MAGMARS: A Melting Model for the Martian Mantle and FeO‐Rich Peridotite

Martian basalts identified by rover in‐situ analyses and the study of meteorites represent a direct link to the melting process in the planet's interior and can be used to reconstruct the composition of the mantle and estimate its temperature. Experimentally calibrated numerical models are powerful tools to systematically search for the mantle compositions and melting conditions that can produce melts similar to primary basalts. However, currently available models are not suitable for modeling the melting of FeO‐rich peridotites. In this study, we present experiments performed at 1.0 and 2.4–2.6 GPa on a primitive Martian mantle with various P2O5 contents. We use the new experiments together with a comprehensive database of previous melting experiments to calibrate a new model called MAGMARS. This model can reproduce the experimental melt compositions more accurately than Gibbs free energy minimization software (e.g., pMELTS) and can simulate near‐fractional polybaric melting of various mantle sources. In addition, we provide an updated thermobarometer that can estimate the P–T melting conditions of primary melts and can be used as a prior step to constrain the input parameters of the MAGMARS melting model. We apply MAGMARS to estimate the source composition of the Adirondack‐class basalts and find that melting a depleted mantle, at 2.3–1.7 GPa (Tp = 1390 ± 40°C) can best explain their bulk composition and K2O/Na2O ratio. MAGMARS represents a fast and accurate alternative to calculate the composition of the Martian primary melts and can be used as a stand‐alone package or integrated with geodynamical models or other independent modeling software.

When sufficient experiments are available, they can be used to parametrize melting models (forward approach), which extend their range of applicability and allow to investigate the melting of mantle sources of contrasting compositions. For example, the Gibbs free energy minimization software pMELTS (Ghiorso et al., 2002) has become highly popular to simulate the partial melting of planetary mantles. However, pMELTS is currently unable to reproduce the composition of Martian primary melts accurately. For example, it systematically overestimates the FeO and MgO contents and underestimates the SiO 2 content (Balta & McSween, 2013;Gross et al., 2011). Some workers apply simple corrections to pMELTS output tables to minimize those offsets (Baratoux et al., 2011;El Maarry et al., 2009;Sautter et al., 2015) but these corrections are usually insufficient (Collinet et al., 2015). Here, we introduce a new melting model, MAGMARS, that was developed as a more reliable alternative to calculate the composition of Martian primary melts more accurately. MAGMARS follows the same basic principles as the Kinzler and Grove (1992) family of models. It was calibrated using experiments from previous studies and new experiments conducted for this study. It is specifically designed to simulate melting of FeO-rich peridotite in the Martian mantle.
Experiments can also be used to create mineral-liquid or liquid thermometers that return the P-T conditions at which primary melts are in equilibrium with mantle minerals (inverse approach) as a function of their chemical composition (Kinzler & Grove, 1992;Putirka, 2008). For Mars, such thermometers have been applied to igneous rocks believed to represent near-primary melts to constrain the thermal state of the mantle (Filiberto & Dasgupta, 2011Lee et al., 2009). We use the same experimental database assembled to parametrize the MAG-MARS melting model to test existing thermobarometers and provide an updated formulation that reproduces the P-T conditions of experiments most accurately.
All previous studies assumed explicitly or implicitly that the Martian mantle was affected by batch melting: the melt is retained in the solid residue during melting. But the extraction of silicate melts during partial melting is efficient (Faul, 2001;McKenzie, 1985;Zhu et al., 2011), and the melting process affecting planetary mantles could more closely reassemble near-fractional melting (i.e., the melt is extracted almost as soon as it forms, above a certain threshold, the critical melt fraction: 0-2 wt.%). In this study, we revisit how both the inverse and forward approaches can be combined to constrain more accurately the melting conditions and source composition of Martian primary basalts. The new thermobarometer can be used to estimate the P-T melting conditions under a batch melting assumption. Those conditions can then be used as initial MAGMARS inputs. By varying those COLLINET ET AL.

10.1029/2021JE006985
3 of 21 initial inputs and the mantle composition, MAGMARS can be used to constrain, for the first time, the P-T melting conditions associated with near-fractional polybaric melting and the melting behavior of various primitive, depleted or metasomatized mantle reservoirs on Mars. In addition to being more accurate, MAGMARS is less computationally intensive than Gibbs free energy minimization software, which makes it better suited for direct implementation in convection models.
In Section 2, we first describe two new series of experiments designed to fill gaps in the experimental database of Martian primary melts. In Section 3, we describe how this database is used to parametrize the new melting model and show that MAGMARS can reproduce the composition of experimental melts more accurately than pMELTS (Ghiorso et al., 2002). We also show how MAGMARS can simulate near-fractional polybaric melting. In Section 4, we use the database to test existing mineral-liquid thermometers and barometers and to provide an updated thermobarometer that reproduces the P-T conditions of experiments most accurately. Finally, in Section 5 we use MAGMARS to constrain the source composition and P-T melting conditions of the Adirondack basalts, some of the more extensively studied Martian primary basalts.

