. The six brass weights (0.2, 0.4, 0.9, 1.2, 1.5, and 3.3 kg, respectively) were suspended by strings in series above a sample container, and the strings were fixed on a rotational shaft. The rotation of the shaft by an ultrasonic motor allowed vertical motion of the weights. The compressional stress in the sample can be varied by changing the number of the weights superimposed onto the sample surface. If assuming that the stress in a powdered sample can be expressed as hydrostatic field, the maximum compressional stress we can control is 18 kPa that is equivalent to the lunar regolith depth of about 6 m. The size of the sample container was 40 mm width, 100 mm length, and 60 mm height. The line heat source sensor described in Section 3.2 was fixed horizontally in the sample container at 20 mm above the bottom. Two stress transducers were mounted on the bottom and lateral planes of the sample container, which enabled direct measurements of the vertical and horizontal stresses in the powder samples. Although their detector planes were placed about 5 mm from the wall into the sample, it does not affect the thermal conductivity measurements because the detector is sufficiently far from the line heat source sensor compared to the heat diffusion length during the thermal measurement. Each sample described in Section 3.1 was poured into the sample container and packed as densely as possible by tapping (not pressing). The sample thickness was controlled to be 40 mm, so that the line heat source sensor was located at 20 mm below the sample’s top surface. The final densities of the samples calculated from the weights and volumes are listed in Table 2. The system was evacuated in a vacuum chamber down to a pressure of 10
4 Pa. At this pressure, the thermal conductivity of gas molecules

Nichrome wire 180 µm

100 mm

Fig. 3. Schematic views of experimental configuration (upper) and sample container with installed line heat source sensor (lower right), and a photo of experimental configuration (lower left) for investigation of stress effect on the thermal conductivity.

of the gas pressure, and the final values represented the stress field in the powder samples. Then, a constant current of 40 mA (corresponding to q = 67.76 mW m
1 ) was supplied to the nichrome wire, and the temperature of the wire was recorded by the thermocouple for 1000 s with sampling frequency of 5 Hz. An example temperature data for the regolith simulant is shown in Fig. 4. The thermal conductivity was calculated from the slope s between 400 and 1000 s as,

k¼

q RI2 ¼ ; 4p s 4p s

ð7Þ

where q ¼ RI2 is heat generation per unit length of the nichrome wire with R being the electrical resistance per unit length and I being the induced constant current. After the first thermal conductivity measurements for the uncompressed sample, the individual weights were loaded one by one on the sample surface, and the thermal conductivity as well as the stresses were measured by the same procedure. After the all weights were put on the sample, a weight was removed and the

N. Sakatani et al. / Icarus 267 (2016) 1–11

30 29 28

Temperature (degC)

27 26 25 24 23 22 21 20

Experimental Data Linear Fitting

1

10

100

1000

Heating Time (s) Fig. 4. Temperature data of the nichrome wire during the thermal conductivity measurements for uncompressed regolith simulant.

conductivity measurements were conducted in order. This cycle was repeated three times for the two glass beads and two times for the regolith simulant. These experiments were conducted at room temperature of 295 ± 2 K. 3.4. Experimental setup for temperature dependence In the temperature-controlled experiment, the thermal conductivities of the three samples were investigated as a function of temperature. These data were utilized to examining the radiative conductivity of the samples. The experimental configuration is shown in Fig. 5. A sample container with an installed line heat source sensor (the same as used in the prior experiment) was put into the vacuum chamber. Temperature of the whole of the system, including the vacuum chamber, was controlled in a thermostatic chamber. The sample was poured into the sample container and tapped. The density of the each sample in this experiment was comparable to that of the prior experiment (Table 2). The sample was evacuated in the vacuum chamber (down to 10
2 Pa, at which

5

the gaseous thermal conduction is negligible) and temperature of the system was raised up to 323 K for the glass beads and 333 K for the regolith simulant. After the temperature was kept in a steady state (it took two days), a constant current of 20 mA was input to the nichrome wire for 1000 s. The method for determining the thermal conductivity was the same as that described before (Section 3.3). Then, the temperature was set to next lower temperature and the thermal conductivity was measured. This procedure was repeated down to a temperature of 248 K with an interval of 25 K for the glass beads, and down to 253 K with an interval of 20 K for the regolith simulant. Since the electrical resistance of the nichrome wire varies with the temperature, the resistance at the temperature T was calculated as RðTÞ ¼ R0 ð1 þ bðT
T 0 ÞÞ, where R0 ¼ 42:35 X m
1 is the resistance per unit length at a reference temperature T 0 ¼ 289 K, and b is the temperature coefficient of the electrical resistance. The value of b for the nichrome wire was determined by measuring the resistance change with sweeping the ambient temperature from 240 to 355 K in a thermostatic chamber. The estimated temperature coefficient was b ¼ 9  10
5 K
1. The calculated resistance ranged from 42.16 to 42.48 X m
1 in the temperature range from 248 to 333 K. 3.5. Measurement error A methodology for the error analysis for the line heat source method was proposed by Presley and Christensen (1997b). The instrumental error relates to errors in the slope angle in the linear regime s, the electrical resistance per unit length of the nichrome wire R, and the electrical current induced in the nichrome wire I, through Eq. (7). We derived measurement errors for all data (see Appendix A). As a result, the slope s was the largest error source for most cases. Some data had the largest error in R. The total relative error in the thermal conductivity was estimated to be less than ±6.8%. Additionally, the finite length and diameter of the heating wire causes the deviation from the perfect line heat source. Especially concerning low thermal conductivity materials with 0.001 W m
1 K
1 order, axial heat loss through the wire can be the largest error source Huetter and Koemle (2012). However, since the ratio of the length to diameter of our heating wire is higher than 500, the sensor we used approximately serves as the ideal line heat source. 4. Experimental results 4.1. Effect of compressional stress

