Corresponding author: R. E. Grimm, Planetary Science Directorate, Southwest Research Institute, 1050 Walnut St #300, Boulder, CO 80302, USA. (email@example.com)
 The Moon's Procellarum KREEP Terrane (PKT) is distinguished by unique geochemistry and extended volcanic history. Previous thermal-conduction models using enhanced radionuclide abundances in subcrustal potassium, rare earth elements, and phosphorus (KREEP) predicted the existence of a contemporary upper-mantle melt zone as well as heat flow consistent with Apollo measurements. Here I show that such models also predict large gravity or topography anomalies that are not observed. If the topography is suppressed by a rigid lithosphere, it is possible to eliminate the gravity anomaly and still match heat flow by completely fractionating the excess radionuclides into a thin crust. This implies that upper-mantle heat sources for mare volcanism were spatially discontinuous or transient and that radionuclides defining the PKT are not necessarily directly related to mare volcanic sources. However, the mantle temperature of a crustally fractionated PKT is insufficient to match the observed electrical conductivity: globally enhanced mantle heating or a thick megaregolith may be required. Alternatively, upper-mantle enrichment in iron, hydrogen, or aluminum can provide the requisite conductivity. Iron is the most plausible: the derived lower limit to the upper-mantle magnesium number 75–80% is consistent with seismic modeling. Regardless of the specific mechanism for electrical-conductivity enhancement, the overall excellent match to simple thermal-conduction models indicates that the lunar upper mantle is not convecting at present.
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 The Procellarum KREEP Terrane, or PKT, may be the primary surface manifestation of asymmetrical thermal evolution of the Moon [Jolliff et al., 2000]. The PKT—encompassing Oceanus Procellarum, Mare Imbrium, and the adjoining mare and highlands—is distinguished by unique geochemistry and spatially grouped, prolonged volcanism. The former is dominated by potassium, rare-earth elements, and phosphorous (KREEP), but enhanced thorium has been one of the defining orbital measurements [Lawrence et al., 1999]. All of these elements are incompatible, implying they were segregated into the last unmelted fraction of the magma ocean [“urKREEP,” Warren and Wasson, 1979]. It is important to understand the thermal and melting record of PKT, the most complex event in lunar history postdating the magma ocean.
 Among the many models for the thermal evolution of the Moon (see Shearer et al.  for a review), two have focused in detail on the role of anomalous subcrustal heating for PKT. Wieczorek and Phillips [2000, hereinafter WP00] assumed that a 10 km thick KREEP-rich layer with radionuclide heating ~300 times chondritic was emplaced beneath the PKT crust at 60 km depth. PKT was represented as a spherical cap in a 2-D axisymmetric conduction model. Partial melting developed to several hundred kilometers depth and persists today, a feature that is suggested to explain the source depth and duration of mare volcanism. The model is also marginally consistent with elevated heat flow observed at the Apollo 15 site. Hess and Parmentier  produced similar heating in a 1-D thermal-conduction model but discounted it because it was geochemically incompatible with the parent magmas of the magnesian suite, it would form an impenetrable buoyant melt layer, and it implied an elastic lithosphere too thin to support mascons. They advocated quicker cooling under a thinner anorthosite crust.
 Remotely observable geophysical properties provide powerful constraints on planetary interiors. Here I analyze the implications of thermal-conduction models with spatially fixed internal heating [Wiezcorek and Phillips, 2000] for the topography, gravity, and electrical conductivity of PKT. I then make some general inferences on the heat sources for mare volcanism and the composition and contemporary thermal state of the lunar mantle.