Experimental Approach
Two series of experiments were conducted in piston cylinder apparatus at the MIT experimental petrology laboratory. The first series consists of five experiments and extends the melting experiments of Collinet et al. (2015) to slightly higher pressure (2.4-2.66 GPa). They were performed using the same starting composition (Dreibus & Wänke, 1985) and the same experimental methods (see below) as described in Collinet et al. (2015).
The second series of experiments consists of eight basalt-peridotite experiments and serve two purposes. First, they represent "reversal experiments" and are used to evaluate how closely the composition of low-degree melts (∼10 wt.%) in direct melting experiments reflects thermodynamic equilibrium. They are similar in concept to "sandwich experiments" in which a liquid of basaltic composition is placed between layers of mantle minerals (Falloon et al., 2001;E. Stolper, 1980) but, instead of layering the different compositions, the basalt and peridotite compositions were mixed. This method was chosen to avoid ambiguities in the ability of sandwich experiments to achieve thermodynamic equilibrium (Falloon et al., 2001;Kinzler & Grove, 1992;Mitchell & Grove, 2015). The rationale behind this type of experiments is that once the thermodynamic equilibrium is achieved, a decrease in the proportion of solid minerals in equilibrium with the silicate melt has no impact on its composition, while the larger melt fraction facilitates its analysis.
The second purpose of basalt-peridotite experiments is to evaluate the role of P 2 O 5 on the melt chemistry by adding different P 2 O 5 concentrations to otherwise identical starting compositions. Phosphorus is believed to be abundant in the Martian mantle but its exact concentration is poorly constrained (Dreibus & Wänke, 1985;Lodders & Fegley, 1997;Yoshizaki & McDonough, 2020). The presence of large concentrations of P 2 O 5 could substantially affect the physical and chemical properties of silicate melts (Kushiro, 1975;Mysen & Richet, 2019;Ryerson, 1985;Ryerson & Hess, 1980;Toplis et al., 1994). First, phosphate and silica tetrahedra can copolymerize. In many cases, a higher P 2 O 5 content increases the silica activity coefficient and therefore decreases the SiO 2 concentration of a melt in equilibrium with olivine and pyroxene (buffered SiO 2 activity). Second, phosphorus can create phosphate complexes with metallic cations and therefore remove some "network modifiers" from the SiO 2 framework. This could also translate into a positive deviation from ideality for SiO 2 (i.e., decrease in SiO 2 concentration) and an increase of FeO and CaO concentrations. In detail, the interactions of phosphorus with silica tetrahedra and metallic cations have been shown to be complex and depend on the melt composition (Mysen & Richet, 2019). Therefore, additional experiments are needed to quantify the effect of phosphorus and decouple it from that of other minor incompatible elements such as alkalis (Hirschmann et al., 1998).
To conduct the basalt-peridotite experiments, two melt compositions analyzed in two previous melting experiments (DW01 from Collinet et al., 2015 and D268 from this study) were first synthesized by mixing oxides (SiO 2 , TiO 2 , Al 2 O 3 , Cr 2 O 3 , Fe 2 O 3 , MnO, MgO), carbonates (Na 2 CO 3 , K 2 CO 3 ), silicates (CaSiO 3 ) and Fe metal. They were ground for four hours in an automatic mortar and decarbonated in air at 850°C. The Fe metal was added last before grinding for another hour. Finally, starting mixes were pre-conditioned at FMQ −0.5 and 1000°C for 72 hr in a gas-mixing furnace. The final O 2 is set to FMQ −2.5 by the graphite capsule during the experiments (Medard et al., 2008), which is close to the average O 2 of the Martian mantle (Nicklas et al., 2021). The basaltic COLLINET ET AL.
The different compositions were loaded in Pt-graphite capsules (10 mg) and residual hydrogen was largely eliminated ( 100 ppm) by drying the capsules at 300°C for 12 hr before welding the outer Pt capsule shut. The capsules were placed in an alumina ring and centered with crushable MgO spacers in a graphite furnace, which was itself inserted in a sintered BaCO 3 cell wrapped in a Pb foil. The experiments were performed in piston cylinder apparatus (12.7 mm pressure vessels) at the same P-T conditions as the corresponding direct melting experiments (DW01 and D268; Table 1). This experimental assembly has been found to require no friction correction through calibration against the reaction Ca-Tschermak pyroxene = anorthite + gehlenite + corundum (Hays, 1966) and the pressure is thought to be accurate to within ±0.05 GPa. The pressure was first increased to 0.8 GPa before raising the temperature to 865°C (100°C/min). The temperature was then held constant for 6 min, during which the pressure was increased to the target experimental pressure. Finally, the temperature was brought to the experimental temperature at a rate of 50°C/min while keeping the pressure constant. The temperature was maintained with a type D thermocouple connected to a Eurotherm 818. A temperature correction of 20°C was applied to account for the temperature difference between the thermocouple location and the capsule interior (hot spot). The temperature reproducibility is estimated at ±10°C. The sample was quenched by simultaneous turning off the power and releasing 200-400 MPa of pressure to limit crystal growth during the quench (i.e., pressure quench).
Experimental run products were analyzed at the MIT Electron Microprobe Facility (JEOL JXA Superprobe 8200) with a current intensity of 10 nA and an accelerating voltage of 15 kV (40 s analysis time). Glasses were analyzed with a broad beam (10 μ m) and for a shorter time (5 s for Na). The CITZAF package was used for matrix correction. Analytical standards include well-characterized glasses, diopside (Si, Ca, Mg), oligoclase (Na, Al), hematite (Fe), rutile (Ti), chromite (Cr), rhodochrosite (Mn), orthoclase and apatite (P). The analytical uncertainties are reported in Table 2 for glasses and Table S5 Table 2 and Table S5, respectively.

Direct Melting Experiments
The five direct melting experiments performed at 2.4-2.66 GPa are in good agreement with previous experiments performed from the same composition (Dreibus & Wänke, 1985) at 0.5-2.2 GPa (Collinet et al., 2015). They contain 5-30 wt.% melt, olivine, orthopyroxene, spinel, and, at low temperature ( 10 wt.% of melting), pigeonite/ sub-calcic augite. All phases are homogeneously distributed in the experimental charges, suggesting limited thermal gradients, and they are homogeneous in composition, suggesting a close approach to the thermodynamic equilibrium. In addition, the oliv-liq Fe−Mg equals 0.365 ± 0.01, close to the expected value calculated with the model of Toplis (2005) and consistent with the average value of 0.35 ± 0.01 for Martian basalts (Filiberto & Dasgupta, 2011). Mass balance calculations indicate that the new experiments could be affected by modest Fe loss of 5-8 wt.% (Table 1), corresponding to 1 wt.% FeO of the 17.9 wt.% bulk FeO content (16.9 wt.% effective bulk FeO). This FeO loss is slightly larger than the FeO loss of the Collinet et al. (2015) experiments but nevertheless acceptable. It could result from a slightly larger fracturing of graphite capsules at higher pressure. The absence of mineral zoning and the oliv-liq Fe−Mg values confirm that the experiments approached a thermodynamic equilibrium. Therefore, they can be understood as representing the melting of a Martian mantle with a slightly higher Mg# (76) than the composition of Dreibus and Wänke (1985). This effect is automatically accounted for during the parametrization of the melting model.
Compared to lower pressure experiments (Collinet et al., 2015), the new 2.4-2.66 GPa experiments contain melts with lower SiO 2 , Al 2 O 3 , and Na 2 O contents, higher FeO and MgO contents, and slightly higher CaO contents at a given melt fraction. A summary of the experiments and the melt compositions is provided in Tables 1 and 2, respectively. The composition of mineral phases are available in Table S5.
At low melt fractions ( 20 wt.% melting), the melt could only be analyzed in cracks within the graphite capsule (Figures 1a-1c). Whether the melt present in such "traps," partly isolated from the main experimental charge, is truly in equilibrium with the solid phases is a common concern in experimental petrology. This concern was one of our motivations for conducting basalt-peridotite experiments.

