Thermochemical evolution of Mercury s interior

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH: PLANETS, VOL. 118, , doi: /jgre.20168, 2013 Thermochemical evolution of Mercury s interior N. Tosi, 1 M. Grott, 2 A.-C. Plesa, 2,3 and D. Breuer 2 Received 27 June 2013; revised 23 August 2013; accepted 28 September 2013; published 11 December [1] A number of observations performed by the MESSENGER spacecraft can now be employed to better understand the evolution of Mercury s interior. Using recent constraints on interior structure, surface composition, volcanic and tectonic histories, we modeled the thermal and magmatic evolution of the planet. We ran a large set of Monte Carlo simulations based on one-dimensional parametrized models, spanning a wide range of parameters. We complemented these simulations with selected calculations in 2-D cylindrical and 3-D spherical geometry, which confirmed the validity of the parametrized approach and allowed us to gain additional insight into the spatiotemporal evolution of mantle convection. Core radii of 1940 km, 2040 km, and 2140 km have been considered, and while in the first two cases several models satisfy the observational constraints, no admissible models were found for a radius of 2140 km. A typical thermal evolution scenario consists of an initial phase of mantle heating accompanied by planetary expansion and the production of a substantial amount of partial melt. The evolution subsequent to 2 Gyr is characterized by secular cooling that proceeds approximately at a constant rate and implies that planetary contraction should be ongoing today. Most of the models predict mantle convection to cease after 3 4 Gyr, indicating that Mercury may be no longer dynamically active. Finally, assuming the observed surface abundance of radiogenic elements to be representative for the entire crust, we determined bulk silicate concentrations of ppb Th, ppb U, and ppm K, similar to those of other terrestrial planets. Citation: Tosi, N., M. Grott, A.-C. Plesa, and D. Breuer (2013), Thermochemical evolution of Mercury s interior, J. Geophys. Res. Planets, 118, , doi: /jgre Introduction [2] After completing three flybys between 2008 and 2009, the Mercury Surface, Space Environment, Geochemistry, and Ranging (MESSENGER) spacecraft was inserted into a polar orbit around Mercury on 18 March Since its first encounter with Mercury, MESSENGER has delivered a wealth of information that has been dramatically advancing the understanding of the geological, chemical, and physical state of the planet. While pre-messenger attempts at modeling the thermal and chemical history of Mercury s mantle and crust could only rely on a limited amount of observational evidence as provided by the Mariner 10 mission [Hauck et al., 2004; Breuer et al., 2007; Redmond and King, 2007], several new findings can now be employed to better constrain the dynamics and evolution of the interior. 1 Department of Planetary Geodesy, Technische Universität Berlin, Berlin, Germany. 2 Department of Planetary Physics, German Aerospace Center, Berlin, Germany. 3 Department of Planetology, Westfälische Universität Münster, Münster, Germany. Corresponding author: N. Tosi, Department of Planetary Geodesy, Technische Universität Berlin, Strasse des 17. Juni Berlin, Germany. (nic.tosi@gmail.com) American Geophysical Union. All Rights Reserved /13/ /jgre [3] Earth-based radar measurements of the planet s spin state allowed Margot et al. [2007] to prove conclusively that Mercury s core must be decoupled from the overlying solid shell and hence be, at least partly, liquid. Low-degree coefficients of the gravity field obtained from the MESSENGER radio science experiment [Smith et al., 2012] combined with the above measurements have been used to determine the normalized polar moment of inertia of the planet (C/MR 2 )as well as the ratio of the moment of inertia of the solid shell overlying the liquid core to that of the whole planet (C m /C) [Margot et al., 2012]. These two parameters provide strong constraints on the internal mass distribution. While previous models of Mercury s thermal history assumed a core radius of 1800 km [Hauck et al., 2004; Grottetal., 2011; Redmond and King, 2007], with C/MR 2 = and C m /C = [Margot et al., 2012; Smith et al., 2012], current interior structure models predict a significantly larger value [Riner et al., 2008; Rivoldini et al., 2009; Hauck et al., 2013]. In particular, Hauck et al. [2013] used a statistical approach based on Monte Carlo techniques to calculate Mercury s interior structure and found that 2020 km and 3380 kg/m 3 represent optimal values for the radius of the liquid core and the mean density of the overlying solid shell, respectively. These values appear to be robust across a broad range of parameters. Furthermore, depending on the composition of the core, the possibility exists that a solid FeS layer forms atop the core-mantle boundary (CMB). Even though

2 the thickness of this layer is difficult to constrain [Hauck et al., 2013], it would further reduce the thickness of the overlying silicate shell. In synthesis, considering the whole range of solutions proposed by Hauck et al. [2013], mantle thicknesses between 300 and 500 km are possible, with intermediate values around 400 km being the most likely. On these grounds, Michel et al. [2013] recently presented a parametric study of 2-D axisymmetric simulations of thermal convection. Among other parameters, they analyzed the effects of different core radii between 1800 and 2100 km on the longevity of mantle convection and concluded that for a sufficiently high rate of internal heat generation, convective heat transport can persist until present day in a mantle at least 400 km thick. [4] As already hypothesized on the basis of early Mariner 10 data [e.g., Robinson and Lucey, 1997], the analysis of images delivered by the MESSENGER cameras confirmed that volcanism, both effusive and explosive, was widespread on Mercury and played a prominent role in shaping its surface [Head et al., 2009; Kerber et al., 2011]. The morphology of volcanic features associated with the Caloris basin [Head et al., 2009] and of the vast plains that cover a large portion of the northern hemisphere [Head et al., 2011] suggests that Mercury was