(6)

(5) (3)

(1)

(2)

(7)

(4)

(1) Samples (2) Line Heat Source System (4) Exhaust System (5) Vacuum Chamber (7) Electronic Devices

(3) Sample Container (6) Thermostatic Chamber

Fig. 5. Experimental configuration for temperature dependence of the thermal conductivity.

Measured vertical and horizontal stresses for the two glass beads and the regolith simulant are shown in Fig. 6. It is known that the stress field in powdered media is heterogeneous in the vertical and horizontal directions, due to internal friction of the particles. We found that both glass beads samples had a ratio of horizontal to vertical stress of about 0.77. On the other hand, the regolith simulant had a much lower value, here the ratio was only about 0.25. This fact implies that the thermal conductivity is also heterogeneous in horizontal and vertical directions, especially for the regolith simulant. Garrett and Ban (2011) performed twodimensional finite element simulations, and stated that the thermal conductivity measured by a line heat source sensor emplaced horizontally in the sample, as is the case for our experiments, is represented by the average of the horizontal and vertical thermal conductivities. Therefore, we treat the average value of the vertical and horizontal stresses as the effective compressional stress applied to the powdered materials, which affects the measured thermal conductivity.

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N. Sakatani et al. / Icarus 267 (2016) 1–11

18 16

Horizontal Stress (kPa)

14 12 10 8 6 4 Glass Beads 90-106 μm Glass Beads 710-1000 μm Regolith Simulant

2 0 0

5

10

15

20

25

30

35

Vertical Stress (kPa) Fig. 6. Relation between vertical and horizontal stresses during the compression.

Fig. 7 shows thermal conductivity of the glass beads and the regolith simulant as a function of the effective compressional stress. We found that the thermal conductivity of all samples increased with the compressional stress. For the initial and uncompressed states, we obtained thermal conductivities of 0.0030, 0.0084, and 0.0069 W m
1 K
1 for the 90–106 lm glass beads, the 710–1000 lm glass beads, and the regolith simulant, respectively. Their thermal conductivity increased up to 0.0077, 0.0160, and 0.0126 W m
1 K
1, respectively, as the compressional stress was applied. During the compression tests, the samples would be densified. After finishing all of the measurements, the sample’s top surface was sank by about 1 mm from the original position. This densification led to an increase in the bulk density by about 3% in consideration of the original height of 40 mm. According to a density dependence model for the solid conductivity by Gundlach and Blum (2012), ksolid / expð5:26qÞ with q being the density, the 3% increase of the density causes about 10% increase of the solid conductivity, which is lower than the observed enhancement of the thermal conductivity. Therefore, we can state that the thermal conductivity variations shown in Fig. 7 are primarily due to the increase of the particle contact area caused by the compressional stress (see Section 4.3) rather than the densification. The regolith simulant we used was also measured by Horai (1981). He reported its thermal conductivity of 0.0088 W m
1 K
1

at the density of 1700 kg m
3. The sample container he used was a cylinder of 6.99 cm height and 2.54 cm diameter. Self-weighted lithostatic pressure at the mid-height of his sample is estimated to be about 0.6 kPa. On the other hand, according to our experiment, the compressional stress of about 8 kPa is required to bring the thermal conductivity of the regolith simulant up to 0.0088 W m
1 K
1 (Fig. 7). Horai (1981) measured the thermal conductivity by the needle probe method, which is almost the same technique as the line heat source method except for the larger diameter of the heater probe and thus a smaller length/diameter ratio. The needle probe method can overestimate the thermal conductivity of insulating materials, because of non-negligible heat losses along the symmetry axis of the thick probe (Huetter and Koemle, 2012). Our line heat source sensor had a thinner diameter and longer length (10 cm  / 180 lm) than the Horai’s probe (6.35 cm  / 860 lm), and therefore, would work as more close to an idealized thermal conductivity sensor, which should have a very high length/diameter ratio.

4.2. Effect of temperature and determination of radiative contribution The thermal conductivity of the same samples as measured in the previous section was also measured as a function of the temperature, using the experimental configuration shown in Fig. 5. Fig. 8 shows the temperature dependence of the thermal conductivity of the two glass beads and the regolith simulant. The thermal conductivity increased with temperature for all samples. The thermal conductivities at about 300 K were lower than those for the uncompressed samples measured in the stress-controlled experiments (Fig. 7). This can be explained by the difference of positions of the line heat source sensors in the sample container. The sensor was placed at 20 mm below the sample surface in the stresscontrolled experiment, whereas it was at 10 mm below the surface in the temperature-controlled experiment shown in this section. The difference of the self-weighted compressional stress corresponding to the depth difference of 10 mm is about 0.15– 0.17 kPa. This small stress difference might have large effect on the conductivity due to microscopic re-arrangement of the particles (that creates new contacts without significant density increase) as suggested by (Pilbeam and Vaišnys, 1973). The solid and radiative conductivities for each sample can be estimated by fitting Eq. (3) to the experimental data, with km ðTÞ of Eq. (4) for the two glass beads and of Eq. (5) for the regolith sim-

0.007

0.016

Thermal Conductivity (W m-1 K -1 )

-1 Thermal Conductivity (W m K

-1

)

0.018

0.014 0.012 0.01 0.008 0.006 0.004 Glass Beads 90-106 μm Glass Beads 710-1000 μm Regolith Simulant

0.002 0 0

5

10

15

20

Compressional Stress (kPa) Fig. 7. Effective thermal conductivity of glass beads and regolith simulant as a function of the compressional stress (average of the horizontal and vertical stresses).

