Lunar heat flow: Regional prospective of the Apollo landing sites

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: PLANETS, VOL. 119, 47 63, doi: /2013je004453, 2014 Lunar heat flow: Regional prospective of the Apollo landing sites M. A. Siegler 1 and S. E. Smrekar 1 Received 6 June 2013; revised 25 November 2013; accepted 26 November 2013; published 13 January [1] We reexamine the Apollo Heat Flow Experiment in light of new orbital data. Using three-dimensional thermal conduction models, we examine effects of crustal thickness, density, and radiogenic abundance on measured heat flow values at the Apollo 15 and 17 sites. These models show the importance of regional context on heat flux measurements. We find that measured heat flux can be greatly altered by deep subsurface radiogenic content and crustal density. However, total crustal thickness and the presence of a near-surface radiogenic-rich ejecta provide less leverage, representing only minor (<1.5 mw m 2 ) perturbations on surface heat flux. Using models of the crust implied by Gravity Recovery and Interior Laboratory results, we found that a roughly 9 13 mw m 2 mantle heat flux best approximate the observed heat flux. This equates to a total mantle heat production of W. These heat flow values could imply that the lunar interior is slightly less radiogenic than the Earth s mantle, perhaps implying that a considerable fraction of terrestrial mantle material was incorporated at the time of formation. These results may also imply that heat flux at the crust-mantle boundary beneath the Procellarum potassium, rare earth element, and phosphorus (KREEP) Terrane (PKT) is anomalously elevated compared to the rest of the Moon. These results also suggest that a limited KREEP-rich layer exists beneath the PKT crust. If a subcrustal KREEP-rich layer extends below the Apollo 17 landing site, required mantle heat flux can drop to roughly 7 mw m 2, underlining the need for future heat flux measurements outside of the radiogenic-rich PKT region. Citation: Siegler, M. A., and S. E. Smrekar (2014), Lunar heat flow: Regional prospective of the Apollo landing sites, J. Geophys. Res. Planets, 119, 47 63, doi: /2013je Introduction [2] Surface heat flow is a fundamental measurement for determining a body s interior composition, structure, and evolution. The partitioning of heat-producing elements such as U, Th, and K can provide tracers of original composition, mixing efficiency in early planetary formation, and distribution of past melt regions (which tend to concentrate incompatible elements). Heat flow from the lunar interior is of fundamental interest as it can reveal details about the bulk composition of these elements as compared to the Earth, can provide evidence of the validity of Earth-Moon forming impact models, and constrain the origin of the material that now composes the deep lunar interior. Heat flow can also reveal information about the density, radiogenic composition, and structure of the lunar crust, providing a window into the processes controlling lunar differentiation. On the Moon, the distribution of radiogenics, crustal thickness, and structure are known to be heterogeneous, especially in a large province known as the Procellarum KREEP Terrane (KREEP is a concentrated 1 Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California, USA. Corresponding author: M. A. Siegler, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, USA. (matthew.a.siegler@jpl.nasa.gov) American Geophysical Union. All Rights Reserved /14/ /2013JE basaltic material rich in chemically incompatible potassium, K, rare earth elements, and phosphorus) [Hubbard et al., 1971]. Regional crustal structure and composition are especially important in this area, which includes both Apollo heat flow measurements. Here we explore how new measurements (topography, gravity, temperature, and radiogenic distribution) can be combined with three-dimensional heat conduction models to aid in deconvolving the nature of lunar heat flow. [3] The Moon is currently the only body other than the Earth for which we have a measured value of internal heat flux (upper limits have been obtained for Io [Matson et al., 2001] and Enceladus [e.g., Spencer et al., 2013]). These values, measured as part of the Apollo Heat Flow Experiment (HFE), differ from each other, with the Apollo 15 measured heat flux of 21 ± 3 mw m 2 and the Apollo 17 values of 15 ± 2 mw m 2 [Langseth et al., 1976]. For comparison, the Earth has means of 65.3 mw m 2 and 100 mw m 2 for the continental and oceanic crust [Jaupart and Mareschal, 2007]. Since the thermal evolution of the Moon has been relatively simple compared to that of the Earth (e.g., due to the lack of plate tectonics), these values should relate directly to the bulk radiogenic concentrations of the Moon. However, the interpretation of what these values mean for the composition of the lunar interior and suggested reasons for their differences has varied greatly. [4] Both Apollo Heat Flow Experiment sites lay at the boundary of several of the Moon s most prominent crustal features, straddling highlands and mare as well as observed crustal 47

2 radiogenic boundaries. The Apollo 15 HFE site (26.13 N, 3.63 E; roughly 800 km from the center of Imbrium) lies at the intersection of Mares Imbrium and Serenitatis and a highly radiogenic highlands mountain chain known as the Apennines. Due to this regional context, it potentially represents one of the most challenging places on the Moon for constraining mantle heat flux. The Apollo 17 HFE site (20.19 N, E; roughly 1600 km from the center of Imbrium) lay within the Taurus-Littrow Valley. The surface has less radiogenic enrichment than that of Apollo 15, but the valley topography and thin mare fill are likely to cause some degree of geometric heat focusing. Both sites lay within a large surface Thorium anomaly that characterizes the Procellarum KREEP Terrane (PKT), which has led to the suggestion that neither represent the global heat flow of the Moon. There is an additional complication of a suggested subcrustal KREEP-rich layer below the crust, which has been proposed to underlie the Apollo 15 and possibly Apollo 17 site [Wieczorek and Phillips, 2000]. [5] A flurry of new data is available to aid in reanalysis of the Apollo Heat Flow Experiment data. Recent crustal thickness models from the Gravity Recovery and Interior Laboratory (GRAIL) and Selene missions have revolutionized our ability to model thickness, crustal density, and crustal structure. Recent reanalysis of Apollo seismic data [Lognonné et al., 2003] and gravity measurements [Ishihara et al., 2009; Wieczorek et al., 2013] has nearly halved past estimates of crustal thickness. As we shall see, each of these aspects can have large effects on observed surface heat flow. Our knowledge of topography, especially important on the regional-scale heat flow, has seen huge advances from stereo photoclinometry and laser altimetry data from the Lunar Reconnaissance Orbiter (LRO) and other missions. [6] Crustal radiogenic composition has been mapped by several gamma ray instruments [e.g., Lawrence et al., 1998] which will be used to identify the extent of the PKT [Jolliff et al., 2000] and serve as a boundary to extrapolate KREEP concentration into the deeper crust. The KREEP (or urkreep) layer is assumed to have formed from the last remaining melt of the global magma ocean which concentrated incompatible elements. If this layer exists, then either it formed into a large melt lens (centered near the Procellarum KREEP Terrane, PKT) [Wieczorek and Phillips, 2000] or remained global and has simply had surface expression in a few areas [Jilly et al., 2011]. While this area contains most of the measured Th increase and about 60% of all lunar volcanism [Wieczorek and Phillips, 2000], areas of surface radiogenic enhancement (and generally associated mare volcanism) exist outside this region, complicating simple interpretation. [7] Several authors have proposed that the measured Apollo HFE values either under represent or overrepresent the global average heat flux by up to a factor of 3 due to the thermal pathways and heat-producing material of the top ~100 km of the Moon [Langseth et al., 1976; Warren and Rasmussen, 1987; Wieczorek and Phillips, 2000; Hagermann and Tanaka, 2006; Saito et al., 2007; Saito et al., 2008]. Equally variable are the reported causes for the differences between global and HFE-measured heat flux. Without a detailed model including a comprehensive examination of near-surface (top ~100 km) effects on measured heat flux, it is difficult to extrapolate these values to constrain the bulk composition of the Moon. [8] Here we combine the proposed causes for the (generally assumed) elevated heat flux at the Apollo 15 and 17 landing sites into a complete, three-dimensional thermal conduction model. This will allow us to weigh the net impact of competing crustal effects on global surface heat flow. We will discuss implications these models would have on the global thermal evolution of the Moon and attempt to show what would be desirable in selecting future measurement location to differentiate between possible variables, such as radiogenic distribution and crustal thickness. 2. Prior Lunar Heat Flow Models [9] Langseth et al. [1976] found that crustal thickness variations should dominate regional differences in lunar heat flow. They modeled the Apollo 15 and 17 heat flow values to be lower than the lunar average. They noted that both Apollo heat flow sites lay near the edge of an area of elevated thorium, up to as much as 7 ppm, based on Apollo 15 orbital Gamma Ray Spectrometer data [Metzger et al., 1974]. They assumed this to be concentrated in a near-surficial, radiogenic-rich layer, giving it a small but measurable heat flux (which primarily affected the Apollo 15 measurement). However, these authors expected the small average crustal radiogenic content, estimated at 1 ppm Th, to dominate heat flow. Using a Th/U ratio of 3.7 and a K/U ratio of 3200, this results in about mw m 2 total heat production per kilometer of crust. Therefore, as Apollo seismic data showed the crust beneath the Apollo sites to be thin as compared to the lunar farside (though their estimated km nearside crustal thickness is nearly twice that estimated by modern studies for the PKT region), they found the global average heat flux (which they estimated as 18 mw m 2 ) should exceed their measured value at the Apollo 17 landing site. They additionally examined the effects of local topography, thermal focusing by local density contrasts, and nearsurface temperature variations but found them to be minor components in the total observed heat flux. [10] Conel and Morton [1975] proposed that crustal density variations would cause the greatest differences between the Apollo sites. In their model, heat fluxes were enhanced by the large-scale contrast in thermal conductivity at the highland/mare boundaries, where both HFE sites lay. By constructing a two-dimensional model of simple cylindrical mare imbedded in low-density crust (similar to that in section A2), they showed heat would preferentially flow into the high conductivity zone, biasing the heat flux near the conductivity boundary. They were responding to an initial HFE claim of mw m 2 [Langseth et al., 1973] that was based on flawed regolith thermal conductivity which was later found to be due to compaction that was not accounted for [Langseth et al., 1976; Grott and Breuer, 2010], so many of their derived values are not comparable to other estimates. The idea, however, was sound, showing that near-surface (1 10 km depth) regolith/bedrock interfaces could cause enhancements inside the denser mare. In area with thin regolith, like the mare, this could cause enhancements of heat fluxes of up to an order of magnitude along the boundary with areas of thicker regolith. Rasmussen and Warren [1985] and Warren and Rasmussen [1987] revised this idea with updated models of regolith structure combined with assumption radiogenic enhancements at both Apollo sites (about 3.3 times the global 48

