New constraints on the thermal and volatile evolution of Mars
Introduction
The thermal and volatile evolution of Mars has important implications for the production of the Martian crust, relaxation of Martian topographic features, evolution of elastic thickness, and generation of the magnetic field. Several models for the thermal evolution of Mars have been proposed. The simplest model for cooling of Mars, the stagnant lid convection model (e.g., Hauck and Phillips, 2002), predicts the crustal production and preservation of the topographic features on Mars (Guest and Smrekar, 2004, Guest and Smrekar, 2005), but does not predict magnetic field generation without superheating of the core. A model that includes an early epoch of plate tectonics followed by stagnant lid convection (Nimmo and Stevenson, 2000, Breuer and Spohn, 2003) can explain the generation of the magnetic field without superheating (Nimmo and Stevenson, 2000), but cannot predict the crustal production within the lifetime of a dynamo. Another type of cooling model for Mars is based on a thermodynamical model of initial magma ocean crystallization (Elkins-Tanton et al., 2003). This model predicts two melt reservoirs, in agreement with the estimated volume and early formation of the Martian crust (e.g., Solomon, 2005).
There is evidence that the Martian volatile content has changed over the history of Mars (Jakosky and Phillips, 2001). The water content in the mantle and crust can change due to volcanism, as was suggested for the Tharsis region during the Noachian to Hesperian, where the formation of valley networks and large outflow channels were formed by an extensive amount of water on the surface, probably released during volcanic activity (e.g., Greeley, 1987, Jakosky and Phillips, 2001, McSween et al., 2001, Phillips, 2001, Solomon, 2005). However, the presence of water in the crust or mantle at 175 Ma (McSween et al., 2001) suggests that the loss of the water was either partial or local.
The thermal and volatile evolution of Mars has not been examined using constraints from the preservation of the Martian dichotomy and estimates of the elastic thickness over time. Each of the thermal evolution models has a different initial thermal state and rate of cooling of the lithosphere and thus has a different effect on the preservation of the topography and elastic thickness evolution. The general cooling history of the lithosphere is preserved in the elastic thickness estimates from the gravity/topography admittance. The loss of water from the interior would influence the strength of the crust and/or mantle and thus influence the relaxation of the topography and evolution of the elastic thickness.
The major topographic feature on Mars, the Martian dichotomy, has been preserved since its formation at 4.13 Ga or earlier (Solomon, 2005, Frey, 2006). The Martian dichotomy is characterized by differences in elevation of ∼5 km (Frey et al., 1998) and in crustal thickness of ∼30 km (Zuber, 2000, Neumann et al., 2004) between the northern and southern hemispheres. The Martian dichotomy was emplaced, when the planet was still relatively hot. It is thus surprising that the dichotomy elevation and crustal thickness difference did not completely relax early in Martian history. Partial modification of the Martian dichotomy, which was suggested for the Ismenius area (Guest and Smrekar, 2005) and other regions (Nimmo, 2005) occurred prior to ∼3.9–3.1 Ga based on geological studies (McGill and Dimitriou, 1990, Tanaka et al., 1992). In any case, insignificant or partial modification of the boundary suggests a relatively rapid cooling of the planet. The evolution of the elastic lithosphere based on the gravity/topography admittance (McGovern et al., 2002, McGovern et al., 2004) shows an increase of elastic thickness from <15 km in the Noachian (∼4 Ga) to at least 110 km at the Hesperian/Amazonian boundary (∼3 Ga) also suggesting relatively fast cooling. In this paper we examine whether preservation of the dichotomy boundary and an increase of the elastic thickness are in accord with any of the three thermal evolution models and whether the volatile evolution played a role.
Here, we use three models of lithospheric cooling based on three different thermal evolution models of Mars to model changes in elastic thickness and the long-term preservation of the Martian dichotomy. We test both wet and dry rheologies in the crust and mantle in order to provide constraints on both the thermal and interior volatile evolution of Mars. We model relaxation across the dichotomy boundary from the Noachian to the Hesperian/Amazonian using semi-analytical viscoelastic numerical modeling with two density and three viscosity layers (Guest and Smrekar, 2005). Our results are compared to the topography across the dichotomy boundary and to elastic thickness estimates derived from gravity and topography data (McGovern et al., 2002, McGovern et al., 2004, Belleguic et al., 2005, Hoogenboom and Smrekar, 2006).