0.006 0.005 0.004 0.003 0.002

Data: Glass Beads 90-106 μm Fitting: Glass Beads 90-106 μm Data: Glass Beads 710-1000 μm Fitting: Glass Beads 710-1000 μm Data: Regolith Simulant Fitting: Regolith Simulant

0.001 0 240

260

280

300

320

340

Temperature (K) Fig. 8. Thermal conductivity of the glass beads and the regolith simulant as a function of the temperature. Three curves represent fitting of Eq. (3) to data of the each sample.

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N. Sakatani et al. / Icarus 267 (2016) 1–11

The solid conductivities of the compressed samples can be estimated by subtraction of the radiative conductivities at 295 K listed in Table 3 from the measured thermal conductivities shown in Fig. 7. Because of the negligible densification of the samples during the compressional tests, we assumed that the radiative conductivity maintains the constant values independent of the compressional stress. Fig. 9 shows the resultant solid conductivity as a function of compressional stress. The solid conductivity deduced from this method depends on the adopted radiative conductivity. By assuming the fitting errors in B values, or errors in the calculated krad at 295 K, as possible ranges in the radiative conductivity of each sample, the upper and lower possible limits of the solid conductivities can be estimated by the subtraction of the lower and upper limits of the radiative conductivities, respectively. In Fig. 9, the possible ranges in the solid conductivity are shown as vertical error bars. We assume the following exponential relation between the solid conductivity and the compressional stress,

ksolid ¼ k0 r x ;

ð8Þ

where r is the compressional stress in kPa, k0 and x are fitting variables. Note that this equation implies that the solid conductivity approaches zero, if no compressional stress including the selfweighted stress is applied. Fitting of Eq. (8) to the nominal values of the solid conductivity in Fig. 9 (deduced from the subtraction of the best estimated values of the radiative conductivity) yields k0 ¼ 0:00263 and x ¼ 0:350 for the 90–106 lm glass beads, k0 ¼ 0:00494 and x ¼ 0:285 for the 710–1000 lm glass beads, and k0 ¼ 0:00595 and x ¼ 0:180 for the regolith simulant. The same fitting to the upper limits of the solid conductivity (upper ends of the vertical error bars in Fig. 9) yielded higher k0 and lower x values. In contrast, the compressional

Solid Conductivity (W m-1 K-1)

0.012 0.01 0.008 0.006 0.004 0.002 Data Fitting of Eq. (8)

0

0

5

10

15

20

Compressional Stress (kPa )

(b) Glass Beads 710-1000 µm 0.014

Solid Conductivity (W m-1 K -1)

4.3. Compressional stress dependence of solid conductivity

(a) Glass Beads 90-106 µm 0.014

0.012 0.01 0.008 0.006 0.004 0.002 Data Fitting of Eq. (8)

0

0

5

10

15

20

Compressional Stress (kPa )

(c) Regolith Simulant 0.014 0.012

Solid Conductivity (W m-1 K-1)

ulant. The fitting curves are also shown in Fig. 8. The values of A and B in Eq. (3) are shown in Table 3. The radiative conductivity or the coefficient B of the 710– 1000 lm glass beads was higher than that of the 90–106 lm glass beads. It has been accepted experimentally and theoretically that the radiative conductivity increases with particle size (Watson, 1964; Merrill, 1969), since radiative energy transfer through the larger void spaces between the larger particles is more efficient. The regolith simulant had a radiative coefficient comparable to that of the 90–106 lm glass beads within the margins of the error. This indicates that typical void sizes of the regolith simulant and the 90–106 lm glass beads are comparable. The value of B for the regolith simulant gives an upper bound of those of the Apollo regolith samples (e.g. Cremers, 1971, 1972), although the solid conductivity was assumed to be independent of the temperature for determining the radiative conductivity of the Apollo samples. Using the values of B, we can estimate the radiative conductivities of the three samples at any temperature. The radiative conductivities at 295 K, at which the stress-controlled experiments were conducted, are listed in Table 3. Change in the radiative conductivity by possible densification of 3% during the compression as mentioned earlier would be negligible according to a model by Gundlach and Blum (2012) (Eq. (15)).

0.01 0.008 0.006 0.004 0.002 Data Fitting of Eq. (8)

0

0

5

10

15

20

Compressional Stress (kPa ) Fig. 9. Solid conductivity as a function of the compressional stress for (a) 90– 106 lm glass beads, (b) 710–1000 lm glass beads, and (c) regolith simulant. These solid conductivities are attained by subtracting the radiative conductivity (Table 3) from the measured effective thermal conductivity in Fig. 7. Solid curves represent fitting results of Eq. (8).

