Apollo lunar heat flow experiment revisited: A critical reassessment of the in situ thermal conductivity determination

Authors


Abstract

[1] Lunar heat flow was determined in situ during the Apollo 15 and 17 missions, but some uncertainty is connected to the value of the regolith's thermal conductivity, which enters as a linear factor into the heat flow calculation. Different approaches to determine the conductivity yielded discordant results, which led to a downward correction of the obtained heat flow values by 30%–50% subsequent to the publication of the first results. We have reinvestigated likely causes for the observed discrepancies and find that neither poor coupling between the probe and regolith nor axial heat loss can explain the obtained results. Rather, regolith compaction and compression likely caused a local increase of the regolith's thermal conductivity by a factor of 2–3 in a region which extends at least 2–5 cm from the borehole wall. We conclude that the corrected lunar heat flow values, which are based on thermal diffusivity estimates sampling a large portion of undisturbed regolith, represent robust results. Future in situ measurements of regolith thermal conductivity using active heating methods should take care to both minimize regolith disturbance during probe emplacement and maximize heating time to obtain reliable results. We find that for the Apollo measurements, heating times should have exceeded at least 100 h, and ideally 200 h.

1. Introduction

[2] Lunar heat flow has been measured at the Hadley Rille and Taurus-Littrow sites during the Apollo 15 and 17 missions and values of 21 and 16 mW m−2 have been obtained [Langseth et al., 1972a, 1972b, 1973, 1976]. Uncertainties for these values are given at ±15%, which mainly stem from the uncertainty connected to the regolith's thermal conductivity, although no rigorous error analysis was presented. Together with independent estimates based on tectonic [Pullan and Lambeck, 1980] and magnetic [Hood et al., 1982] data, these measurements represent the basis for estimates of the current thermal state of the Moon. Correcting for local heat flow focusing effects, Rasmussen and Warren [1985] and Warren and Rasmussen [1987] have calculated the globally averaged lunar heat flow, which was found to be 12 mW m−2. From this value, the bulk uranium content of the Moon has been estimated at about 20–21 ppb [Warren and Rasmussen, 1987].

[3] However, some uncertainty is connected to the obtained heat flow values. First of all, a long-term subsurface temperature drift, which was originally thought to stem from astronaut induced perturbances [Langseth et al., 1976], was found to persist over the entire duration of the mission and did not develop into a new equilibrium thermal state as expected [Wieczorek and Huang, 2006; Saito et al., 2007]. If this temperature drift substantially influenced the subsurface thermal gradient at the time of the Apollo measurements, then the derived heat flow values could be in question. In addition to astronaut disturbance, different causes for the temperature increase including the 18.6 year precession cycle and topographic effects have been considered [Wieczorek and Huang, 2006; Saito et al., 2008], but so far no conclusion concerning the cause of the perturbation has been reached. Additionally, some uncertainty is connected to inconsistencies concerning the in situ thermal conductivity determination, and thermal conductivity values derived using different methods yielded discordant results. Therefore, some skepticism concerning the merits of the Apollo heat flow measurements exist.

[4] Heat flow probes employed during the Apollo experiments were equipped with eight platinum resistance temperature detectors and four thermocouples, and a schematic sketch of the deployed probes is shown in Figure 1. In addition to the temperature sensors, four active heating elements, which were operated like classical line heat sources [Langseth et al., 1972a, 1973], were placed on the outside of the probes. Names of the sensors in Figure 1 are given according to the following naming convention: The first letter designates the type of sensor, where T stands for resistance temperature detectors (as compared to TC for thermocouples). This is followed by the designation G for a gradient bridge sensor (R for a ring bridge sensor), the number of the probe (1 or 2), the segment of the probe (1 for the top segment, 2 for the bottom segment) and A or B for the upper and lower sensor of the respective bridge arrangement.

Figure 1.

Schematics of the deployed Apollo heat flow probes. The probes consist of two segments, each of which is equipped with four resistance temperature detectors, and a cable equipped with thermocouples. Note that the target depth was not reached during the Apollo 15 experiments. Sensor names are given according to the naming convention explained in the text. Heating elements are situated around the gradient bridge sensors designated TG and are shaded in dark gray. The probes were inserted into predrilled holes, which were fitted with a fiberglass borestem.

[5] Probes were emplaced into predrilled holes which were fitted with a fiberglass borestem. Due to difficulties encountered during the drilling process the target drilling depth of 250 cm was not reached during the Apollo 15 experiments. Only after a mechanical redesign of the borestem could the holes be successfully drilled to the designated depth during the Apollo 16 and 17 missions [Heiken et al., 1991]. Unfortunately, the Apollo 16 experiment did not return any data due to a broken cable connection.

[6] With this setup it was possible to estimate the thermophysical properties of the lunar regolith using four different methods [Langseth et al., 1972a, 1973]: (1) active heating experiments were carried out by operating the heating elements as line heat sources; (2) the thermal re-equilibration of the borestem was monitored as a function of time after initial insertion of the probes; (3) the decay of the periodic temperature perturbations induced by the annual temperature waves was analyzed as a function of depth; and (4) the propagation of Astronaut induced thermal disturbances was evaluated as a function of depth.