characterized by extensive volcanic activity that postdated at least the end of the late heavy bombardment (LHB). Furthermore, X-ray spectrometric measurements of the surface materials revealed higher Mg/Si and lower Al/Si and Ca/Si ratios than typical terrestrial or lunar crusts [Nittler et al., 2011]. These ratios are indicative of compositions intermediate between basaltic and ultramafic rock types, possibly comparable to terrestrial komatiites and thus hinting at an origin from partial melts at melt fractions as high as 20 30%. On the base of early MES- SENGER data taken during the flybys, Grott et al. [2011] were the first to use the information on the prolonged phase of volcanic activity to constrain the thermochemical history of the interior. While previous parametrized models predicted a very short phase of crustal production concentrated in the first few ten million years of the evolution [Hauck et al., 2004], Breuer et al. [2007] and Grottetal.[2011] found that the presence of a thin, low-conductivity layer like a highly fractured megaregolith acts as an efficient insulator that can help to significantly extend the period of melt production as required by the observations. [5] Prior to MESSENGER observations, the planet s bulk silicate composition was only poorly constrained, and the relative abundance of heat-producing elements could only be estimated by invoking a particular formation scenario. Several models have been proposed that can account for the unusually large metal-to-silicate ratio of Mercury, such as a late evaporation of silicates [Fegley and Cameron, 1987], a high-temperature equilibrium condensation of nebular materials [Lewis, 1973], and a giant impact [Cameron et al., 1988] or meteoritic mixing of refractory and volatile condensates [Morgan and Anders, 1980]. Measurements of the average surface abundance of K ( ppm), Th ( ppb), and U (90 20 ppb) performed by the MESSENGER Gamma-ray spectrometer [Peplowski et al., 2011, 2012] finally shed some light on possible formation scenarios. The relatively high elemental ratios of K/Th and K/U hint at a volatile content similar to other terrestrial planets and hence permit to rule out at least vaporization and condensation models as these require a nearly complete loss of volatiles. However, at the relatively low-oxygen fugacities of Mercury s interior, which can be inferred from the high-s and low-feo content of the surface [Zolotov et al., 2013], elements such K, Th, and U, which are usually lithophile, can become more siderophile [Malavergne et al., 2010]. McCubbin et al. [2012] thus pointed out that it is still uncertain whether Mercury s mantle is enriched in volatiles or not. In addition, up to 10% of Mercury s U inventory, and possibly also a significant amount of Th, may have partitioned into the core if its formation took place at low oxygen-fugacity [McCubbin et al., 2012]. [6] More important in the context of thermal evolution modeling is the fact that the measured surface abundances of radioactive elements can be used to constrain the planet s thermal history by inferring the degree to which K, Th, and U originally in the bulk silicate mantle have fractionated into the crust. Michel et al. [2013] systematically analyzed the effects of varying the total amount of heat sources by assuming that the mantle contains between 20% and 50% of the observed surface abundance of radiogenic elements. When considering the highest heat production rates, they observed not only extended periods of mantle convection but also a prolonged phase of partial melt production at high meltfractions in accord with MESSENGER s X-ray observations [Nittler et al., 2011]. However, the lack of a self-consistent treatment of heat sources partitioning between mantle and crust as well as of latent heat consumption upon melting in the energy budget renders the conclusions of Michel et al. [2013] liable to further sharpening and improvement. [7] Finally, it has long been recognized that the surface of Mercury is characterized by a global system of compressional tectonic landforms the so-called lobate scarps which are generally interpreted to be the result of periods of planetary contraction [Strom et al., 1975]. Estimates of the contractional strains associated with lobate scarps as imaged both by Mariner 10 and by MESSENGER during its flybys implied a limited decrease of planetary radius of up to 0.8 km only [Watters et al., 1998, 2004, 2009; Watters and Nimmo, 2010]. As shown by Grottetal.[2011], obtaining such small values of global contraction represents a severe constraint for evolution models which can only be met by assuming the contribution of core freezing to be negligible for most of the planet s history. However, as pointed out by Watters et al. [2009], the identification of lobate scarps, and thus the accompanying contraction estimates, can be strongly biased by the illumination conditions at which the images were taken. Indeed, data collected during MESSENGER s orbital phase at high-sun incidence angles allowed Di Achille et al. [2012] to identify a number of previously undetected structures. By mapping an area covering 21% of the surface and extrapolating the results to the entire planet, the authors estimates of contractional strain are significantly higher than previous ones. These imply an overall radius decrease of km when considering lobate scarps only and of km if wrinkle ridges are also taken into account, with the upper end of these ranges including the possibility of additional contractional features that may have been erased by large impacts. [8] To summarize, in order to fully exploit the latest constraints posed by MESSENGER observations, models of the thermochemical evolution of Mercury s interior should sat- 2475