3 mean at Apollo 15 and 1.2 times at Apollo 17) were not simply near surface (as was assumed in Langseth et al. [1976]). Using their revised thermal conductivity models, they found both sites should have an expected upward heat flux bias and the global mean to be around 12 mw m 2, dropping the estimated Uranium content of the bulk Moon to ppb, more consistent with terrestrial mantle material. [11] Warren and Rasmussen [1987] also included an enhancement of crustal heat flux at the Apollo sites in the form of extra radiogenic material throughout a 70 km thick crust. They noted, however, that the origin of the measured surface enhancement of radiogenics, a concentrated basaltic material known as KREEP was likely to have been delivered from a deeper source either by impact [Ryder and Wood, 1977; Haskin, 1998; Hagermann and Tanaka, 2006] or volcanic origin [Spudis, 1978; Ryder, 1994]. However, the location, depth, and concentration of this presumed KREEP reservoir varies dramatically, from local origin in the Apennine Bench [Spudis, 1978] to a global layer at the base of the crust [Warren and Wasson, 1979]. [12] Hagermann and Tanaka [2006] surveyed explanations of the Apollo heat flow measurements with near-surface radiogenic enhancement. In their model they explore the hypothesis that a KREEP-rich ejecta blanket, which has been partially exposed in the PKT region, could explain the difference between the Apollo sites [Haskin, 1998]. They modeled an ejecta blanket around the center of the Imbrium impact (37.5 N, 19 W) [Wieczorek and Zuber, 2001] with local thickness as modeled by Haskin [1998], resulting in km for Apollo 15 and km for Apollo 17. They assume the radiogenic component of this blanket to be determined by the Lunar Prospector Gamma Ray Spectrometer (LP-GRS) data (with and 2.74 ppm Th, respectively, at Apollo 15 and 17). As more central ejecta originate from greater depths, this distribution was attributed to a deep Th-rich layer having been excavated by the impact [Warren, 2001]. These ejecta would therefore decrease in radiogenic concentration with distance from the impact point. Spudis [1978] and Ryder [1994] suggest impact-induced volcanism could account for the surface KREEP enhancement, which should result in a similar shallow crustal emplacement of heat-producing elements; however, the thermal consequences of this distribution have not been explored. [13] Alternatively, Wieczorek and Phillips [2000] proposed the Apollo heat flux values were due to the deep, Imbrium ejecta source region. They assumed all the radiogenic enhancement of the PKT to be subsurface. They modeled the effect of a localized (10 km thick, 40 radius spherical cap about the center of the PKT) subsurface KREEP layer at the base of a 60 km thick crust. They found an area of this extent could retain heat within the cooling mantle long enough to keep a melt body that could cause relatively recent (< 900 Ma) mare volcanism and a measured seismic discontinuity at 500 km depth. More central to our paper, they also found such KREEP spherical cap (called KREEP disk here for shorthand) could explain the drop in heat flow between the Apollo 15 and 17 landing sites. As we discuss later, their ability to model the Apollo measurements depends both on crustal thickness and the extent of this KREEP cap. Though this model provides predicable results for measurements in given localities, it does not include near-surface radiogenic enhancement or effects of variations in crustal thickness and density. [14] In summary, the measured values at Apollo HFE sites are generally, though not universally, assumed higher than the global lunar average. Additionally, the Apollo 15 site, which is located closer to the center of the Thorium-rich PKT, is found to have a higher heat flux than that of Apollo 17. The magnitude of this variation has been proposed to result from the following: (1) crustal thickness variations [Langseth et al., 1976], (2) focusing of heat flow at thermal conductivity boundaries [Conel and Morton, 1975; Rasmussen and Warren, 1985], (3) shallow enhancements of KREEP-like radiogenics [Hagermann and Tanaka, 2006], or (4) deep concentrations of KREEP-like radiogenic enhancement [Wieczorek and Phillips, 2000]. Each of these authors has shown their assumed cause could dominate the resulting measured heat flow, substantially impacting the resulting extrapolation to unmeasured locations. 3. Model Approach [15] We examine each of these possible effects in a three-dimensional configuration, using actual lunar topography. We explore the effects of plausible values of crustal thickness, density, thermal properties, and radiogenic concentration. In this section, we will describe a nominal model that serves as a basis for comparison for exploration of these parameters. We choose to study a region km region surrounding the two Apollo HFE sites and containing the large geographic features (namely, Mare Imbrium and Mare Serenitatis) that are most likely to impact measured heat fluxes. We extend the model to 150 km depth, which should be deep enough to allow for lateral conduction within the upper mantle, but shallow enough that no melting would occur. We generally refer to the heat flux at 150 km depth as the mantle heat flux, as the mantle layer between 150 km and the base of the crust only accounts for about 0.5 mw m 2 (with our assumption of Wm 3 mantle heat production) which is small compared to the error of the Apollo HFE measurements. To examine the full three-dimensional heat conduction effects of each variable, we combine currently available data using the commercially available COMSOL Multiphysics package. This software was chosen as it allows us to create an accurate representation of irregular surface and crust/mantle topographic boundaries and compute the effects of lateral heat flow with unprecedented detail. [16] The topography of the Moon is now well known. Here we begin with a 16 pixel/degree resolution grid of the Lunar Orbiting Laser Altimeter (LOLA) data set to act as a surface boundary. Figure 1 shows the topography used in our study region. Grid elements are vertically thin near the topographic surface boundary to a resolution of 1 km to accommodate model dramatic changes in density near the surface as will be discussed in section 4.2. Layers below this boundary layer increase in vertical thickness by a factor of 1.2 for the first 10 layers (1.0 km, 1.2 km, 1.44 km, 1.72 km, etc.). Grid resolution is scaled based on geometry and thermal properties (thin layers require smaller elements) and an internal check that COMSOL can come to a unique steady state solution, creating high-resolution layers near thermal property boundaries. Distant from boundaries, elements are allowed to grow larger, up to about 35 km per side, though most crustal elements are less than 10 km per side. Horizontal resolution varies between roughly 5 km in vertically thin regions (like the mare) and 49

