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 The present day elastic lithosphere thickness at the Martian north pole has recently been constrained to De > 300 km and this is a factor of 3–4 larger than elastic thickness estimates for other Amazonian surface features like the Tharsis volcanoes. Here we present a model for the Martian elastic lithosphere thickness which takes the locally varying crustal thickness, the local concentration of heat-producing elements, as well as variations of strain rate into account. The model predicts De = 196 km at the north pole today, whereas elastic thicknesses at the Tharsis volcanoes are best compatible with middle to late Amazonian loading ages. Therefore, although a large degree of spatial heterogeneity can be explained by the presented model, large elastic thicknesses in excess of 300 km cannot be reproduced. In order to fit all elastic thickness estimates derived from observations, mantle heat flux at the north pole needs to be reduced by 35%. However, this can only be reconciled with a bulk chondritic concentration of heat-producing elements in the Martian interior if the excess heat is deposited elsewhere. Therefore, this argues for the presence of recently active mantle plumes, possibly underneath Tharsis. The size and strength of such a plume can be constrained by the elastic thickness at the Tharsis Montes and maximum average heat flux between 8 and 24 mW m-2, corresponding to a central peak heat flux of 40 to 120 mW m−2, is consistent with the observations. Such a plume would leave a clear signature in the surface heat flux and should be readily detectable by in situ heat flux measurements.
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 Processes in the Martian interior have important implications for the geological activity we observe at the surface and directly bear on phenomena like tectonism and volcanism. In particular, the style and vigor of mantle convection strongly influence the thermal state of the lithosphere and govern the formation of partial melt zones. On Mars, mantle energy transport is currently in the stagnant lid regime of mantle convection [Solomatov and Moresi, 1997; Reese et al., 1998] and the convecting mantle is covered by a rigid layer not taking part in the convective motion.
 On a one plate planet like Mars lithospheric properties are usually assumed to be fairly homogeneous. This is because heat transport in the stagnant lid is dominated by heat diffusion and this process acts to level out heterogeneities stemming from the convecting mantle. As the main factor controlling the rheology of lithospheric rocks is temperature, no large spatial variations of lithospheric properties are expected to be present. However, recent estimates of the present day elastic thickness De indicate that De is larger than 300 km at the north pole today [Phillips et al., 2008], a value three to four times larger than elastic thickness estimates for the Tharsis volcanoes [McGovern et al., 2004; Belleguic et al., 2005].
 It has been suggested that the observed large degree of spatial heterogeneity of lithospheric properties could be explained by the presence of hot mantle plumes [Kiefer and Li, 2009] and such plumes have also been invoked to explain the formation of the Tharsis bulge [Harder and Christensen, 1996; Li and Kiefer, 2007; Jellinek et al., 2008; Kiefer and Li, 2009]. Plumes represent thermal anomalies which act to locally thin the lithosphere over geological timescales as heat carried by the plumes slowly diffuses into the stagnant lid. Kiefer and Li  have estimated the elastic thicknesses above the upwelling plumes and cold downwellings and in their models the elastic thickness can spatially vary by a factor of 2 to 2.5. However, plumes are difficult to sustain under recent mantle conditions. Parameterized thermal evolution models suggest that today's temperature difference across the core-mantle boundary layer is small [e.g., Hauck and Phillips, 2002; Breuer and Spohn, 2006; Schumacher and Breuer, 2006; Grott and Breuer, 2009] and convection models show that plumes cease or cannot even develop under these conditions [e.g., Spohn et al., 2001]. Therefore, it remains controversial whether a Tharsis forming plume could still be active today.
 The model by Kiefer and Li  demonstrates one possible mechanism to generate large elastic thickness differences in terms of mantle upwellings and downwellings, but special assumptions regarding the locations of the cold downwellings need to be made. Furthermore, although the difference between maximum and minimum De in their models is close to the range derived from observations, De > 300 km at the north pole [Phillips et al., 2008] and the small elastic thicknesses at the volcanoes [McGovern et al.; Belleguic et al., 2005] cannot be explained simultaneously.
 As temperature is the main factor controlling elastic thicknesses [Hyndman et al., 2009] we will consider sources for lateral temperature heterogeneity in the following. The local crustal thickness is one factor controlling lithospheric temperatures, because radioactive elements are enriched in the crust and the thermal conductivity of crustal rocks [Clauser and Huenges, 1995; Seipold, 1998] is reduced with respect to the mantle [Hofmeister, 1999]. This results in an effectively insulating layer and the interior is heated up [Schumacher and Breuer, 2007]. As crustal thickness is predicted to vary between 5 and 100 km on Mars [Neumann et al., 2004], crustal thickness variations are expected to have a huge influence on the variability of the elastic thickness.
 Furthermore, the local abundance of heat-producing elements also influences lithospheric temperatures. Abundances have been found to vary by up to a factor of five on the surface [Taylor et al., 2006] and this will be taken into account in the following.
 Finally, strain rate represents the most important non thermal parameter influencing rock deformability and rocks are more resistant to plastic deformation at large strain rates. On Mars, deformation at the north pole and the emplacement of volcanic loads are expected to act on vastly different timescales: Deformation at the poles is likely driven by the deposition of the polar caps [Phillips et al., 2008], which is believed to occur on the timescale of the Martian obliquity cycles [Laskar et al., 2004], whereas emplacement of the volcanic loads is expected to be much slower and possibly linked to the timescale of mantle convection [Wilson et al., 2001].