Table 3 Solid and radiative coefficients in Eq. (2) of the glass beads and the regolith simulant. Sample

A (10
3 )

ksolid at 295 K (10
3 W m
1 K
1)

B (10
11 W m
1 K
4)

krad at 295 K (10
3 W m
1 K
1)

Glass beads 90–106 lm Glass beads 710–1000 lm Regolith simulant

1.34 ± 0.05 1.89 ± 0.51 1.67 ± 0.05

1.48 ± 0.06 2.09 ± 0.56 3.54 ± 0.11

2.76 ± 0.27 13.9 ± 2.62 3.23 ± 0.40

0.71 ± 0.07 3.57 ± 0.67 0.83 ± 0.10

N. Sakatani et al. / Icarus 267 (2016) 1–11


rc ¼

3 1
m2 FRp 4 E

1=3 ;

ð9Þ

where Rp is particle radius, m and E are Poisson’s ratio and Young’s modulus of the solid material, respectively (Timoshenko and Goodier, 1951). If the stress r is applied on powders, the force F is proportional to cross section of the particle as, F / R2p r (the proportionality constant differs by packing structure of spheres), and we obtain,

r c / Rp r1=3 :

ð10Þ

Thus, the contact radius between the spheres is proportional to the compressional stress with an exponent of 1/3. For the two glass beads, we found that the solid conductivity is depends on the compressional stress with an exponent of 0.285–0.350, close to 1/3. The consistency of the 1/3 power law of the contact radius and the solid conductivity implies that the solid conductivity is proportional to the contact radius. This is theoretically supported by Chan and Tien (1973), who showed that the thermal conductance between two contacting spheres is proportional to the radius of the contact spot. Pilbeam and Vaišnys (1973) stated that the compressional stress dependence of the solid conductivity of powdered materials with wide range of the particle size and irregular particle shapes, such as the regolith simulant we measured, is greater than that of uniform-sized spheres packing, since not only the interparticle contact size but also the number of the contacts can increase by the compressional stress. However, this feature was not found for the regolith simulant. One of the causes of the weaker compressional stress dependence for the regolith simulant is that the solid conductivity had a finite value even if the compressional stress infinitesimally reduces to zero. It is known that lmsized small particles can make contacts with finite contact area by inter-particle adhesive force even without the external compressional force (Johnson et al., 1971). The solid conductivity attributing only the adhesive force could be the reason for the offset value at compression zero in Fig. 9(c). Its contribution cannot be estimated since the adhesive force and resulting contact size are unpredictably affected by particle size distribution, macroscopic particle shape, or microscopic roughness on individual particles (Perko et al., 2001). Its possible maximum value for the regolith simulant would be 3:54  10
3 W m
1 K
1, which was obtained from the uncompressive temperature-controlled experiment (Table 3). When we use this solid conductivity as an offset value, the resulting best-fit k0 and x are 2:65  10
3 and 0.352, and their possible ranges are 2:55  10
3 –2:76  10
3 and 0.343– 0.362, respectively.

Table 4 Values of k0 and x of Eq. (8). Sample

k0 (10
3 ) Nominal

Glass beads 90–106 lm 2.63 Glass beads 710–1000 lm 4.94 Regolith simulant 5.95

5. Vertical thermal conductivity distribution in lunar regolith layer As mentioned in Section 1, the thermal conductivity of the lunar subsurface regolith has not been verified by laboratory experiments and/or theoretical modeling. The self-weighted compressional stress is an essential parameter to constrain vertical structure of the thermal conductivity of a planetary regolith layer. In this section, we estimate the thermal conductivity distribution in depth direction in the lunar regolith layer using our experimental results for the lunar regolith simulant. We assume that the compressional stress at a given depth of the regolith layer is represented by lithostatic pressure. Bulk density of the lunar regolith varies with depth. Core samples collected in Apollo missions gave the following empirical hyperbolic relation between density and depth (Carrier et al., 1991),

qðzÞ ¼ 1920

z þ 0:122 ; z þ 0:18

ð11Þ

where q is the density in kg m
3, and z is the depth in meters. The lithostatic pressure is calculated as,

Z

rðzÞ ¼

z 0

qðz0 Þg dz0

ð12Þ

where g (=1.6 m2 s
1) is gravitational acceleration at the lunar surface. Porosity distribution can be calculated as uðzÞ ¼ 1
qðzÞ=qs , where qs is the true density of the regolith materials. We use the true density of qs ¼ 3100 kg m
3 as Carrier et al. (1991) recommended. The estimated stress and porosity distributions in the lunar regolith layer along the depth are shown in Fig. 10. In the following discussion, we assume that thermophysical properties, beside the true density, of the lunar surface regolith are the same as those of the lunar regolith simulant used in this study. A simple model of the lunar thermal conductivity is directly deduced from our experimental results as,

k¼

km ðTÞ k0 r x þ BT 3 ; km ðT 0 Þ

ð13Þ

where km ðTÞ is given by Eq. (5), k0 ¼ 5:95  10
3 , x = 0.180, and B ¼ 3:23  10
11 are experimentally determined constants for the lunar regolith simulant, and T 0 = 295 K. A factor of km ðTÞ=km ðT 0 Þ in the first term of the right hand is for the correlation of the temperature effect on the solid conductivity. Note that r in this