4 Figure 3. Surface temperature boundary condition from Diviner thermal models. Figure 1. Topography of regional study in kilometers above a 1738 reference radius sphere. The km 150 km dimensions. The cross marks the Apollo 15 and circle Apollo 17 linked by a linear transect. The four circles in increasing size are the thicker mare of the Imbrium transient cavity, Mare Serenitatis, Mare Imbrium, and a plausible KREEP-rich region. especially important in our study region which contains some of the thinnest crust on the Moon and the largest local variations in crustal thickness. Wieczorek et al. [2013] have created several models based on assumed mantle density and crustal porosity (GRAIL crustal thickness models are available at Thickness Archive/GRAILCrustalThicknessArchive.html). For our nominal model we have chosen the thickest of the Wieczorek et al. [2013] models (model #4), which has a globally averaged crustal thickness of 43 km, 38 km crustal thickness at the Apollo 12 site (where it is tied to seismic crustal thickness values), 7% crustal porosity, and a higher 3300 kg m 3 average mantle density. Figure 4 shows this model (with depth in respect to a 1738 km radius reference sphere) which serves as the bottom boundary of our crustal layer and top of our mantle layer (though here we change only crustal thickness, not density or porosity). [19] The crust contains radiogenic material which will add to surface heat flux. Assuming values of 0.14 ppm U, 0.53 ppm Th, and 480 ppm K [Wieczorek and Phillips, 2000], we add a constant background heat production of W m 3 to standard 2550 kg m 3 crustal material (models with variable density used the equivalent W kg 1). In addition to this background heat production, we include a representation of additional crustal radiogenic material based on data from the Lunar Prospector Gamma Ray Spectrometer [Lawrence et al., 1998]. Using these measured Th values and assumed 3.7 Th/U and 2500 K/Th ratios [Warren and Wasson, 1979] and a heat generation model [e.g., Turcotte and Schubert, 2002; Grott and Breuer, 2010], a pure KREEP is 12.4 ppm Th which would result in W m 3. To extrapolate surface radiogenic concentration into the subsurface, we choose an e-folding model with surface concentration c0 and decreasing concentration of again nearly 35 km in regions far from any boundary. Models have been verified against standard analytic solutions using simplified geometries (planar surface and crust/mantle boundaries and 2-D models in Appendix A). Figure 2 shows the tetrahedral finite element grid used for study calculations with 840,086 elements (the surface boundary layer and mare gridding are too thin to be seen here). [17] Surface temperature is also well known. The Diviner Lunar Radiometer Paige et al. [2010a], also aboard LRO has now mapped global surface temperatures to roughly 100 m resolution. These data have been used to constrain global thermal properties models (at the scale of available topography), which allow for a calculation of temperatures below the lunar surface [Paige et al., 2010b]. All models presented here assume a steady state, meaning we must use temperatures that do not change with time. By a depth of roughly 2 m, diurnal temperature cycles will have damped to a constant temperature. Though this model temperature is itself dependent on an assumed heat flux (15 mw m 2 in this model), the model temperature at 2 m depth will serves as a reasonable, timeinvariant surface temperature boundary condition, which falls in temperature with increasing latitude (as seen in Figure 3). [18] Recent crustal thickness, based on data from the recent Gravity Recovery and Interior Laboratory (GRAIL) mission [Zuber et al., 2013] and reanalysis of Apollo seismic data [Lognonné et al., 2003], has revolutionized our understanding of the lunar crust and interior. Crustal thickness models based on these measurements have shown the crust to be substantially thinner and less dense [Wieczorek et al., 2013] than previous models [Wieczorek and Phillips, 1998]. This is z cðzþ ¼ c0 exp H (1) where z is depth (in kilometers) and H is the e-folding scale, or depth at which concentrations will drop by 1/e. Later, we Figure 2. Three-dimensional tetrahedral grid used for regional thermal model created in COMSOL containing 840,086 elements. The largest elements are ~35 km per side, the smallest, ~ 1 km. Figure 4. GRAIL-modeled crustal depth of study region in kilometer depth below 1738 km. 50

5 Figure 5. GRS-measured Th converted into near-surface volumetric heat production. will also use this formulation to approximate quantities that increase or decrease dramatically with depth (e.g., radiogenic mixing and density profiles). It is sensible for a surface deposit that has been mixed into the upper crust over several billion years and corresponds well with mixing that might be expected from crustal density models. In the nominal model, we will use a e 5km scaling depth, shown in Figure 5. [20] Density, and its effect on thermal conductivity, varies between primary lithosperic components (crust, mare, and mantle). Specific heat capacity of all materials is arbitary as we are assuming the Moon has reached a thermal steady state. Even with precise gravity data, the density structure of the lunar crust is uncertain. The mare represents flood basalt emplaced sometime later than the primary crust, and they are generally believed to be denser than the surrounding older highlands crust. For our nominal model we assume a constant density crust of 2550 kg m 3 [Wieczorek et al., 2013]. The crust is given a thermal conductivity of 2 W m 1 K 1 [from Wieczorek and Phillips, 2000]. [21] Based on the mare models of Solomon and Head [1980] combined with mare thickness constraints from Thomson et al. [2009], we chose a model for the mare, assuming several cylindrical bodies embedded within the crust. The upper boundaries of these cylinders are determined by the surface topography. The main Imbrium and Serenitatis basins are represented by mare reaching to 5km depth below a 1738 km radius lunar sphere, with the upper surface defined by LOLA topography (this resulted in most of these mares being about 3 km thick). Using available crustal thickness data and topographic expressions of the mare and previous models [e.g., Hikida and Wieczorek, 2007], we chose to represent the main Mare Imbrium as a disk 550 km in radius centered at (34.5 N, 14.5 W), and Mare Serinatatis as a disk 295 km in radius, centered at (27 N, 18 E). Within the Imbrium basin; previous authors found a deeper area of mare filling the Imbrium transient cavity. To represent this area of increased mare thickness, we constructed a 270 km radius disk reaching to 7km depth centered at (37.5 N, 20.5 W), the center of the Imbrium impact (based on aligning the disk with the thinnest crust). Mares are given a density of 3000 kg m 3 and thermal conductivity of 2 W m 1 K 1 [from Wieczorek and Phillips, 2000]. We chose a mare heat production based on an average of Apollo 15 mare samples (an average of 15,058 and 15,555), which showed radiogenic compositions similar to our nominal crustal model, adding a constant background heat production of Wm 3 to this material [Meyer, 2010; W. Kiefer, personal communication, 2013]. The model densities are shown in Figure 6. The edges of these mares are denoted as vertical lines in the transect figures (e.g., Figure 8) throughout this paper. [22] We also chose to include a section of mantle material in this model. This is important as heat can flow laterally through the mantle between two areas of the crust. The main constraints on the depth of mantle we can include lie in the assumption of conduction and our planar geometry (going too deep will require us to model a spherical, rather than planar Moon). Examining these constraints, we choose to extend the base of our model to a depth of 150 km. Here we give the mantle a density of 3200 kg m 3 and thermal conductivity of 3 W m 1 K 1 [Wieczorek and Phillips, 2000]. The heat production per unit volume, also Wm 3 (all models use a constant density mantle), is too small for this roughly 100 km thick section of mantle to have an appreciable effect on surface heat flux (adding at most ~0.5 mw m 2 ). [23] Finally, we need to choose a nominal value for lower boundary heat flux, which is the primary objective of this study. By adjusting this value, we can then see the mantle radiogenic contribution required for a given crustal model to match the Apollo HFE results. Langseth et al. [1976] found a uniform mantle heat flux of 4 mw m 2 best explained the observed Apollo HFE values. This value is equivalent to a heat production of Wm 3 over the 1588 km column between 150 km depth and the center of the Moon (neglecting the core). Based on the values of mantle radiogenics in Wieczorek and Phillips [2000], the mantle should have a heat production of Wm 3,orabout7mWm 2 at 150 km depth. From initial model runs we found both these models under predicted Apollo HFE values and found a 10 mw m 2 to be more appropriate for our nominal model value. Assuming that the Moon is in a thermal steady state, 10 mw m 2 would represent about Wm 3 average lunar mantle heat production, which is roughly 70% the heat production rate of the Earth s mantle[turcotte and Schubert, 2002; Jaupart and Mareschal, 2007]. If the Moon is not in thermal equilibrium, less radiogenic material might plausibly give the same heat flux, but this will require a thermal evolution model including the effects of the low-density crust (such models are reviewed in Zhang et al. [2013]and Laneuville et al. [2013]). [24] Assuming an infinite plane with periodic boundaries along the four remaining sides of our model, we can now calculate the steady state heat conduction and temperatures required by our geometry, internal heat production, thermal properties, and boundary conditions. Figure 7 shows the resulting heat flux in the vertical direction resulting from our model assumptions. Here we can see that the highest heat flux values occur near the near-surface radiogenic concentrations surrounding Imbrium basin. The mares, which are lower in heat production and over areas of thinner crust, show somewhat lower surface heat flux. Figure 6. Nominal model crustal density based on Wieczorek and Phillips [2000] and Wieczorek et al. [2013]. 51