 Therefore, although mantle plumes are attractive candidates to explain at least part of the observed large spread of elastic thicknesses, it remains to be investigated how important their contribution to the observed lithospheric properties really is. In particular, the question whether mantle plumes need to be active on Mars today will be addressed. We will first investigate how lateral variations of crustal thickness, the distribution of heat-producing elements and strain rate influence the local Martian elastic thickness. Model results will then be compared to elastic thickness estimates available in the literature and the necessary contribution of hypothetical mantle plumes will be quantified. Furthermore, the maximum size and strength of mantle plumes compatible with observations will be determined.
2. Modeling Approach
 We calculate the Martian elastic lithosphere thickness De(θ, ϕ) as a function of latitude θ and longitude ϕ assuming the thermal evolution of Mars to be governed by global heat loss driven by mantle convection in the stagnant lid regime [Solomatov and Moresi, 1997; Reese et al., 1998; Grasset and Parmentier, 1998]. The thermal evolution then provides globally averaged values of the heat flux ql out of the mantle into the stagnant lid and a stagnant lid thickness Dl, which we assume to be independent of position.
 The bulk content of heat-producing elements in the Martian interior is assumed to be essentially chondritic, as suggested by the analysis of the SNC meteorites [Morgan and Anders, 1979; Treiman et al., 1986; Wänke and Dreibus, 1994]. Although some models derive larger concentrations of radioactive elements [Lodders and Fegley, 1997], the resulting large heat production rates would have resulted in the topographic relaxation of large isostatically supported structures on Mars by lower crustal flow [Grott and Breuer, 2008b] and are incompatible with the Martian crustal evolution [Hauck and Phillips, 2002]. Therefore, they are not considered here. As data from gamma ray spectroscopy [Taylor et al., 2006] imply that the model by Wänke and Dreibus  is the one best compatible with the observed distribution of K and Th on the surface this will be the model used here. For an average crustal thickness of 57 km, Taylor et al.  estimate that about half the planet's inventory of heat-producing elements is concentrated in the crust, corresponding to a crustal enrichment factor Λ of 9 with respect to the undepleted mantle. For the mean crustal thickness of 45 km adopted here, 40% of heat-producing elements will be concentrated in the crust, consistent with the model of Taylor et al. .
 Once the thermal evolution of Mars has been calculated, the lithospheric temperature profile is calculated locally on a one by one degree grid using estimates of the local crustal thickness and radiogenic heat production rates. Our model uses local crustal thicknesses Dc(θ, ϕ) given by the model of Neumann et al. , which assumes constant crustal densities of 2900 kg m3 and derives crustal thicknesses using Bouguer inversion.
 The local concentration of heat-producing elements is given by the crustal concentration maps of K and Th provided by Boynton  and we assume that the observed surface concentrations are representative for the respective crustal columns. Taylor et al.  argue that this is a good assumption for three reasons: (1) The similar geochemical behavior of K and Th implies that the K/Th ratio at the surface should reflect the ratio in the mantle source region and the SNC meteorites argue that there was no systematic change of magma composition with time. (2) The high rate of volcanic intrusions with respect to extrusions will act to average out a possible depth dependence of the K and Th concentrations. (3) Impact gardening will have mixed the crust to considerable depth, especially for large basin forming impacts.
 However, it cannot be ruled out that heat-producing elements have been moved to the surface by, e.g., hydrothermal processes. Indeed, there are some indications that the Martian crust is locally stratified and an enriched upper crust exists [Ruiz et al., 2009]. If this is the case, the concentration of heat-producing elements in the mantle is underestimated here, resulting in an overestimation of spatial heterogeneity. Therefore, the results presented in the following will reflect the maximum possible spatial variability of the elastic thickness.
 Using the concentration maps of Boynton  and deriving the concentration of U by assuming a ratio of Th/U of 3.5 [Wänke and Dreibus, 1994], the local crustal heat production rate Qc is given by
where CU(θ, ϕ), CTh(θ, ϕ) and CK(θ, ϕ) are the local concentrations of U, Th and K, H238U, H235U, H232Th and H40K are the heating rates of 238U, 235U,232Th and 40K (in W kg−1), and τ238U, τ235U, τ232Th and τ40K are the half lives of 238U, 235U,232Th and 40K, respectively. ρcr is the crustal density.
 The mantle heat generation rate Qm is then calculated from mass balance constraints, using CK,bulk = 305 ppm, CTh,bulk = 56 ppb and CU,bulk = 16 ppb [Wänke and Dreibus, 1994] as the bulk silicate concentrations. Concentrations of K, Th and U in the mantle are calculated by integrating their respective concentrations over the crust and subtracting this amount from the bulk concentration:
where Vsi and ρsi are the volume and average density of the silicate fraction, Vcr(θ, ϕ) is the crustal volume of the respective grid cell, Vm is the mantle volume and ρm the mantle density. The index i stands for K, Th and U and the sum extends over all 1 × 1 degree grid cells. Qm is then calculated using equation (1) with CK,m, CTh,m and CU,m.
 In the presented model we assume a homogeneous distribution of heat-producing elements in the lithospheric and convecting mantle. Furthermore, equal concentrations of heat-producing elements in the lithospheric and convecting mantle are assumed. However, the extraction of partial melt from the mantle will leave behind a mantle component likely resistant to mantle flow [Hirth and Kohlstedt, 1996], thus accumulating depleted material in the lithosphere. The degree of depletion is difficult to estimate, but the lithospheric mantle is expected to be more severely depleted beneath regions of thick crust. Although this is a small effect, it would act to reduce spatial heterogeneity, such that the presented model again results in an overestimation of spatial heterogeneities stemming from crustal thickness and concentration differences.