10

Compressional Stress Porosity

0.6

8

0.55

6

0.5

4

0.45

2

0.4

Porosity

stress dependences of the lower limits of the solid conductivity of these samples can be expressed by lower k0 and higher x values. These fitting results are shown in Table 4. We found that the exponent x of the regolith simulant was lower than those of the two glass beads. If two spheres are compressed by an external normal force, the radius of the circular contact area between the spheres follows the Hertzian elastic theory, which predicts the following expression for the contact radius rc ,

Compressional Stress (kPa)

8

x (Range)

Nominal (Range)

(2.57–2.70) 0.350 (4.26–5.62) 0.285 (5.85–6.06) 0.180

(0.344–0.356) (0.259–0.317) (0.177–0.182)

0

0

0.5

1

1.5

2

0.35 2.5

Depth (m) Fig. 10. Compressional stress (or lithostatic pressure) and porosity distributions in the lunar regolith.

9

N. Sakatani et al. / Icarus 267 (2016) 1–11

equation has an unit of kPa. This equation is applicable to the regolith with the porosity of 0.43 (Table 2). The porosity of the lunar regolith decreases with the depth (Fig. 10), which also affects the thermal conductivity distribution in addition to the effect of the compressional stress. The solid conductivity increases with decreasing the porosity. Fountain and West (1970) measured the thermal conductivity of crushed basaltic powders (37–62 lm in diameter) under vacuum condition with changing the density from 790 to 1500 kg m
3. By measuring the temperature dependence of the thermal conductivity of each sample, they estimated the solid and radiative conductivities in a similar way as applied in this study. Fig. 11 shows their results on the solid conductivity as a function of the porosity. We assumed the true density of 2990 kg m
3 (the same as our basaltic regolith simulant) for calculating the porosity from the bulk density, since we cannot find the true density of the basalt powders they used. Fitting of an exponential function to their data shown in Fig. 11 yields the following empirical relationship between the solid conductivity and porosity,

ksolid / expð
5:02uÞ:

ð14Þ

This dependence agrees with a model by Gundlach and Blum (2012). Theoretical works predicted that the radiative conductivity increases with porosity, because void size between the particles becomes larger for the higher porosity. In this work, a theoretical expression by Gundlach and Blum (2012) for the porosity dependence of the radiative conductivity as,

krad / u=ð1
uÞ;

ð15Þ

is applied. Although some different porosity dependences were suggested (see Merrill, 1969), the choice does not significantly affect the thermal conductivity of the lunar regolith underground, since the solid conductivity is the dominative contribution. Using the porosity dependences of the solid and radiative conductivities of Eqs. (14) and (15) and our empirical model of Eq. (13), the porosity-dependent solid and radiative conductivity models for the lunar regolith are given as

ksolid ðu; T; rÞ ¼

expð
5:02uÞ km ðTÞ k0 r x ; expð
5:02u0 Þ km ðT 0 Þ

ð16Þ

and

u=ð1
uÞ 3 BT ; u0 =ð1
u0 Þ

ð17Þ 0.012

Thermal Conductivity (W m-1 K -1)

krad ðu; TÞ ¼

0.01 Fountain and West (1970) k solid = 0.0202 exp (-5.02 φ)

Solid Conductivity (W m-1 K -1)

respectively, where u0 ¼ 0:43. The effective thermal conductivity is given as the sum of Eqs. (16) and (17). Using the above thermal conductivity model (Eqs. (16) and (17)), we can estimate the thermal conductivity of the lunar regolith in terms of the compressional stress, porosity, and temperature. In Fig. 12, thermal conductivity distribution in the lunar regolith layer (Model 1) is calculated using the porosity and compressional stress distributions shown in Fig. 10. Our model predicts that the thermal conductivity raises to 0.0085 W m
1 K
1 at the depth of 1 m. For comparison, a model with a constant compressional stress of 0.1 kPa (corresponding to the lunar depth of 5 cm) is shown in Fig. 12 (Model 2). The thermal conductivity increase in depth direction is due to only the effect of porosity. This model has a thermal conductivity of 0.0049 W m
1 K
1 at 1 m depth. Comparing models 1 and 2, it is found that the effect of the compressional stress has great importance for controlling the thermal conductivity distribution in the lunar regolith layer. A model with a constant porosity of 0.43 and with the compressional stress being the only a variable is also plotted (Model 3). Effect of the porosity on the thermal conductivity at 1 m depth is less than 0.001 W m
1 K
1. Thus, the porosity (or density) variation is a secondary factor for determining the thermal conductivity of the lunar subsurface regolith. In Fig. 13, our model (Model 1 in Fig. 12) is compared to the insitu measurements in the Apollo Heat Flow Experiments (Langseth et al., 1973, 1976). We found that the modeled thermal conductivity is lower than any in-situ measurements. The heater activated measurement values (Langseth et al., 1973) are higher by several factor than our model, which is interpreted as the result of the regolith compression near the borestems during the drilling process. If the regolith was densified up to 2000 kg m
3 by the drilling, the compressional stress of 10–600 kPa is needed to enhance the thermal conductivity to the in-situ measurement values (0.014– 0.03 W m
1 K
1). The annual wave measurements (Langseth et al., 1976) are also higher than our model. The relative difference is larger at shallower depth. One of the possible reasons is the heat penetration from the lunar surface through the borestems and/or heat flow probes themselves. However, Keihm and Langseth (1973) predicted the conductivity around 0.01 W m
1 K
1 below 10 cm depth, consistent with the results of Langseth et al. (1976). Another examination would be that the near-surface density distribution is not accurately described by the model by Carrier

0.001

0.01 0.008 0.006 0.004 0.002 0

Model 1; σ and φ = variable Model 2; σ = 0.1 kPa, φ = variable Model 3; σ = variable, φ = 0.43

0

0.5

1

1.5

2

2.5

Depth (m) 0.0001 0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

Porosity Fig. 11. Solid conductivity of basalt powders of 37–62 lm in diameter as a function of the porosity. These data are from Fountain and West (1970). For calculating the porosity from the bulk density, the true density of 2990 kg m
3 was assumed.