We use three models of the cooling of the lithosphere: stagnant lid, early plate tectonics and mantle overturn (Fig. 1). The three thermal models represent distinct lithospheric cooling histories. The stagnant lid model starts with the coolest lithosphere, but does not cool as efficiently as the other two models. The early plate tectonics and mantle overturn models cool with similar efficiency, even though the mantle temperature evolutions differ significantly.
The stagnant lid model assumes cooling of the interior through an immobile lid on top of a convecting mantle. It is based on a stagnant lid coupled thermal–magmatic convection model for Martian mantle and crust (nominal model of Hauck and Phillips, 2002). Key elements of the Hauck and Phillips (2002) model are the inclusion of the energetics of melting, a wet (weak) mantle rheology, self-consistent fractionation of heat-producing elements to the crust, and a near-chondritic abundance of those elements.
In the early plate tectonics model, the planet's interior cools efficiently during the active plate tectonic regime and warms up after the transition to the stagnant lid regime. The early plate tectonics model is based on convection models of Breuer and Spohn (2003) and Spohn et al. (2001) that study the temperature evolution in the Martian interior assuming an early epoch of plate tectonics followed by single-plate tectonics with stagnant lid mantle convection. When plate tectonics stops at 4 Ga, the initially thin lithosphere significantly thickens during the first 1 Gyr causing cooling of the lithosphere and warming of the mantle. The transition from plate tectonics to a stagnant lid regime is assumed to occur at 4 Ga when the magnetic field is estimated to have shut down (Connerney et al., 1999, Breuer and Spohn, 2003).
Temperature for the mantle overturn model is based solely on the thermodynamic data that predict an inverse thermal gradient (hot on top, cold at the bottom) following mantle overturn in the Martian interior. The model is based on the thermodynamic calculations of Elkins-Tanton et al. (2003) corrected for the heat of fusion during melting (Elkins-Tanton et al., 2005). This model predicts a temperature distribution in the Martian interior after mantle overturn that is caused by unstable cumulate density stratification related to the crystallization of a magma ocean that forms due to the energy of accretion. Mantle overturn brings the hot temperature close to the surface and cold temperature to the bottom of the mantle almost instantaneously. Precise timing of this model is unclear. However, the latest work by Elkins-Tanton et al. (2005) indicates that the crust created by the overturn is emplaced within 50 Myr after accretion of the planet. Based on the convection model of Zaranek and Parmentier (2004), the mantle will cool conductively for 100–300 Myr after the overturn; mantle convection may or may not initiate subsequently depending on the initial viscosity. The limiting viscosity may be on the order 1018 Pa s or less. If mantle convection does occur, our results are valid only up to 100–300 Myr.
We determine cooling of the lithosphere by solving the heat conduction equation in the lithosphere using the initial and boundary conditions based on constraints from the original thermal evolution models (see also Guest and Smrekar (2005) for details of the thermal modeling). These constraints are the evolution of the thickness of the thermal lithosphere, defined as a conductive lid, and the evolution of the temperature at the base of the lithosphere. We model cooling of the lithosphere for 1 Ga, based on an estimate of the maximum time needed for relaxation and on the estimated age of the southern highlands for which elastic thickness estimates exist (McGovern et al., 2002, McGovern et al., 2004). Unless otherwise noted, the average global crustal thickness in each model is 62 km.
For the stagnant lid model, we use the nominal model of Hauck and Phillips (2002) that best fits the observations for Mars. In this model, 75% of the 62 km thick crust forms by ∼4 Ga which is used as a starting point in our model. This model includes radiogenic heating. The initial thermal gradient in the crust of ∼18 K/km at 4 Ga (Fig. 1a) is determined by solving the heat conduction from 4.5 to 4 Ga. From 4 to 3 Ga, the thermal gradient decreases from 18 to 13 K/km. The mantle cools only about 60 K, but because of the conductive lid thickening, the temperature at 100 km cools by 300 K. The cooling rate is constant in time.