2.1. Thermal Evolution Model
 The thermal evolution of Mars is calculated solving the energy balance equations for the mantle
and the core
where ρm and ρc are the density, and cm and cc the heat capacity of mantle and core, respectively. Vl is the volume of the convecting mantle and given by Vl = 4/3π(Rl3−Rc3), where Rl and Rc are the stagnant lid and core radii. Al is the corresponding surface area given by Al = 4π Rl2. Vc and Ac are the volume and surface area of the core, Tm is the upper mantle temperature and Tc is the temperature at the core-mantle boundary. ϵm is the ratio between the average and upper mantle temperatures and ϵc is the ratio between the average core and core-mantle boundary temperatures. ql is the heat flux from the convecting mantle into the base of the stagnant lid, qc is the heat flux from the core into the mantle and t is time.
where ∂T/ and Tl are the thermal gradient and the temperature at the base of the stagnant lid, respectively. km is the mantle thermal conductivity.
 The efficiency of mantle energy transport is largely determined by the mantle viscosity, which in turn is a strong function of the mantle water content and the operating deformation mechanism. The effective viscosity η = σ/2 , where σ is shear stress and the strain rate, can be estimated from flow laws and is given by
where μ is the shear modulus, A an experimentally determined constant, d the grain size, b the Burgers vector, E the activation energy, V the activation volume, pref the pressure, R the gas constant, n the stress exponent, m the grain size exponent and T the temperature [Karato and Wu, 1993].
 Thermal evolution models are then calculated starting from an initial temperature profiles that varies adiabatically in the core and mantle and the core is assumed to be superheated with respect to the mantle by 300 K [Stevenson, 2001; Breuer and Spohn, 2003]. Cooling of the planet is driven by convective energy transport in the mantle which governs the thickness of the thermal boundary layers, stagnant lid growth and core cooling. Details of the model are given in Grott and Breuer [2008a].
2.2. Lithospheric Temperature Profile
 Once the thermal evolution of the planet has been calculated, lithospheric temperature profiles can be constructed. Given the heat flux ql out of the convecting mantle into the stagnant lid the heat flux qm out of the lithospheric mantle into the crust is determined by
where Rcr is the crustal radius given by Rp−Dc(θ, ϕ) and Rl the lithospheric radius. It is therefore given by the sum of the lithospheric heat flux and the heat flux generated by the decay of radioactive elements in the lithospheric mantle. Furthermore, the surface heat flux qs can be calculated from
where Qc(θ, ϕ) is the local crustal heat generation rate and Rp is the planetary radius. Surface heat flux is therefore given by the sum of qm and the heat flux generated by the decay of radioactive elements in the crust.
 A sketch of the different heat fluxes, temperatures and interfaces between crust, lithospheric and convecting mantle is shown in Figure 1. The lithospheric temperature profile is defined piecewise and crustal temperatures are given by
where Ts is surface temperature, kc is crustal thermal conductivity and z is depth. Temperatures in the lithospheric mantle are given by
where Tcr(θ, ϕ) is the temperature at the base of the crust and km is mantle thermal conductivity. On average, the temperature at the base of the lithospheric mantle will equal the stagnant lid temperature calculated using the thermal evolution model.
2.3. Elastic Thickness Calculation
 Once the lithospheric temperature profile has been calculated, the mechanical thickness of the lithosphere can be determined for given crustal and mantle rheologies. The mechanical thickness of the lithosphere corresponds to a rheological boundary and is usually defined by the depth below which the strength of the rocks drops below a given bounding stress σB [McNutt, 1984; Burov and Diament, 1995]. The onset of ductile deformation is highly temperature-dependent and governed by a flow law of the form
where Q is the creep activation energy.
 Given the bounding stress below which the individual layers loose their mechanical strength, equation (11) may be used to determine the temperature associated with ductile failure which is given by
where the rheological parameters A, n and Q correspond to Adia, ndia and Qdia in the diabase crust and Aol, nol and Qol in the olivine mantle. These are summarized in Table 1.
Table 1. Parameters Used for the Calculation of Elastic Thicknesses
 The mechanical thickness Dm corresponding to the isotherms defined by equation (12) will always be equal to or larger than the effective elastic thickness De and the two quantities can be related to one another if the lithospheric curvature is known [McNutt, 1984; McNutt et al., 1988]. For small curvatures and bending moments, elastic and mechanical thickness will be similar, while large bending moments result in larger differences between the two. For the geological features considered here, large elastic thicknesses and thus small curvatures have been reported [McGovern et al.; Belleguic et al., 2005] and we will assume De ≈ Dm. In the following De will therefore be identified with a rheological boundary. Note that for smaller elastic thicknesses and larger curvatures this approximation is no longer valid and although the model presented here can in principle also be applied to such features the resulting mechanical thickness values would need to be corrected for yielding effects to obtain the effective elastic thickness. In other words, only upper bounds on De would be obtained using the presented method. This limits the applicability of our model to the Amazonian period and although we will briefly discuss the application of our model to Hesperian and Noachian features in section 3, these values need to be interpreted with some caution.