Fig. 12. Model 1 is the estimated thermal conductivity distribution in the lunar regolith from Eqs. (16) and (17) with the stress and porosity distributions of Fig. 10. Model 2 and 3 are the calculated thermal conductivity with compressional stress and porosity being constant values of 0.1 kPa and 0.43, respectively. The temperature is fixed at 250 K as a typical temperature of the lunar regolith underground (Langseth et al., 1976).

10

N. Sakatani et al. / Icarus 267 (2016) 1–11

Thermal Conductivity (W m -1 K -1)

0.03 0.025 0.02 0.015 0.01 Model Model with K 0 = 0.25 (lower) and 5.0 (upper) Apollo heater activated measurements Apollo annual wave measurements

0.005 0

0

0.5

1

1.5

2

2.5

Depth (m) Fig. 13. Comparison between our thermal conductivity model (Model 1 in Fig. 12) and in-situ measurements in Apollo Heat Flow Experiments. Lower and upper gray curves are given with coefficient of lateral stress K 0 being 0.25 and 5.0, respectively (see text in detail). Data of the heater activated measurements are from Langseth et al. (1973), and the annual wave measurements are from Langseth et al. (1976) and Grott et al. (2010).

Our model Vasavada et al. (2012)

0.008

reff ¼

0.006

2000

-3

Density (kg m )

-1

-1 Thermal Conductivity (W m K )

0.01

0.004

0.002

1600 1400 1200

0

0.5

1

1.5

2

0.5

1

3

¼

1 þ 2K 0 3

Z 0

z

qðz0 Þg dz0 :

ð18Þ

2.5

6. Summary

Depth (m)

0

rv þ 2r h

As an example for the effect of K 0 on the lunar regolith thermal conductivity, the models with K 0 = 0.25 and 5.0 are shown in Fig. 13. If K 0 ¼ 5:0, the thermal conductivity becomes 25% higher than for the model with K 0 ¼ 1:0. Therefore, we suggest that K 0 of the lunar subsurface regolith layer is also one of the important parameters controlling the thermal conductivity.

1800

1000

0

It should be noted that we assumed that the stress field in the lunar subsurface regolith is represented by lithostatic pressure. However, the stress could be heterogeneous in vertical and horizontal directions, as seen in our compressive experiment (Fig. 6). Its heterogeneity is represented by the coefficient of lateral stress, K 0 , defined by the ratio of horizontal stress rh to vertical stress rv . In powdered media infinitely extended in horizontal directions, the vertical stress is expressed as lithostatic pressure. The above analysis implicitly assumed K 0 = 1. In reality the value of K 0 depends on history of regolith compression on the lunar surface. If the regolith is normally consolidated, which means that the maximum stress in the past applied on the regolith layer is represented by the present stress or self-weight, K 0 is estimated to be lower than unity. In this study, we obtained K 0 = 0.25 for the regolith simulant (see Section 4.1). In this case, the compressional stress effective on the thermal conductivity would be lower than the lithostatic pressure. On the other hand, when the maximum past stress applied on the regolith was higher than the present stress (such a state is called over-consolidated), K 0 possibly becomes higher than unity an may even reach values of 3–5 (Carrier et al., 1991). Therefore, the effective compressional stress can be higher than the lithostatic pressure, and correspondingly the thermal conductivity also would become higher. Unfortunately, the K 0 value of the lunar regolith has not been measured in-situ and in the laboratory. A simple expression on the compressional stress effective on the thermal conductivity is given as the average of the triaxial stress values as,

1.5

2

2.5

Depth (m) Fig. 14. Thermal conductivity profile of lunar regolith layer using a density model given by Vasavada et al. (2012) (inset graph at the right bottom). The dashed curve represents a thermal conductivity profile estimated by Vasavada et al. (2012).

et al. (1991) of Eq. (11). The higher density at the near-surface could make the conductivity higher. Indeed, the density model of Keihm and Langseth (1973) showed more rapid density increase with depth than the Carrier’s model we used. Vasavada et al. (2012) estimated the thermal conductivity distribution of the near-surface regolith at equatorial region. Their density model is expressed as, qðzÞ ¼ 1800
500 expð
z=0:06Þ. Using this density model, we can also estimate the thermal conductivity distribution by the same manner as done above. Fig. 14 shows the comparison between our thermal conductivity model using the Vasavada’s density model and the conductivity model estimated by Vasavada et al. (2012). The thermal conductivity is consistent with each other; the difference is less than 0.002 W m
1 K
1. The most notable difference appeared in this figure is that our model predicts gradual increase in the thermal conductivity even below the depth where the density is saturated at a maximum value (1800 kg m
3 for their model). This is the contribution of the self-weighted compressional stress on the thermal conductivity that we verified in this study. Thus, the compressional stress is one of the key parameter for the theoretical modeling of the thermal conductivity of lunar subsurface regolith.