2 Data and Methods
2.1 Thermal Model
 In order to reproduce the principal features of the WP00 model and extend it to alternate heating configurations, Comsol Multiphysics (v3.5a, www.comsol.com) was used to model 2-D, axisymmetric, time-dependent thermal conduction with the finite-element method. PKT was assigned to the symmetry (vertical) axis with a half-width of 40°. Crust, mantle, and the differentiated KREEP layer were assigned unique thickness, thermal conductivity, density, heat capacity, and concentrations of radioactive U, Th, and K (Table 1; also see Table 2 in WP00). The U and Th abundances in the KREEP layer are ~300 times chondritic, whereas the K abundance is 7× chondritic. The crust is somewhat enhanced in U and Th, whereas the mantle is depleted. Note that bulk heat production in the Moon is still uncertain by up to a factor of 2 [Wieczorek et al., 2006]. For simple interaction with the Comsol graphical user interface, a single exponential function was fit to radioactive heating for each of the materials. Based on geochemical arguments, WP00 suggested that PKT contains the equivalent of 30 or 40 km of KREEP basalt, but all of their models use just 10 km, either applied to the base of the crust or distributed within it. The same 10 km equivalent KREEP thickness was adopted here. For the remainder of this paper, the full complement of radionuclides in the 10 km KREEP layer of WP00 is denoted H0, and fractional abundances thereof are denoted H′ = HPKT/H0.
 No melting algorithm was implemented in the present study, which produces somewhat larger temperature excursions compared to WP00. However, an empirical adjustment of ~20% to the overall heating (pre-exponential factor) resulted in an extant temperature profile through the middle of PKT and heat flow across the Moon that is in good agreement with WP00 (Figure 3). Anomalous contemporary temperatures ΔT throughout the model were computed by subtracting the radial temperature profile 90° from PKT and were converted to density anomalies as Δρ = −ρ0αΔT, where ρ0 = 3400 kg m−3 and α = 3 × 10−5 K−1 [Turcotte and Schubert, 1982]. Note that this approach does not distinguish the crust and mantle, but the absolute error is <15% for the crustal density and the mantle contribution is several times that of the crust anyway.
 Two sets of models are reported here. The “underplating” model was the nominal case, in analogy with WP00: KREEP was distributed beneath PKT as a 10 km thick layer underlying a 60 km or 30 km thick crust. The larger crustal thickness was chosen simply for consistency with WP00, reflecting earlier consensus, and the smaller value was considered to be the minimum for the nearside [Wieczorek et al., 2006; Lognonné and Johnson, 2007]. The starting temperature was taken everywhere to be 1450 K. This approximates an adiabat following WP00, assumed to represent the state just after crystallization of a convecting magma ocean. The second set of models distributed the anomalous heat sources uniformly across the 30 or 60 km thick crust. The initial temperature was taken to lie on the mantle solidus [Ringwood, 1976]: Ts = −1.5 × 10−4z2 + 0.65z + 1430 K, where z is the depth in kilometers. This was chosen to maximize temperatures throughout lunar history in a model that would otherwise see sharp decreases due to upward fractionation of radionuclides. The solidus would represent the state just after crystallization of a nonconvecting magma ocean.
 Because PKT is a large feature of the Moon, coarse topography and gravity are adequate for general analysis. I used the one-degree grids generated by Wieczorek et al.  from the Clementine topography of Smith et al.  (model GLTM2C) and the Lunar Prospector gravity of Konopliv et al.  (model LP150Q). PKT is topographically characterized by regional negative relief of order −2 km that is closely associated with the western maria. Because the isostatic topographic response to anomalous heating is uplift, the observed topography has the opposite sign to the prediction. Less heating or a thick elastic lithosphere will inhibit the uplift, but the sign discrepancy will always exist. Therefore, the relevant model constraint is simply to minimize uplift, and it is not necessary to plot the observed topography here.
 The predicted isostatic topography h was calculated by force balance with the vertical integral of the anomalous density Δρ as a function of radius r:
where ρc = 2900 kg m−3, a is the Moon's radius, and W(r) is a weighting factor. The weighting factor (r/a)3 accounts for the change in surface area (r/a)2 and the change in buoyancy due to the change in gravity within a uniform sphere (r/a). The integral was limited to depth L = 600 km for consistency with the subsequent gravity calculations: the bulk of density variations are above this depth, but it allows curvature to be neglected and the gravity calculated in cartesian coordinates.
 Bending and membrane stresses in an elastic lithosphere will suppress uplift. Here I simply treat the end-members in which the lithospheric rigidity D = 0 and D = ∞. The former allows the maximum isostatic uplift. The gravity is minimized but nonzero due to the incompletely canceling effects of the topography and compensating subsurface Δρ. The latter implies zero uplift and a maximized gravity anomaly given by the influence of all subsurface Δρ. Using the method of Turcotte et al. , it can be shown that lunar elastic lithospheres 150–200 km are essentially rigid, as deflections are <10% of the zero-lithosphere case.