[7] Of these approaches, 1 and 2 gave broadly consistent results, with k ranging from 0.0141 to 0.0295 W m−1 K−1, depending on probe location and depth. On the other hand, 3 and 4 also yielded consistent results, but in the range 0.009 to 0.013 W m−1 K−1 (see Table 1 for a compilation of results from methods 1 and 3). Furthermore, using approaches 3 and 4, it was found that the regolith's thermophysical properties vary only little with depth, contrary to the results obtained using methods 1 and 2. It was concluded at that time that the values obtained by methods 3 and 4 were more reliable, because the small volumes of regolith sampled by methods 1 and 2 may have been thermally altered during the drilling process [Langseth et al., 1976].

Table 1. Thermal Conductivities k as a Function of Depth z at the Apollo 15 and 17 Probe 1 and 2 Sitesa
SiteActive HeatingWave Attenuation
z (m)k (W m−1 K−1)z (m)k (W m−1 K−1)
  • a

    A15, Apollo 15; A17, Apollo 17; P1, probe 1; P2, probe 2. Results are displayed as deduced from the active heating experiments [Langseth et al., 1973] as well as the attenuation of the annual temperature wave [Langseth et al., 1976]. For the calculation of thermal conductivities from thermal diffusivities a heat capacity of 670 J kg−1 K−1 has been assumed. Average densities at the Apollo 15 and 17 sites were 1825 kg m−3 and 1960 kg m−3, respectively [Langseth et al., 1973].

  • b

    High uncertainty measurement due to a broken borestem.

A15, P10.350.01410.35–1.380.0107
0.830.0211  
0.910.0160  
1.380.0250  
A15, P20.490.01460.49–0.960.0091
A15, P2b0.960.0243  
A17, P11.300.02500.14–1.850.0132
1.770.0172  
1.850.0179  
2.330.0295  
A17, P21.310.02060.16–1.860.0116
1.780.0236  
1.860.0264  
2.340.0224  

[8] Supporting evidence for low in situ thermal conductivities comes from thermal conductivity measurements of returned samples, which were found to be even lower than the value determined in situ [Cremers, 1975; Horai et al., 1977]. Even if self-compaction was taken into account, thermal conductivities did not exceed 0.01 W m−1 K−1 [Horai, 1981].

[9] However, apart from a possible alteration of thermophysical regolith properties by the emplacement of the probes, other explanations concerning the shortcomings of the active heating experiment have been put forward. These include possible problems concerning the thermal coupling between probe and regolith [Hagermann and Tanaka, 2006] as well as axial heat loss in the borestem, both of which could potentially lead to apparently increased thermal conductivity values.

[10] With the interest in the Moon renewed, new opportunities for the in situ determination of lunar heat flow are arising through missions like the International Lunar Network [Science Definition Team for the ILN Anchor Nodes, 2009], and it is due time to assess the lessons learned from Apollo. Therefore, we will reinvestigate the Apollo active heating experiment data and analyze the plausible causes for the obtained discrepancies. We will show that soil compaction is a likely cause for the observed mismatch and prove this hypothesis by following a three step approach: First, we will build a finite element model of the probe-regolith system to match the Apollo measured temperature variations during the heater-activated experiments in section 2. The robustness of the model with respect to variations of contact resistance between probe and regolith and with respect to uncertainties in the model parameters will be investigated in section 3. It will also be shown that axial heat transfer cannot explain the discrepant results. Second, we will derive reasonable models of the extent of the perturbation region disturbed by the drilling process and estimate the associated conductivity perturbation in section 4.1. This will be done by incorporating laboratory data and soil mechanics constraints. Third, we will repeat the data fitting process in section 4.2 to match the Apollo heater-activated experiment results, but fixing the conductivity values derived for the undisturbed regolith in the model, and solving for the extent and conductivity of the compacted, higher-conductivity material adjacent to the borestem. In this way we will be able to show that feasible ranges of the compacted layer thickness and elevated conductivity could have produced the (erroneous) Apollo results. Finally, we will investigate the influence of heating time on the Apollo results in section 5 and derive measurement requirements that should have been fulfilled by Apollo.

2. Finite Element Model

2.1. Model Setup

[11] To test the different hypotheses concerning the cause for the high thermal conductivity determined from the active heating experiment data we have built a finite element model which captures the main aspects of the Apollo experiment. These include the dimensions of the heater and probe, the finite heat capacity and thermal conductivity of the probe, as well as the finite contact resistance between probe and regolith. We assume that the regolith is in thermal equilibrium prior to the activation of the heaters and effects of changing insolation are disregarded. This is justified because of the small heating times and large emplacement depths of the sensors considered here. The borestem itself is not modeled and its effect on heat exchange between probe and regolith is implicitly taken into account in the contact resistance. Furthermore, we do not include radiative heat transfer in the open borehole above the heaters, as radiative contributions to axial heat transfer are expected to be small. This is due to the small view factor of the probe to regions of differing temperature.

[12] The model is implemented using the commercial finite element package COMSOL Multiphysics. The two dimensional axisymmetric model encompasses the probe, heater and regolith and a sketch of the model setup is given in Figure 2. The model allows for splitting the regolith into a compacted portion close to the probe and an undisturbed portion at greater distance to allow for a study of the influence of soil compaction on the obtained results. A compacted region of regolith will be incorporated into the model in section 4, but for the present discussion we will assume thermal conductivity to be constant in the radial direction. The quasi-cylindrical computational domain extends 50 cm into the vertical and radial directions and the element size was chosen to be 0.2 and 1 cm for elements inside and outside the probe, respectively. In this way, boundary effects are minimized and heat transport inside the probe is well resolved. Note that in the finite element model, the gap indicated by the contact conductance H in Figure 2 is infinitesimally small.