For the early plate tectonics model, our starting point is 4 Ga when plate tectonics has stopped. The early plate tectonics model fails to generate the observed crustal thickness of at least 50 km during the stagnant lid regime and, therefore, requires that the entire crust was emplaced during the plate tectonics epoch. We adopt the model from Spohn et al., 2001, which is similar to the model EPT21 from Breuer and Spohn (2003), with an initial mantle viscosity of ∼1021 Pa s. As an initial condition, we assume a linear increase of temperature with depth in the 62-km-thick crust from 220 to 1728 K (a thermal gradient of 24.3 K/km). The mantle warms up by 230 K after the start up of the stagnant lid regime (see also Lenardic et al., 2004) in the convection model, but because of the stagnant lid thickening, the temperature at a depth of 100 km cools by 800 K (Fig. 1b), faster than in the stagnant lid model without plate tectonics (Hauck and Phillips, 2002). Therefore, the end of plate tectonics is reflected in the shallow lithosphere (∼50 km) by an increased cooling rate. Cooling is fastest during the first 250 Myr.
For the mantle overturn model, we assume a conductive cooling of the reversed temperature profile after mantle overturn which may have occurred as early as 50 My after accretion (Elkins-Tanton et al., 2005). As an initial condition in the crust, we assume a linear temperature increase with depth from 220 to 1790 K (thermal gradient of 25.3 K/km). Temperature cools by about 200 K in the mantle and by 750 K at 100 km during 1 Gyr (Fig. 1c). Cooling is very fast during the first 250 Myr. The cooling of the lithosphere in this model is similar to the cooling in the early plate tectonics model.
Our models of lithosphere cooling are simplified versions of the thermal models, especially the plate tectonics and mantle overturn models, for which we make assumptions about crustal thickness, radiogenic heating, and initial thermal gradient. We use a crustal thickness of 62 km. This value is predicted by the stagnant lid model and is within the model uncertainties of the other two thermal models and thus allows for a comparison of the models results. The distribution of the radiogenic heating is not specified for the plate tectonics and mantle overturn models and so we neglect it in our modeling. Therefore, the initial thermal gradient in the crust is not specified for these models and we assume the simplest option, the linear thermal gradient. As a consequence, our crust is cooler than it would be with the radiogenic heating included. The initial thermal gradient is high because of the initially thin lithosphere. In order to test for uncertainties in the model parameters, for each of the three thermal models (in the following text termed “warm mantle” variant) we construct a temperature profile with a 200 K cooler mantle temperature (termed “cold mantle” variant). This variation in mantle temperature results in a 100–200 K difference in the temperature at the base of the crust, and simulates possible variations in thermal gradient and radiogenic heating.
We model the topographic and crustal relaxation of the dichotomy boundary in the 4500 km long cross-section through the boundary (Fig. 2). The vertical dimension of the model is 3000 km. Our model consists of two layers of different density materials, crust and mantle, and three viscosity layers. The initial topography and crust–mantle boundary relief, in the Cartesian coordinate system, are described using geometric functions of sin and cos and are thus dependent only on the horizontal distance and time.
We simulate topographic relaxation using a semi-analytical model (Guest and Smrekar, 2005) in which we solve the same equations as in the model of gravity driven relaxation of a topographic load at the surface of a density-stratified incompressible fluid (Grimm and Solomon, 1988, Appendix A), except for the coordinate system (Cathles, 1975). In this method, the horizontal variations of stress and displacement are transformed to the frequency domain with a Fourier transform (Cathles, 1975) while the vertical variations are integrated numerically using a fourth order Runge–Kutta method thus allowing for vertical variations of viscosity. We solve the time-dependence of the coupled topographic decay on the surface and the crust–mantle boundary by integrating velocity at the boundaries over the time step. We solve for two characteristic deformation modes associated with two density interfaces. We update the boundary topographies after each time step and solve the equilibrium and constitutive equations again. Such a time stepping allows for accommodation of viscosity changes with time. The viscosity variations with depth and time must be input in the semi-analytical model a priori at each time step.
In order to incorporate time changes in viscoelasticity, a Laplace transform is applied on the constitutive equation of the incompressible viscous fluid (Zhong, 1997, Eq. (A1)). The transformed equation has the same mathematical form as the constitutive equation for the incompressible viscous fluid if the viscosity that is input in the solution is expressed as:where ηs is the viscosity dependent on the Laplace transform variable, η the viscosity calculated using only creep strain rate (representing a viscous fluid), s the Laplace transform variable, and τ is the relaxation time defined in our calculations as τ = η/3G, where G is rigidity. The Laplace transform variable s is equivalent to the reciprocal value of time t, which is numerically integrated over the time of calculations (0–1 Gyr) in seconds. If t is significantly smaller than τ, then the value of ηs is decreased in comparison to the value η (see Eq. (1)). This decrease in viscosity can be understood as being caused by the increased importance of the elastic strain rate, and thus for small times we obtain a viscoelastic solution. If t is near or higher than τ, then value of ηs is near the value η, and viscous solution is obtained. In this way, our numerical solution for viscous fluid becomes a solution for the viscoelastic fluid. This method was tested against benchmarks of Zhong (1997) (see Guest and Smrekar, 2005).