3 isfy the following requirements: (1) consider a mantle with a thickness between 300 and 500 km; (2) allow for at least 1 Ga of volcanic activity (i.e., well beyond the end of the LHB); (3) predict high melt-fractions (between 20 and 30%); (4) form a crust enriched in radiogenic elements compatible with the observed surface abundances of K, Th, and U; (5) account for the possibility that part of the heat sources may have partitioned into the core; (6) result in a net global contraction not larger than 3 4 km; and (7) allow for the freezing of only a small inner core in order to simultaneously provide a source of compositional buoyancy able to drive a dynamo and not to induce an excessive planetary contraction. [9] Here we present two sets of thermochemical evolution models aimed at fulfilling the above constraints. First, we carried out a large set of Monte Carlo simulations (40,000 in total) using 1-D parametrized models in line with those of Grottetal.[2011]. Second, we chose a representative 1-D model to conduct finite-amplitude numerical simulations of thermochemical convection in 2-D cylindrical and 3-D spherical shell geometries aimed at verifying the limits of validity of the parametrized models and gaining additional insight into the spatiotemporal evolution of mantle convection. [10] In section 2 we describe the main features of our parametrized and numerical models, the set of parameters that were varied, and the conditions employed to establish whether a run could be considered admissible or had to be rejected on the basis of the observational constraints. Section 3 presents the results of the Monte Carlo simulations based on the parametrized approach and of one representative evolution model, which has also been investigated in detail through fully dynamic numerical solutions. Discussion of the results and conclusions are presented in sections 4 and 5. An appendix containing a detailed description of the 2-D and 3-D models completes our work. 2. Theory and Models 2.1. One-Dimensional Parametrized Models [11] A convenient way to simulate the interior evolution of planetary bodies is to resort to one-dimensional models in which appropriate scaling laws are employed to quantify convective heat transport in the mantle and to determine the thickness of the thermal boundary layers across which heat is transported by conduction [e.g., Grasset and Parmentier, 1998; Reese et al., 1998]. Such models are computationally inexpensive and allow one to span a broad range of parameters, which could hardly be covered using two- or three-dimensional simulations. In the first part of the paper (section 3.1), we show results from parametrized models of Mercury s thermochemical evolution that closely follow those of Grott et al. [2011] and Morschhauser et al. [2011], to which we refer for a detailed description. Here we limit ourselves to recall their main features and highlight a few important differences. [12] We solve numerically the time-dependent energy balance equations for the core, mantle, and stagnant lid, which deliver respectively the evolution of the core-mantle boundary temperature, the mantle temperature below the lid, and the thickness of the lid itself. Boundary layer theory [Turcotte and Schubert, 2002] is used to determine the 2476 thickness of the upper and lower thermal boundary layers from a Nusselt-Rayleigh scaling relationship as appropriate for stagnant lid convection [Grasset and Parmentier, 1998]. With the above information, the interior evolution is obtained by constructing a radial thermal profile for the entire planet assuming that the temperature increases linearly in the boundary layers and adiabatically in the mantle and core. [13] As far as the core is concerned, the solidification of its inner part would lead to a significant volume decrease, and hence to a large amount of global contraction. Assuming a pure Fe composition, freezing of the entire core would cause a radius change as large as 17 km [Solomon, 1976] or more in the presence of a light-alloying element such as sulfur. Grottetal.[2011] showed that already the growth of a relatively small inner core would lead to a global contraction that easily exceeds the amount observed at the surface, even if the revised estimates of Di Achille et al. [2012] were considered. Therefore, we assume here that only a small solid inner core can form towards the end of the evolution, so that the contribution of its solidification to both core energy balance and global contraction is negligible to first order. An implicit consequence of this assumption is that the amount of S present in the core must be sufficiently large (typically & 6%) to allow for its partial solidification only toward the end of the evolution [Hauck et al., 2004; Williams et al., 2007; Grott et al., 2011]. For small S concentrations, in fact, most of the core would freeze rapidly, rendering unfeasible to match the contraction constraint. [14] Energy conservation in the mantle accounts for the consumption of latent heat of melting. Melt fractions are computed by comparing the mantle temperature to the solidus of dry peridotite [Takahashi, 1990]. Within the small pressure range of Mercury s mantle (approximately up to 5 GPa, assuming a 400 km thick silicate shell), the density of the molten phase is lower than the density of the surrounding matrix [Suzuki et al., 1998]. Therefore, melt is not expected to be trapped or sink to the bottom of the mantle. The solidus of the mantle residue is increased linearly with the degree of depletion [Morschhauser et al., 2011], and we assume a total maximum solidus change of 150 K between undepleted and depleted peridotite, i.e., harzburgite [Maaløe, 2004], at a maximum melt fraction of 40%. [15] The model also accounts for crust formation and partitioning of radiogenic elements. Melt is produced when the mantle temperature rises above the peridotite solidus. This melt is extracted to form the crust, being distributed uniformly across the planetary surface. Since the thermal conductivity of volcanic and plutonic rocks is generally lower than that of pristine peridotite [Clauser and Huenges, 1995], the crust is assigned a lower conductivity than that of the underlying mantle. We varied crustal conductivity between 1.5 and 4 W/mK; the latter being the value used for the mantle. Furthermore, the uppermost part of the crust (down to a depth of up to 5 km) is assigned a conductivity of 0.2 W/mK, thus simulating the effect of an insulating megaregolith layer. This layer of fractured bedrock could have been created by large impacts, and its composition is the same as that of the underlying crust. Finally, fractionation of heat-producing elements between mantle and crust is obtained according to a fixed enrichment factor ƒ (see section 2.4).