 The one-degree gridded gravity was reprojected in azimuthal equal-area format around the center of PKT, and the Serenitatis, Imbrium, and Humorum mascons were masked by assigning a constant regional value. A cosine-tapered filter centered at 840 km wavelength was applied following a Fourier transform. The resulting map (Figure 1) reveals a quasi-annular low associated with the outer parts of PKT surrounding a gravity high centered on the Copernicus region. The total range in the long-wavelength gravity is less than ±100 mGal. If the topography is largely suppressed, then the thermal anomaly must produce a negative gravity anomaly, so I ignore the observed central high and focus on the surrounding lows. The median of all the negative values in PKT is just −30 mGal in the filtered map and −50 mGal in the unfiltered map. I therefore take −50 mGal as an upper limit to the PKT regional negative gravity anomaly.
 In principle, basalts filling the PKT maria could generate a positive gravity anomaly that masks the negative anomaly from elevated temperatures at depth. This can be ruled out because (1) the average mare thickness on the nearside is generally less than 400 m (see review by Wieczorek et al. ), which would result in a maximum (uncompensated) gravity anomaly of +50 mGal—even this will be seen to be small compared to the expected negative anomalies from the upper mantle—and (2) the mantle and surface contributions to gravity do not neatly superpose. Mantle temperature anomalies are both diffuse and low-pass filtered, and so the resulting gravity anomalies extend far beyond the mare boundaries of PKT.
 The gravity anomaly Δg is calculated from the modeled thermal anomaly using vertically distributed mass sheets in the spectral domain [after Turcotte and Schubert, 1982]:
where k is the wavenumber, G is the gravitational constant, Δz is the constant layer thickness, and Δρi, zi, and Ai = (1 − zi/a)2 are the density anomaly, depth, and geometric correction of the ith layer, respectively (a is the Moon's radius). Again, this cartesian summation was carried out to a depth of 600 km. Results were spot-checked against a direct integration of equivalent point masses on a sphere. Where topography h is present, its contribution is 2πGρch(k).
2.4 Electrical Conductivity
 The electrical conductivity σ(z) of the lunar interior was estimated from the time-dependent magnetic transfer functions between the Explorer 35 satellite and the Apollo 12 site (see Dyal et al.  and Sonett  for reviews). I have converted the “high-frequency” Apollo-Explorer transfer functions [Sonett et al., 1972] to frequency (f) dependent apparent conductivities (Figure 4) following equation (4) of Grimm and Delory . The apparent conductivity σa(f) is the conductivity of a uniform halfspace having the same electromagnetic (EM) response as the target under test. Because σa is dimensionally the same as σ, it provides a more intuitive representation of the lunar EM response.
 The EM field attenuates to 1/e in one skin depth δ (km) = 0.5/√σaf, where f is the frequency in hertz. Because the horizontal resolution is comparable to a skin depth [e.g., Vozoff, 1991], the high-frequency Apollo 12 data are sensitive to both lateral and vertical scales of several hundred kilometers. This implies that the upper mantle beneath PKT is emphasized, as the Apollo 12 site is within several hundred kilometers of the center of the ~2400 km wide region. Note that Hood et al.  emphasized the deeper, global structure recovered from lower frequencies.
 Next consider the controls on electrical conductivity in silicates (see Tyburczy  and Yoshino  for reviews). Mobile point defects due to impurity substitution determine the electrical conductivity of Earth's mantle. There are three main conduction mechanisms in rocks [Yoshino, 2010]. Each follows a Boltzmann distribution and is therefore strongly temperature dependent. Above ~1800 K, migration of cation vacancies (“ionic conduction”) dominates. From ~1300 to 1800 K, hopping of electron holes between ferrous and ferric iron (the “small polaron”) is the most important. Although ferric iron is effectively absent mineralogically under the highly reducing conditions attributed to the lunar mantle [e.g., Papike et al., 1991], only trace quantities are required electrically. This is evident from the measured dependence on iron content of the conductivity of olivine [Cemič et al., 1980] and garnet [Romano et al., 2006] at or below the reducing iron-wüstite buffer. Furthermore, early laboratory studies relevant to the Moon [Huebner et al., 1979] revealed that orthopyroxene (opx) conductivity increases with the contents of aluminum and chromium, trivalent cations that can substitute for Si4+. This substitution is also hypothesized to cause a charge-compensating increase in ferric iron content, i.e., production of a small polaron. Note that recent attempts in the planetary literature to develop general formulae for conductivity as a function of bulk oxide content [Verhoeven et al., 2005; Khan et al., 2006] may capture the effect of iron (in olivine) but do not yet account for the substitution of aluminum in orthopyroxene.