 The thicknesses De,c(θ, ϕ) and De,m(θ, ϕ) of the elastic cores of crust and mantle are thus determined by the depth at which the lithospheric temperatures calculated using equations (9) and (10) exceed the yield temperatures of the respective materials. If crustal temperatures are below the diabase yield temperature throughout, De,c(θ, ϕ) equals the crustal thickness Dc(θ, ϕ) and the two elastic layers are coupled. Otherwise, they are separated by a layer of incompetent crust. It has been shown that the existence of a decoupling layer of incompetent crust between the elastic cores of crust and mantle can significantly reduce the elastic thickness of the compound system and this model has been successfully applied to continental lithosphere on Earth [McNutt et al., 1988; Burov and Diament, 1992]. Multilayer lithospheric plates behave similar to a leaf spring and the elastic thickness De(θ, ϕ) of the compound system is given by
 If, however, De,c(θ, ϕ) equals the crustal thickness and no layer of incompetent crust exists, De(θ, ϕ) is simply given by the sum of the individual components which then act as a single plate and
 The parameter most sensitively influencing the thermal evolution calculations is the assumed mantle rheology, which is a strong function of mantle water content and the viscosities of water-saturated and anhydrous olivine vary by more than 2 orders of magnitude [Karato and Wu, 1993]. Although it has been argued that Mars may not have retained any water during its accretion and water has only been added at a later evolutionary stage as a late veneer [Dreibus and Wänke, 1987], a dry mantle rheology is incompatible with the low elastic thickness values derived for the early Martian evolution [Guest and Smrekar, 2007; Grott and Breuer, 2008a]. Furthermore, geochemical evidence suggests that small amounts of water are present in the Martian mantle [McSween et al., 2001; Médard and Grove, 2006]. As olivine is the most easily deformable mantle mineral, we will assume diffusion creep law parameters appropriate for the deformation of wet olive [Karato and Wu, 1993] in our model. However, for completeness, the influence of a dry mantle rheology will also be briefly discussed in section 3.
 Another important parameter influencing the elastic thickness is strain rate. Large stresses are necessary to initiate fast deformation and the mechanical thickness will be larger for slower deformation. We will generally assume strain rates of 10−17 s−1 [e.g., McGovern et al.], appropriate for deformation associated with the timescale of mantle convection. However, for comparing our results to the elastic thicknesses reported for lithospheric loading at the polar caps, = 10−14 s−1 will be used [Phillips et al., 2008]. This strain rate is appropriate for deformation acting on the timescale of the Martian obliquity variations [Laskar et al., 2004], which are believed to drive polar cap deposition [Phillips et al., 2008].
 The elastic lithosphere thickness also depends on the definition of the bounding stress σB below which the lithosphere looses all mechanical strength. However, due to the exponential dependence of the stresses necessary to induce ductile deformation on temperature, results are fairly insensitive to the exact choice of σB as long as σB < 50 MPa [McNutt, 1984]. Here we adopt σB = 10 MPa, which results in slightly elevated elastic thickness values. The parameters used to determine the elastic thicknesses are summarized in Table 1.
 Using the model presented in section 2.1, we have calculated the thermal evolution of Mars assuming mantle energy transport in the stagnant lid regime and an essentially chondritic concentration of heat-producing elements in the Martian interior. Radioactive element are assumed to be enriched in the crust and a crustal enrichment factor Λ of 9 with respect to the undepleted mantle is assumed. Figure 2a shows the results of this calculation where the surface heat flux qs, the mantle heat flux qm, and the heat flux into the stagnant lid ql are shown as a function of time for the entire Martian evolution. The globally averaged surface heat flux is large during the early evolution, but declines to ∼40 mW m−2 at the beginning of the Amazonian at 2000 Myr. At present, the average surface heat flux is predicted to be 21 mW m−2. The heat flux provided to the base of the lithosphere is 14.5 mW m−2, corresponding to 70% of the surface heat flux and accounts for the loss of heat produced by the decay of radioactive elements as well as the loss of primordial heat by secular cooling.
Figure 2b shows the thickness of the stagnant lid Dl as a function of time. The stagnant lid grows as the planet cools and the lid thickness reaches values of 140 km at the beginning of the Amazonian. It keeps growing almost linearly and reaches values of 265 km today. Note that the temperature at the base of the stagnant lid Tl is given by [Davaille and Jaupart, 1993; Grasset and Parmentier, 1998]
such that for Tm = 1600 K, corresponding to the upper mantle temperature at the end of the evolution, the lid temperature is approximately 1400 K. This is well above the temperatures associated with ductile failure of wet (920 K) as well as even dry olivine (1065 K), such that the stagnant lid thickness can be viewed as an upper limit on the mechanical thickness of the lithosphere.
 Using the stagnant lid heat flux ql and lid thickness Dl from the thermal evolution model, the local lithospheric temperature profiles and resulting elastic thicknesses have been calculated for three times at 2000, 3500, and 4500 Myr corresponding to the early, middle and late Amazonian periods [Hartmann and Neukum, 2001], respectively. Figure 3a shows a color-coded Hammer projection map of the elastic lithosphere thickness De overlaid over a shaded relief map of MOLA topography, where the heat flux provided to the base of the stagnant lid ql and the stagnant lid thickness Dl have been evaluated at t = 2000 Myr after core formation.