4 2.2. Planetary Contraction [16] Given the interior temperature profile as a function of time, changes of the planetary radius can be computed from volume changes associated with thermal expansion/contraction of the core and mantle, from partial melting and accompanying differentiation and from the solidification of the inner core [Grottetal., 2011]. As discussed above, the latter contribution is here neglected under the assumption that the amount of light element (e.g., sulfur) alloyed with iron is sufficiently large for inner core growth to start toward the end of the evolution and not to contribute significantly to the global radius balance. Temporal radius variations due to core and mantle heating or cooling are obtained from R th (t) = c(t c (t) T c,0 ) R3 c + 1 3R 3 p R 2 p Z Rp R c m Tm (r, t) T m,0 (r) r 2 dr, (1) where R c and R p are the core and planetary radii, T c and T m (r) the CMB temperature and the mantle temperature profile, and c and m are the coefficients of thermal expansion of the core and mantle, respectively. The subscript 0 refers to initial values. [17] The change of planetary radius associated with mantle differentiation and melt extraction is a consequence of the density difference between undepleted and depleted mantle material and is given by R md (t) = 1 f ıv V (D cr(t) D cr,0 ), (2) where f is the maximum volume fraction of extractable crustal components, D cr the crustal thickness, and ıv/v the volume change upon differentiation. The latter can be expressed as [Grott et al., 2011] ıv V = f (1 )+ m f cr 1. (3) d Here, is the ratio of extrusive to intrusive volcanism, and m, d,and cr are the density of the undepleted mantle, depleted mantle, and crust, respectively. The total radius change is then given by the sum of equations (1) and (2) Thermal Expansivity of The Liquid Core [18] The contribution of mantle and core cooling to planetary contraction (equation (1)) depends crucially on the respective coefficients of thermal expansion m and c. While the former is not expected to experience large variations across the thin mantle of Mercury [Li et al., 2007; Tosi et al., 2013], the latter can vary substantially with pressure across the large core and, in addition, depends on chemical composition [Rivoldini et al., 2009; Williams, 2009]. Both Hauck et al. [2004] and Grottetal.[2011] used a thermal expansivity of K 1 for the core, a value that is likely attained only at the largest pressures near the center of the planet where c is considerably smaller than at the core-mantle boundary. In order to provide more realistic estimates of the global contraction due to core cooling, we adopt here the approach of Rivoldini et al. [2009] to determine c for a liquid FeS mixture. The pressure and temperature dependence of the coefficient of thermal expansion is approximated as follows [Poirier, 2000]: c(p, T )= c,0(t ) K T,0 (T ), (4) K T (P, T ) where K T is the isothermal bulk modulus and the subscript 0 indicates that a given quantity is computed at reference ambient pressure. The reference thermal expansivity is obtained from its thermodynamic definition, i.e., 1 c,0 (T )= 0 (T 0 (T (5) where 0 (T ) is the reference density of the FeS mixture, which can be expressed as the sum of the fractional weighted densities of each component (i.e., liquid Fe and liquid FeS): 1 FeS 0 (T )= 0,Fe (T ) + 1 FeS. (6) 0,FeS (T ) 2477 In equation (6), FeS denotes the weight fraction of liquid FeS, which is given by S /0.3647, with S being the weight fraction of S. Finally, assuming equation of state parameters for liquid Fe from Anderson and Ahrens [1994] and liquid FeS from Kaiura and Toguri [1979], we calculate the temperature dependence of 0,Fe and 0,FeS and the isothermal bulk modulus needed in equation (4) using a third-order Birch-Murnaghan equation of state [e.g., Poirier, 2000]. [19] If the core is vigorously convecting, its temperature profile is adiabatic, and dt (P, T )T = dp K S (P, T ), (7) where is the Grüneisen parameter, which can be obtained from K T (P, T ) and its pressure derivatives [Anderson, 2000; Rivoldini et al., 2009]. The quantity K S denotes the adiabatic bulk modulus, which is related to K T through the thermodynamic relation [Poirier, 2000] K S (P, T )=K T (P, T )(1 + (P, T )(P, T )T ). (8) [20] Assuming an outer core radius of 2040 km, interior structure models predict that pressure ranges from approximately 5 GPa at the CMB to 40 GPa at the planet s center [Hauck et al., 2013]. We used equation (4) for different values of the CMB temperature and sulfur contents to calculate the pressure variation of the core thermal expansivity along an adiabatic temperature profile as given by equation (7). Figure 1a shows the profile of c for T c = 1800 Kand S =6%, while Figure 1b gives a contour plot of the volume average of c as a function of CMB temperature and sulfur content assuming a pressure interval of 5 40 GPa. For a broad temperature range and expected sulfur concentrations not larger than 10% [Rivoldini et al., 2009; Grottetal., 2011], c is systematically higher than K 1. Here, we adopt a single expansivity value to determine radius changes due core cooling of c = K 1, which results from averaging the profile of Figure 1a over the core volume Parameters and Admissible Models [21] Thermochemical evolution models are sensitive to a number of poorly constrained parameters, which need to be varied systematically. Using parametrized models, we ran four series of Monte Carlo simulations, each consisting of 10,000 runs. Within the error bars of the estimated moment of inertia factors, interior structure models are compatible with mantle thicknesses D m ranging from about 300 to 500 km [Hauck et al., 2013], and we distinguished the four sets of simulations according to this parameter. In