Figure 2.

Model setup for the Apollo active heating experiment in cylindrical symmetry. The heat flow probe is modeled as a cylinder including a heater of length L = 1.7 cm which is thermally coupled to the regolith with a contact conductance H. The regolith is modeled using two layers corresponding to a section of compacted regolith with thermal conductivity kcom close to the probe and the undisturbed regolith with thermal conductivity kreg further away. The heater is supplied with the power q = 0.002 W for 36 h, and the temperature rise at the center of the heater is recorded.

[13] To solve the model, thermophysical properties need to be specified for each domain (heater, probe and compacted/undisturbed regolith) and model temperatures are then calculated by solving the heat conduction equation

equation image

where T is temperature, t is time, r is radius, cp is specific heat capacity, ρ is density, and k is thermal conductivity. Furthermore, the thermal coupling between probe and regolith needs to be prescribed. This is done by defining the contact conductance H between probe and regolith which is given by

equation image

where Tl and Tr are the temperatures on the left and right hand side of the infinitesimally small gap and F is the heat flux between the interfaces. Contact conductance H is given in units of W m−2 K−1.

[14] At the beginning of the simulation the L = 1.7 cm long heater is energized with 0.002 W [Langseth et al., 1972a, 1973] and temperatures at the center of the heater are measured to determine the self-heating curve. The rate of temperature increase at the heater is then inversely proportional to the ability of the ambient medium to conduct the heat and the method is similar to the classical line heat source method [e.g., Presley and Christensen, 1997; Hütter et al., 2008; Hammerschmidt and Sabuga, 2000]. Heating is continued for 36 h, as was done during the Apollo experiments.

[15] Due to the small length to diameter ratio of the applied heater, the measurements cannot be inverted for the thermal conductivity of the ambient regolith using standard methods. Traditionally, active heating methods utilize a long, linear heat source, and the slope of the temperature versus ln(t) curve is then equivalent to Q/4πk, where Q is the heating power per unit length [Carslaw and Jaeger, 1959; Hammerschmidt and Sabuga, 2000]. Instead, we adopt the method developed by Langseth et al. [1972a], who established the empirical relation

equation image

between the temperature rise ΔT at the heater and time t, valid for t > 1000 min. The functional dependence of ΔT on ln(t) is similar to that for a classical line heat source and for a given model setup C1 and C2 are constants. Langseth et al. [1972a] found that C1 and C2 mainly depend on the properties of the probe-borestem system, the thermal conductivity k of the surrounding regolith, and the thermal contact resistance between probe and regolith. Furthermore, Langseth et al. [1972a] concluded that C1 is essentially independent of H, whereas C2 is independent of k. Assuming that the probe thermophysical properties are known, k and H can therefore be determined by fitting of the heating curve for large t.

[16] Curve fitting is then achieved in a two-step process: In a first step the thermal conductivity of the regolith is determined by repeatedly running the model to match the slope of the ΔT versus ln(t) curve at t > 1000 min by adjusting the regolith conductivity k. Then, using the derived best fit conductivity, the contact conductance H is determined by matching the amplitude of the temperature rise. Provided that all heat paths are adequately taken into account, this procedure represents a robust way to invert the self-heating curves for the thermal conductivity of the surrounding regolith.

[17] Complications arise due to the imperfect knowledge of the thermophysical properties of the probe which influence heat transfer in the axial direction. Axial heat transfer has been identified as a major source of error by Langseth et al. [1973] who estimate the uncertainty of their results at 15 percent. However, the thermal properties of the probe are not well documented in the available literature and we need to make simplifying assumptions concerning density, heat capacity and thermal conductivity of the probe/heater system. In this context the most critical parameter is the probe thermal conductivity, which can be estimated from the conductivity of the fiberglass borestem, which Langseth et al. [1972a] give as 0.23 W m−1 K−1. The conductivity of the probe can then be estimated from the fact that thermal gradients in the probe are slightly smaller than those in the surrounding borestem, indicating a slightly increased thermal conductivity of the probe as compared to the borestem [Langseth et al., 1973]. Here we use a thermal conductivity of kProbe = 0.35 W m−1 K−1. For other thermophysical properties standard values have been adopted and these are summarized in Table 2. We choose ρp = ρh = 1650 kg m−3 for the density of the probe and heater and cp,p = cp,h = 750 J kg−1 K−1 for their heat capacity, respectively. For the thermal conductivity of the heater we use the conductivity of Constantan (20 W m−1 K−1). Density and heat capacity of the regolith used in our model are identical to the values used by Langseth et al. [1973], i.e., ρ = 1800 kg m−3 and cp = 670 J kg−1 K−1, as appropriate for lunar regolith at 250 K [Hemingway et al., 1973].

Table 2. Parameters Used in the Finite Element Model
VariablePhysical MeaningValueUnits
cp,Probeheat capacity of the probe750J kg−1 K−1
kProbethermal conductivity of the probe0.35W m−1 K−1
ρProbedensity of the probe1650kg m−3
cp,Heaterheat capacity of the heater750J kg−1 K−1
kHeaterthermal conductivity of the heater20W m−1 K−1
ρHeaterdensity of the heater1650kg m−3
cp,Regolithheat capacity of the regolith670J kg−1 K−1
ρRegolithdensity of the regolith1800kg m−3
qheating power0.002W
Llength of the heater0.017m
dradius of the probe0.01m

2.2. Results

[18] Results of the model runs are compared to data obtained during the Apollo heat flow experiments. These are available in digital form at NASA's National Space Science Data Center [Peters, 2010], but the heating experiment data is fairly noisy. Therefore, we have decided to scan heating curves from the original publications [Langseth et al., 1972b, 1973].