 Elastic thicknesses De are low and around 30 to 40 km in the southern Highlands, whereas locations of thinned crust like the northern lowlands and the giant impact basins exhibit De > 80 km. Low elastic thickness values are associated with the presence of an incompetent layer at the base of the crust and the pronounced increase of elastic thicknesses in the transition regions is caused by the vanishing of this layer.
Figure 3c shows a histogram of the elastic thickness distribution given in the map (Figure 3a), where the percentage of the cumulative surface area for the respective elastic thickness values is shown. The data has been gathered into 50 equally sized bins and the distribution of De is clearly bimodal with one peak centered around De = 40 km and a second peak around 85 km. These peaks correspond to the southern highlands and northern lowlands, respectively, where a layer of incompetent crust is present or absent at the crustal base.
Figure 3b is similar to Figure 3a, but shows the distribution of De at 3500 Myr after core formation; that is, lid heat flux ql and stagnant lid thickness Dl have been evaluated at that time. Elastic thicknesses vary between 55 km near Arsia Mons to ∼105 km in the southern highlands and ∼125 km in the northern lowlands and the Hellas basin. Driven by planetary cooling the region in which an incompetent crustal layer is present has shrunken to the center of Tharsis, where the largest crustal thicknesses have been reported.
 The histogram of the respective elastic thickness distribution is shown in Figure 3d and exhibits three peaks centered around 65, 105 and 125 km. The peak at De = 65 km corresponds to the center of Tharsis, where large crustal thicknesses result in the presence of a decoupling layer between the crustal and mantle lithospheres. The peak at De = 105 km corresponds to the southern highlands, where large crustal thicknesses result in smaller De as compared to the northern lowlands, which correspond to the peak around De = 125 km.
 For the present day elastic thickness, we have considered two different strain rates of = 10−17 and 10−14 s−1 corresponding to deformation on the timescale of mantle convection and deformation on the timescale of the Martian obliquity cycles, which are believed to drive polar cap deposition. The results of these calculations are shown in Figure 4, where Figures 4a and 4c correspond to = 10−14 and Figures 4b and 4d to 10−17 s−1, respectively. For = 10−14 s−1 almost no decoupling layers exists and elastic thicknesses vary between 170 km in the highlands and 200 km in the lowlands and impact basins. Minimum De is reached in at Arsia Mons, where a decoupling layer of incompetent crust causes elastic thicknesses of only 95 km. Other areas in central Tharsis exhibit De around 160 km. For = 10−17 s−1 an incompetent crustal layer remains even today around Arsia Mons and De exhibits a spatial variability between 70 and 170 km.
 This is also expressed in the histograms for these respective cases. Figure 4c exhibits a bimodal distribution of De values with peaks around 180 and 200 km, corresponding to the highland and lowland regions. For = 10−17 s−1 three peaks at 80, 140 and 160 km are visible, where the small peak at De = 80 km corresponds to the decoupling region around Arsia Mons. However, the most important effect of the change of strain rate lies in a shift of elastic thicknesses toward larger values for lower strain rates and De increases by ∼40 km between = 10−17 and 10−14 s−1.
 A more quantitative analysis of the modeled elastic thickness values is summarized in Table 3, where the elastic thickness values derived in the literature for Amazonian features are compared to the De values modeled here. Table 3 lists the loading ages, elastic thickness estimates and modeled elastic thicknesses in the early, middle and late Amazonian, respectively (Values in brackets correspond to a model using a dry mantle rheology, see below). Note that for the estimates at the poles values derived for = 10−14 s−1 are given, whereas the values derived for the volcanoes use = 10−17 s−1.
Table 3. Elastic Thickness Estimates Given in the Literature and Results Obtained in This Studya
De (km) 2000 Myr
De (km) 3500 Myr
De (km) 4500 Myr
Values in brackets correspond to a model assuming a dry mantle rheology. The mantle heat flux and stagnant lid thicknesses were evaluated at 2000, 3500, and 4500 Myr corresponding to the early, middle, and late Amazonian (recent) periods, respectively. Strain rates for the elastic thickness estimates were 10−17 s−1 at the volcanoes and 10−14 s−1 at the poles.
 Elastic thicknesses modeled at the poles fall short of the values reported by Phillips et al.  by about 100 km, although they are consistent with the estimate given by Wieczorek . For the volcanoes modeled elastic thicknesses are generally compatible with those derived from flexure studies and De at Olympus, Ascraeus, and Pavonis Mons is best reproduced for middle to late Amazonian loading ages. Although the general trend of De being smallest at Arsia Mons is reproduced by the model presented here, the small elastic thickness of only 20 to 35 km can only be modeled if an early Amazonian loading age is assumed.
 The present day elastic thicknesses modeled here vary by almost a factor of three between the Tharsis Montes and the north pole, but we fail to model the extremely large present day De > 300 km at the north pole. Furthermore, elastic thicknesses at the Tharsis Montes seem to favor relatively old loading ages, which seems ad odds with the volcanic activity ongoing well into the Amazonian period and possibly even today [e.g., Neukum et al., 2004].
 Therefore, other sources of spatial heterogeneity need to be considered. A prime candidate as a source for further heterogeneity is a spatially varying lithospheric heat flux ql(θ, ϕ), which would be expected to be larger than average near the volcanoes and possibly reduced at the poles. Physically, this would represent a hot mantle upwelling underneath Tharsis and/or a cold downwelling at the poles, and this will be investigated in section 4.