5 a b c [10 5 K 1 ] s [%] P [GPa] layer T c, thermal conductivity of the crust k cr, thickness of the regolith layer d reg, volume change upon differentiation ıv/v, and enrichment factor of radiogenic elements ƒ. We varied the above parameters according to a uniform random distribution across the (absolute) intervals reported in Table 1. Table 2 lists instead the parameters that have been held constant. [22] In all models, a primordial crustal thickness of 5 km was considered in which heat sources are enriched by a factor ƒ with respect to the primitive mantle. For a given value of ƒ, heat production in the primitive mantle, expressed in W/kg, is given by H 0 = 1 ƒ 3X H i exp( i t), (9) i=1 where H i are the heat production rates associated with all relevant radioactive species (i.e., K, Th, and U) using the abundances of Peplowski et al. [2012] (see Table 2), i are the corresponding decay constants, and t = 4.5 Gyr. From mass-balance considerations, the heat production in the mantle can then be calculated from H m = 1+ R3 p R3 cr cr R 3 cr (1 ƒ) R3 c m! H 0, (10) T c [K] c [10 5 K 1 ] Figure 1. (a) Thermal expansivity of liquid FeS as a function of pressure for T c = 1800 Kand S = 6% and (b) volume-averaged thermal expansivity as a function of S concentration and CMB temperature assuming a core pressure range of 5 40 GPa. the first three sets, we employed a mantle thickness of 300, 400, and 500 km. The last set consisted of models with D m = 400 km but 10% U and Th were assumed to be partitioned into the core [McCubbin et al., 2012]. In all sets, we used random combinations of the following parameters: initial mantle temperature T m,0, mantle reference viscosity ref, initial temperature jump across the bottom thermal boundary Table 1. Parameters Varied in The Monte Carlo Simulations where R cr is the radius of the base of the crust. [23] The enrichment factor is held constant throughout the simulation. This assumption will be relaxed in the 2-D simulations, in which the enrichment of the secondary crust is calculated in a self-consistent manner according to a fractional melting model (section 3.3 and Appendix A). [24] In the Monte Carlo simulations, admissible models compatible with the observations were required to satisfy two criteria: (1) The total amount of global contraction accumulated after the end of the LHB (3.8 Gyr ago) must be less then 5 km and (2) a minimum amount of 5 km of secondary crust must be produced. The second requirement, albeit somewhat arbitrary, simply represents a conservative choice based on the evidence that a preliminary inversion of gravity and topography data predicts an average crustal thickness of 50 km [Smith et al., 2012] and that the observation of widespread volcanism [Head et al., 2011] necessarily implies the production of a non-negligible amount of secondary crust Two- and Three-Dimensional Numerical Models [25] In order to complement and validate the parametrized models, we carried out additional selected simulations of Mercury s thermochemical evolution using finite-amplitude calculations in 2-D cylindrical and 3-D spherical shell Symbol Description Numerical Range D m Mantle thickness 300, 400, or 500 km T m,0 Initial mantle temperature K ref Reference viscosity at T = 1600 K Pa s T c Initial core excess temperature K k cr Crustal thermal conductivity W m 1 K 1 d reg Regolith thickness 0 5 km ıv/v Volume change upon differentiation 1 5% ƒ Enrichment factor of radiogenic heat sources 2 10 U-Th c Percentage of U and Th in the core 0 or 10% 2478

6 Table 2. Parameters Held Constant in All Simulations TOSI ET AL.: THERMOCHEMICAL EVOLUTION OF MERCURY Symbol Description Numerical Value R p Planetary radius 2440 km g Surface gravity 3.7 m s 2 T s Surface temperature 440 K T ref Reference temperature 1600 K cr Crustal density 2800 kg m 3 m Mantle density 3500 kg m 3 c Core density 7200 kg m 3 c p,cr Magma heat capacity 1000 J kg 1 K 1 c p,m Mantle heat capacity 1200 J kg 1 K 1 c p,c Core heat capacity 850 J kg 1 K 1 E Activation energy Jmol 1 k reg Regolith thermal conductivity 0.2 W m 1 K 1 k m Mantle thermal conductivity 4 W m 1 K 1 m Mantle thermal expansivity K 1 c Core thermal expansivity K 1 L Latent heat of melting Jkg 1 D cr,0 Primordial crust thickness 5 km f Fraction of extractable crust 0.4 C Th Surface concentration of Th 155 ppm C U Surface concentration of U 90 ppm C K Surface concentration of K 1288 ppm geometries. Details of the mathematical and numerical aspects are presented in Appendix A. Here we briefly describe the main features of these models. We employed the finite-volume code Gaia [Hüttig and Stemmer, 2008] and the extended Boussinesq approximation [e.g., Christensen and Yuen, 1985] to solve the conservation equations of mass, linear momentum, thermal energy (including adiabatic and shear heating), and composition for an incompressible fluid with infinite Prandtl number and strongly temperaturedependent viscosity. The upper and lower boundaries of the computational domain are free slip and isothermal. An additional energy equation is solved to compute the time evolution of the CMB temperature assuming the core to be an isothermal medium with constant density and heat capacity. A random perturbation was superimposed on the initial temperature field to trigger the onset of convection. [26] In 2-D geometry, we used the particle-in-cell approach [Plesa et al., 2012] to account for partial melting, crustal production, and enrichment as well as mantle depletion according to the procedure described in Appendix A. It is worth recalling here that a fixed enrichment factor was used to determine the initial heat source content of the mantle and primordial crust but that the subsequent partitioning of radioactive elements was obtained in a self-consistent way according to a fractional melting model. We applied a geometric rescaling of the core radius to guarantee that the ratio of the CMB to planetary surface area of the cylindrical shell is equal to that of the corresponding spherical shell [van Keken, 2001]. Finally, we used a structured grid with a uniform resolution of 4 km in the radial and horizontal directions at mid-mantle depth, resulting in a total of grid cells, each of which containing 20 particles. [27] In 3-D geometry, we ran instead a purely thermal simulation with no particles. While keeping the overall heat source content of the planet equal to that of the 2-D case, we assumed from the beginning of the simulation a fixed crustal thickness of 20 km. The crust was enriched in radiogenic elements by a factor of ƒ (see section 3.2) and assumed to have a thermal conductivity that takes the presence of a 5 km 2479 thick insulating regolith layer into account. In the 3-D case, we used a regular icosahedral grid with a uniform resolution of 12.5 km in the radial direction and an average lateral resolution of 40 km, resulting in a total of grid cells. 3. Results 3.1. Monte Carlo Simulations [28] Results of the Monte Carlo simulations are presented in Figure 2, where histograms of the fraction of successful models as a function of the crustal enrichment factor ƒ for different core radii are shown. Figures 2a and 2b refer a b c Fraction of Models [%] Fraction of Models [%] Fraction of Models [%] R > 5 km R > 3.5 km Figure 2. (a) Histograms of the fraction of successful models as a function of the crustal enrichment factor for R c = 2040 km and no heat sources in the core, (b) same as Figure 2a, but considering 10% U and Th partitioned into the core, and (c) R c = 1940 km and no heat sources in the core. Green and red bars refer to models exhibiting global radial contraction at the end of the evolution <5and 3.5 km, respectively.