6 considered twice the thickness of the GRAIL crustal models (red, also shown in Figure 8), would not on its own cause more than a few milliwatt per square meter error. The small dips and rises are mainly due to small-scale topography (such as the dip at Apollo 17 associated with the transect crossing the Taurus-Littrow valley). Figure 7. Vertical heat flux resulting from nominal model assumptions. Note the decreased heat flux in regions with thin crust and increased heat flux where surface radiogenics are present. 4. Variations on 3-D Topographic Model [25] The nominal model presented in section 3 cannot explain differences observed between the Apollo 15 and 17 HFEs. Here we examine which of the four effects presented could potentially cause the differences observed between the Apollo sites and how large a variation might plausibly explain these observations. Assuming each effect is the sole cause of the heat flow difference, we show the maximum possible effect of each parameter that would be consistent with the Apollo HFE values. We then present arguments in support of our preferred model parameters. [26] Another way to view model differences is with a surface transect. In Figure 8, we illustrate a cross section that passes through the two Apollo HFE sites. This transect is seen as a diagonal line in crossing Figures 1 7. Examining the model output this way will allow us to compare it directly to the Apollo HFE constraints. Apollo 15 measured values are marked with a cross and the Apollo 17 measured values are marked by a circle, each with error bars from Langseth et al. [1976]. We found Langseth et al. s [1976] 4 mw m 2 mantle heat production appears to be much too low, so we chose a basal (mantle) heat flux of 10 mw m Effects of Crustal Thickness [27] The thickness of the lunar crust is a fundamental parameter controlling the loss of heat from the interior. Thicker crust will generally provide a higher surface heat flux due to radiogenic material in the crust (see examples in section A1). Here we examine the implications of newly revised and reduced estimates of crustal thickness and porosity. [28] In Figure 8 we compare the two end-member GRAIL models [Wieczorek et al., 2013]. This figure illustrates heat flux along a surface transect through the two Apollo HFE sites (the straight line in Figures 1 7). Subsequent figures will compare possible models. All the models in this section are given a basal heat flux of 10 mw m 2. Vertical lines in Figure 8 represent mare edges or changes in mare thickness. [29] In Figure 8 we show our nominal model (blue) and results from the thinnest plausible GRAIL crustal thickness model (green, which has a globally averaged crustal thickness of only 34 km, 29.9 km crustal thickness at the Apollo 12 site, 12% crustal porosity, and 3220 kg m 3 average mantle density). This illustrates the range of error of our model fits due to our choice of plausible crustal thickness model and shows that reasonable crustal thickness assumptions will have little effect on model outcome. Older crustal models [e.g., Wieczorek and Phillips, 1998], which can be roughly 4.2. Effects of Crustal Density [30] Horizontal differences in crustal density will cause heat to flow laterally to find a more efficient pathway to the surface (see examples in section A2). If crustal density also varies as a function of depth, it will affect the efficiency of this lateral heat conduction. Though gravity measurements from GRAIL are extremely high resolution, they do not provide a unique solution for the density of the lunar crust as a function of depth. Wieczorek et al. [2013] found that the porosity of the crust is likely to extend to depths of 30 km and could be modeled by an exponentially dependent (similar to equation (1)) model, which is consistent with the decreased admittance as a function of spherical harmonic degree. That study also excludes our region of interest due to mare density variations. Based on seismic studies [e.g.,toksöz et al., 1978], considerable crustal porosity is expected to > 10 km depth [Hörz et al., 1991]. Apollo regolith density cores can also be extrapolated to depth (where density, ρ =1390z kg m 3 for z in meters) to provide a rough estimate of large-scale crustal densification and are seen in cyan in Figure 9 [Carrier, 1974; Heiken et al., 1991, here called the Lunar Sourcebook Power Law model]. [31] Based on these studies, we propose a series of plausible density profiles, in addition to the constant 2550 kg m 3 [Wieczorek et al., 2013] of the nominal model, which are also shown in Figure 9. These models are constrained to have a surface density of 1700 kg m 3. This should be equivalent to the density at ~2 m depth [Carrier et al., 1991, Lunar Sourcebook], the same depth as our surface temperature condition below the diurnal thermal perturbations. We then create an exponentially increasing density profile (here either increasing density by e z/1 km kg m 3, 1km e-folding ; e z/5 km kg m 3, 5km e-folding ; or e z/10 km kg m 3, 10 km e-folding with z in kilometers). We constrain these models to have the same Figure 8. Vertical heat flux (watt per square meter) cross sections from nominal model assumptions with plausible crustal thickness models. Blue (GRAIL nominal model, which has a 38.0 km average crustal thickness), green (thinnest GRAIL model, which has a 29.9 km average crustal thickness), and red (twice GRAIL thickness model, 76.0 km average crustal thickness). 52

7 Figure 9. Plausable crustal density model with 1 km (in blue), 5 km (green), or 10 km (red) e-folding depth as compared to a Lunar Sourcebook Power Law density model (cyan). total column density of a 30 km crustal section with the GRAIL measured 2550 kg m 3 density. These profiles roughly bound the curvature of the Lunar Sourcebook model and are consistent with previous models and the GRAIL data [Wieczorek et al., 2013]. The mares, which postdate the large-scale crustal mixing of early lunar bombardment, are assumed to be constant density (3000 kg m 3 ). Crustal radiogenic content increased or decreased linearly with density in these models (relative to a standard 2550 kg m 3 density with our assumed volumetric heat production). [32] Density also will have an effect on thermal conductivity. Many models connecting density to thermal conductivity exist [e.g., Shoshany et al., 2002; Smoluchowski, 1981]. For simplicity we adopt the model of Smoluchowski [1981], which states thermal conductivity as λ = λ o (1 φ 2/3 ), where λ o is the thermal conductivity of solid rock (2.5 Wm 1 K 1 in our model) and φ is porosity (1 [porous material density] / [solid material density of 2700 kg m 3 ]). In Figure 10 we show three plausible thermal conductivity models. The surface value of ~1 W m 1 K 1 value is representative of thermal conductivities at the centroids of our surface layer volume elements, or about 1 km depth. Although thermal conductivity is potentially orders of magnitude lower in the top few meters, this extremely near surface layer should have little effect on our regional-scale models. The primary effect of the very near surface regolith on heat flow is to set the surface temperature boundary condition shown in Figure 3. [33] Figure 11 shows the resulting surface heat fluxes across our surface transect as compared to the Apollo sites and our nominal model values. The low-density, low-conductivity, near-surface layer tends to pipe heat away from the lowdensity crust and into the higher-density mare, (see 2-D models in section A2). This implies that if the highlands crust follows such a density profile, more background (or crustal radiogenic) heat flow will be required to reach the measured Apollo values. [34] As highlighted in section A2, the effect of a crustal density change will be most dramatic near a horizontal thermal conductivity boundary. Figure 11 shows this effect has the largest impact for the strongest density contrasts. At every boundary (marked by the vertical dotted lines) the 1 km e-folding model shows the strongest effect, giving a high spike in heat flux. This is a result of heat being pulled from the low-conductivity layer into the high conductivity mare nearby. This effect is strongest when the mare is of similar thickness to the underdense, low-conductivity layer. This occurs because the higher-density material below allows heat to flow laterally to the high-conductivity mare, where it can escape more efficiently (again, see examples in section A2). This creates a larger, lower (than the nominal model) thermal conductivity zone surrounding the mare, which prevents efficient lateral conduction from the highlands crust. This heat is then trapped at the mare edge, causing a spike in heat flux along this boundary. We also see in Figure 11 that the nominal density model can be thought of as an end-member model, having similar effects to a very gradually increasing density crust. [35] This illustrates that the depth of gardening in the upper crust can have a substantial effect on observed heat flux, but only directly at thermal properties boundary as lateral heat transport is still relatively inefficient. As both landing sites were near such boudaries, this effect is likely important in our understanding of the Apollo HFE values. Our mare boundaries are likely not set with sufficient accuracy to address the entirety of this effect and may require localized landing site scale models. [36] A low-density megaregolith will cause substantial amounts of heat to be funneled into higher-density areas, such as the lunar mare. This process is most efficient when the low-density layer is thin and overlies a higher-density layer. If this layer is too thick, it will prevent laterally flowing heat into regions of high density. Though heat must take different pathways to escape, it is still able to escape. However, if the entire surface of a body were insulating, this effect can serve to trap extra heat inside [Warren and Rasmussen, 1987; Haack et al., 1990; Zhang et al., 2013]. Figure 10. Plausable thermal conductivity models with 1 km (in blue), 5 km (green), or 10 km (red) e-folding. Figure 11. Effect of e-folding crustal density/thermal conductivity models (as in Figures 9 and 10) on surface heat flux (milliwatt per square meter). 53

8 Figure 12. Lunar Prospector Gamma Ray Spectrometer (LP-GRS) Th data (parts per million) plotted as a function of distance from the Imbrium impact point (37.5 N, 20.5 W) Effects of a KREEP-Rich Ejecta Blanket [37] Hagermann and Tanaka [2006] modeled the heat flow at the Apollo sites based on the theory that the PKT represented small surface exposure of a KREEP-rich ejecta blanket from the Imbrium impact [Haskin, 1998]. Such an impact may have dredged up past KREEP-rich lower crustal material. Modeling column radiogenic abundance, they found that the differences between the Apollo 15 and 17 heat flux measurements could be due to a near-surface ejecta blanket but found it had to be quite radiogenic rich or very thick. Their model did not include the effects of lateral heat conduction (see examples in section A3). In examining how to implement a similar model, we begin with the surface constraint of the LP-GRS Th data [Lawrence et al., 1998]. Plotting this data as a function of distance from the Imbrium center (37.5 N, 20.5 W), there is a symmetric rise in Th concentration (Figure 12), which gives credence to the impact ejecta origin of the surface KREEP [Haskin, 1998]. We found a roughly ppm Th km 1 slope approximated the average increase outside of a relatively radiogenic free central crater 372 km in radius [Wieczorek and Phillips, 1998]. As this ejecta is assumed to primarily lie below a surface layer, this radiogenic ejecta is in addition to that shown in Figure 5. The resulting volumetric heat production is shown in Figure 13. [38] This model gives a maximum Th concentration of 5 ppm along the inner rim of the impact basin. To extrapolate this modeltodepthweagainuseequation(1),shownase-folding profiles of either 5e z/5 km ppm or 5e z/20 km ppm (for z in kilometers) in Figure 14 (two models similar to those advocated by Hagermann and Tanaka [2006]). This exponential decrease in concentration with depth would be a natural outcome Figure 14. Concentration of Th versus depth for a location in a mixed ejecta blanket that has a current surface concentration of 5 ppm Th. of megaregolith mixing from impacts. We can constrain the maximum subsurface concentration using McGetchin et al. s [1973] crater ejecta model. For the Imbrium crater, we find the maximum ejecta thickness at the crater rim would be limited to approximately 2 km. If the ejecta were initially pure KREEP (12.4 ppm Th) [Wieczorek and Phillips, 2000], this would imply a maximum column abundance of Th-rich ejecta at this location. Integrating equation (1), we find a net concentration as a function of depth: h z 0 cz ð Þdz ¼c 0H 1 exp z i H which is shown in Figure 15. This value should represent the total amount of Th that the initial ejecta blanket had. When compared to the maximum value (for 2 km thick ejecta at crater rim) of 24.4 ppm km total column abundance, we find (2) Figure 13. Our model of added heat production from a Th-rich ejecta blanket (here with 5 km e-folding depth). Scaled 0 6e 7 Wm 3. Figure 15. Net concentration of Th versus depth a location in a mixed ejecta blanket that has a current surface concentration of 5 ppm Th. 54