In this study, we measured the thermal conductivity of two glass beads samples and lunar regolith simulant with varying applied compressional stress up to 20 kPa, equivalent to about 6.5 m depth in the lunar regolith layer. By combining the radiative conductivity determined from the temperature-dependent thermal conductivity data of the same samples, the compressional stress dependence of the solid conductivity was quantified. We found that the solid conductivity increases with the compressional stress, and it is proportional to approximately 1/3 power of the stress for the glass beads. The value of 1/3 is supported by the combination of Hertzian contact theory and thermal conductance theory of the contacting bodies. The experimental results for the regolith simulant showed weaker compressional stress dependence of the solid conductivity than the glass beads. This indicates that additional contact forces other than the externally applied forces may act between the regolith simulant particles. Using the experimental data on the regolith simulant and a theoretical work of the porosity dependence of the solid and radiative conductivities, a semi-empirical model for the thermal conductivity of the lunar regolith was developed. Considering the density distribution in the lunar regolith and lunar surface gravitational acceleration, we estimated the thermal conductivity distribution along the depth. This is the first study that estimates the subsurface thermal conductivity on the ground of the experimental evidence. Our model predicted that the thermal conductivity gradually increase with the depth and that it is lower than the value estimated by the Apollo 15 and 17 Heat Flow Experiments

N. Sakatani et al. / Icarus 267 (2016) 1–11

(Langseth et al., 1973, 1976). However, the stress field in the lunar regolith layer is uncertain and depends on the K 0 value, or history of compression and relaxation of the regolith by impact cratering processes, micrometeorite bombardments, and seismic shaking. Future in-situ or laboratory measurements to determine K 0 will help us to understand the thermal conductivity distribution in the lunar regolith. Note that our model was based on experimental data for the lunar regolith simulant. The data sets of the thermal conductivity of the actual lunar regolith samples as a function of compressional stress by our experimental technique will produce more feasible models. Acknowledgments The authors appreciate Dr. Rie Honda for fruitful discussions. We thank to Dr. Taizo Kobayashi for the discussion on mechanical properties of the lunar regolith. The authors are grateful to Dr. Matthew Siegler and an anonymous reviewer for their constructive comments on this paper. N.S. is supported by a research fellowship from the Japan Society for the Promotion of Science (JSPS) for Young Scientists (266845). Appendix A. Error analysis Error of the thermal conductivity measurement by the line heat source method was deduced in accordance with a method proposed by Presley and Christensen (1997b). If all human errors are assumed to be random, the maximum relative error of the thermal conductivity can be deduced from Eq. (7) as,

      dI dR ds dk ¼ 2  þ   þ  : k I R s

ðA:1Þ

Uncertainty of the electric current is determined by performance of the constant current source. When the induced current is 40 mA, dI=I ¼ 0:05%, and dI=I ¼ 0:03% when I ¼ 20 mA. They are negligible compared to the other errors below. We determined electrical resistance of the nichrome wire per unit length, R, by measuring the resistance as a function of the length (at room temperature of 298 K) and fitting a linear function to a plot of the resistance versus length. The slope of the linear function corresponds to R, whose fitting error was 0.24%. In addition, the resistance can vary with the temperature. Temperature increase of the nichrome wire during the line heat source measurements also causes the error. If the temperature difference of DT was appeared between the beginning and end of the measurement, the relative change of the resistance can be evaluated as dR=R ¼ bDT, where b ¼ 9  10
5 K
1 is temperature coefficient of the electrical resistance. In our experiments, DT was less than 10 K, which causes the maximum relative error of 0.09%. Adding the measurement uncertainty and its relative change during the measurements, the total relative error of the resistance is less than 0.33%. The error in the slope s has two sources. One is the deviation of the data from a straight line. It can be evaluated as the fitting error. The other is how well the linear regime of the curve of temperature versus natural logarithm of time is resolved. Presley and Christensen (1997b) suggested a way to verify this error, by dividing the linear regime of the curve into smaller segments and comparing their slopes with the original one. We divided the linear regime (400–1000 s) into three segments (400–600, 600–800, and 800–1000 s), so that the error can be evaluated as,