7 a c D cr [km] t con [Myr] b log 10 ref [Pa s] R p > 5 km R p > 3.5 km R p > 5 km R > 3.5 km p Time [Myr] d T c [K] Figure 3. Successful models with R c = 2040 km and no heat sources in the core: (a) crustal thickness at the end of the evolution, (b) reference mantle viscosity, (c) duration of mantle convection, all as a function of the crustal enrichment factor, and (d) time evolution of the CMB temperature. to simulations with R c = 2040 km with 0 and 10% U and Th partitioned into the core, respectively, while 2c refers to a set of simulations with R c = 1940 km and no heat sources in the core. For R c = 2140 km, no admissible models were found. Although models with large core radii can be compatible with the contraction constraints, they need to be started at very cold mantle temperatures and thus tend to produce negligible amounts of crust. On the other hand, when assuming high initial temperatures, significant crustal volumes can be produced, but the radial contraction systematically exceeds the constraints. For R c = 2040 km, about 1% of the models are successful when requiring global contraction to be smaller than 5 km (green bars). This amount reduces to 0.1% when the stricter constraint of 3.5 km is to be matched (red bars). With R c = 1940 km, we found that 2% and 0.15% of the models can satisfy the 5 and 3.5 km constraints, respectively. [29] For R c = 2040 km and R c = 1940 km, admissible models have crustal enrichment factors between 2.5 and 3.7, and between 2.5 and 4.5, respectively. The latter range corresponds to bulk silicate concentrations of ppb Th, ppb U, and ppm K. Furthermore, as predictable because of the small mantle-core volume ratio, the presence of 10% U and Th in the core does not significantly influence the results (compare Figures 2a and 2b). A contraction smaller than 3.5 km can be obtained when smaller values of ƒ are used, as these correspond to larger concentrations of mantle heat sources that tend to retard secular cooling while heat released close to the surface is quickly radiated to space [30] In Figure 3 we present the results for R c = 2040 km and no heat sources in the core for successful models. The figure shows the thickness of the crust at the end of the evolution after 4.5 Gyr (Figure 3a), the reference mantle viscosity (Figure 3b), the duration of mantle convection (Figure 3c), and the time evolution of the CMB temperature (Figure 3d). Most of the models produce between 5 and 15 km of secondary crust (note that all models were initialized with a primordial crust of 5 km); only few cases that meet the contraction constraint exhibit larger thicknesses. Crustal production lasts between 2 and 3.5 Gyr (not shown here). A high-reference mantle viscosity in excess of Pa s or higher and indicative of a dry rheology is generally preferred over a weaker rheology which would tend to increase convective vigor, promote more rapid cooling, and hence a large global contraction that would exceed the observational constraints. Mantle convection typically ceases after 3 4 Gyr. Only in few models convective heat transport lasts until present; these are characterized by small enrichment factors (i.e., a large mantle heat source content) and by a significant insulation obtained through relatively large regolith thicknesses. Present-day CMB temperatures are mostly confined between 1700 and 1900 K, and the average cooling rate of the core is K/Gyr, following an initial phase of mantle heating Representative Evolution Model [31] Figure 4 shows the details of the thermochemical evolution of a representative model with R c = 2040 km and no heat sources in the core. The parameters used in this

8 a c b d Figure 4. Representative thermochemical evolution model obtained using parameters listed in Table 3. (a) Time evolution of mantle temperature T m and CMB temperature T c ; (b) time evolution of the planetary radius change due to thermal expansion/contraction of the mantle and core R th (solid line), due to mantle differentiation R md (dashed line) and of the sum of the two contributions R P (dash-dotted line); (c) time evolution of the surface heat flux q s (solid line), mantle heat flux q m (dash-dotted line), and core heat flux q c (dashed line); (d) time evolution of the thickness of the secondary crust (dashed line), of the stagnant lid (solid line), and of the region where partial melting occurs (shaded gray). simulation are listed in Table 3. Figure 4a displays temporal changes of the mantle and CMB temperatures. Since the core is initially superheated by 205 K with respect to the mantle, heat is extracted from it, resulting in a decrease of its temperature during the first 300 Myr. At the same time, the relatively large concentration of heat sources considered (ƒ =2.7) causes the mantle temperature to increase rapidly for 500 Myr. As shown in Figure 4b, this phase is characterized by an overall expansion of the planet (dash-dotted line) to which mantle heating (solid line) and differentiation associated with the production of secondary crust (dashed line) contribute in similar ways. Furthermore, reduction of T c andgrowthoft m both result in a decline of the core heat flux (dashed line in Figure 4c) and in an increase of the mantle heat flux into the stagnant lid (dash-dotted line in Figure 4c). As observed in Figure 4d, the bulk of the secondary crustal volume is generated during the first 2 Gyr of evolution (dashed line), even though partial melt production, which involves nearly the entire mantle below the stagnant lid (solid line), continues for 2.7 Gyr. The subsequent evolution is characterized by a relatively rapid cooling of mantle