Martian thermal state and evolution depend principally on the radiogenic heat-producing element (HPE) distributions in the planet's crust and mantle. The Gamma-Ray Spectrometer (GRS) on the 2001 Mars Odyssey spacecraft has mapped the surface abundances of HPEs across Mars. From these data, we produce the first models of global and regional surface heat production and crustal heat flow. As previous studies have suggested that the crust is a repository for approximately 50% of the radiogenic elements on Mars, these models provide important, directly measurable constraints on Martian heat generation. Our calculations show considerable geographic and temporal variations in crustal heat flow, and demonstrate the existence of anomalous heat flow provinces. We calculate a present day average surface heat production of 4.9 ± 0.3 × 10−11 W · kg−1. We also calculate the average crustal component of heat flow of 6.4 ± 0.4 mW · m−2. The crustal component of radiogenically produced heat flow ranges from <1 mW · m−2 in the Hellas Basin and Utopia Planitia regions to ∼13 mW · m−2 in the Sirenum Fossae region. These heat production and crustal heat flow values from geochemical measurements support previous heat flow estimates produced by different methodologies.
 The incompatible radiogenic isotopes of K, Th, and U provide the most important source of heat in the terrestrial planets. Normally these elements are preferentially sequestered into a planet's crust during differentiation [Taylor and McLennan, 2009], and this is especially true for Mars, which possesses a thick and mostly ancient crust that is proportionally large with respect to the planet's total volume compared to Earth. The Gamma-Ray Spectrometer (GRS) instrument on board the 2001 Mars Odyssey spacecraft has been used to map the K and Th abundances across nearly the entire Martian surface [Boynton et al., 2007]. Here we present the first detailed Martian surface heat production and crustal heat flow maps based on unambiguous orbital geochemical measurements that show significant geographic variation in the crustal thermal reservoir. These results are valuable for better understanding Martian geodynamics, crust-mantle evolution, the cryosphere, formation and history of geologic provinces, and many other varied applications.
 Orbital GRS data are of lower spatial resolution than most other orbital remote sensing instruments and, therefore, are best suited for global or large, regional-scale studies, rather than detailed, local analyses of geographically small features and landforms. Elemental abundance data for K and Th have been compiled into smoothed maps with 5° × 5° per pixel resolution [Boynton et al., 2007; Taylor et al., 2006a, 2006b]. At this time, it is not possible to generate a useful map of U abundances at comparable resolution (due to U concentrations too low to provide the necessary gamma-ray detection counting statistics). Fifty percent of incident gamma rays collected by the GRS are from a footprint of 440 km diameter. The penetration depth is on the order of a few tens of centimeters, depending on the density of the underlying material (see Boynton et al.  for technical specifications). Data were collected and summed over 2 mapping periods from June 2002 through April 2005 and from April 2005 through March 2006. Although the GRS instrument can determine K and Th abundances at high latitudes, extremely high concentrations of water ice (e.g., very near the poles) dilute these elemental signatures. Also, H concentrations have a strong influence on the correction techniques used for determining other elemental abundances. To compensate for these complexities, we have excluded data pole-ward of ∼45°–60° latitude in both hemispheres using a cut-off based upon water equivalent hydrogen concentration – the H-mask described by Boynton et al. .
2. Surface Heat Production
 Using smoothed GRS global K and Th maps where the data have been binned into 5° × 5° pixels, we determined the radiogenic 40K and 232Th surface abundances for each GRS pixel based on well-determined isotopic fractions. Currently, 232Th is 100% of total Th abundance with a heat release constant of 2.64 × 10−5 W·kg−1; 235U and 238U are 0.7204% and 99.2742% of total U abundance with heat release constants of 5.69 × 10−4 W·kg−1 and 9.46 × 10−5 W·kg−1, respectively; and 40K is 0.012% of total K abundance with a heat release constant of 2.92 × 10−5 W·kg−1 [Turcotte and Schubert, 2001]. Uranium abundances (235U and 238U) were calculated using an assumed Th/U ratio of 3.8; a canonical cosmochemical value thought to be representative of most planetary bodies and that also agrees with analyses of most Martian meteorites [Meyer, 2003].
 The GRS instrument measures elemental abundances in the top-most tens of centimeters of the Martian surface, and thus is strongly influenced by near-surface soils, ice and dust deposits. These sediments broadly represent the bulk chemistry of the Martian upper crust when renormalized to a volatile-free basis [Taylor and McLennan, 2009] and as such, K and Th values must be renormalized to a H2O-, S-, and Cl-free basis to better reflect bulk crustal values. H2O and Cl surface abundances are obtained by using smoothed 5° × 5° GRS maps. Although the GRS instrument has not yet mapped surface S abundances at the required resolution, we use preliminary GRS estimates of a global S/Cl weight ratio of 5 to calculate the S content of each individual pixel. This ratio is also within the range of values measured in Martian soils at various landing site locations [Brückner et al., 2003; Rieder et al., 2004; Gellert et al., 2004]. These renormalizations are modest and equate to a 7–14% correction to the K, Th, and U abundances.