 Below ~1300 K, proton migration is the largest contributor to electrical conductivity. Water is the most obvious source of protons for the Earth, but it is more likely that protons would be derived from H2 in the interior of the Moon [Sharp et al., 2012]. Consequently, H2 and H2O can be treated equivalently, with H/Si in forsterite equal to 16 times the weight fraction of H2O. Laboratory relationships for conductivity as a function of temperature and water content were developed for Mg-rich olivine by Yoshino et al.  and Poe et al. . The laboratory conductivities were measured at the Mo-MoO2 buffer, which should be representative of the expected low oxygen fugacity of the lunar interior. These measurements were independent yet in good agreement: the water abundances required to match a specified conductivity at some fixed temperature differ only by about a factor of 2. Alternative experiments for the electrical conductivity of olivine [Wang et al., 2006] call for H2O abundances 2 orders of magnitude smaller at the same temperature and conductivity. Similar low water content was obtained by these researchers for orthopyroxene [Dai and Karato, 2009]. A recent measurement of orthopyroxene by Yang et al.  has even higher conductivity. Yoshino et al.  suggested that the samples of Wang et al.  contained interstitial fluid, which would increase conductivity and mask proton hopping.
 I modeled the composition- and temperature-dependent electrical conductivity of the lunar mantle using present-day geotherms from the thermal models described above and published conductivity formulae and data for relevant minerals, assuming depth-independent composition. The weak pressure dependence was ignored. The baseline uses olivine (Fo90) in the anhydrous limit of Poe et al. . The variation in olivine conductivity as a function of iron content was scaled from Cemič et al. : in order to assure a consistent end-member at Fo90, a ratio applied to the results of Poe and coworkers was preferred over using the absolute values of Cemič and colleagues. Hereafter I treat bulk iron content in olivine in terms of the complementary magnesium number XMg, which is equivalent to the forsterite mole fraction. Alternatively assuming that orthopyroxene electrically dominates the lunar upper mantle, I used the measurements of Huebner et al.  at alumina weight fractions of 0.14%, 1.9%, and 6.8%. Finally, the effect of H2O (or H) on Fo90 was modeled using the formulae given by Yoshino et al.  and Poe et al. . The conductivity curves of these two teams have similar shapes due to similar derived activation energies, so any such curve can be essentially described by lower and upper bounds in water content appropriate to the two research groups. Implications of alternative measurements introduced above are addressed in section 4. The presence of melt would greatly increase electrical conductivity, but it is neglected because it will be seen that satisfactory thermal profiles do not approach the solidus.
 The EM response of a layered sphere [Grimm and Delory, 2012: Equations 7–11] was used to calculate σa(f) from σ(z). No inversion for σ(z) was performed: the overall σa(f) plots produced from the thermal models sufficiently matched the data a priori to allow large-scale interpretation.
 Density anomalies predicted from representative thermal models are shown in Figure 2. The implementation of the nominal KREEP underplating from WP00 (H′ = 1, first row in Table 2) yields temperature and heat flow in good agreement with that work (Figures 3a and 3b). Contemporary temperatures beneath PKT at depths ~300–600 km exceed the solidus (recall that melting is not explicitly modeled: this is identified only for tracking). Heat-flow q peaks at 32 mW/m2 beneath PKT. This is on the high end of what is consistent with the Apollo 15 measurements [Warren and Rasmussen, 1987] on the flank of PKT but is a good match with the Apollo 17 measurements approaching the background. However, the unconstrained topographic uplift from this heating configuration is ~7 km (Figure 3c). Predicted gravity signatures (Figure 3d) also cannot be reconciled with the data: when the topography is completely suppressed, an anomaly exceeding −800 mGal appears.