9 4.4. Effects of Subcrustal KREEP Layer [40] The concept that the Imbrium impact brought KREEP-rich material up to the lunar surface implies that there is a subsurface reservoir of KREEP-rich material. To explain the enhanced nearside volcanism, basin relaxation [Wieczorek and Phillips, 1998], and the previously noted Th enhancements, Wieczorek and Phillips [2000] hypothesized that rather than being a global layer, this deep KREEP had pooled preferentially beneath the PKT. Using surface LP-GRS measured Th as a guide, they approximated the localized deep KREEP as a disk 40 (~1200 km) in radius about the Imbrium impact center (37.5 N, 19.5 W). This resulted in the KREEP disk ending roughly halfway between the Apollo 15 and 17 sites. Due to the assumed thick, 60 km crust, lateral heat conduction was large enough that Apollo 15 was lower than the 35 mw m 2 that this model implies at the Imbrium center and Apollo 17 was elevated above the background 11 mw m 2 predicted far afield from the PKT. Their modeled KREEP layer was ~10 km thick and at the base of a 60 km, uniform thickness crust. [41] In this section, we revisit this model by assuming a 10 km thick layer (rather than a flat disk) which follows the contours of the GRAIL modeled crust-mantle boundary. The layer is just above this boundary, displacing material we had formerly designated as crust. The layer also extends ~40 (1200 km) from the Imbrium center and has been seen in all mapped figures (e.g., Figures 1 7 and 13) as a large circumference circle reaching beyond the model borders. This deep KREEP layer is given a thermal conductivity of 2 Wm 1 K 1 and a volumetric heat production of Wm 3 [after Wieczorek and Phillips, 2000]. With the ability to create a relatively pure KREEP layer of unknown thickness at depth, this model does not suffer the limitations in total radiogenic volume as the ejecta blanket model. Figure 17 illustrates that a 10 km thick, 100% KREEP layer at the base of the thin crust has a very localized impact on heat flux. Figure 18 illustrates the effect of placing a 10 km thick layer of 25% KREEP composition at the same location. This is approximately equivalent to a pure KREEP layer one fourth the thickness (2.5 km) as KREEP has a substantially higher heat production rate than the crustal material this layer displaced. [42] Upon first implementation of this model, we found far less lateral heat conduction than the previous model, resulting from the far thinner crust measured by GRAIL. This creates a much sharper boundary than previously found [Wieczorek and Phillips, 2000], placing Apollo 15 well within the area of elevated heat flux and Apollo 17 well outside, creating too large of a difference in heat fluxes between the two sites (see the green line in Figure 19). The plots use 10 mw m 2 mantle flux as opposed the ~7 mw m 2 equivalent value from Wieczorek and Phillips [2000]. This result shows that unless the KREEP layer extends under the Apollo 17 site; this location is likely representative of typical lunar heat flow values. [43] In Figure 19, we see that a 10 km 100% KREEP composition layer dramatically overpredicts the heat flux at the Apollo 15 landing site (green line). This differs from the conclusion of Wieczorek and Phillips [2000], who used a 2-D Figure 17. Surface heat flux given a 10 km thick layer of pure KREEP at the base of the crust (scaled 0 30 mw m 2). Figure 18. Surface heat flux given a 10 km thick layer of one-fourth KREEP composition at the base of the crust (scaled 0 30 mw m 2). Figure 16. Surface heat flux (watt per square meter) resulting from a 5 km e-folding of an ejecta blanket (green) or a 20 km e-folding of an ejecta blanket (red). that an e-folding model exceeding 5e z/5 km is likely not possible (i.e., a 20 km e-folding depth would require either a thicker initial ejecta blanket or one with supra-kreep concentration). This imposes a strict limit on the amount of heat that can be derived from an eject blanket source. Figure 14 also illustrates how deep material could be mixed to with each e-folding scenario before exceeding the total column abundance of Th of the original ejecta blanket. [39] Figure 16 shows our transect through the Apollo sites. In both models (5 km e-folding in green and 20 km e-folding in red), there is a rise in heat flow nearer to Imbrium. This is consistent with the Apollo 15 site having a higher heat flux with the Apollo 17. In order to create the difference between the Apollo sites with an ejecta blanket alone is easily accomplished by the 20 km e-folding model (red), but this places an implausible amount of radiogenic material in the initial ejecta blanket. The maximum plausible 5 km e-folding model can explain the difference, but just at the limit of the measurement errors. Therefore, we find that a radiogenic-rich ejecta blanket alone is likely not a plausible explanation for the heat flux difference between the two Apollo sites. 55

10 Figure 19. Surface heat flux (watt per square meter) resulting from a 10 km layer of 100% KREEP composition (green), 50% KREEP (red), or 25% KREEP (cyan) with 10 mw m 2 background mantle heat flux. model, with a thicker, higher-density crust (similar to that in section 2D). Halving the concentration of KREEP (Figure 19, red line), which has a nearly identical effect to making it half as thin but does not require regridding the mesh, was found to more closely match the heat flux difference between the sites (as does quartering the concentration, Figure 19, cyan). From this plot, it would seem that the Moon with our nominal (2 Wm 1 K 1 ) crust, 10 mw m 2 heat flux (at ~150 km depth), and a 10 km thick layer of one-fourth KREEP composition (or equivalently a 2.5 km pure KREEP layer) could explain much of the difference between the Apollo sites. Were this the lone cause of radiogenic enhancement of the Apollo sites, we could rule out a pure KREEP layer > 5 km. This is consistent with the findings of recent thermal evolution models, which find a KREEP layer thicker than 2 5 km to be inconsistent with the observed gravity signature of the PKT [Grimm, 2013]. However, Grimm [2013] also found that such a thin layer of KREEP would underpredict heat flow for the Apollo sites. We find that there is in fact good agreement if we include the thin crust implied by GRAIL Combined Sources [44] In this study we have seen that each of the proposed causes of heat flux variation, (1) crustal thickness variations, (2) crustal thermal conductivity variations, (3) near-surface radiogenic (KREEP) enrichment, and (4) deep radiogenic (KREEP) enrichment, can have substantial effects on surface heat flow. Cause 4 has the most potential to cause substantial differences between the Apollo sites, while causes 1 3 have either a localized or relatively limited ability to cause the observed differences. However, none of the models are likely to act alone, leaving us with a nonunique, but testable, set of plausible models for the heat production at Apollo HFE sites. [45] Each of these models will have different implications for what the Apollo HFE values mean for the concentration of heat-producing elements within the mantle and the global thermal evolution of the Moon. We will report required values of our free parameter, the mantle heat flux (at 150 km depth) to produce the Apollo HFE values based on our model assumptions. As there are only two Apollo HFE locations each with substantial error in heat flow, we report only on models consistent with the Apollo values, rather than providing an exhaustive multivariable fit. Such fits will be more appropriate with future data analysis (i.e., of GRAIL data) that may provide greater constraints on crustal density structure and presence of a KREEP layer. [46] First, we examine if it is possible to approximate the Apollo HFE measurements without an underlying subcrustal KREEP layer. In Figure 20, we examine the effect of our maximum 5 km e-folding ejecta blanket, the maximum concentration of gardened ejecta blanket we found plausible. An e 5km ejecta blanket has a fairly weak effect on surface heat flux, raising the model Apollo 15 value by roughly 3 mw m 2 and the Apollo 17 by only 1 mw m 2. By increasing the mantle heat flux to 13 mw m 2 (with nominal 2 Wm 1 K 1 crust), this difference can be made to approximate the Apollo values, just within the lower bound of Apollo 15 and upper bound of Apollo 17 (Figure 20, green). Adding an assumption of an e 5km, density crust changes the intensity of heat flux, especially near the Apollo 15 site, but we found 13 mw m 2 still presented an adequate fit (Figure 20, red). [47] Adding a subcrustal KREEP layer is the most effective way to raise surface heat flux. A thin, highly enriched or even pure KREEP layer can be placed at the base of the crust and not be in conflict with any currently existing data. In Figure 21 we examine different plausible scenarios for the addition of a 10 km thick layer with 50% or 25% concentration KREEP (approximately equivalent to a 5 or 2.5 km thick 100% KREEP layer, respectively). Figure 20. Surface heat flux (watt per square meter) that is consistent with Apollo HFE values from a model without subsurface KREEP. Figure 21. Surface heat flux (watt per square meter) that is consistent with Apollo HFE values from a model with a constant density crust and a subcrustal KREEP layer. 56