and core at a rate of 50 K/Gyr which leads to the cessation of mantle convection approximately at 4 Gyr and to a continuous reduction of the planetary radius, which eventually leads to a total global contraction of 3.5 km following the end of the LHB Thermochemical Evolution From 2-D and 3-D Models [32] Using the same parameters employed for the representative model described in the previous section (see Table 3), we ran two finite-amplitude simulations of Mercury s evolution in 2-D cylindrical geometry and in 3-D spherical shell geometry. [33] In Figure 5, we present snapshots of the temperature field after 1 Gyr of evolution. Figure 5a refers to the 2-D simulation, while Figures 5b and 5c refer to the 3-D case, with Figure 5b showing a vertical slice through the model domain and panel Figure 5c a 3-D view of the 1930 K temperature isosurface. Both in two and three dimensions, mantle Table 3. Parameters Used in the Reference Model Symbol Parameter Numerical Value D m Mantle thickness 400 km T m,0 Initial mantle temperature 1685 K ref Reference viscosity Pa s T c Initial core excess temperature 205 K k cr Crustal thermal conductivity 3.75 W m 1 K 1 d reg Regolith thickness 5 km ıv/v Volume change upon differentiation 3% ƒ Enrichment factor of radiogenic heat sources 2.7 U-Th c Percentage of U and Th in the core 0% 2481

9 a b c Figure 5. Snapshots of the temperature field after 1 Gyr of evolution. (a) Temperature distribution from a 2-D cylindrical simulation with geometric rescaling of the core radius. Black areas below the surface indicate the crust, and white contours delimit regions of partial melting; (b) vertical slice of the temperature distribution from a 3-D simulation and (c) 3-D view of the 1930 K isosurface. convection is characterized by a regular small-scale pattern with several cells of roughly unitary aspect ratio. Partial melting (white contours in Figure 5a) occurs within upwelling regions; above which, under the assumption adopted here of vertical melt extraction, the thickness of the crust is largest (black areas below the surface in Figure 5a). Note that in the 3-D case, as discussed in section 2.5, we did not model crustal production in a self-consistent way but assumed a constant crustal thickness of 20 km. Because of the low thermal Rayleigh number (Ra = in these simulations, see Appendix A for its definition), the convection planform established at the beginning of the simulation tends to be conserved throughout the evolution with time dependence being introduced by secular cooling only. The three-dimensional structure of the temperature field (Figure 5c) is not characterized by traditional columnar, mushroom-shaped plumes but by several horizontally stretched upwellings, which do not exhibit any preferential orientation. [34] From a quantitative point of view, the temporal evolution obtained from these calculations closely follows that predicted by the reference one-dimensional model described in section 3.2. In Figure 6, we compare the evolution of the CMB temperature (Figure 6a) and of the crustal thickness (Figure 6b) obtained from the 1-D (black), 2-D (red), and 3-D (blue) models. The initial decrease in the CMB temperature exhibited by the 1-D model is not observed in the 2-D and 3-D models. This is due to the fact that while the former is assumed to be convecting vigorously from the beginning of the evolution and can thus lose heat rapidly because of the superheated core, the latter needs about 50 Myr before convection sets in. During this period, heat is not lost efficiently, and mantle and core heat up. A systematic difference (25 K at most) in the CMB temperatures obtained from parametrized and numerical models is thus present until 2 Gyr. Afterward, the three runs show satisfactory agreement, although during the second half of the evolution, the cooling rate predicted by the 2-D simulation is slightly larger than that obtained form the 1-D and 3-D simulations. As far as the volcanic history is concerned, Figure 6b shows that even though in the 2-D case the crust grows sooner and more quickly than in the 1-D case, the average a T c [K] D 2D 3D Time [Myr] b Crustal Thickness [km] D 2D 5 3D Time [Myr] Figure 6. Comparison of the time evolution of (a) the CMB temperature and (b) the average crustal thickness for the representative model obtained with 1-D (black lines), 2-D (red lines), and 3-D (blue lines) models. 2482

10 thicknesses obtained at the end of the evolution ( 17 km) agree extremely well. In the 2-D and 3-D models, convective heat transport ceases around 3 and 3.7 Gyr, respectively, in reasonable agreement with the prediction of nearly 4 Gyr of the parametrized model (see Figure 4a). 4. Discussion [35] The wealth of new information provided by the MESSENGER mission prompted us to reinvestigate the thermochemical evolution of Mercury in view of the latest available findings in terms of structure, composition, and tectonics of the planet. To this end, we conducted a systematic analysis of the interior evolution using a large number of Monte Carlo simulations based on one-dimensional parametrized models, which we complemented with twoand three-dimensional simulations. For a given model to be considered admissible, we required that the global contraction after the end of the LHB does not exceed 5 km and that at least 5 km of secondary crust are produced. Within the broad range of tested parameters, the fraction of models that can satisfy these requirements is generally small: 1% when assuming a mantle thickness D m of 400 km, which appears to be a robust estimate across a variety of different interior configurations [Hauck