 Although the model used here can most reliably be applied to the Amazonian period, we can still compare predictions for the Noachian and Hesperian mechanical thicknesses to the values derived from observations. Elastic thickness estimates for these periods are summarized in Table 4 and the modeled mechanical thicknesses Dm are also given (Values in brackets correspond to a model using a dry mantle rheology, see below). In general, elastic thicknesses in the Noachian are satisfactorily reproduced, with Dm being slightly larger than De for most features. This might be expected as Dm/De can reach a factor of 2 for curvatures in excess of 5 × 10−7 m−1, as has for example been reported at the Coracis Fossae rift system [Grott et al., 2005]. The elastic thickness at the Hellas impact basin is overestimated by our model, but this can be attributed to the isostatic compensation setting in after the impact event, which is not modeled here.
Table 4. Elastic Thickness Estimates Given in the Literature and Mechanical Thickness Dm Obtained in This Studya
Values in brackets correspond to a model assuming a dry mantle rheology. The mantle heat flux and stagnant lid thicknesses were evaluated at 1500, 1000, 750 and 500 Myr for the Amazonian-Hesperian, Hesperian, Hesperian-Noachian and Noachian periods respectively. Strain rates for the elastic thickness estimates were 10−17 s−1.
 The elastic thickness reported for Isidis Planitia has been derived by fitting the locations of the Nili and Amenthes Fossae graben systems to loading models, and 100 km < Dl < 180 km has been reported by Ritzer and Hauck . They also note that these relatively large elastic thicknesses better correspond to Hesperian-Amazonian than to early Noachian loading ages, which are suggested by the early Noachian formation age of the Nili Fossae. This inconsistency might at least partially be removed if the deformation leading to the formation of the graben has been fast, as might be expected if the tectonics are related to basin formation. If the Noachian heat flux is scaled such that ql is compatible with Dl > 300 km at the north pole today, strain rates in excess of = 10−15 s−1 are sufficient to be consistent with Dl > 100 km.
 Modeled values for the Hesperian and Hesperian-Noachian periods generally agree with the values derived from flexure studies. Also, estimates for the volcanoes exhibiting an Amazonian-Hesperian surface age are generally consistent with the values derived from the observations. However, elastic thicknesses at Valles Marineris are significantly underestimated.
 Further support for the model presented here is provided by the analysis of wrinkle ridge spacing in the northern lowlands as compared to spacing in Solis and Lunae Planum in the southern highlands [Montési and Zuber, 2003]. Both sets of wrinkle ridges exhibit Hesperian surface ages, but the ridge spacing indicates that the depth to the brittle-ductile transition zone is about a factor of two larger in the northern lowlands than in the southern highlands [Montési and Zuber, 2003]. This is consistent with a mechanically incompetent crustal layer being present at the time of ridge formation in the highlands, whereas such a layer was absent in the northern lowlands at that time [Montési and Zuber, 2003]. Our model predicts decoupling in the lowlands to set in around 500 Myr after core formation, whereas decoupling persists until 3000–3500 Myr in Solis Planum, consistent with the results of Montési and Zuber .
 If the rheology of the Martian mantle is assumed to be dry rather than wet, present day elastic thicknesses obtained by our model are increased by 35 km on average (compare Table 3, values in brackets). This facilitates the reproduction of large elastic thicknesses at the north pole and larger heat fluxes of 11 mW m−2 would be compatible with Dl > 300 km there. However a dry mantle rheology results in a large overestimation of Noachian elastic thicknesses by up to a factor of four due to the early vanishing of the incompetent crustal layer (NE Terra Cimmeria, NE Arabia Terra, Amazonis and Arcadia Planitia, Amenthes Rupes and Acheron Fossae, compare Table 4, values in brackets). Therefore, a dry mantle rheology can be ruled out.
4. Plume Model
 We have shown that the spatial variability caused by crustal thickness variations, the distribution of heat-producing elements on the Martian surface and the differences in strain rate cause a huge degree of spatial variability of the elastic lithosphere thickness De, with values ranging from 70 to 230 km today. However, the large elastic thicknesses observed at the north pole are incompatible with the presented models.
 The present day heat flux provided to the base of the lithosphere by mantle convection is 14.5 mW m−2 (compare Figure 2), but a reduction of ql by about 35% to 9 mW m−2 would result in De = 300 km at the north pole today. However, a globally reduced mantle heat flux of only 9 mW m−2 is only compatible with the thermal evolution if the bulk content of heat-producing elements in the Martian interior is reduced with respect to the chondritic composition assumed here.
 A subchondritic concentration of heat-producing elements in the Martian interior is difficult to reconcile with geochemical analysis of the SNC meteorites [Kiefer and Li, 2009] which implies essentially chondritic concentrations of radioactive elements [Wänke and Dreibus, 1994; Treiman et al., 1986]. Furthermore, if the concentration of heat-producing elements in the Martian interior is indeed reduced, the resulting low interior temperatures would probably inhibit partial mantle melting and magmatism [Grott and Breuer, 2009]. However, geological evidence suggests that Mars has been volcanically active in the recent past [e.g., Neukum et al., 2004]. Therefore, in order to be compatible with both a chondritic composition and thermal evolution models, the excess energy needs to be transported to other locations.
 One way to achieve this is energy transport by mantle plumes and we will investigate the properties of such a hypothetical plume in this section. If it exists and is still active today, the plume would most likely be located underneath Tharsis and it has been speculated that the long-lasting volcanism observed there may indeed be caused by plume activity [Harder and Christensen, 1996; Li and Kiefer, 2007; Kiefer and Li, 2009]. Furthermore, morphological and structural data of the tectonics in the Tharsis region are widely compatible with a plume in the Noachian [Mège and Masson, 1996] and the demagnetization observed in central Tharsis has also been attributed to lithospheric heating caused by a mantle upwelling [Jellinek et al., 2008].