11 Figure 22. Surface heat flux (watt per square meter) that is consistent with Apollo HFE values from a model with a constant density crust, near-surface ejecta, and a subcrustal KREEP layer. [48] In Figure 21, we assume our nominal crust with constant density (2550 kg m 3 ) and constant thermal conductivity (2 W m 1 K 1 ). With this model, we can obtain the better approximation by placing a one-fourth concentration, 1200 km radius KREEP disk at the base of the crust with 12 mw m 2 of heat from the mantle (green). A slightly poorer approximation was found for a one-half concentration KREEP layer by dropping the basal heat flux to 10 mw m 2. This decrease in heat flux was required to lower the heat flux to within the error of the Apollo 15 measurements but cannot be dropped too low in order to remain consistent with the Apollo 17 measurements. [49] The required mantle heat flux can slightly decrease if there is also an ejecta blanket. Figure 22 illustrates the effect of adding a e 5km mixed ejecta blanket to the one-fourth and one-half concentration KREEP models in Figure 21. The main lever arm we have to adjust is the heat flux at Apollo 17, which remains relatively unaffected by the presence of the KREEP layer as modeled. However, the e 5km ejecta blanket model only adds roughly 1 mw m 2 heat production at the Apollo 17 location. Therefore, this can only be considered a minor perturbation on our baseline subcrustal KREEP models, lowering the plausible mantle heat flow values to 11 and 9 mw m 2. [50] These models assume of a uniform density, uniform thermal conductivity crust, which as seen in section 4.2 and Figure 11, provides a limiting case for the effect a very gradual (> 10 km e-folding) increase in crustal density. Seismic studies [e.g., Hörz et al., 1991] and now GRAIL [Wieczorek et al., 2013] data have shown that an intermediate density model is likely closer to reality. In Figure 23, we examine the effect of placing our modeled one-fourth and one-half KREEP composition layers at the base of a variable density crust. As GRAIL models become more refined, actual crustal density profiles may become possible, but for now our e 5km model appears to be reasonable and consistent with available data. As seen in Figure 23, both one-fourth and one-half KREEP models (which are equivalent to a 2.5 or 5 km layer of pure KREEP) vary more dramatically than the models in Figure 21 and could have large effects on heat flow near thermal properties boundaries. Despite this increased variability, we find roughly the same best fit heat fluxes as with the nominal crust (12 and 10 mw m 2, respectively). Adding an e 5km ejecta blanket to this model will, as shown in Figure 22, simply lower the required heat flux by roughly 1 mw m 2 (to11and 9mWm 2,respectively). 5. Discussion [51] These models imply the lunar mantle heat flux lies somewhere between 9 and 13 mw m 2. The results presented in this section are summarized in Table 1. As GRAIL results suggest that crustal density will vary with depth [Wieczorek et al., 2013], the e 5km crust models may be considered more likely, demanding closer examination of the local mare thickness at the Apollo sites. These examples show that by accounting for likely effects of the crust on measured heat flow, the thinner KREEP layer of Grimm [2013] can be made consistent both with gravity and Apollo HFE data. Our favored mantle heat flow values are higher than past studies (e.g., 4 mw m 2 for Langseth et al. [1976] and ~7.5 mw m 2 for Wieczorek and Phillips [2000]). A Moon composed of chondritic material would only result in about 6.6 mw m 2 [Turcotte and Schubert, 2002]. However, these values are generally only slightly below with the Moon uniformly composed of Earth mantle material (which we estimate as 13.3 mw m 2 assuming a uniform sphere with Wkg 1 heat production, 3400 kg m 3 density, and diameter [ ] = 1584 km radius, neglecting the core) [Turcotte and Schubert, 2002; Jaupart and Mareschal, 2007]. Our estimated values of mantle heat production of 9 13 mw m 2 are consistent with the interior composition of the Moon is not being significantly depleted in radiogenic elements as compared to the Earth s mantle [Ringwood, 1986]. [52] These higher values are near the upper values proposed by Langseth et al. [1976] and fall within reasonable values obtained by Warren and Rasmussen [1987]. The general increase over past models comes, namely, from the effect of the newly revised models of a thinner, lower density lunar crust [Lognonné et al., 2003; Ishihara et al., 2009; Wieczorek et al., 2013]. New models put crustal thickness at approximately half of previous models [e.g., Wieczorek and Phillips, 1998] and about 85% the density or lower near the surface [e.g., Wieczorek and Phillips, 2000]. As the crust is assumed to concentrate more radiogenic elements than the mantle, thinning the crust demands greater heat production in the mantle, or a more widespread subsurface KREEP layer. As the volumetric heat production of the crust is linearly related to density this will also demand more heat originating below the crust. Additionally, the thinner, less dense (and therefore less thermally conductive) crust inhibits horizontal heat flow. Figure 23. Surface heat flux (watt per square meter) that is consistent with Apollo HFE values from a model with an exponentially decreasing density crust (as e 5km ) and a subcrustal KREEP layer. 57

12 Table 1. Mantle Heat Flux and Temperature for Models Consistent With the Apollo HFE (Presented in Section 4) Including Model Predicted Temperatures at 150 km Depth Below the Apollo 15 and 17 Landing Sites Heat Flux A15: T at 150 km A17: T at 150 km One-Fourth KREEP One-Half KREEP e 5km Ejecta One-Fourth KREEP + e 5km Ejecta One-Half KREEP + e 5km Ejecta Nominal crust 12 mw m 2 10 mw m 2 13 mw m 2 11 mw m 2 9mWm K 1100 K 1053 K 1051 K 1048 K 986 K 872 K 1046 K 932 K 819 K e 5km crust 12 mw m 2 10 mw m 2 13 mw m 2 11 mw m 2 9mWm K 1047 K 1016 K 1007 K 999 K 919 K 816 K 974 K 871 K 769 K [53] Additionally, Table 1 shows examples of temperature at the base of our model (150 km below the 1738 km reference sphere), and Figure 24 shows the range of temperature profiles found as a function of depth at both Apollo Heat Flow Experiment sites. Examples of the most extreme cases are plotted in color, showing that the simplified Wieczorek and Phillips [2000] model is likely an overestimate of subsurface temperatures (for the Moon assumed to be in thermal equilibrium). These models clearly show that if a subcrustal KREEP layer is present only below Apollo 15, temperatures there will be far warmer as a function of depth than at Apollo 17. Assuming this difference existed in the past, this should be expected to have a dramatic expression in elastic thickness differences between the two sites. We note that temperatures are consistent with a lack of present-day melting in the upper crust, which should occur at ~1500 K in the top 150 km [Ringwood and Kesson, 1976]. [54] If our favored basal heat flux values (9 13 mw m 2 )are taken as global phenomena, they have several interpretations. This leads to a total heat production of W from the lunar mantle and W when the 34 km average thickness crust is included (representative of the nominal, KREEP-free model). A 1200 km radius, 2.5 km thick cylindrical KREEP layer (fit with12mwm 2 basal heat flux) would add W of total heat production. This study cannot distinguish whether these values representative of the entire Moon or only the PKT region. If these values are representative of the Moon as a whole, this implies that interior is more radiogenic than previously believed. Our range would be equivalent to the Moon made with material times as radiogenic of the Earth s mantle. Such heat flux would be consistent with a slightly enriched Moon as compared to chondritic material, having formed largely of Earth mantle material. [55] This radiogenic enhancement may be in the crust, rather than in the mantle. Global concentrations of surface Th from LP-GRS data average about 1.4 ppm (nearly a factor of 3 higher than the crustal 0.53 ppm Th assumed in this and the Wieczorek and Phillips [2000] model). However, impact crater ejecta hint that Th concentrations decrease with depth in the crust, bringing this estimate down to ~0.98 ppm Th [Warren, 2001]. Future models combining orbital GRS data with a more detailed understanding of impact excavation of material from the lower crust could help probe radiogenic concentration as a function of depth. [56] Additionally, it is plausible that the Moon is not in thermal equilibrium. The thick (~20 km) low-density layer implied by GRAIL data acts to insulate the Moon and potentially lose its initial heat very slowly. Haack et al. [1990] found that such a megaregolith layer could slow heat loss by a factor of 10 in small bodies. This could imply that some of the mantle heat flux we are ascribing to radiogenic heat production is actually a result of this disequilibrium. Such a blanket might retain heat both from the formation of the Moon and mantle radiogenics much longer than previous models [e.g., Konrad and Spohn, 1997]. Models of thermal evolution with nonuniform distributions of heat-producing elements suggest that heat flow may be higher in the KREEP region today than elsewhere on the Moon [e.g., Laneuville et al., 2013]. [57] Alternatively, the high modeled heat flow values at Apollo 15 and 17 sites may reflect a regional heat flux increase, such as would be caused by localized, subcrustal KREEP layer. The thinner crust implied by GRAIL [Wieczorek et al., 2013] and reinterpreted seismic measure- Figure 24. Subsurface temperatures (Kelvin) below (a) Apollo 15 and (b) Apollo 17 that are consistent with both Apollo HFE values. 58