    3 ds 1 X sn
s ¼  ; s 3  n¼1 s 

ðA:2Þ

11

where s is the original slope, and sn (n = 1, 2, 3) is the slope of the each segment. The total error in the slope is given as the sum of the two errors. In most cases of our experiments, the slope’s error was the largest one of the three sources in Eq. (A.1). References Carrier, W.D., Olhoeft, G.R., Mendell, W., 1991. Physical properties of the lunar surface. In: Heiken, G.H., Vaniman, D.T., French, B.M. (Eds.), Lunar Sourcebook, A User’s Guide to the Moon. Cambridge University Press (Chapter 9). Carslaw, H.S., Jaeger, J.C., 1959. Conduction of Heat in Solid, second ed. Oxford University Press, London. Chan, C.K., Tien, C.L., 1973. Conductance of packed spheres in vacuum. J. Heat Transfer 95 (3), 302–308. Cremers, C.J., 1971. Density, pressure, and temperature effects on heat transfer in Apollo 11 fines. AIAA J. 9 (11), 2180–2183. Cremers, C.J., 1972. Thermophysical properties of Apollo 12 fines. Icarus 18, 294– 303. Cremers, C.J., Birkebak, R.C., Dawson, J.P., 1970. Thermal conductivity of fines from Apollo 11. In: Proceedings of the Apollo 11 Lunar Science Conference, vol. 3, pp. 2045–2050. Desai, P.D. et al., 1974. Thermophysical Properties of Selected Rocks. Tech. Rep., Center for Information and Numerical Data Analysis and Synthesis, Purdue University, Indiana. Fountain, J.A., West, E.A., 1970. Thermal conductivity of particulate basalt as a function of density in simulated lunar and martian environments. J. Geophys. Res. 75 (20), 4063–4069. Garrett, D., Ban, H., 2011. Compressive pressure dependent anisotropic effective thermal conductivity of granular beds. Granul. Matter 13, 685–696. Grott, M., Knollenberg, J., Krause, C., 2010. Apollo lunar Heat Flow Experiment revisited: A critical reassessment of the in-situ thermal conductivity determination. J. Geophys. Res. 115, E11005. Gundlach, B., Blum, J., 2012. Outgassing of icy bodies in the Solar System – II: Heat transport in dry, porous surface dust layers. Icarus 219, 618–629. Halajian, J.D., Reichman, J., 1969. Correlation of mechanical and thermal properties of the lunar surface. Icarus 10, 179–196. Horai, K., 1981. The effect of interstitial gaseous pressure on the thermal conductivity of a simulated Apollo 12 lunar soil sample. Phys. Earth Planet. Interiors 27, 60–71. Huetter, E.S., Koemle, N.I., 2012. Performance of thermal conductivity probes for planetary applications. Geosci. Instrum. Methods Data Syst. 1, 53–75. Johnson, K.L., Kendall, K., Roberts, A.D., 1971. Surface energy and the contact of elastic solids. Proc. R. Soc. London, Ser. A, Math. Phys. Sci. 324, 301–313. Keihm, S.J., Langseth, M.G., 1973. Surface brightness temperature at the apollo 17 heat flow site: Thermal conductivity of the upper 15 cm of regolith. In: Proceedings of the Fourth Lunar Science Conference, vol. 3, pp. 2503–2513. Langseth, M.G. et al., 1972. Heat-flow experiment. In: Apollo 15 Preliminary Science Report (Chapter 11). Langseth, M.G., Keihm, S.J., Chute, J.L., 1973. Heat-flow experiment. In: Apollo 17 Preliminary Science Report (Chapter 9). Langseth, M.G., Ruccia, F.E., Wechsler, A.E., 1974. Thermal conductivity of evacuated glass beads: Line source measurements in a large volume bead bed between 225 and 300 K. Proc. Heat Transm. Meas. Therm. Insul. 1, 256–274. Langseth, M.G., Keihm, S.J., Peters, K., 1976. Revised lunar heat-flow values. In: Proceeding of the Seventh Lunar Science Conference, pp. 3143–3171. Merrill, R.B., 1969. Thermal Conduction through an Evacuated Idealized Powder Over the Temperature Range of 100° to 500 °K. Tech. Rep., National Aeronautics and Space Administration, Washington, D.C. Perko, H.A., Nelson, J.D., Sadeh, W.Z., 2001. Surface cleanliness effect on lunar soil shear strength. J. Geotech. Geoenviron. Eng. 127, 371–383. Pilbeam, C.C., Vaišnys, J.R., 1973. Contact thermal conductivity in lunar aggregates. J. Geophys. Res. 78 (23), 5233–5236. Piqueux, S., Christensen, P.R., 2009. A model of thermal conductivity for planetary soils: 1. Theory for unconsolidated soils. J. Geophys. Res. 114, E09005. Presley, M.A., Christensen, P.R., 1997a. Thermal conductivity measurements of particulate materials: 1. A review. J. Geophys. Res. 102 (E3), 6535–6549. Presley, M.A., Christensen, P.R., 1997b. Thermal conductivity measurements of particulate materials: 2. Results. J. Geophys. Res. 102 (E3), 6551–6566. Sakatani, N., Ogawa, K., Iijima, Y., Honda, R., Tanaka, S., 2012. Experimental study for thermal conductivity structure of lunar surface regolith: Effect of compressional stress. Icarus 221, 1180–1182. Timoshenko, S.P., Goodier, J.N., 1951. Theory of Elasticity. McGraw-Hill Book Company, Inc., New York. Vasavada, A.R. et al., 2012. Lunar equatorial surface temperatures and regolith properties from the diviner lunar radiometer experiment. J. Geophys. Res. 117, E00H18. Watson, K., 1964. I. The Thermal Conductivity Measurements of Selected Silicate Powders in Vacuum from 150°–350 °K, II. An Interpretation of the Moon’s Eclipse and Lunation Cooling as Observed through the Earth’s Atmosphere from 8–14 lm. Ph.D. Thesis, California Institute of Technology, Pasadena, California. Wechsler, A.E., Glaser, P.E., Fountain, J.A., 1972. Thermal properties of granulated materials. In: Lucas, J.W. (Ed.), Thermal Characteristics of the Moon, Progress in Astronautics & Aeronautics, vol. 28. MIT Press (Chapter 3a).