 The size and strength of an active plume underneath Tharsis is limited by the elastic thicknesses observed at the Tharsis Montes. However, hard constraints for the plume size are difficult to pin down, as elastic thickness estimates at the Tharsis Montes vary over a considerable range. Here we will use De > 50 km as a constraint for Ascraeus and Pavonis Mons (compare Table 3) which represents a lower limit on the elastic thickness at these features. This implies that the plume strength determined in the following must be viewed as an upper limit. Higher plume heat fluxes would result in even smaller elastic thicknesses, thus clearly violating the constraints posed by even the lowest estimates.
 Given the size of the plume's footprint at the surface Rpl and the heat flux at the plume center qcen, the heat flux at a distance r from the center of the plume may be calculated by [Jellinek et al., 2003, 2008]
The energy carried by such a plume is
and an analytical integration of this expression yields qcen = 5pl, where pl is the average plume heat flux and Epl = πRpl2pl. This energy is connected to the plume conduit radius Rcon and the plume excess temperature ΔTex by [Turcotte and Schubert, 2002]
where cp is specific heat, g is gravitational acceleration and α is the thermal expansion coefficient. The plume viscosity ηpl is given by equation (6) evaluated at the plume temperature Tpl = Tb + ΔTex, where Tb is the temperature at the base of the mantle given by
Here, ΔR is the thickness of the convecting mantle given by ΔR = Rl − Rc − δu − δl and δu and δl are the thickness of the upper and lower thermal boundary layer, respectively.
 The location of the plume center is usually assumed to be close to the center of the Tharsis bulge, but estimates do vary for different approaches. Here we follow Jellinek et al.  and assume the plume center to be located at 6°S, 95°W, i.e., in the center of the region of demagnetized crust. This location implies a distance of 1100 km between the plume center and the Tharsis Montes and using the model presented in section 2, the maximum admissible plume center heat flux qcen compatible with De > 50 km can be determined. We find that qpl must not exceed 16 mW m−2 at the location of the Tharsis Montes and Figure 5a shows the plume center heat flux qcen as well as the total plume energy Epl as a function of plume footprint size Rpl for qpl(1100 km) = 16 mW m−2.
 Maximum plume central heat fluxes qcen vary between 40 and 120 mW m−2, corresponding to pl = 8 to 24 mW m−2. The energy carried by these plumes corresponds to 7–11% of the total mantle heat flux, with Etot = 4πRp2ql = 2100 GW. Note that although plumes with larger footprints need to have lower central heat fluxes they can carry more energy and still be compatible with De > 50 km at the Tharsis Montes as the energy deposited by the plume is proportional to the surface area covered by the footprint. If the global background mantle heat flux ql is assumed to be 9 mW m−2, such a plume could carry up to 30% of the excess energy.
Figure 5b shows the plume conduit radius Rcon as a function of plume excess temperature ΔTex for plumes satisfying De > 50 km and two footprint sizes Rpl of 1500 and 3000 km, respectively. Plumes below the lines corresponding to Rpl = 1500 and 3000 km are compatible with the constraint of De > 50 km. Estimating the size and strength of the Tharsis plume, Jellinek et al.  have calculated a plume excess temperature of 205–240 K during the Noachian and using their model we calculate a plume conduit radius of ∼190 km at that time. Assuming that the excess temperature has declined as the core cooled but the conduit radius remained constant, excess temperatures of a few tens of K would be sufficient to transport the maximum admissible energy. Note that this excess temperature is of the same order as today's temperature drop typically found across the core-mantle boundary in thermal evolution models [Grott and Breuer, 2009]. Furthermore, the dimensions of the plume conduit and footprint are compatible with values derived from mantle convection models [Redmond and King, 2004].
Figure 6a shows a color-coded Hammer projection map of the surface heat flux qs overlaid over a shaded relief map of MOLA topography. The spatial variation of qs is caused by differences of the local crustal thickness and the concentration of heat-producing elements in the crust and surface heat fluxes vary between 17 mW m−2 in Hellas and 28 mW m−2 at Arsia Mons and in Terra Cimmeria, where large crustal thicknesses and high concentrations of heat-producing elements are present.
 As a comparison, Figure 6b shows the surface heat flux in the presence of a hot spot underneath Tharsis where a plume footprint radius of 3000 km has been assumed. The plume energy is the maximum energy compatible with the elastic thicknesses observed at the Tharsis Montes and the plume center heat flux is 40 mW m−2. If a plume is present underneath Tharsis it dominates the spatial distribution of surface heat fluxes and the whole of Tharsis exhibits qs in excess of 25 mW m−2, with central values reaching 60 mW m−2. In comparison, other regions remain unaffected and typical highland heat fluxes remain at 20 to 25 mW m−2 while the northern lowlands have qs around 15 to 20 mW m−2. Minimum heat fluxes are found at the north pole and in the Hellas basin. Note that the color scale has been shifted in Figure 6b with respect to Figure 6a to increase the color contrast.