[19] Results obtained for the thermal conductivity k are summarized in Table 3, where k and H are compared to the original results korg, Horg obtained by Langseth et al. [1972b, 1973]. Results obtained in this study agree to within a few percent with the results by Langseth et al. [1972b, 1973], except for the measurement at TG12A, which is off by 23 percent. Best fit values for the contact conductance H are also satisfactorily reproduced, which is slightly surprising given the relatively crude estimates which had to be used for the thermophysical probe properties.

Table 3. Results of Fitting the Self-Heating Curves at the Designated Sensorsa
MissionSensorz (m)k (W m−1 K−1)H (W m−2 K−1)korg (W m−1 K−1)Horg (W m−2 K−1)
  • a

    Emplacement depth of the sensors z, best fit regolith thermal conductivity k, and contact conductance H obtained in this study are given along with the results korg and Horg obtained by Langseth et al. [1973].

A15TG22A0.490.01540.800.01460.5
A15TG12A0.910.01971.080.01600.9
A15TG12B1.380.02471.220.02501.0
A17TG22A1.860.02591.770.02641.5

[20] To test the accuracy of the obtained results with respect to the random noise of the digitized data, we have fitted the heating curves for three different time intervals between 17 h < t < 22 h, 18 h < t < 23 h, and 30 h < t < 35 h. Results were found to be consistent to within 15 percent, such that fitting of the entire curve can be regarded as reliable.

[21] Best fitting curves for the sensors TG22A of the Apollo 15 mission and TG22A of the Apollo 17 mission are given in Figure 3. The modeled heating curves agree extremely well with the data at t > 17 h, emphasizing the validity of the adopted approach. The model even fits the data satisfactorily at smaller times, indicating that the chosen probe thermophysical properties are reasonable.

Figure 3.

(a) Model heating curve measured in the center of the heater as a function of time as compared to the Apollo 15 heating experiment at sensor TG22A. Best fit thermal conductivity k and contact conductance H are 0.0154 W m−1 K−1 and 0.8 W m−2 K−1, respectively. Original fit by Langseth et al. [1973] yielded korg = 0.0146 W m−1 K−1 and Horg = 0.5 W m−2 K−1. (b) Same as Figure 3a but for the Apollo 17 heating experiment at sensor TG22A. Best fit thermal conductivity k and contact conductance H are 0.0259 W m−1 K−1 and 1.77 W m−2 K−1, respectively. Original fit by Langseth et al. [1973] yielded korg = 0.0264 W m−1 K−1 and Horg = 1.5 W m−2 K−1.

3. Axial Heat Transfer

[22] To test the sensitivity of the obtained results to axial heat transfer along the probe we have varied probe thermophysical properties with respect to the nominal values. kProbe and cp,Probe were varied within a factor of two and the results of the computations are shown in Figure 4, where a contour plot of the deviation of the obtained regolith thermal conductivity with respect to the nominal value is given in percent for the TG22A sensor of the Apollo 17 mission.

Figure 4.

Contour plot of the change of the obtained regolith thermal conductivity k with respect to the nominal model in percent as a function of probe heat capacity cp,Probe and thermal conductivity kProbe. Nominal assumed values for probe heat capacity and thermal conductivity are 750 J kg−1 K−1 and 0.35 W m−1 K−1, respectively.

[23] The range of heat capacities considered here covers values from the heat capacity of copper around 385 J kg−1 K−1 to that of most plastics around 1500 J kg−1 K−1 and is centered around the heat capacity of fiberglass around 800 J kg−1 K−1. The thermal conductivity considered is centered around the conductivity of epoxy around 0.3 W m−1 K−1, with most plastics having conductivities of 0.2 to 0.25 W m−1 K−1. The higher thermal conductivities considered take the contributions of the electrical connection lines into account, which in the case of the Apollo heat flow probes were manufactured from Evanohm wire, a material similar to Constantan with a thermal conductivity around 20 W m−1 K−1.

[24] Varying the heat capacity cp,Probe within these bounds changes the obtained thermal conductivity of the regolith by ±10%. The influence of the probe's thermal conductivity is slightly larger and variations of kProbe within the specified bounds lead to regolith conductivity estimates which differ by ±15%–20%. Taken together, these variations result in changes of kreg of <30%.

[25] This implies that even if the thermophysical properties of the probe had not been known better than to within a factor of two the regolith thermal conductivities determined using the active heating experiments would remain unchanged to within 30%. Given that regolith conductivities determined using the different methods differ by more than 55% (compare Table 1), it seems unlikely that an underestimation of axial heat transport accounts for the conductivity differences obtained using the different methods. Furthermore, it can be assumed with some confidence that the thermophysical properties of the heat flow probes were known much better than to within the generous factor of two considered here.