 We calculate a present day average surface heat production of 4.9 ± 0.3 × 10−11 W · kg−1. Heat production varies significantly across the Martian surface (Figure 1), ranging from 2.5 ± 0.2 × 10−11 W · kg−1 in the Hellas Basin and Solis Planum regions to 7.5 ± 0.5 × 10−11 W · kg−1 in the Acidalia Planum region. Of the elemental abundances measured by GRS, K and Th most closely correlate with one another compared to other measured elements (e.g., Fe, Cl, Si, or H) [Taylor et al., 2006b]. In addition, Th is presently the largest heat source among the radiogenic isotopes (Figure 2). Consequently, present-day surface heat production closely correlates with Th surface abundances measured by the GRS (as U is also a function of Th abundance) (see Figure S1 of the auxiliary material).
3. Crustal Heat Flow
 Using the crustal thickness model of Neumann et al. , smoothed to the same spatial resolution as the GRS 5° × 5° pixels, and an average uniform crustal density of 2,900 kg·m−3 [Zuber, 2001; Spohn et al., 2001], we calculate the average crustal component of heat flow of 6.4 ± 0.4 mW · m−2 and ranging from <1 mW · m−2 in the Hellas Basin and Utopia Planitia regions to ∼13 mW · m−2 in the Sirenum Fossae region (Figure 3). Crustal thicknesses proportionately vary more across Mars than do heat producing elemental abundances. Therefore, although there are some significant deviations, the crustal component of heat flow generally correlates with crustal thickness – i.e., a thicker crust represents higher crustal heat flow (see Figure S1). Note that Neumann et al.'s  crustal thickness model presents a reasonable average crustal thickness of 45 km. The heat flow estimates presented here would need to be scaled appropriately for future models suggesting a different average crustal thickness.
 It has been estimated that as much as 50% or more of the Martian planetary budget of heat producing elements has seen sequestered into the crust during planetary differentiation due to their incompatibility in igneous processes [Taylor et al., 2006b; McLennan, 2001]; a process that mostly took place very early in Martian geological history [Carr and Head, 2010]. As such, the crustal component of heat flow represents as much as half of the total planetary radiogenic heat generation. We estimate this total to be ∼13 mW · m−2. This value agrees well with estimates from previous works based on separate and independent cosmochemical or lithospheric cooling models which range from approximately 8–18 mW · m−2 [Montési and Zuber, 2003]. For comparison, in the terrestrial continental crust, average heat flow is 65 mW · m−2 and 48 mW · m−2 from tectonically stabilized crust with the crustal radiogenic component estimated to be in the range of 25–35 mW · m−2 [Taylor and McLennan, 2009; Jaupart and Mareschal, 2006; McLennan et al., 2006].
 These calculations assume a vertically homogeneous crust with no vertical change in the concentrations of K, Th and U with crustal depth. This is in contrast to the Earth's continental crust that typically exhibits an approximately exponential decrease in the abundances of the heat producing elements with depth (Figure 4). This fractionation is due to intracrustal melting and tectonic processes that further concentrate the incompatible elements into the more-evolved, terrestrial upper crust [Taylor and McLennan, 2009]. However, these processes of intracrustal differentiation that so affect the terrestrial continental crust are unlikely to be active on Mars to any significant degree. Although it has been suggested, there is no definitive evidence for current or past plate tectonic processes, nor have orbital remote sensing instruments detected any major provinces of the highly-evolved crustal products that we see on Earth or on the Moon [Jolliff et al., 2000]. GRS data show no suggestion that ejecta from the largest (and deepest) impact basins have compositions that differ from surrounding areas [Boynton et al., 2007]. Although there are localized regions of quartz-bearing igneous rocks [Bandfield et al., 2004; Bandfield, 2006; Christensen et al., 2005], previously observed regions of relatively high-silica, igneous “andesite” (Surface Type 2) have been more recently reinterpreted to reflect surface coatings of a silica-bearing secondary mineral phase [Wyatt and McSween, 2002; Rogers and Christensen, 2007]. Also, extensive impact gardening is thought to have vertically homogenized a significant portion of the Martian crust, so measured surface abundances can be reasonably inferred to be representative of concentrations from lower depths [Hartmann et al., 2001]. However, if Mars has a vertically heterogeneous crustal chemistry with upward enrichment of incompatible elements, our calculations for heat flow would represent an upper limit.
 It is important to note that surface heat production and heat flow calculated from orbital GRS measurements only constrain the crustal component of radiogenically-produced heat flow and provide no information about heat flow from the mantle. Mantle heat flow cannot be constrained through this method since the radiogenic heat producing element abundances are not observed and measured. This paper makes the first order assumption that the total radiogenic heat from the Martian mantle is approximately equal to the radiogenic heat produced in the crust based on estimated of bulk planetary chemistry and the behavior of the incompatible heat producing elements during the formation of the crust (i.e., 50% of the radiogenic elemental budget is sequestered in the Martian crust) [Taylor et al., 2006a, 2006b; Driebus and Wänke, 1985; Wänke and Driebus, 1988]. Variations in mantle heat flow cannot be constrained at all from GRS observations. Global or regional variations in mantle heat flow would have important consequences for total heat flow and regional geothermal gradients [Grott and Breuer, 2010]. Also, while present-day Martian planetary heat generation is dominated by heat produced through radiogenic decay, other sources of heat would have more significance earlier in Martian history (e.g., primordial heat of formation and secular cooling). Thus, the results reported here only provide constraints on one source of total Martian heat flow, albeit, a geochemical source of great importance.