Basalt-Peridotite Experiments
Basalt-peridotite experiments contain the same mineral phases as in the corresponding low-temperature direct melting experiments (DW01 and D268): olivine, orthopyroxene, pigeonite and spinel. They contain a larger fraction of liquid in equilibrium with this mineral assemblage which, in addition to being present in capsule cracks, also pooled on the sides of the capsules where it could be analyzed in the direct vicinity of solid phases ( Figure 1d). As with the direct melting experiments, the mineral textures and oliv-liq Fe−Mg values are consistent with equilibrium conditions, but basalt-peridotite experiments were also likely affected by modest FeO loss, as suggested by mass balance calculations (Table 1).
The melt composition of basalt-peridotite experiments is in good agreement with the melt of the corresponding direct melting experiments, performed at the same P-T conditions. At 1.0 GPa, the only minor differences are the slightly lower SiO 2 content (−1.5 wt.%) and higher FeO and MgO content (+1 wt.%) of the melt of basalt-peridotite experiments. At 2.6 GPa, the only significant difference is the lower CaO content (−1.5 wt.%) of the melt of basalt-peridotite experiments. Those differences could result from the inherent difficulties associated with the analysis of the glass in very low melt fraction experiments ( 15 wt.%). But overall, we conclude that those differences are not significant and that the melt of all experiments, direct melting and basalt-peridotite, is representative of the equilibrium conditions and can be used for the calibration of the MAGMARS melting model.
The basalt-peridotite experiments also allow us to investigate the effect of phosphorus on the concentrations of the main oxides of silicate melts. At 1.0 GPa, experiment B1384 with 3.3 wt.% P 2 O 5 has 3.5 wt.% less SiO 2 compared to experiment B1371 with 0.15 wt.% P 2 O 5 , conducted at the same temperature. Phosphorus-rich melts also have higher MgO and FeO contents and slightly higher CaO contents (Table 2). Finally, they are in equilibrium with a larger fraction of pyroxene and a smaller fraction of olivine (Table 1). These results are in agreement with prior experimental studies suggesting that phosphorus copolymerizes with silica tetrahedra and acts as a network-forming cation while also forming complexes with divalent cations. This decreases the SiO 2 concentration and increases the FeO, MgO and CaO concentrations of melts in equilibrium with olivine and pyroxene (Kushiro, 1975;Ryerson, 1985;Toplis et al., 1994). These interactions also increase the primary phase volume 7 of 21 of pyroxene at the expense of olivine. We refer to the experimental study of Payré and Dasgupta (2021, under review) for additional discussion of the role of phosphorus on the composition of Martian basalts. The MAGMARS calibration database includes our new experiments and the ones of Payré and Dasgupta (2021, under review) performed on an alkali-rich Martian mantle composition (Lodders & Fegley, 1997) to capture most accurately the effect of various P 2 O 5 contents on the composition of silicate melts, which is discussed further in the next section.