13 Figure 25. Surface heat flux (watt per square meter) with 5km e-folding crust that are consistent with Apollo HFE values from a model with a 1200, 1500, or 1800 km radius subcrustal, 10 km thick one-half KREEP disk. ments [Lognonné et al., 2003] limits lateral heat flux. This would require a subcrustal KREEP layer to extend very near to the Apollo 17 site to explain the difference between the two sites. The subcrustal KREEP region need not necessarily end at radius of 40 (1200 km) as was proposed in Wieczorek and Phillips [2000]. Their 10 km thick KREEP layer would represent roughly 60% of the radiogenic heat production of the entire Moon (0.25 of their total of 0.42 TW). Such a large, or even larger, KREEP body is not inconsistent with the heat production a lunar volume composed of Earth mantle material (neglecting the presence of a core) that would produce 0.42 TW of power (again, roughly 13.3 mw m 2 ). [58] A larger radius KREEP region could also substantially affect the heat flux at Apollo 17 (which is 1600 km from the center of Imbrium). Figure 25 illustrates the effect of a KREEP disk 1500 and 1800 km in radius from the Imbrium center. Given a 1500 km radius one-half KREEP disk, the Apollo 17 site falls on the transition zone and the required mantle heat flux would drop to 10 mw m 2 (9 mw m 2 if a e 5km ejecta layer is added). An 1800 km radius disk of the same composition underlying Apollo 17 could drop the required mantle heat flux to as low as 8 mw m 2 (7 mw m 2 if a e 5km ejecta layer is added). This means that if a KREEP layer is found to extend beyond the Apollo 17 landing site, the lunar mantle could be considered to be chondritic in composition. This emphasizes the importance of having a future measurement of heat flux that is clearly distant from the PKT region. [59] GRAIL Bouguer gravity shows evidence of a positive anomaly (~ mgal) over much of the PKT region [Andrews-Hanna et al., 2013]. This roughly pentagon-shaped anomaly is on the order of magnitude increase that would be implied by a simple calculation of a ~1600 km radius, 5 10 km thick disk of KREEP density material underlying a 30 km thick crust [Turcotte and Schubert, 2002]. This anomaly could be the gravity signature of the subcrustal KREEP layer heat source suggested by Wieczorek and Phillips [2000]. Grimm [2013] calculates that this layer, if present, must be thinner than ~5 km to avoid excess heating of the mantle (which would lower the Bourguer anomaly or even cause it to be negative). This evidence seems generally consistent with our models of an underplating by a 2.5 to 5 km KREEP layer being a likely cause for the increased heat flux at the Apollo sites. Interestingly, the pentagon-shaped Bouguer anomaly reached roughly to the Apollo 17 landing site, suggesting a larger radius KREEP disk model may be consistent with available data. Definitively, identifying the extent of this plausible layer would have dramatic effects on our interpretation of global heat production. [60] A second, but related, localized phenomena is the fossilized remnant of an asymmetric thermal evolution of the Moon. Models have shown [e.g., Laneuville et al., 2013] that a KREEP-rich area will have a feedback on the loss of heat from the mantle below. The subcrustal KREEP layer would act as a hot lid, slowing heat loss from the mantle and causing higher temperatures below the PKT region. In the distant lunar past, this could have led to degree-1 convection [e.g., Zhong et al., 2000] and may retain a substantial present-day impact on lunar interior temperatures. Wieczorek and Phillips [2000] examined a similar scenario to show that recent, or even present-day, melt may be plausible below the PKT. An alternate model of lunar thermal evolution shows that mantle overturn that concentrates heat-producing elements near the core predicts both elevated temperatures under the KREEP region and global contraction consistent with observations [Zhang et al., 2013]. These scenarios can also be further investigated by coupling the models presented here with a model of global thermal evolution. [61] Our study has taken advantage of the wealth of new data available for the Moon to carry out a new analysis of lunar heat flow data. We have reassessed a variety of effects that have a very significant effect on heat flow interpretation. Although current computing capabilities have allowed us to develop much more detailed models than was feasible at the time the Apollo measurements were made, still more detailed models are possible. Our model makes the simplification that the mare basins are constant density disks. This model places both Apollo HFE sites just outside of the modeled mare that leads to a diminished heat flow at both of these locations. In reality, both the Hadley Rille and Taurus-Littrow Valley sites do lie outside the central mare but were filled with local thin ( ~100 m) tongues of mare basalt reaching into highland terrane [e.g., Spudis et al., 1988; Spudis and Pieters, 1991, Lunar Sourcebook]. Additionally, we have used relatively simple density assumptions, such as constant density mare and the globally averaged crustal density increase from GRAIL. Future work could use gravity data to investigate the local density structure at Apollo 15 and 17 in detail. We plan to incorporate plausible local density and geological details into regional-scale modeling in a future study. 6. Conclusions [62] The Apollo heat flow measurements provide important constraints on the composition and thus the origin of the Moon. Three-dimensional heat conduction models such as those presented here represent an important link between localized heat flow measurements and global internal heat production. Our ability to interpret these heat flow measurements is greatly enhanced by new data sets that provide information on crustal thickness and density structure, high resolution topography, and surface temperature and composition. [63] The revised crustal thickness estimates [Lognonné et al., 2003; Ishihara et al., 2009; Wieczorek et al., 2013] and decrease in density over a large portion of the crust lead to a very different interpretation of heat flow than prior studies. 59