[26] Another factor acting to increase heat transfer in the axial direction is the potentially poor thermal coupling between probe and regolith. A high thermal contact resistance would result in higher probe temperatures, increasing the loss of heat in the axial direction and fitting of the self-heating curves would then result in an apparently increased regolith conductivity. The minimum thermal contact between probe and regolith is given by a purely radiative coupling and the contact conductance can be estimated from

equation image

where σ = 5.67 × 10−8 W m−2 K−4 is the Stefan-Boltzmann constant. At a temperature of 250 K, Hmin ∼ 0.875 W m−2 K−1. We have varied H between 0.65 and 6 W m−2 K−1 and found that the influence on the obtained regolith conductivity is <5%. Therefore, the potentially poor contact between probe and regolith also fails to explain the large differences between the obtained regolith conductivities and other possibilities need to be investigated.

[27] Finally, we have tested whether heat dissipation along the electrical connection wires inside the probe had any significant effect on the results. To estimate this, we assumed that thermal conductance along the probe was solely due to heat conduction along the electrical connections and we calculated the Joule heating along these wires by estimating their electrical resistance and sharing the heating power between the wires and the 1000 Ω heaters energized at 0.002 W. However, Joule heating was found to have a negligible influence.

4. Regolith Compaction

[28] When the discrepancies between obtained regolith conductivities became first apparent, Langseth et al. [1976] concluded that the thermal properties of the small volume of regolith sampled by the heat pulse may have been altered during the drilling process. This would imply that the apparent differences obtained using the different methods indeed reflected different material properties.

[29] Due to lateral heat transport, the amplitudes of the diurnal and annual temperature waves depend on regolith properties in a region which extends to a distance rz, where z is the measurement depth. On the other hand, active heating methods sample regions to the penetration distance

equation image

where κ is the regolith thermal diffusivity and t is the heating time. Given regolith diffusivities of 10−8 m−2 s−1 and a heating time of 35 h, regolith to a distance r ≈ 3.5 cm was sampled during the active heating experiments.

[30] Therefore, regolith disruption and compaction is a plausible explanation for the results obtained in the active heating experiments, but this claim has never been substantiated. In particular, the extent and severity of the alteration have never been quantified, and we will turn to this problem here. To this end we will first develop a model for the thermal conductivity of the compacted region and determine the plausible range of compacted thermal conductivities as well as the extent of the compacted region in section 4.1. In a second step we will then investigate if the amount of regolith disruption estimated from this model is sufficient to explain the increased thermal conductivities derived from the Apollo heating curves. Readers not interested in the details of the compaction model may skip section 4.1 and go directly to section 4.2, where the influence of compaction on the Apollo heating curves will be investigated.

4.1. Compaction Model

[31] Thermophysical properties of the lunar regolith are expected to be a complex function of bulk density, relative density and the stresses exerted on soil particles and we will start off by considering the influence of compaction on regolith conductivity. The lunar regolith is relatively loosely packed in the near-surface layers, but relative density rapidly increases with depth until compaction increases only little below depths of ∼50 cm [Carrier et al., 1973; Carrier, 1974]. On the other hand, Langseth et al. [1976] found that regolith thermal diffusivity was surprisingly uniform over a considerable depth range and given that regolith heat capacity does not vary by any significant amount from sample to sample [Hemingway et al., 1973], Keihm and Langseth [1975] derived a model for the regolith density and thermal conductivity. They assumed that for depths z below 2 cm

equation image

where z is in units of cm. Similarly, conductivity was assumed to be

equation image

where kd = 8.25 × 10−3, ks = 6 × 10−4, and k2 = 3.78 × 10−11 W m−1 K−1 are empirical constants. At constant temperature, thermal conductivity therefore varies with density according to

equation image

Note that this dependence of conductivity on density is in good agreement with the laboratory measurements of Horai [1981], who found that the thermal conductivity of simulated Apollo 12 lunar soil increased from 0.0088 W m−1 K−1 to 0.0109 W m−1 K−1 when the density was increased form 1700 to 1850 kg m−3, i.e., ∂k/∂ρ ≈ 1.4 × 10−5 W m2 K−1 kg−1.

[32] A change of the ambient stress can have large effects on the regolith conductivity without noticeably changing the regolith density. This change of conductivity is caused by increasing the contact area between grains, such that conductive coupling becomes more efficient. The dependence of the thermal conductivity on stress has been analyzed by Pilbeam and Vaisnys [1973] for irregularly packed aggregates and thermal conductivity was found to vary with pressure p according to k(p) ∼ p3/5.

[33] Taking contributions of density increase as well as pressure into account, the thermal conductivity of compacted regolith is therefore given by

equation image

where kreg is the thermal conductivity of the unstressed material, p0 is the ambient pressure (the lithostatic pressure at the respective depth) and ρcom is the compacted density.

[34] Given the porosity ϕ = 1 − ρ/ρs of the regolith, where ρs = 3100 kg m−3 is the specific (mineral) density of the regolith [Carrier et al., 1973; Heiken et al., 1991], the influence of compaction on thermal conductivity can be evaluated. At a depth of 50 cm, the average lunar regolith has a porosity of 0.43, which decreases to 0.38 at 100% relative density. A porosity below this value can only be achieved by crushing of grains, such that a range of porosities between ϕ = 0.34 and ϕ = 0.42 covers loosely packed as well as compacted and crushed material. For ambient pressures between 1 and 5 times the lithostatic pressure and an undisturbed thermal conductivity of 0.01 W m−1 K−1, the plausible range of compacted conductivities kcom then covers the range from kcom = 0.02 W m−1 K−1 for moderately compressed and compacted regolith to kcom = 0.04 W m−1 K−1 for highly compressed/crushed and compacted material.