Modeling Approach
The MAGMARS melting model uses the same approach as the Kinzler and Grove (1992) family of models, inspired by the Gibbs method (Spear et al., 1982). Previous models were designed to simulate partial melting in a wide range of terrestrial environments from mid-ocean ridges basaltic magmatism (Kinzler, 1997;Kinzler & Grove, 1992;Krein et al., 2020), subduction zone magmatism (Mitchell & Grove, 2015), intra-plate and high-K magmatism (Grove et al., 2013;Till et al., 2012). MAGMARS is a similar model but primarily designed to simulate melting in the Martian mantle.
The thermodynamic equilibrium constraint is represented by the pressure-dependent melting equations of Collinet et al. (2015) for a spinel lherzolite. The composition of the partial melts in equilibrium can be calculated by fixing a sufficient number of compositional and intensive variables: the pressure, the Mg# of the liquid, and the (a-c) Direct melting experiment D268 conducted at 1380°C and 2.6 GPa. (d) Corresponding reversal experiment D314 conducted at 1390°C and 2.6 GPa (phosphorus-rich). The melt was quenched as a glass but could only be analyzed in cracks within the graphite capsule in D268. In D314, the glass could be analyzed on the sides of the capsule and was found to be of nearly identical composition ( (1) with l and s the concentration of the incompatible element i in the liquid and solid, respectively, F the melt fraction, p i the average partition coefficients weighted by the coefficients of the melting reactions, and D i the bulk partition coefficient (see Section S1.1 in Supporting Information S1).
Once those variables are fixed, the Gibbs phase rule states that the variance of the system (degree of freedom, f) is decreased sufficiently to uniquely determine the concentration of major elements in the melt (SiO 2 , CaO, FeO and MgO) and the temperature at which such a melt would be in equilibrium with the mantle. In Equation 2, Φ represents the number of phases and C the number of independent chemical components (oxides). The concentrations of SiO 2 , CaO, FeO and MgO, as well as the temperature of the melt, are calculated with a set of multiple linear regressions, which uses the pressure, the Mg# and incompatible element concentrations as predictor variables (see Section S1.2 in Supporting Information S1). Alternatively, the concentration of FeO can be calculated from the concentration of MgO and appropriate distribution coefficients oliv-liq Fe−Mg , the ratio of FeO and MgO concentrations in the liquid divided by the same ratio in olivine Fe ol ∕Mg ol Fe liq ∕Mg liq (Toplis, 2005).
The linear regressions are parametrized with a calibration database containing experiments from a large number of studies, with more weight placed on the experiments from Collinet et al. (2015) and this study, which were performed with the bulk composition of Dreibus and Wänke (1985). Selected experiments performed on different Martian mantle compositions are included, with relatively less weight, to account for the influence of variable Na 2 O, K 2 O and P 2 O 5 concentrations, and the effect of pressure up to 5 GPa (Agee & Draper, 2004;Bertka & Holloway, 1994;Ding et al., 2020;Matsukage et al., 2013;Payré & Dasgupta, 2021, under review). Finally, experiments performed to model melting in the Earth's mantle are included to anchor the model at higher Mg# (Hirose & Kushiro, 1993;Hirschmann et al., 1998;Kinzler, 1997;Kushiro, 1996;Laporte et al., 2004;Pickering-Witter & Johnston, 2000;Robinson et al., 1998;Wasylenki et al., 2003), and are assigned the least weight. The detailed database and linear regressions are available in Table S5 and Section S1.2 in Supporting Information S1, respectively.
MAGMARS can be used to reproduce several melting processes. First, we consider near-fractional polybaric melting (NFP). This process represents decompression melting that occurs over a range of pressures and during which the melt increments are extracted rapidly and do not have time to re-equilibrate with the mantle during ascent. Therefore, the final composition of the aggregate melts, the sum of all melt increments pooled in the crust, reflects the full range of P-T conditions of the melting zone. In NFP mode, MAGMARS calculates the composition of small melt increments (0.1 wt.%) along mantle adiabats, at progressively lower pressure (Figure 2a). The temperature of the solidus is not pre-determined. It is calculated from the composition of the first melt increment that would appear at any given pressure if the mantle-of specific chemical composition-was melting. Therefore, whether melting has begun or not, the composition of a hypothetical 0.1 wt.% melt increment is calculated to determine the temperature of the effective solidus at the pressure of interest (Section S1.2 in Supporting Information S1). Initially, this temperature is higher than the temperature of the mantle adiabat. But because solidus temperatures decrease faster than the temperature of mantle adiabats with decreasing pressure, the mantle temperature eventually exceeds the effective solidus ( Figure 2d). Once melting begins, all the melt produced above a fixed critical melt fraction (0.4-2 wt.%) is progressively removed and the bulk composition and mineral mode of the melting residue are continuously updated. The temperature of the mantle is also decreased to account for the loss of latent heat. The solidus temperature continually increases due to the decrease in incompatible element concentrations and the increase in Mg# (i.e., the mantle becomes depleted). Finally, all melt increments produced along the adiabat (i.e., instantaneous melt) are pooled together in the crust (i.e., aggregate melt).
The melting model can also be used by increasing the temperature at constant pressure, while extracting the melt progressively (near-fractional isobaric melting, "NFI mode") and can approximate batch isobaric melting by 9 of 21 calculating the temperature at which aggregate melts would be in equilibrium with the mantle at a given pressure, if they were produced as batch melts (Figure 2e). This approximation is justified because the compositions of aggregate melts and batch melts are largely analogous. Compared to batch melts, aggregate melts are only slightly enriched in incompatible elements (up to + 1 wt.% in Al 2 O 3 and Na 2 O) and slightly depleted in FeO and MgO (up to − 1 wt.%, Figure S10 in Supporting Information S1). NFI melting occurs at higher temperatures than batch melting due to the progressive removal of the incompatible elements with the melt. The NFI mode can also be interpreted as another end-member process of decompression melting for which all melt increments re-equilibrate completely at a single pressure (Figures 2c and 2e). For NFP, our working assumption is that the melt increments are extracted rapidly and do not have time to re-equilibrate with mantle minerals during ascent. However, if the ascent of the melt increments was sufficiently slow, the melt could react and re-equilibrate with mantle minerals, up to the base of the lithosphere. In this case, although melting still occurs over a range of pressures, the lowest pressure of the melting zone overprints the melt chemistry.  Dreibus and Wänke (1985) begins to melt at the pressure where the adiabat-of given mantle potential temperature (T p = 1410°C)-intersects the temperature of the effective solidus (see Section 3). Melts are continuously produced and extracted, up to the base of the lithosphere along the color gradient (∼22 wt.% total melt fraction). Melt increments do not re-equilibrate with the mantle at shallower depth and are pooled directly within the crust. (e) P-T melting conditions during isobaric melting. MAGMARS simulates near-fractional isobaric melting: melt is produced at increasing temperature but constant pressure and is continuously extracted. The mantle T p required to produce the same amount of melt (∼22 wt.%) is ∼100°C higher (1515°C). The NFI mode can be used to approximate batch melting (b) as the composition of aggregate melts are nearly identical to batch melts (see Section 3.3 and Figure S10 in Supporting Information S1). In (d) and (e), the gray open square represents the temperature of the melt if it had been produced as a batch melt at 2.05 GPa (batch melting temperature). The T p required to reach a similar melt fraction (∼22 wt.%) by batch melting is intermediate between the NFP and NFI modes (1450°C). The NFI mode can also be understood as representing a melting process occurring over a range of pressure (c), while the melt is continuously extracted and each melt increment re-equilibrates with the mantle at the shallowest depth. The dashed line in (e) would represent the P-T melting conditions in this case.

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The model uses different systems of equations to predict the composition of silicate melts in equilibrium with lherzolite and harzburgite (i.e., following the disappearance of clinopyroxene; Section S1.2 in Supporting Information S1). Following the disappearance of orthopyroxene (dunite), melting essentially stops because the solidus temperature increases dramatically. But at very high temperature ( 1700°C), dunite melting is in principle possible and the composition of incremental melts corresponds to the composition of olivine of increasing Mg# (up to pure forsterite at 100% melting).