14 Using the model of crustal density implied by GRAIL results [Wieczorek et al., 2013] and a thin subcrustal KREEP-rich layer, we found that a roughly 9 13 mw m 2 mantle heat flux would be a most likely case to produce the surface heat flux observed at both Apollo sites. Variations in crustal thickness and the possibility of a near-surface radiogenic-rich ejecta blanket were found to have only minor (< 1.5 mw m 2 ) effects on surface heat flow. [64] These values of internal heat flow are larger than most prior models owing to either global or local effects. If heat flux is elevated globally, this could be due primarily to the insulating effect of the low-density and low-conductivity megaregolith suggested by GRAIL rather than simply higher radiogenic abundance. The presence of such a layer strongly suggests that the Moon has retained a significant portion of its early heat. Future studies will need to consider this possibility and assess the global heat budget related to thicker farside crust and higher conductivity mare bodies. [65] Alternatively, or in addition, the high heat flow values may be a local (restricted to the PKT) effect due to the presence of subsurface KREEP [Wieczorek and Phillips, 2000]. However, new interpretations of crustal thickness mean that a KREEP layer must be very close to or underlie both Apollo 15 and 17. If a KREEP layer does underlie both landing sites, mantle heat flows as low as roughly 8mWm 2 (7 mw m 2 if including the effects of surface radiogenic rich ejecta) are feasible. New gravity analysis [e.g., Andrews-Hanna et al., 2013] may help constrain the extent of such a high-density layer. Subsurface KREEP could also lead to past subsurface melt [Wieczorek and Phillips, 2000] and possibly to asymmetric convection [Laneuville et al., 2013]. [66] Distinguishing between global or local influences requires heat flow measurements away from the PKT, where subcrustal, KREEP-rich material may dominate measured heat flow. The Apollo 17 site is ambiguously close to the edge of this region and, if it is not underlain by KREEP, may imply a high global heat flux. More remote from this region, Diviner Lunar Radiometer polar surface temperature measurements may provide non-pkt constraints on heat flux [Paige et al., 2010c; Siegler et al., 2012]. Although interpreting this data requires detailed three-dimensional modeling, similar to that in this paper, the initial results suggest regional heat flux enhancement in the PKT relative to other areas of the Moon. Further examination of this data will better refine the constraints on non-pkt heat flow. Both the presence of a positive Bouguer anomaly in the PKT region [Andrews-Hanna et al., 2013] and Diviner results [Paige et al., 2010c; Siegler et al., 2012] point toward a regional effect from subsurface KREEP on the Apollo measurements. Future modeling of the longterm thermal evolution in the presence of a low-density crust and detailed analysis of gravity data at the Apollo (and Diviner) heat flow sites will allow us to better access the influence of the global, low-density crustal layer. Finally, future landed measurements in areas distant from the PKT will allow for a detailed understanding of the lunar interior [e.g., Cohen et al., 2008; Neal et al., 2010]. Appendix A: Model Variations in 2-D [67] The previous models to explain the differences between the Apollo HFE measurements can be summarized into four classes: (1) crustal thickness variations, (2) crustal density and thermal conductivity variations, (3) near-surface radiogenic (KREEP) enrichment, and (4) deep radiogenic (KREEP) enrichment. Large temperature changes at depth (~50 K) over short distances (~10 km) will also affect heat flow but are only an important factor in the polar regions of the Moon. [68] Here we will briefly illustrate the effect these four variations can have on a simple two-dimensional model. The models here represent an idealized crustal cross section 100 km across and 60 km deep. The surface is held at a constant 250 K, a typical temperature at ~1 m depth near the equator, and side boundaries reflected to assume an infinite horizontal extent. The lower boundary has been set to 4 mw m 2 ;a suggested value for mantle heat flux [Langseth et al., 1976]. For these simple models, thermal properties are arbitrarily the following: thermal conductivity of 1 W m 1 K 1,density of 2000 kg m 3, and heat capacity of 1000 J kg 1 K 1 (which is not actually important in steady state) unless otherwise noted. All computations were carried out using the commercially available COMSOL finite element package. A1. Model 1: Crustal Thickness Variations [69] As highlighted in Langseth et al. [1976], variation in crustal thickness can be anticipated to dramatically affect surface heat flow. Figure A1a shows an example of a radiogenicfree crustal section, with a mantle heat flux of 4 mw m 2 (the mantle heat flux value derived by Langseth et al. [1976]) along the entire lower boundary. Mantle material with thermal conductivity of 2 W m 1 K 1 is placed in the lower right corner of this model. Here we see that due to the fact that the right column has a higher thermal conductivity, heat is transferred from the left side of the model to the right (Figure A1b, solid red line). A second model with an exaggerated higher mantle thermal conductivity (5 W m 1 K 1 ), which increases this effect, is also shown. Langseth et al. [1976] noted that radiogenic material concentrated in the crust would also enhance surface heat flux. Their estimated 1 ppm Th concentration can be modeled to roughly result in a heat production of Wm 3. Figure A1b (solid black line) shows the change such a crustal heat production would have on heat flux throughout the volume and at the surface. The distance of the peak heat flux from the crustal thickness boundary and its maximum value will be affected by the sharpness on that boundary, so this sharp contrastislikelyanextreme.the5wm 1 K 1 mantle thermal conductivity model shows that this effect can be nearly eliminated if the conductivity difference between the lower layer (crust and mantle) is high enough. Such high thermal properties difference between solid crust and mantle materials is not likely, but such thermal conductivity differences are not unreasonable if the crust is highly fractured or porous. A2. Model 2: Crustal Conductivity Variations [70] Conel and Morton [1975] and later Warren and Rasmussen [1987] highlighted that heat flow can be focused by differences in thermal conductivity. The largest thermal conductivity differences in the lunar crust are at the border between battered anorthasitic highland material and the relatively dense mare basalts. Unfortunately, both the Apollo 15 and 17 landing sites lay at such a boundary. These earlier studies showed that a high-thermal conductivity mare would reroute 60

15 Figure A1. (a) Two-dimensional model with thickness variation without radiogenics. The lower right quadrant is mantle material, given twice the thermal conductivity of the crustal material. A 4 mw m 2 is supplied at the model base. The color scale indicates vertical heat flux in watt per square meter. (b) Heat flux along the top boundary of the model in Figure A1a and several variations, including a mantle quadrant with 5 times crustal thermal conductivity and Wm 3 heat production for crustal materials, approximately equivalent with 1 ppm Th concentration. Figure A2. (a) Two-dimensional model with thermal conductivity variation without radiogenics. The color scale indicates vertical heat flux in watt per square meter. (b) Heat flux along the top boundary of the model in Figure A2a and several variations, including a rounded bottom boundary (marked by the dotted line) and Wm 3 heat production, approximately equivalent with 1 ppm Th concentration. heat from the low-thermal conductivity highlands. This results in a higher than average surface heat flux in the interior edge of the mare and a lower heat flux just outside. Figure A2 shows this effect for a mare 10 km thick with a conductivity of 2Wm 1 K 1 with the crust having 1 W m 1 K 1. This model contains no radiogenic contributions beyond a 4mWm 3 heat flux at the bottom boundary. We can see a sharp shunting of heat at the mare edge. If the mare is more gradual (bowl shaped) or buried, the maximum heat flux peak will become less sharp. Warren and Rasmussen [1987] claimed the megaregolith (roughly the upper 2 10 km of the crust) would have substantially lower thermal conductivity. Figure A3. (a) Two-dimensional model with surface radiogenics reaching from 0 to 50 km in x. This might represent a radiogenic-rich ejecta deposit that has been gardened into the underlying crust by eons of impacts. The color scale indicates vertical heat flux in watt per square meter. (b) Heat flux along the top boundary of the model in Figure A3a and several variations, looking at various e-folding depths as in equation (1). 61

16 Figure A4. (a) Two-dimensional model with a subsurface KREEP layer at the base of the crust reaching from 0 to 50 km in x. This layer is a proposed remnant of a past magma ocean and postulated to lie beneath the PKT. The color scale indicates vertical heat flux in watt per square meter. (b) Heat flux along the top boundary of the model in Figure A3a and several variations, looking at various KREEP layer thicknesses. Figures A2a and A2b show that in this case, a factor of 10 in thermal conductivity (0.1 W m 1 K 1 crust instead of 1 W m 1 K 1 ) can result in about a factor of 50% increase in heat flux just on the mare side of the boundary. In this 1:20 case (0.1 W m 1 K 1 crust, 2 W m 1 K 1 mare), there is also a doubling of the background heat flux inside the mare boundary and a drop to less than one fourth of the background heat flux just outside. A3. Model 3: Near-Surface KREEP Variations [71] Hagermann and Tanaka [2006] suggested an ejecta blanket similar to that of Haskin [1998] would be mixed into the regolith, forming an exponentially decreasing concentration with depth. Here we adopt that same model by placing material with KREEP radiogenic composition ( Wm 3 heat production) at the surface, concentration c 0, and having concentration decrease as in equation (1). Later, we will also use this formulation to approximate quantities that increase or decrease dramatically with depth (e.g., radiogenic mixing, density, and thermal conductivity profiles). [72] In Figure A3, with a surface KREEP layer from 0 50 km in x, we can see that this has a dramatic effect on surface heat flow which increases as the mixed layer thickness increases (essentially because there is more radiogenic material). There is a substantial amount of lateral heat flux through the crust below the radiogenic-rich layer, causing the heat flux gradient at the boundary to soften as radiogenics reach greater depths. This example is exaggerated as surface concentration of pure KREEP, as illustrated here, is not likely to occur as KREEP would be mixed with less radiogenic material. In the full model surface concentration, c 0, will be set by the measured [Lawrence et al., 1998] radiogenic concentration at the lunar surface. A4. Model 4: Subcrustal KREEP Variations [73] The final model we examine here follows Wieczorek and Phillips [2000], who suggested a subsurface pure KREEP layer remaining from the solidification of an early magma ocean. [74] This model offers a way to include more radiogenic material in a crustal column than the surface ejecta, as this pure KREEP layer could be rather thick (Wieczorek and Phillips [2000] suggested up to two thirds of the crustal column could be composed of KREEP). In Figure A4, we can see that placing a KREEP layer from 0 to 50 km in x (the horizontal dimension) with various thicknesses can also have a dramatic effect on surface heat flux and lateral variations. As the heat must conduct through the entire crustal column, the heat flux gradient across the compositional boundary is less sharp than in the case of surface KREEP. This sharpness will depend heavily on total crustal thickness, which is now proposed to be quite thin (~20 30 km) in the PKT area [Wieczorek et al., 2013] [75] Each of these four primary models can be the cause of lateral variations in surface heat flux, and each has been claimed by past authors to be the dominant cause for the difference between the Apollo HFE sites. However, none of these authors had access to the wealth of knowledge about the lunar crust as exists today, and none combined all of these effects, which are not mutually exclusive, in a realistic scenario. [76] Acknowledgments. Thank you to Mark Wieczorek for providing quick access to GRAIL crustal thickness models, which greatly enhanced this paper s relevance. Thank you to Mark Wieczorek, Jonathan Besserer, and Walter Kiefer for their detailed and extremely helpful reviews. Thank you to Pierre Williams, Paul Warren, David Paige, and Christophe Sotin for their useful discussion and aid. This work was supported in part by the Brown/MIT node of the NASA Lunar Science Institute. The research was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration (Copyright 2013 California Institute of Technology). Government sponsorship is acknowledged. References Andrews-Hanna, J. C., et al. (2013), Ancient igneous intrusions and early expansion of the Moon revealed by GRAIL gravity gradiometry, Science, 339(6120), Carrier, W. D. (1974), Apollo drill core depth relationships, Moon, 10, Carrier, W. D., G. R. Olhoeft, and W. 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