Viscosity is updated in the model before each time step (, where is the strain rate based on the creep laws shown in Table 1, and σ is the stress). Because strain rate is dependent on temperature and stress, viscosity varies continuously with depth and time. While temperature is based on the three thermal models, it is impossible to predict the stress variations for all depths over 1 Gyr because stress evolves during the relaxation depending on the rheology used. Based on comparison of the semi-analytical modeling with the finite-element modeling (details in Guest and Smrekar, 2005), we simplify the depth dependence of viscosity into three layers, initially consisting of the viscosity of the upper crust, lower crust and mantle. The depth of the boundary between the crust and mantle is fixed, whereas the boundary between upper and lower crust evolves with time depending on the depth location of viscosity of 1029 Pa s. The later boundary divides the elastic and viscoelastic part of the lithosphere, initially located in the crust. When this boundary reaches the mantle due to lithospheric cooling, the three layers become elastic crust, elastic mantle (both with the same viscosity) and viscoelastic mantle. Therefore, the time evolution of only four parameters is needed for our calculations: viscosity of the elastic layer, viscosity of the lower crust, viscosity of the mantle and depth of the elastic layer. The viscosity of the elastic layer, represented by a value of 1029 Pa s, is constant with time. The time evolution of the viscosity for lower crust and mantle is based on the stress estimates that are described in brief below. The description of the location of the depth of the elastic thickness is given later in the chapter.
The relaxation of stress in the lower crust is difficult to predict. For strong rheologies (dry, cold and warm temperature), we assume zero strain rate changes in the layer. The stress evolution is then solved in small time steps from time 0 till calculation time t assuming an exponential decay of stress in time (Turcotte and Schubert, 2002) and temperature change with time. For weak rheologies (wet, cold, and warm temperature), we assume a strong lower crustal flow, similar to the flow in the mantle. The viscosity in the lower crust is then the same as viscosity in the mantle. Only the stress at the bottom of the layer is necessary to predict for our calculations, because the viscosity at the bottom of the lower crust successfully represents the viscosity of the whole lower crust (shown in Guest and Smrekar, 2005). Stress in the mantle is essentially constant (0.12 MPa) and mantle viscosity changes mainly due to temperature changes. The viscosity in the mantle is represented by the viscosity value just below the base of the thermal lithosphere for the stagnant lid and plate tectonics models. For the mantle overturn model, we use the depth of 150 km because the base of the thermal lithosphere, defined as a conductive lid, would be thicker than 250 km which is too deep to influence the relaxation of the dichotomy boundary. The depth of 150 km is close to the depth of the thermal lithosphere in the other two models during first 300 Myr when most of the relaxation happens. Therefore, the mantle viscosity is higher and relaxation is smaller than if we used a 250 km thick thermal lithosphere. This choice of depth does not influence the elastic thickness modeling.
The values of mantle and lower crustal viscosities input in the relaxation models are shown in Fig. 3. These values are taken at the bottom of the conductive layer (as shown on Fig. 1) for first two models and at depth of 150 km for the mantle overturn model. The viscosity of the upper elastic layer is always 1029 Pa s. The mantle viscosity at the beginning of calculations is the same for the stagnant lid and early plate tectonics models and is one order of magnitude higher for the mantle overturn model. Mantle viscosity increases by a factor of five during 1 Gyr for the stagnant lid model. For the early plate tectonics model, mantle viscosity decreases by two orders of magnitude during 1 Gyr due to heating of the planet's interior. For the mantle overturn model, mantle viscosity increases by one order in 100 Myr and at least by ten orders in 1 Gyr. The viscosity in the lower crust for dry rheology increases by four to seven orders during first 100 Myr in all three models. This is caused by fast relaxation of stress in the layer. We use the same viscosity in the lower crust and in the mantle if wet rheology is used.