 We have presented a model to estimate the elastic lithosphere thickness De as a function of local crustal thickness, concentration of heat-producing elements in the crust and strain rate and found that crustal thickness is the controlling factor on De. The concentrations of heat-producing elements do not vary over a large range on Mars and the surface concentrations of K and Th as derived from gamma ray spectroscopy range between 2000 to 6000 and 0.2 to 1 ppm, respectively [Taylor et al., 2006]. This is a relatively small spread if compared to crustal thickness variations, which range from 6 km at Hellas to 102 km at Arsia Mons [Neumann et al., 2004]. Therefore, the distribution of heat-producing elements in the crust can be considered a second-order effect.
 If present, a mechanically incompetent layer in the lower crust has a large influence on the elastic thickness and De abruptly increases as such a layer vanishes. The presence of incompetent layers is controlled by the thermal state of the lithosphere and the crustal thickness and we find that during the early Amazonian essentially all the highland lithosphere comprises a two layered system. As the planet cools the extent of these regions shrinks to areas with large crustal thicknesses but an incompetent crustal layer may be present even today around Arsia Mons. Therefore, although chemically different to continental lithosphere on the Earth, the mechanical behavior of the Martian lithosphere is quite similar to continents on the Earth as far as mechanical layering is concerned [McNutt et al., 1988; Burov and Diament, 1992, 1995].
 Strain rate was found to also play an important role in terms of the absolute value of the elastic thickness. An increase of from 10−17 to 10−14 s−1 results in elastic thicknesses which are larger by about 40 km. As the mechanisms governing deformation at the Tharsis Montes must be assumed to be significantly different from the processes responsible for polar cap deposition, this effect also adds to the spatial heterogeneity of the elastic thickness. Together, these sources of spatial heterogeneity result in present day elastic thicknesses which range from 70 km at the Tharsis Montes to 200 km at the north pole.
 However, taking into account contributions from the spatially varying crustal thickness, concentration of heat-producing elements and strain rates, the spread of De derived from observations at the Tharsis Montes and the north pole cannot be reproduced. In particular, the large De > 300 km at the north pole cannot be explained using the presented model. Even if were reduced to 10−12 s−1 the elastic thickness at the north pole would not exceed De = 225 km. Only if the heat flux provided to the base of the lithosphere is reduced to 9 instead of 14.5 mW m−2, De > 300 km is obtained. However, this value cannot be globally representative as it is at odds with a Martian bulk composition which is essentially chondritic [Wänke and Dreibus, 1994; Treiman et al., 1986; Kiefer and Li, 2009] and ongoing volcanic activity [Neukum et al., 2004; Grott and Breuer, 2009].
 It has been debated if the recent volcanism in the large volcanic provinces like Tharsis and Elysium could have been sustained by longstanding and stable mantle plumes [Kiefer, 2003]. Although the generation of the bulge of the large volcanic regions by mantle plumes is generally accepted [e.g., Spohn et al., 2001], plumes are difficult to sustain under recent mantle conditions. Thermal plumes develop from instabilities at the core-mantle boundary and as the interior cools this thermal boundary layer weakens strongly with time, resulting in a decrease of the plume strength. Parameterized thermal evolution models suggest that the initial temperature difference across the core-mantle boundary layer rapidly decreases below 50 K [e.g., Hauck and Phillips, 2002; Breuer and Spohn, 2006; Schumacher and Breuer, 2006; Grott and Breuer, 2009] and convection models show that plumes cease or cannot even develop under these conditions [e.g., Spohn et al., 2001]. However, an increase of the mantle viscosity with depth may allow a longer lifetime of plumes up to several billion years [Buske, 2006] and the results presented here corroborate the view that weak active mantle plumes are indeed present on Mars today.
 The size and strength of such plumes is limited by the elastic thicknesses of 50 km derived for the Tharsis Montes and we have constructed plumes compatible with this constraint. We found that plumes can carry up to 11% of the total energy transported in the mantle, corresponding to 35% of the energy difference between the global background mantle heat fluxes of 9 and 14.5 mW m−2. Average and peak central heat fluxes of these plumes range from 8 to 24 mW m−2 and 40 to 120 mW m−2, respectively. A combination of plumes at Tharsis and downwellings at the north pole or other plumes elsewhere could therefore explain the range of elastic thicknesses derived from flexure studies. Furthermore, plumes would leave a clear signal in the surface heat flux which should be detectable by in situ heat flux measurements.
 Finally, the crustal thickness model used here derives crustal thicknesses on the assumption of constant crustal densities. However, there are some indications that crustal densities at the Martian volcanoes are around 3200 ± 100 kg m−3 [Belleguic et al., 2005], whereas the crustal density in the southern highlands is smaller than 3020 ± 70 kg m−3 [Pauer and Breuer, 2008]. If the high densities found at the volcanoes are representative for the northern lowlands it seems likely that Pratt isostasy is operating and responsible for at least part of the elevational difference between the lowlands and highlands. Therefore, crustal thicknesses would be expected to be more similar in these two regions, making it even more difficult to reproduce De > 300 km at the north pole. More homogeneous crustal thicknesses would furthermore imply that spatial heterogeneity of the elastic thickness and surface heat flux will be dominated by the distribution of heat-producing elements rather than crustal thickness differences, such that heat flux will be enhanced in Acidalia and Utopia Planitia. Again, this should be detectable by in situ measurements.
 We wish to thank two anonymous reviewers and the Associate Editor for their comments, which helped to improve this manuscript. This research has been supported by the Helmholtz Association through the research alliance “Planetary Evolution and Life.”