Model Uncertainties and Comparison of Experiments With Isobaric MAGMARS Calculations
As a first step to validate the applicability of the new melting model, we compare the calculated melt compositions (mode NFI) to the melts of the experiments performed by Collinet et al. (2015) using the Dreibus and Wänke (1985) composition, and the direct melting experiments of this study (Figure 3). Both the experimental temperature ( Figure 3a) and the melt composition (Figures 3c and 3d) can be reproduced accurately with MAG-MARS (see also Figure S6 in Supporting Information S1). The model might appear to overestimate the FeO content by 1-2 wt.% at 2.5 GPa, but the lower FeO content of the experimental melts at 2.4-2.66 GPa actually results from the small Fe loss that the higher pressure experiments suffered. The FeO concentration of those experiments is slightly underestimated but this does not affect the calibration of the model (Section 2.2.1). Next, the model can be used to predict how the solidus temperature and melt composition of various mantle compositions differ. A mantle with a higher concentration of alkalis (Lodders & Fegley, 1997) starts to melt at a 20-40°C lower temperature compared to the Dreibus and Wänke (1985) composition ( Figure S8d in Supporting Information S1). This is within error of the effect of alkali elements observed by Ding et al. (2020). The partial melts are richer in alkalis, slightly richer in SiO 2 , and slightly poorer in FeO, MgO and CaO. Still using MAGMARS, a terrestrial mantle with a KLB-1 composition (Hirose & Kushiro, 1993) starts to melt 60-80°C above the Dreibus and Wänke (1985) solidus (Figures S8a and S8b in Supporting Information S1), due to its higher Mg# and lower alkali content, which is also in agreement with the solidus parametrization of Ding et al. (2020). Finally, a mantle with 0.32 wt.% P 2 O 5 , twice the amount of the Dreibus and Wänke (1985) composition, melts at a 30-50°C lower temperature compared to a mantle with no P 2 O 5 and produces melts poorer in SiO 2 (by 3-6 wt.%) at low melt fraction ( Figure S9 in Supporting Information S1), in overall agreement with Payré and Dasgupta (2021, under review). Therefore, the melting model appears to adequately account for the effect of varying source compositions, including the effect of phosphorus, and even for drastically different compositions corresponding to terrestrial lherzolites ( Figure S7 in Supporting Information S1).
The model uncertainties can be quantified by propagating the errors associated with the experimental database and the linear regressions. The uncertainties related to the experimental database include the uncertainties on the P-T experimental conditions, small inhomogeneities in the chemical composition of the melts (i.e., small deviations from equilibrium conditions) and the uncertainty of the microprobe analyses. The latter two are accounted for by calculating the mean standard deviations of microprobe analyses. The uncertainties on the P-T conditions are taken as 0.1 GPa and 20°C. These values, which are twice as large as the estimated experimental errors of this study, were chosen to accommodate possible inter-laboratories discrepancies. For incompatible elements (i.e., TiO 2 , Al 2 O 3 , Na 2 O, K 2 O, and P 2 O 5 ), model-specific errors are estimated by comparing the concentrations calculated with Equation 1 to the experimental melts and by calculating the mean absolute error. For major elements, we calculate directly the mean absolute error of the different linear regressions. Finally, the model, experimental and analytical uncertainties are added in quadrature to obtain the total uncertainties. For lherzolite melting, we obtain the following errors on the liquid composition: ±1.3 wt.% SiO 2 , 0.21 wt.% TiO 2 , 0.38 wt.% Al 2 O 3 , 0.11 Cr 2 O 3 , 0.85 wt.% FeO, 0.06 MnO, 0.75 wt.% MgO, 0.61 wt.% CaO, 0.28 wt.% Na 2 O, 0.12 wt.% K 2 O, 0.12 wt.% P 2 O 5 . The average uncertainty on the temperature is 27.6°C. For harzburgite melting, the errors are identical for incompatible elements but slightly larger for major elements and the temperature: (±)1.35 wt.% SiO 2 , 0.9 wt.% FeO, 0.82 wt.% MgO, 0.8 wt.% CaO and 29.3°C.

Near-Fractional Polybaric MAGMARS Calculations
The accuracy of MAGMARS having been demonstrated, we can use the model to calculate the composition of melts produced by near-fractional polybaric melting. The composition of NFP melts is nearly identical to the one of NFI melts produced at a constant pressure corresponding to the average pressure of the melting zone. For example, the composition of the melts produced in Figures 2d and 2e are both similar to the Adirondack basalts (Table S7). The average P-T conditions of melting are identical to the P-T conditions at which the corresponding batch melt would be produced, as previously discussed by Asimow and Longhi (2004). In the NFP case, mantle melting must continue up to a shallower depth, implying that the lithosphere is thinner than for the isobaric cases (batch and NFI). Another difference is that because melting is allowed to continue up to a shallower depth, at which the solidus temperature is low, the mantle T p required to produce a melt of given composition is lower in the NFP case. To produce a primary melt similar to the Adirondack basalts from a Dreibus and Wänke (1985) mantle composition, a T p of 1415°C is sufficient (vs. 1450°C for batch melting and 1520°C for NFI melting). The implications of MAGMARS calculations for the origin of the Adirondack basalts is discussed in Section 5.

Comparison of MAGMARS Calculations With Gibbs Free Energy Minimization Models
Thermodynamic models based on Gibbs free energy minimization are powerful tools that are becoming increasingly popular for constraining various metamorphic and igneous processes, including crystallization and partial melting. One model that is frequently used to simulate the partial melting of the Martian mantle is pMELTS (Ghiorso et al., 2002). It has been used to constrain the formation conditions of Martian igneous rocks and minerals (Balta & McSween, 2013;Payré et al., 2020;Sautter et al., 2015), entire volcanic provinces (Baratoux 12 of 21 et al., 2011(Baratoux 12 of 21 et al., , 2013(Baratoux 12 of 21 et al., , 2014El Maarry et al., 2009) and igneous processes in geodynamical models (Zeff & Williams, 2019). However, the composition of primary melts calculated with pMELTS significantly differs from the melts produced in experiments. Most notably, pMELTS systematically overestimates the FeO content and underestimates the SiO 2 content of partial melts. Using the experiments available at the time (Bertka & Holloway, 1994), El Maarry et al. (2009 suggested that this issue can be circumvented by applying a constant correction to the composition of partial melts: FeO (−3 wt.%), SiO 2 (+3 wt.%) and CaO (+1 wt.%). However, additional experiments later showed that the amplitude of the pMELTS offsets are pressure-dependent and much larger than previously anticipated, and that the corrections of El Maarry et al. (2009) were insufficient (Collinet et al., 2015). In addition, Balta and McSween (2013) also showed that pMELTS systematically underestimates the pressure at which Martian basalts are multiply saturated with olivine and pyroxene due to this error.
Predicting the FeO content of partial melts accurately is particularly important due to the large influence of iron on the density of silicate melts and mantle minerals. Relative to MAGMARS calculations, pMELTS overestimates the FeO content by 4-9 wt.% between 1.0 and 5.0 GPa ( Figure 4) and can only match the composition of experimental melts at 0.5 GPa. This leads to a systematic overestimation of the density of the partial melts (Lange & Carmichael, 1990) relative to the residual mantle (by up to 150 kg/m 3 , Figure S14 in Supporting Information S1). In addition, pMELTS overestimates the MgO content by up to 5 wt.%, underestimates the SiO 2 content by up to 7 wt.%, and does not account for the increase of Na 2 O compatibility in pyroxene with pressure (Figures S11-S13 in Supporting Information S1).
The new solution models and equations of state of Holland et al. (2018) and Tomlinson and Holland (2021) appear to be better suited to simulate partial melting in the Martian mantle. However, to simulate partial melting, and near-fractional melting in particular, these models must be implemented in a Gibbs free energy minimization software, such as Perple_X (Connolly, 2009). The composition of partial melts calculated with Perple_X, including the FeO content, is much closer to the experimental melts and MAGMARS calculations (Figure 4c and Figures S11-S13 in Supporting Information S1). While those initial tests are encouraging, Perple_X calculations are computationally intensive and relatively cumbersome for inexperienced users. We believe that MAGMARS represents a faster and more user-friendly alternative to calculate the composition of Martian primary melts. It is also more suitable for integration with geodynamical software packages or other independent modeling algorithms.