Table 2. Predictions of Thermal-Conduction Models
 Reducing the subcrustal KREEP heating reduces the mantle thermal anomaly, which in turn reduces the gravity anomaly (Table 2; all results for an infinitely rigid lithosphere). However, subcrustal heating must be effectively eliminated (H′ < 0.1) in order to reduce the gravity anomaly to an acceptable value. But the heat flow is insufficient for H′ < 0.25, and partial melting ceases earlier than 2.4 Ga for H′ < 0.6. The latter is intended to capture the tail of mare volcanism [Hiesinger et al., 2000]; a limit at 1 Ga [Spudis, 1999] requires H′ > 0.7.
 There is little improvement if KREEP is underplated to a 30 km crust (Table 2). H′ = 0.2 now produces an acceptable gravity anomaly, but the heat flow is still too low, and temperatures still fall below the solidus early in the Moon's history.
 The topography and gravity anomalies are muted if KREEP is distributed throughout the crust, a consequence of reduced temperature and density contrasts at depth (e.g., Figure 2b). There is still no consistent solution between gravity anomaly and heat flow for a 60 km thick crust: when the heating is low enough to fall with the gravity bounds, the heat flow is too low. However, for the 30 km thick crust, there is sufficient heat flow at H′ = 0.5 to be marginally consistent with Apollo, and the gravity anomaly for zero topography is just −30 mGal. On the other hand, there is virtually no melting from such shallow and distributed heat sources. The “hot start” was introduced for this series of models to examine the retention of deep melt during secular cooling. Table 2 shows that at 3.6 Ga, near the peak of mare volcanism [Hiesinger et al., 2000], the depth to the solidus is already approaching the 500 km limit of the mare basalt source region (see Shearer et al.  for a review). However, the inferred melt regions remaining today in these models have radii ~500 km, which is consistent with the 480 km radius outer-core melt zone derived from seismology by Weber et al. .
 Electrical conductivity constrains the thermal state of the PKT upper mantle. The nominal crustal underplating at H′ = 1 produces the hottest contemporary temperatures. Even the most poorly conducting mineral (anhydrous Fo90) produces apparent conductivities in excess of Apollo data at these temperatures (Figure 4). Indeed, all H′ > 0.6 are too conductive. Conversely, crustal-underplating models with lower KREEP abundance and all of the models with KREEP distributed throughout the 30 km crust are cold enough that “dry,” Mg-rich olivine is insufficiently conductive to match the Apollo data. The case H′ = 0.5 (optimum for gravity and heat flow) is illustrated. The seismically derived temperatures of Khan et al.  and Kuskov and Kronrod  are colder than the background temperature in the thermal models and hence even less conductive (Figure 4).
 There are several ways in which the electrical conductivity of the optimum thermal model can be raised and still satisfy gravity (for zero topography) and heat flow. First, the mantle radionuclide abundance can be increased threefold (Figure 4). A global change like this would not affect the PKT gravity anomaly (which depends on lateral differences), and it is still in-bounds for the heat flow. It is, however, a very large excursion from the accepted range described by Wieczorek et al. . Second, the upper 5 km of the Moon can everywhere be represented as a megaregolith with thermal conductivity 0.2 Wm−1 K−1. The thermal conductivity follows that suggested by Warren and Rasmussen , but the thickness is about double that suggested by those workers. Third, increasing the iron, aluminum, or hydrogen content of the mantle will raise the conductivity yet have no affect on the other modeled geophysical observables. An anhydrous olivine-dominated upper mantle with XMg = 75–80% satisfies the Apollo data (Figure 5), as does olivine at XMg = 90% and 200–500 ppmw H2O (H/Si = 0.3–0.8%). If orthopyroxene dominates the upper-mantle conductivity, the weight fraction of Al2O3 must exceed 1.9% and possibly approach 6.8%.