et al., 2013], and 2% when considering D m = 500 km, which however is only marginally compatible with the moment of inertia constraints [Hauck et al., 2013]. The model series featuring a thin silicate mantle of only 300 km, which could be obtained if a solid FeS layer of 100 km thickness formed on top of the liquid core [Smith et al., 2012; Hauck et al., 2013], did not generate any admissible solution because of the lack of any significant volcanic activity. [36] It should be noted that Michel et al. [2013] reported results of 2-D axisymmetric simulations indicating that widespread and long-lived magma production is possible even when using this mantle thickness together with a no-slip boundary condition at the CMB. The reason of discrepancy with our study is probably due to the fact that Michel et al. [2013] did not use the contraction constraint to limit the set of admissible models. The thickness of the FeS layer is poorly constrained, but present-day CMB temperatures compatible with its formation (i.e., K) are not difficult to achieve. Therefore, our results do not rule out the possibility that this layer can actually exist but suggest instead that assuming 400 km to be the most likely value for the thickness of the whole solid shell above the CMB (i.e., mantle and FeS layer), the FeS layer may need to be significantly thinner than 100 km in order for the production of a significant amount of secondary crust to be possible. Nevertheless, further study on this aspect appears necessary. In fact, the formation and subsequent contraction or expansion of a solid FeS layer, which is not considered in our models, may lead to different predictions of global radius changes. Furthermore, the existence of such a layer would require a core in which, beside sulfur, also silicon is alloyed with iron [Smith et al., 2012; Hauck et al., 2013], in agreement with the reducing conditions at which Mercury s precursory materials seem to have formed [Malavergne et al., 2010; Nittler et al., 2011]. Finally, although the presence of a relatively small amount of radiogenic elements in the core does not impact the number nor the distribution of the admissible models (Figures 2a and 2b), it could render the solidification of the top of the core as a consequence of radiogenic heating more difficult. [37] Limiting ourselves to consider model runs with D m = 400 km and assuming that the present-day concentration of radiogenic elements observed at the surface is representative of the average crust, we found that admissible models are characterized by crustal enrichment factors between 2.5 and 3.5. These values imply concentrations in the primordial mantle of ppb Th, ppb U, and ppm K. The low Th/K ratio of the surface material already hints at a composition more similar to that of the other terrestrial planets than previously thought [Peplowski et al., 2012]. Our results confirm this finding by establishing in addition possible limits on the actual concentration of heatproducing elements in the mantle source regions of magmas on Mercury. For example, the standard composition usually assumed for bulk silicate Mars is 56 ppb Th, 16 ppb U, and 305 ppm K [Dreibus and Wänke, 1987], i.e., similar to the concentrations predicted by our calculations. [38] The analysis of the MESSENGER imagery indicates that volcanism was widespread on the planet at least until the end of the LHB 3.8 Gyr ago [Head et al., 2011], with single regions associated with impact basins characterized by volcanic activity as young as 1 Gyr [Prockter et al., 2010]. Consistent with this scenario, our simulations predict the phase of magma production to last between 2 and 3.5 Gyr during which the bulk of secondary crust is produced. For most of the models, crustal thicknesses between 5 and 15 km are obtained (Figure 3a), although isolated solutions that imply larger thicknesses up to 75 km are possible. However, these comply only with the weaker constraint of 5 km contraction. The thin crust that emerges as a relatively robust feature of our Monte Carlo simulations is not in agreement with the average value of 50 km recently reported on the basis of gravity and topography data [Smith et al., 2012]. However, Smith et al. [2012] assumed an unusually low-density contrast of only 200 kg/m 3 between mantle and crust. Assuming a higher-density contrast and a lower mean thickness closer to our estimates would imply a smaller amplitude of the crustal undulations. Whether a good fit to the observed topography and gravity data can be achieved under these conditions is not clear at present and needs further investigation. [39] The need to understand whether mantle convection is currently ongoing or ceased in the past is related to the interpretation of the gravity field and topography in terms of the interior structure of the planet. This issue, on which Redmond and King [2007] and Michel et al. [2013] focused their studies, is important since if the mantle is still convecting, deep-seated density anomalies and flow-induced dynamic topography of the surface and CMB may contribute to the observed fields. Although present-day convection is possible, the majority of our models with D m = 400 km predict its cessation after 2.5 to 4.2 Gyr of evolution (see Figures 3c and 4a). In this view, it appears difficult to reveal a dynamic signature due to the convecting mantle in the observed gravity and topography data. [40] As already pointed out in previous studies of Mercury s evolution [Hauck et al., 2004; Grott et al., 2011], a relatively small amount of contraction is only obtained if a large reference viscosity is assumed, which in most of 2483