[35] What can we say about the extent of the compacted region? Given that probe emplacement was achieved using a rotary percussion drill system, we can assume that some fraction of the material was displaced in the radial direction, while another was removed by the drill. If forces acting on the regolith during probe emplacement are assumed to act on the cylindrical surface of the borehole, they will decay as d/r with distance r from the symmetry axis, where d is the radius of the borehole. On the other hand, if forces are primarily exerted on the tip of the drill, pressure will decay as d2/r2 in the radial direction. These two scenarios cover the range of functional dependencies for the pressure decay and will be investigated in the following.

[36] Given the radius of the borehole d, the density of the soil ρ, as well as the maximum compacted density ρmax, and assuming density to fall off in a way similar to the exerted forces, the compaction radius rcom can be computed from mass balance considerations. In case of a d/r decay, the compaction radius is given by

equation image

where f is the fraction of radially displaced material with respect to the material removed from the hole by the rotary percussion drill system. On the other hand, if ρcomd2/r2, compaction radius is given by

equation image

The compacted density is then given by

equation image

where n = 1 for the d/r and n = 2 for the d2/r2 decay, respectively.

[37] Adopting the model for regolith compaction with depth by Carrier et al. [1973] and assuming that little crushing of grains occurs in the compacted region, the minimum porosity of the compacted regolith is 0.38, corresponding to a maximum density of 1920 kg m−3. Figure 5a shows the radius of the compacted region rcom as a function of the fraction of displaced material f for three different depths z and ρcomd/r. Porosity is larger at shallower depth, such that the region affected by compaction becomes increasingly larger at greater depth for the same amount of displaced material. Furthermore rcom increases proportional to the amount of material that has to be displaced. Inspecting Figure 5a, we find that the likely range for the radius of the compacted region spans 2.5 to 6 cm for fractions of displaced material of 30%.

Figure 5.

(a) Radius of the compacted region close to the borestem as a function of the fraction of displaced material f for three different depths z, each corresponding to a different undisturbed porosity ϕ. Maximum compaction is assumed to correspond to 100% relative density. Stresses and compacted densities are assumed to decay as d/r. (b) Same as Figure 5a but for stresses and compacted densities which decay as d2/r2.

[38] Figure 5b shows the radius of the compacted region for the case ρcomd2/r2. Due to the faster decay of density with distance from the borehole, the compacted region needs to extend further from the borehole wall than in the case ρcomd/r. At shallow to intermediate depth the radius of the compacted region is between 2 and 4 cm for fractions of radially displaced material around 15%. At larger depth, where packing is already very dense to begin with, rcom can easily exceed 10 cm for f = 0.15.

[39] To induce plastic deformation, stresses at r = rcom need to exceed the shear strength of the material. This places constraints on the pressure distribution in the soil and therefore on the pressure contribution to the thermal conductivity increase. Lunar regolith at high relative density can withstand shear stresses in excess of a factor of 2 to 3.5 with respect to the ambient pressure before deforming plastically [Heiken et al., 1991] and assuming pressure to drop off as dn/rn we find that

equation image

where n = 1, 2 is the decay exponent, pmax is defined such that p = 2.5p0 at r = rcom, and p0 is the ambient pressure.

[40] For simplicity, we have setup the finite element model such that a single compacted conductivity needs to be prescribed for the compacted region. Therefore, we need to determine the average compacted conductivity, which is given by the harmonic mean of the individual compacted layers [Beardsmore and Cull, 2001]:

equation image

Average undisturbed conductivities at the two sites considered here range from kreg = 0.0091 to 0.0116 W m−1 K−1, and the associated uncertainty is given at 15% [Langseth et al., 1976]. Taking the best fit values as a basis for further analysis, we find that most likely average compacted thermal conductivities range from 0.021 to 0.026 W m−1 K−1 for ρcomd/r. For ρcomd2/r2, equation imagecom is found to range from 0.025 to 0.031 W m−1 K−1. In the following, we will drop the bar over equation imagecom for readability such that kcom refers to the average compacted conductivity in the compacted region. Note that the higher average thermal conductivities obtained for the model assuming faster pressure decay are caused by the increased pressure contribution to the thermal conductivity: In order to exceed the shear strength at the larger compacted radius rcom, models assuming ρcomd2/r2 need to have larger pmax.

4.2. Extent of the Compacted Region

[41] As demonstrated in section 4.1, the region affected by compaction can be expected to reach 2–10 cm from the borehole wall and its extent depends on the fraction of displaced material f with respect to the material removed from the hole by the rotary percussion drill system. Compacted thermal conductivities range from 0.021 to 0.031 W m−1 K−1 on average and we now move on to quantify the amount of compaction compatible with the Apollo data.

[42] To estimate the extent of the compacted region close to the borestem we have rerun our model taking the presence of compacted regolith within a radius rcom into account (compare Figure 2). Using data from the TG22A sensors of the A15 and A17 missions and assuming the appropriate undisturbed regolith conductivities of 0.0091 and 0.0116 W m−1 K−1 as determined from the annual wave analysis [Langseth et al., 1976], we have determined the best fit average conductivity of the regolith using the method described in section 2. Results of these computations are given in Figure 6 for three different compacted conductivities kcom as a function of the radius of the compacted region rcom.