 The KREEP radionuclide abundance and crustal underplating modeled by Wieczorek and Phillips  ensure long-lived, regional partial melting of the lunar upper mantle, thus providing a simple explanation for the duration of mare volcanism, its mean source depth, and its general association with PKT. This model is also in marginal agreement with the Apollo 15 heat flux measured on the flank of PKT. I find that the radionuclide abundance can be reduced by a third (say to 200× chondritic) and still produce deep-seated partial melting until the end of the main phase of mare volcanism between 2 and 3 Ga [Hiesinger et al., 2000]. This reduced heating is also in better agreement with the Apollo 15 heat flow. However, the large temperature excursion and resulting buoyancy should result in a large topographic uplift and gravity anomaly. If the uplift is suppressed by a rigid lithosphere, the gravity anomaly increases. These geophysical anomalies can be largely eliminated by distributing the KREEP radionuclides throughout a thin PKT crust while still matching the heat flow. In this case, a long-lasting upper-mantle partial-melt zone cannot be sustained. WP00 recognized that radionuclide upward redistribution would shut down melting prematurely.
 It is reasonable to infer that virtually any static configuration of spatially continuous, strong radioactive heating at depth will produce major gravity or topographic signatures that are not observed. Therefore, the heat source for mare volcanism must have been discontinuous, transient, or both. It is important to recall that >1/3 of mare resurfacing is outside of PKT (see tabulation by Whitford-Stark ). Mare basalts could have been generated from numerous, smaller regions of anomalous radionuclides at established source depths of 100–500 km. Alternatively, thermal or compositional convection could have been responsible for mare volcanism, and gravity-evident upper-mantle density contrasts dissipated as long as convection ceased one or two billion years ago. Specialized initial conditions in thermo-chemical convection simulations indicate that degree-1 (hemispheric) asymmetry can be caused by downwelling of post-magma-ocean cumulates [Parmentier et al., 2002] or subsequent upwelling of sunken, reheated cumulates [Zhong et al., 2000]. The former requires that the cumulate layer has a very low viscosity (in contrast to expectations), and the latter requires that the lunar core is very small (see Shearer et al.  for discussion). Published results do not indicate if the duration of mare volcanism can be accounted for by these transient mechanisms. Note that the long-lived degree-1 asymmetry produced in the convection model of Laneuville et al.  is due to choosing the same initial conditions as WP00, i.e., the results are dominated by conduction from the PKT KREEP basalt layer.
 The electrical conductivity depends on both the thermal state and composition of the mantle. It is significant that the apparent electrical conductivity is never overpredicted by thermal models that satisfy the gravity data and that under such conditions reasonable fits could be obtained simply by increasing the overall conductivity via mantle heat production, regolith thickness, or abundance of elements that lead to electrically conductive point defects. The relative changes with depth are reasonably fit by the exponential dependence on temperature, so a purely conductive thermal model is a very good representation for the electrical conductivity. In other words, solid-state convection at present can be ruled out to the maximum EM skin depth ~800 km due to the strongly superadiabatic geotherms inferred from the high-frequency electrical-conductivity data. This is not surprising, as even early thermal models [Toksöz et al., 1978] predicted that present-day convection would be restricted to depths >700–800 km.
 For an anhydrous mantle with electrical conductivity dominated by olivine, the best fitting magnesium number lies in the range 75–80%. Although this is applied to the whole mantle in the model, the “high-frequency” Apollo EM sounding data are most sensitive to the depth range 400–600 km. Seismic modeling by Kuskov and Kronrod  yielded XMg from 270–500 km as low as 73.5%, with a whole-mantle value ~83%. In contrast, Khan et al.  used the seismic data to derive a whole-Moon XMg = 83%, with little or no depth variation in the mantle. Furthermore, recent analysis of lunar meteorites [Warren, 2005] suggests an Earth-like whole-Moon XMg = 88–89%. Therefore, enhancement of electrical conductivity by ferric iron in olivine alone is plausible but at the lower limit of other published results.