Liquid Thermobarometer
The new melting model can be used to estimate the P-T conditions of melting and the source composition of primary basalts using a forward approach. The composition of the mantle and the P-T conditions of melting are varied until one or several solutions can be found to reproduce the composition of the target primary melt. However, this approach can be time consuming and how the input parameters must be varied to obtain successful simulations can be difficult to determine. It is often desirable to be able to estimate the P-T melting conditions first using an inverse method (Filiberto & Dasgupta, 2011. Here, we re-evaluate how accurately previously used liquid thermobarometers can predict the P-T of our experimental database. We then propose a new set of equations that can be used to approximate the P-T melting conditions within ±50°C and ±0.5 GPa (Figure 5), and which can serve as a first step to constraining the mantle source of specific igneous rocks with MAGMARS.
The most precise liquid thermobarometers are calibrated from experimental melts in equilibrium with a lherzolite mineral assemblage (ol, opx, cpx, and an aluminous phase). Ideally, different equations should be used for plagioclase, spinel and garnet-bearing peridotites (Grove et al., 2013;Kinzler & Grove, 1992;Krein et al., 2020;Till et al., 2012). However, for Martian basalts, in the absence of a clear geologic context, the mineralogy of the mantle source can be difficult to determine a priori. Other liquid thermobarometers have been developed based on experiments containing various mineral assemblages. In principle, they can be applied to any primary melt, regardless of the mineralogy of the source (Filiberto & Dasgupta, 2011;Lessel & Putirka, 2015;Lee et al., 2009;Putirka, 2008), but at the cost of a decreased accuracy. Filiberto and Dasgupta (2011) compared the predictions of these thermo-barometers to experiments performed with Martian basalt compositions. They concluded that Equations A and B from Putirka (2005) could reproduce the experimental temperature within 50°C and in most cases within 10°C. However, when using Putirka (2005) to calculate model temperatures for our larger experimental database, we find that it only reproduces the experimental temperatures within ±100°C and with a mean average error (MAE) of 50°C. In addition, the offsets are not random and the Putirka (2005) thermometer tends to underestimate the experimental temperatures at low temperature but to overestimate them at high temperature ( Figure S2 in Supporting Information S1). Using our experimental database to re-calibrate the model sightly lowers the offsets (MAE = 30°C) and produces a thermometer of comparable accuracy as the one of Lessel and  (Filiberto et al., 2008;Filiberto, Dasgupta, et al., 2010;Monders et al., 2007). The black dots are the experiments on terrestrial compositions from Hirschmann et al. (1998), Hirose and Kushiro (1993), Kinzler (1997), Kushiro (1996), Laporte et al. (2004), Pickering-Witter and Johnston (2000), Robinson et al. (1998), andWasylenki et al. (2003). See complete database in Table S6. MAE is the mean absolute error for each set of experiments. The dashed lines in (a) and (b) represent a ±50°C and a ±0.5 GPa interval, respectively.

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14 of 21 Putirka (2015) (Figure S5 in Supporting Information S1). To estimate the pressure, Filiberto and Dasgupta (2011) found that Equation 2 of Lee et al. (2009) was the best suited one. Here, we find that the barometer of Lee et al. (2009) reproduces the experimental pressure of most experiments within ±0.5 GPa, but overestimate more significantly the pressure of some P 2 O 5 -rich melts ( Figure S4 in Supporting Information S1). It was re-calibrated to include a P 2 O 5 dependent term: (GPa) = ln(Si4O8) − 4.24 + 0.0185 Fe4Si2O8 + 0.0109 Ca4Si2O8 + 0.009 P16∕5O8 −800 −1 + 0.0075 1∕2 with Si 4 O 8 the "silica index" as defined by Lee et al. (2009) and the melt composition recalculated using their Appendix A. Compared to Lee et al. (2009), the temperature was changed to °C for consistency with Equation 4 and the MAGMARS model.
Because this barometer is a function of the temperature, the accompanying thermometer must be independent of the pressure-which is not the case of the thermometers used in the MAGMARS model (see Section S1.2 in Supporting Information S1). After an extensive search, we find that the following thermometer can predict the experimental temperature within ±50°C with a mean absolute error of 22°C for the experiments of Collinet et al. (2015) and this study: with all oxides in wt.% and Mg# equals to the Mg/(Fe + Mg) ratio of the melt, with Mg and Fe in atom.%.
Equations 3 and 4 can thus be used to approximate the input P-T conditions of the MAGMARS model. The P-T conditions of melting are then randomly varied around this initial guess, while also adjusting the mantle composition. While the use of this thermobarometer is absolutely not required to run MAGMARS, it reduces the size of the parameter space that needs to be explored before finding mantle compositions able to produce basalts of appropriate composition.