 It is beyond the scope of this report to describe in detail the implications of regional-scale variations in heat production and heat flow. However, below we discuss two examples that serve to illustrate such approaches.
 A number of regions of Mars show anomalous crustal heat flow (Figure 3). Despite having a relatively thick crust, the Solis Planum region is unusually depleted in K and Th, resulting in an anomalously low crustal heat flow compared to surrounding terrains. Thus earlier suggestions of higher heat flow at Solis Planum could suggest a larger proportional contribution of mantle heat flow to compensate for lower crust production [Ruiz et al., 2006]. These chemical anomalies are not well understood for Solis Planum, a large terrain with unusual geophysical and morphological characteristics associated with the Tharsis formation [Carr, 2006].
 Conversely, regions in the Northern Plains are highly enriched in K and Th (and Fe) with respect to the planetary average and thus yield a high surface heat production. Most workers attribute incompatible element enrichment to primary igneous composition and not to near-surface secondary sedimentary processes (although secondary processes cannot be entirely disregarded) [Taylor et al., 2006b; Karunatillake et al., 2006; Hahn et al., 2007]. Despite high abundances of heat producing elements, these regions show very low crustal heat flow due to a thin crust.
 Consequently, future studies concerned with the thermal or geodynamic properties of a particular Martian geologic province should take care to use more geographically refined estimates of heat flow, such as those mapped and reported here, for modeling purposes, rather than previously reported global or large-scale regional averages.
 Because the respective half-lives of 40K, 232Th, 235U, and 238U are well-known, we can calculate the average crustal component of heat flow over geologic time (Figure 2). Furthermore, temporal regressions of the mapped data show crustal heat flow evolution. Crustal heat flow increases back through time and most dramatically before 2 Ga when the decay of 40K becomes the dominant source of radiogenic heat (see Figure S2). Average crustal heat flow at 4 Ga is almost five times that of present day (∼28 mW · m−2), with some regions, specifically in some sections of the Southern Highlands, showing crustal heat flow of over 50 mW · m−2. Note that these heat production and heat flow regression calculations assume that the K and Th abundances in the Martian crust have not been significantly modified due to past igneous processes. While Boynton et al.  showed K and Th enrichment in some regions of the relatively young northern lowlands, Hahn et al.  showed that over all GRS coverage, K and Th vary only subtly with apparent surface age. This suggests that past crustal resurfacing and the formation of past geologic provinces, on average did not form with dramatically different K and Th and, therefore, heat production values. This also assumes no significant change in crustal thickness; a reasonable assumption, as it is estimated that 80 ± 10% of the Martian crust has been in place since ∼4 Ga [Taylor and McLennan, 2009].
 Previously, total surface heat flow has been calculated from estimates of the thickness of the mechanical lithosphere based on gravity and topography data and estimates of the depth to the brittle-ductile transition using wrinkle ridge morphologies to constrain fault depths [Montési and Zuber, 2003; McGovern et al., 2002; Ruiz et al., 2006]. Such calculations provide heat flow estimates at the time of formation of features being considered. Since the age of a particular geologic province can be roughly determined from cratering history; the results of this study can also be used to determine the crustal thermal characteristics of these geologic regions at the time of formation. For example, Solis Planum, with a present day crustal heat flow range of 4.8–5.2 ± 0.3 mW · m−2, would have had a crustal heat flow range of approximately 8.8–9.8 ± 0.6 mW · m−2 at time of formation in the late Hesperian [Tanaka et al., 1992]. Similarly, the Opportunity rover landing site at Meridiani Planum, with a present day crustal heat flow value of 5.6 ± 0.3 mW · m−2, would have had a crustal heat flow of approximately 19.2 ± 1.2 mW · m−2 at time of formation in the late Noachian or early Hesperian [Hynek et al., 2002].
 The maps produced here show the considerable variation and complexity of radiogenic crustal heat flow across the present day Martian surface. As a direct orbital instrument observation, these heat production and crustal heat flow values from geochemical measurements support previous heat flow estimates produced by different methodology. The specific thermal histories for individual locations that can be calculated with these data will be a valuable aid for landing site selections, refining geologic timelines, and understanding the evolution of buried sedimentary deposits.
 We thank the Mars Odyssey science and engineering teams and William V. Boynton for the measurements that made this paper possible. Maps were prepared by GMT by P. Wessel and W. F. Smith. This work was supported by NASA through the 2001 Mars Odyssey Participating Science Program and through the Mars Data Analysis Program (NNX07AN96G).
 The Editor thanks Laurent G. J. Montesi and an anonymous reviewer for their assistance evaluating this manuscript.