The location of the elastic boundary in the crust, defined by viscosity of 1029 Pa s, is based on the thermal evolution as shown on Fig. 1 and the stress distribution in the southern hemisphere of the model. The initial deviatoric stress distribution in the crust is caused by the elevation and crustal thickness differences between the northern lowlands and southern highlands (calculation time 0). The initial values of 2 MPa at the surface to 20 MPa at the base of crust are based on finite element modeling. Early in time, the stress in the lower crust (defined as having a viscosity <1029 Pa s) relaxes and redistributes to the elastic part of the crust. The viscosity difference due to the stress increase in the elastic crust is, however, insignificant in comparison to changes resulting from the temperature gradient and, therefore, the elastic thickness does not change. When cooling becomes significant (around 10–100 Myr), the base of the elastic crust migrates downward. The stress distribution stays locked within the elastic upper crust and does not change, but in the meantime, stress decreased via viscous relaxation in the lower crust. The stress drop in the lower crust results in a more rapid increase of viscosity causing a more rapid increase in the thickness in the elastic layer than would result from the decrease in temperature alone. When the bottom of the elastic layer reaches the mantle, the thickness of the layer increases due to a change of the rheological law and a stress drop with temperature being a minor effect.
When wet crustal rheology is used, two separate elastic layers, consisting of a layer in the crust and a layer in the mantle, will develop. In such a case the effective elastic thickness is determined using (Burov and Diament, 1995):where Te is the effective elastic thickness, h1 the depth of the elastic thickness in the crust, hc the crustal thickness and h2 is the depth of the elastic thickness in the mantle. If the elastic thickness in the crust (h1) is the same as the elastic thickness in the mantle (h2 − hc) then Te = 1.26h1. The thickening of the elastic layer is shown on Fig. 4 and discussed in more details in Section 2.
The uncertainties in stress evolution introduce uncertainties into the viscosity predictions. Using a simple viscosity profile also results in small errors in prediction of topographic relaxation (Guest and Smrekar, 2005). For this reason, we will distinguish only three stages of relaxation in our results: no relaxation, partial relaxation and complete relaxation, even though our technique provides more precise results. Despite the challenges inherent in analytic models of viscosity evolution, our approach newly includes the time evolution of viscosity and elastic thickness, and is, therefore, better suited for modeling of the effects of cooling of the lithosphere than the other studies (Zhong, 1997, Nimmo and Stevenson, 2001, Nimmo, 2005).
Input parameters that influence the relaxation of the dichotomy boundary are the highland elevation, dichotomy boundary slope, crustal thickness, and creep laws. We assume an initial highlands elevation of 5 km that is isostatically compensated by a crustal root of 24 km (Fig. 2). The initial slope of the dichotomy boundary is 2°, slightly higher than present-day observations (Frey et al., 1998). The average crustal thickness for Mars is estimated to be greater than 45 km (Neumann et al., 2004) and lower than 80 km (Nimmo and Stevenson, 2001). In our models, we use an average crustal thickness of 62 km, yielding values of 47 and 77 km in northern plains and southern highlands, respectively. We use this value because it is predicted by the stagnant lid thermal model and is also consistent with the other two thermal models. Our rheology is based on creeps laws determined during laboratory experiments (Table 1). For a wet crust, we use the creep law determined with undried specimens of diabase (Caristan, 1982), while for dry crust samples are dried prior to deformation (Mackwell et al., 1998). Similarly for mantle flow laws, water-free and water-saturated conditions were used on olivine, the most abundant and probably the weakest mineral of peridotites. (Karato and Wu, 2001). The density of crust is 2900 kg/m3, and the density of mantle is 3500 kg/m3 (Zuber, 2000). The Young's modulus and Poisson's ratio are 1.0e11 Pa and 0.5, respectively, for both crust and mantle.