Application: The Mantle Source of the Adirondack Basalts
In this section, we describe how MAGMARS can be used to calculate the composition of silicate melts identical to primitive Martian basalts and how the model can be used to constrain the composition and temperature of their mantle source. As an example, we focus on the Adirondack basalts, which are olivine-phyric basalts analyzed at Gusev crater by the Mars Exploration Rover Spirit (McSween, Ruff, et al., 2006). The Adirondack basalts are thought to represent primary melts, or near-primary melts affected by minor ( 10 wt.%) olivine fractionation, and have received much attention for their potential to constrain the melting process in the Martian mantle (Collinet et al., 2015;Filiberto & Dasgupta, 2011;Filiberto et al., 2008;McSween, Wyatt, et al., 2006;Monders et al., 2007).
Whether the Adirondack basalts represent true primary melts has been debated. Monders et al. (2007) showed experimentally that the Adirondack basalts were multiply saturated with olivine and orthopyroxene at the P-T conditions of the upper mantle (1.0 GPa and 1340°C). This observation, and the similarities in composition with the experimental melts of Bertka and Holloway (1994), were interpreted as evidence that the Adirondack basalts represent primary melts. However, another experimental study (Filiberto et al., 2008) found that the Adirondack basalts could instead have been in equilibrium with olivine and pigeonite at slightly higher P-T conditions in the upper mantle (1.3 GPa and 1380°C), instead of olivine and orthopyroxene. Because a mantle source made of olivine and pigeonite was deemed unlikely, Filiberto et al. (2008) concluded that the Adirondack basalts do not represent primary melts, but were affected by 10 wt.% olivine fractionation. The lower P-T conditions of the MSP of Monders et al. (2007) were suggested by Filiberto et al. (2008) to result from a higher concentration of dissolved H 2 O (0.8 wt.% vs. 0.13 wt.%) in melts and the slightly lower FeO content of the bulk material. Assuming that 10 wt.% of olivine fractionation had occurred prior to emplacement, Filiberto and Dasgupta (2011) later used a liquid thermobarometer to calculate new P-T conditions of melting: 1.7-1.9 GPa and 1470-1480°C (T p = 1535-1590°C).

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15 of 21 The melting experiments of Collinet et al. (2015) confirmed that melts similar in composition to the Adirondack basalts, after correction for minor olivine fractionation (0-10 wt.%), could be produced by melting a primitive Martian mantle at P-T conditions of 1.5-2.0 GPa and 1400-1450°C (corresponding to 20-25 wt.% of melting). The apparent discrepancies between the MSP studies of Monders et al. (2007) and Filiberto et al. (2008) can easily be reconciled by inspection of the peritectic melting reactions at 1.5-2.0 GPa (Collinet et al., 2015).
Lherzolite melting reaction: Harzburgite melting reaction: In MSP studies, the melt can only be saturated with the phases that are on the left-hand side of the melting reactions (reactant side). The Adirondack basalts are similar in composition to the primary melts produced exactly at the lherzolite-harzburgite transition (Collinet et al., 2015). Therefore, minor differences in the starting compositions of both MSP studies (and in the water and FeO contents in particular) could explain the contrasting mineralogy observed by Monders et al. (2007) and Filiberto et al. (2008). Even if a basaltic melt is multiply saturated with pigeonite and olivine but not orthopyroxene, this does not signify that orthopyroxene was absent from the source but could simply mean that melting was incongruent (Equation 5). We conclude that the Adirondack basalts could represent primary melts in equilibrium with a FeO-rich mantle (Mg# 75-76). However, we cannot rule out that the Adirondack basalts had been affected by olivine fraction and that the mantle source was initially poorer in FeO as suggested by Filiberto and Dasgupta (2011).
All of these previous studies assumed, implicitly or explicitly, that the Adirondack basalts were produced by batch melting. However, it is generally recognized that melt extraction is very efficient during decompression melting and retaining a 20 wt.% melt fraction in the residual mantle is likely unrealistic (Faul, 2001;McKenzie, 1985;Zhu et al., 2011). In Section 3.3 and Figure 2, we show that the major-element composition of the Adirondack basalts can also be produced by near-fractional polybaric melting of a Dreibus and Wänke (1985) mantle composition between 3.3 and 0.8 GPa, with a mantle T p of 1415°C (Table 3 and Table S7). However, this result is not unique and, in this section, we discuss the possibility that the mantle source of the Adirondack basalts had been depleted by prior melting events and was therefore significantly distinct from the Dreibus and Wänke (1985) composition. This is supported by the low K 2 O content of the Adirondack basalts (Schmidt & McCoy, 2010) compared to melts produced by melting of the Dreibus and Wänke (1985) mantle (0.08 vs. 0.19 wt.%), at a constant Na 2 O content of 2.2 wt.% (Table 3 and Table S7). Because K 2 O is more incompatible than Na 2 O in partial melts, the K 2 O/Na 2 O ratio of the residual mantle decreases over time with successive melting events. Therefore, the low K 2 O/Na 2 O ratio of the Adirondack basalts could be best explained by re-melting a depleted mantle from which silicate melt had been extracted during prior melting stages.
First, the composition of the melts produced by melting a primitive mantle similar to the Dreibus and Wänke (1985) model at an average pressure of 2.0 GPa was calculated with MAGMARS. We recorded the composition of the residual mantle after 10 wt.% melt removal and used it as the starting composition for a second set of melting simulations. The TiO 2 , Al 2 O 3 , FeO, MgO, CaO, Na 2 O, K 2 O and P 2 O 5 concentrations were then modified randomly around the average value to run five hundred simulations and the simulations that produced melts similar to the Adirondack basalts (13) were identified (Table 3). The melt fraction of those simulations is much smaller (4-8 wt.%) than the ∼22 wt.% necessary to produce Adirondack-like compositions from a primitive mantle (Table 3  and Table S7). In order to produce similar compositions, the average pressure of the melting zone must remain unchanged (∼2 GPa). Therefore, melting starts at lower pressure and ends at higher pressure ( Figure 6b) than when the mantle is more fertile (Figure 6a). Because a much smaller amount of melt is produced, the amount of latent heat lost during melting is lower, and the initial mantle potential temperature (T p ) is also lower (by 30°C).
Finally, as mentioned above, we cannot rule out that the mantle source was less FeO-rich than the Dreibus and Wänke (1985) composition. Geophysical, geochemical and petrological evidence all strongly suggest that the mantle of Mars is FeO-rich but a range of average Mg# (75-79) can account for most observations (Dreibus & Wänke, 1985;Khan et al., 2018;G. J. Taylor, 2013;Yoshizaki & McDonough, 2020). The mantle sources of shergottites were also relatively poorer in FeO (Mg# = 80-86), suggesting that the Mg# could also reflect local mantle heterogeneities (Collinet et al., 2017;Filiberto, 2017;Musselwhite et al., 2006). It is therefore important to evaluate the effect of a mantle source with a higher Mg# on the P-T conditions of melting.