 The abundance of water in the lunar mantle—formerly thought to be a closed case at essentially zero—became controversial again when Saal et al.  detected H2O in some lunar volcanic glasses using modern analytical techniques. They estimated that the source region contained a minimum of 260 ppmw. Several other studies suggest 2–200 ppmw H2O: see the review by Elkins-Tanton and Grove . However, these workers argued that the bulk lunar interior contained <10 ppmw H2O based on a geochemical fractionation model of the magma ocean as well as the well-established low oxygen fugacity. Sharp et al.  described how, if H2 is the dominant phase in the H-O system, oxygen fugacity and the H2O/H2 ratio would be buffered by iron metal, therefore allowing relatively slow loss of H2.
 The best fitting H/Si ratio ~0.5% is equivalent to ~300 ppmw H2O, comparable to the minimum value derived by Saal et al. . The distinction is that the hydrogen content inferred from electrical conductivity is that of the lunar upper mantle today, whereas sample analysis gives the water content (or hydrogen equivalent) early in lunar history. It should be noted that if the alternative laboratory relationships for the electrical conductivity of olivine [Wang et al., 2006] and orthopyroxene [Dai and Karato, 2009] are adopted instead, the best fitting H/Si is ~30–50 ppm, equivalent to just 2–5 ppmw H2O. However, these latter results represent fits and extrapolations of just a handful of different measurements at H2O concentrations far above the derived range. Nonetheless, the present lack of convergence in the literature of the electrical conductivity of water-bearing silicates requires that the hydrogen content inferred from interpretation of EM sounding data is treated as an upper bound.
 For orthopyroxene-dominated mantle conductivity, Hood et al.  found that 1.9–6.8 wt % Al2O3 was consistent with thermal models of that time. This conclusion is reinforced by the optimum thermal model presented here, which is not far above the global background. However, such high alumina is inconsistent with the Moon's composition. The Al2O3 content of the lunar upper mantle (from which the crust differentiated) is likely just a few weight percent [Shearer et al., 2006], which can be accommodated in a comparable fraction of spinel or, at greater depth, several percent garnet. Such abundances of spinel and garnet are inferred from modeling of lunar seismic velocities [Khan et al., 2007; Kuskov and Kronrod, 2009]. Several percent alumina in the pyroxene would translate to a few bulk percent Al2O3—equaling or exceeding the estimated total—in turn calling for an unrealistically high partition coefficient. Therefore, electrical conduction generated by aluminum substitution is the least likely of the mechanisms discussed here.
 Heating from a thick KREEP layer below the PKT crust would create very large gravity or topography anomalies that are not observed. However, KREEP distributed throughout a thin crust can match the Apollo heat flow without producing gravity or topography signatures. These results imply that the heat source for mare volcanism may not be directly related to surficial radionuclides that define PKT.
 Temperature profiles produced by thermal conduction yield good matches to the Apollo EM sounding data. However, some enhancement in electrical conductivity is required over a nominal anhydrous Fo90. A magnesium number of 75–80% for the lunar upper mantle fits the EM data. Alternatively, proton hopping can supply the required enhanced conductivity, using water abundances comparable to the Earth's upper mantle. This seems doubtful given the highly reducing conditions of the lunar interior: H2 is preferred over H2O. Enhanced mantle heating or a thick megaregolith would further reduce the need for enhanced electrical conductivity using either iron or hydrogen. High conductivity due to aluminum substitution in orthopyroxene can be ruled out due to the large quantities and high partition coefficients necessary.
 A lunar geophysical network with seismic, electromagnetic, and heat-flow experiments could resolve contrasts between PKT and the rest of the Moon, with joint inversion separating thermal versus compositional differences.
 The Gravity Recovery and Interior Laboratory (GRAIL) mission recently found that the porosity of the lunar crust is ∼12%, consistent with brecciated rocks (Wieczorek et al., submitted to Science, 339, 671, 2013). The low thermal conductivity for megaregolith advocated by Warren and Rasmussen  may therefore be representative of much of the crust. Hence, higher interior temperatures due to thermal insulation by an extensively fractured crust may be largest factor contributing to the high electrical conductivity of the lunar mantle.
 This work was partially supported by the NASA LASER Program and the ARTEMIS mission. I am grateful to Barb Cohen, Steve Mackwell, Brent Poe, Roger Phillips, and Jim Tyburczy for helpful discussions. Francis Nimmo, Mark Wieczorek, Olivier Verhoeven, and an anonymous reviewer provided very constructive comments.