We use estimates of elastic thickness for local areas of different ages (McGovern et al., 2002, McGovern et al., 2004, Hoogenboom and Smrekar, 2006, Belleguic et al., 2005) as a general constraint on elastic thickness with time. McGovern et al., 2002, McGovern et al., 2004 use admittance to calculate elastic thickness for 15 regions located mostly on southern hemisphere of Mars. These regions have surface ages ranging from the Noachian to Amazonian (Fig. 4), and are either located in the old cratered terrain of the southern highlands or consist of large volcanic provinces. Although these regions have had distinct geologic histories, they show a trend of increasing elastic thickness with time that very likely reflects the overall cooling of the planet (e.g., Solomon and Head, 1990, McGovern et al., 2002). Elastic thickness estimates in the northern lowlands, except Elysium Rise and Alba Patera, are problematic and were not included in the analysis of McGovern et al., 2002, McGovern et al., 2004. The elastic thickness for four regions in the northern plains in the Noachian was estimated by Hoogenboom and Smrekar (2006). Their values range from 0 to 45 km with an overlap in 10–25 km. The values for Alba Patera and Elysium Rise in the Hesperian–Amazonian were recalculated by Belleguic et al. (2005) with an improved method that gives slightly higher values that those given by (McGovern et al., 2004). The age of the regions is constrained only roughly.
In a later section, we compare our estimates of elastic thickness from thermal models to those derived from admittance. To avoid confusion we refer to values derived from gravity and topography data as ‘admittance estimates’ to distinguish them from the estimates predicted by our thermal models. The areas which represent each epoch and that we try to fit with our modeling are: the Noachian terrains with an elastic thickness up to 25 km, Solis Planum of Hesperian age and elastic thickness of ∼30 km, and Hesperian–Amazonian chasmata with elastic thicknesses higher than ∼60–100 km. All these regions are located in the vicinity of Tharsis in the southern hemisphere and should, therefore, experience similar thermal evolution. The fit to Alba Patera and Elysium Rise, located in the northern hemisphere, will be discussed separately as well as the fit to the South Hellas Rim. It should be noted that most of the observations were made on the southern hemisphere, where the crustal thickness is higher than our average thickness (but this is appropriate for our models).
A few caveats are worth considering when attempting to fit ‘admittance elastic thickness’. One possible uncertainty is that the surface age might not reflect the time of loading. For example, one hypothesis is that a broad scale plume caused removal of the lower crust under the northern plains (Zhong and Zuber, 2000). In such a case, the elastic thickness may have been reset without altering the surface age. Or, the time of loading could be older than surface age. Areas of very significant volcanic activity, such as Elysium and Tharsis, might have reset the elastic thickness due to the sustained volcanic activity, however, modeling of such a problem confirms that the elastic thickness reflects the general decrease of heat flux predicted for Mars by thermal models (Dombard and Phillips, 2005). We concur with McGovern et al.'s (2002) assessment that the increase in elastic thickness with time reflects global cooling. However, we do not try to match their values of elastic thickness precisely due to the caveats above. The features with the largest admittance elastic thickness estimates are the Valles Marineris Chasmata and the South Hellas Rim with intrusion that are all modeled with bottom loading. The bottom loading tends to give larger values that top loading models (e.g., Petit and Ebinger, 2000) but certainly can’t be responsible for the very large differences in the values.
Section snippets
Elastic thickness
For each thermal model, we predict the thickening of the elastic lithosphere using combinations of wet or dry mantle and crustal rheologies, and warm or cold mantle temperature variations. We do not consider the combination wet crust/dry mantle because we are uncertain about the prediction of stresses in the relaxation modeling due to the lack of the experience with finite element modeling for this case. However, we treat this possibility in the Discussion section.
The predicted elastic
Discussion
The comparison between the modeled and admittance elastic thicknesses has major implications for the rheology during the Noachian and the Hesperian in the southern hemisphere.
In the Noachian, the elastic thickness due to wet crustal rheology fits best the admittance elastic thickness of ∼15 km in the Noachis, Cimmeria and Hellas regions, located in the central southern hemisphere. The elastic thickness is better fit with wet than dry crustal rheology for all three thermal models. The exception
Conclusions
We have used three thermal evolution models for Mars, stagnant lid, early plate tectonics followed by stagnant lid, and mantle overturn, calculated with two different mantle temperatures along with wet and dry rheologies, to predict the temperature and viscosity evolution of the lithosphere. These viscosities are then used to predict the relative amount of relaxation of the global dichotomy and elastic thickness values from the Noachian to the Hesperian/Amazonian.
The thermal models predict
Acknowledgements
This work was supported by a grant from the Mars Data Analysis Program. We thank Sean Solomon for his comments on Martian rheology. We thank Linda Elkins-Tanton and Patric J. McGovern for constructive comments that helped to improve the manuscript.
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