Figure 6.

(a) Best fit thermal conductivity k as a function of compaction radius rcom for three different compacted thermal conductivities kcom. Thermal conductivity of the undisturbed regolith kreg is 0.0091 ± 0.0014 W m−1 K−1. The shaded area corresponds to the range of conductivities determined from the Apollo 15 active heating experiment at sensor TG22A. (b) Same as Figure 6a but for an undisturbed regolith conductivity kreg of 0.0116 ± 0.0017 W m−1 K−1. The shaded area corresponds to the range of conductivities determined from the Apollo 17 active heating experiment at sensor TG22A.

[43] Figure 6a shows the result for the A15 measurement, for which Langseth et al. [1976] estimated an undisturbed thermal conductivity of 0.0091 ± 0.0013 W m−1 K−1. Fitting of the self-heating curves at sensor TG22A yielded a best fit thermal conductivity of 0.0146 ± 0.0022 W m−1 K−1 [Langseth et al., 1976]. This range of best fit values is marked by the shaded area in Figure 6a, and fitting of the self-heating curves including a compacted region are consistent with the lower bound of the Apollo result (0.0124 W m−1 K−1) for a minimum compaction radius of 2.6–3.2 cm, i.e., extending 1.6–2.2 cm away from the borestem. To obtain best fit conductivity values closer to the upper bound of 0.0168 W m−1 K−1, rcom needs to be 4–4.5 cm.

[44] Figure 6b shows the result for the A17 TG22A measurement, for which Langseth et al. [1976] estimated an undisturbed thermal conductivity of 0.0116 ± 0.0017 W m−1 K−1. Fitting of the self-heating curves yielded k = 0.0264 ± 0.0040 W m−1 K−1 [Langseth et al., 1976]. For this A17 measurement, the compacted region must extend further away from the borestem to be compatible with the data, because the difference between measured compacted and undisturbed conductivity is larger than for the A15 measurement: The minimum radius of the compacted region is 5 cm for kcom = 0.03 W m−1 K−1 and needs to be 6.5 cm for kcom = 0.025 W m−1 K−1. For this measurement kcom = 0.02 W m−1 K−1 is incompatible with the observations.

[45] Comparing these results to the compaction model presented in section 4.1, we find that for the Apollo 15 measurement at a depth of 49 cm the fraction of displaced material f needs to exceed 20% to result in a compaction radius of 3 cm (compare Figure 5). On the other hand, results for the Apollo 17 measurement at a depth of 186 cm are compatible with the compaction model if at least 10% of the material have been laterally displaced (Figure 5). These results are in line with the higher porosity of the material at shallow depth, where larger fractions of material can be displaced into the pores.

[46] The presented calculations indicate that the results obtained by the Apollo active heating experiment are compatible with the compaction model presented in section 4.1. The compacted region probably exhibits thermal conductivities which are enhanced by a factor of 2 to 3 with respect to the undisturbed values and extends at least 2 to 5 cm from the borehole wall.

[47] Further support for this conclusion is provided by the fact that discrepancies between methods were largest at depths below 50 cm (compare Table 1). This is expected because porosity is significantly larger at shallow depth [Carrier et al., 1973] and less pressure needs to be exerted on the regolith to emplace the probes. Given that pressure increase is the major contribution to the compacted thermal conductivity (equation (9)), kcom can be expected to be closer to the undisturbed conductivity kreg at shallow depth.

5. Heating Time

[48] If regolith compaction was indeed responsible for the discordant conductivity values obtained using different methods, concordant results could have been obtained if the region of sampled regolith would have been large enough. This implies that the heating time would have needed to be sufficiently long (compare equation (5)), and we will address the question of what sufficiently long means in the following.

[49] To this end, we have calculated synthetic heating curves for heating times of up to 1000 h assuming compacted conductivities of 0.02 and 0.03 W m−1 K−1. In these calculations, three different compaction radii of 3, 5, and 7 cm have been assumed and the curves have been fitted at different time intervals each spanning 5 h. The results of these computations are shown in Figures 7a and 7b for the low- and high-compaction case, respectively. Assuming a background thermal conductivity of the undisturbed regolith of 0.01 W m−1 K−1, the deviation of the obtained conductivity from the undisturbed value is given as a function of the time t at the beginning of the fitting interval.

Figure 7.

(a) Error of the obtained thermal conductivity as a function of heating time for three different compaction radii rcom. Undisturbed regolith thermal conductivity kreg is 0.01 W m−1 K−1, and compacted regolith conductivity is kcom = 0.02 W m−1 K−1. (b) Same as Figure 7a but for kcom = 0.03 W m−1 K−1.

[50] In the lower-conductivity case, heating times in excess of 30, 100, and 200 h would be necessary to obtain results which are accurate to within 10% of the undisturbed conductivity for compaction radii of 3, 5 and 7 cm, respectively (Figure 7a). If, however, compaction is more severe and kcom is three times larger than the undisturbed conductivity, heating times in excess of 50 and 200 h would be necessary to reach this accuracy for rcom = 3 and 5 cm, respectively. In this case, rcom = 7 cm would require heating times longer than 600 h.

[51] Considering the results for the A15 TG22A measurement presented in section 4.2, heating times of 30–60 h would have been necessary to obtain results concordant with those determined using the annual wave analysis. For the more severely compacted A17 TG22A measurement, heating times of at least 100–200 h would have been necessary. However, given that this is still less than the duration of a lunar day, this is certainly technically feasible and should be considered when designing future experiments. Ultimately, lab tests will be necessary to estimate the amount of regolith compaction due to the probe emplacement process and measurement requirements for the heating times should be derived on this basis.

6. Discussion and Conclusions

[52] The in situ determination of the thermal conductivity of lunar regolith during the Apollo 15 and 17 missions yielded discordant results when employing different methods [Langseth et al., 1976] and we have re-evaluated data from the Apollo active heating experiments to investigate different hypotheses concerning the origin of this mismatch. We find that neither poor coupling between the probe and regolith nor axial heat loss are likely candidates to explain the observed discrepancies. Rather, a modification of the thermophysical properties of the regolith due to the emplacement of the probes by the rotary percussion drill system seems to have caused the higher-thermal-conductivity values determined using the active heating experiments, as originally concluded by Langseth et al. [1976]. Therefore, the corrected lunar heat flow values, which are based on thermal diffusivity estimates sampling a large portion of undisturbed regolith, represent robust results.

[53] The elevated thermal conductivity in the disrupted region is likely a factor of two to three larger than the conductivity of the undisturbed regolith and the compacted region extends at least 2 to 5 cm from the borehole wall. Furthermore, we find that measurements at greater depth are more severely affected by regolith compaction. Depending on the functional dependence of the decay of compaction with distance from the borehole, our models indicate that at least 30% and 15% of the material in the borehole have been laterally displaced in the drilling process if ρcomd/r and d2/r2, respectively.

[54] The presence of a highly conductive layer close to the borestem will result in additional shunting of heat in the axial direction and the temperature gradient measured at the gradient and ring bridge sensors will differ from that of the undisturbed regolith. Shunting effects have been taken into account by Langseth et al. [1972a], but no compacted region of regolith was considered in their analysis. We have estimated the additional shunting caused by the compacted region using the model presented in section 2 and prescribing boundary conditions resulting in a thermal gradient of 1 K m−1. We find that Langseth et al. [1972a, 1972b] have probably underestimated the thermal gradient by about 3%. However, considering other uncertainties connected to the thermal gradient measurement, this correction can be considered as negligible.

[55] The presence of a compacted annulus around the borestem could also potentially increase the downward propagation speed of the diurnal and annual temperature waves, but the small radial extent of the compaction region argues against this being a significant effect. Due to the long periods of the diurnal and annual perturbations, lateral heat transport will act to mitigate any effects of the compaction region. The diffusion time scale across the annulus is close to a few days, and no significant temperature difference is thus expected to build up in the radial direction. While a small effect might be visible for the diurnal wave, shunting should be unobservable in the annual signal.

[56] It should have been possible to match up results obtained using different methods if the heating times had been longer, and we estimate that heating times of 100–200 h would have been necessary to sample enough undisturbed regolith. Future thermal conductivity experiments should take care to both minimize regolith disturbance and maximize heating time to obtain reliable results. Furthermore, lab tests concerning the influence of regolith compaction due to the probe emplacement process need to be carried out to properly define the measurement requirements for the necessary heating times.

[57] A possible drawback of extended heating times could be the time required for the regolith to reach its preheated state after completion of the experiment. Due to the induced thermal disturbances, monitoring of the diurnal and annual temperature cycles might need to be interrupted during this time period. However, this can probably be accommodated within the lifetime of most planned missions. In particular, extended heating times should not pose a problem for missions like the International Lunar Network [Science Definition Team for the ILN Anchor Nodes, 2009], which aims at deploying a long-lived network of geophysical stations on the Moon.

[58] Another caveat concerning extended heating times is connected to increased heat loss in the axial direction: The obtained heating curves will deviate considerably from the standard line heat source solution at large times and measurements cannot be analyzed using standard methods. As a consequence, heat loss in the axial direction will need to be carefully modeled to obtain reliable results.

[59] The combination of the small heating power used in the Apollo experiments together with the low sampling rate of the data inhibited a meaningful estimate of the change of slope of the dT versus ln(t) curve. This would have been useful in that a piecewise fit of the heating curves could have allowed for a direct observation of the influence of soil compaction on the obtained results. If a higher heating power had been used in the experiments a better signal-to-noise ratio would have been obtained, potentially allowing for the above mentioned piecewise fit. Future heating experiments should be laid out such that the small temperature increase at large times >100 h is well above the noise level of the applied electronics.

[60] As discussed by Langseth et al. [1976], radiative coupling and heat transfer along the probe-borestem axis could have contributed to errors in the conductivity estimates. However, due to the small view factor of the top of the probe to space this effect is expected to be small for the active heating experiment. Furthermore, radiative heat loss would be expected to be even less important for sensors further down the borehole and since observed discrepancies were largest for these sensors we conclude that radiative coupling can be disregarded for the purpose of the present analysis. Considering the design of future experiments, this conclusion might need to be reevaluated and radiative coupling will indeed play a role in the analysis of transient temperature waves. One possible way to circumvent these complications and guarantee only conductive coupling would be closing of the borehole by, e.g., filling or collapsing the hole subsequent to probe emplacement.

Acknowledgments

[61] We wish to thank Steve Keihm and an anonymous reviewer for their comments, which helped to improve this manuscript. This research has been supported by the Helmholtz Association through the research alliance Planetary Evolution and Life.

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