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16 of 21 We consider a mantle source with a Mg# of 80-81 as a FeO-poor mantle end-member. In order for the Adirondack basalts to be in equilibrium with such a FeO-poor mantle source, substantial olivine fractionation would have had to occur (14-17 wt.% olivine). Random MAGMARS calculations were performed to identify the P-T conditions required to produce primary melts corresponding to this composition. Due to the higher olivine component of Note. DW85 is the mantle composition from Dreibus and Wänke (1985) for comparison. The DW85 composition was varied randomly to run 1000 MAGMARS simulations and P is the average mantle composition of 90 simulations which produced melts similar to the Adirondack basalts. A DW85 residual mantle following 10 wt.% melt removal at 2.0 GPa was used for a second set a MAGMARS calculations (500), while also randomly varying the starting composition. D is the average mantle composition of 13 runs which produced melts similar to the Adirondack basalts. σ is the standard deviation based on all successful MAGMARS simulations. Oxides in italic were not systematically varied but simply taken from the DW85 composition. Lower section: composition of the partial melts produced by melting the P and D average mantle compositions compared to the Adirondack basalts (+2.5 wt.% olivine; in equilibrium with olivine Fo 77 ). F is the melt fraction, T p is the mantle potential temperature required to produce the melt, P 1 and P 2 are the pressures at the bottom and top of the melting zone, respectively.
17 of 21 those primary melts, they are characterized by a higher MgO content. For example, with 17 wt.% olivine fractionation, the primary melt would have had a MgO content ∼4 wt.% higher (16.3 wt.%; Table S7) to the primary melts derived from a mantle with Mg# 76. The average P-T conditions (T p = 1480°C; average P = 2.8 GPa) and the extent of melting (F = 11.5 wt.%) required to match this composition are significantly higher than for a source with a lower Mg# (Figure 7 and Table S7).
Our preferred mantle source for the Adirondack basalts corresponds to a depleted mantle with a Mg# of 77, intermediate within the range of commonly accepted average Martian mantle Mg# (75-79; D in Table 3). The depleted nature of this mantle has the advantage of accounting for the low K 2 O content of the Adirondack basalts. Figure 6. P-T conditions required to produce primary melts of composition corresponding to the Adirondack basalts by near-fractional polybaric melting. A primitive mantle (a) as well as more depleted mantle (b) can produce primary melts of nearly identical composition. The average pressure of melting controls the composition of aggregate melts and is identical in both cases. The lithosphere is thicker in (b) and the melting zone narrower. The latent heat lost in (b) is lower relative to (a) due to the lower degree of melting. The mantle potential temperature required to match a given composition is therefore lower when the mantle is more depleted, despite the higher solidus temperature (b). The mantle adiabat, P-T trajectory of melting, and solidus from (a) are represented in gray in (b) for comparison. The corresponding mantle compositions can be found in Table S7 and are similar to the composition P and D of Table 3. Figure 7. P-T melting conditions of the mantle source of the Adirondack basalts, assuming that the mantle source was depleted to account for the low K 2 O/Na 2 O (see text), with (81) and without (76) olivine fractionation prior to emplacement. The circled numbers represent the Mg# of the mantle source. If no olivine fractionation occurred (76), the primary melt was identical in composition to the Adirondack basalts. The case with Mg# = 81 corresponds to 17 wt.% olivine fractionation prior to emplacement (i.e., the primary melt is 17 wt% more olivine-normative than the Adirondack basalts). Those P-T melting conditions likely represent lower and upper limits. The composition of the corresponding primary melts and mantle sources can be found in Table S7. COLLINET ET AL.

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18 of 21 In this scenario, the Adirondack basalts would represent a 6-7 wt.% aggregate melt produced by near-fractional polybaric melting between 1.8 and 2.3 GPa, and which subsequently fractionated 2.5 wt.% of olivine. The corresponding mantle potential temperature (1390 ± 40°C) is much lower than the T p calculated by Filiberto and Dasgupta (2011) and Filiberto (2017Filiberto ( ) (1522. This ∼150°C discrepancy has three sources: (a) the tendency of the Putirka (2005) AB thermometer to overestimate the temperature at high temperature (50-70°C; Figure S2 in Supporting Information S1), (b) the higher latent heat required to produce ∼25 wt.% melt (vs. 6-7 wt.% if the mantle is depleted), and (c) the assumption of isobaric melting and associated higher melting temperatures (Figures 2d and 2e, Table S7). This new T p of 1390°C is probably close to the minimum T p required to form primary melts similar to the Adirondack basalts. As discussed above, and summarized in Table S7, different assumptions regarding the nature of the melting process and composition of the source lead to a range of possible T p (1375-1480°C).

Summary and Conclusion
In this study, we have performed melting experiments at 1.0 and 2.4-2.6 GPa on a primitive Martian mantle with varying P 2 O 5 content. We use these new experiments together with a comprehensive database of previous melting experiments to calibrate a new model called MAGMARS. This model is found to compare favorably with other petrological modeling programs such as Perple_X and pMELTS and can accurately reproduce the experimental melt compositions, as well as predict the melt compositions from different mantle sources produced by different melting processes (e.g., batch melting and near-fractional polybaric melting). We also provide an updated thermobarometer that can estimate the P-T melting conditions of primary melts and can be used as a preliminary step to constrain the input parameters of the MAGMARS melting model. As an example, we re-investigated the P-T melting conditions and source compositions of the Adirondack-class basalts, assuming that the mantle source was either primitive or depleted. Depending on which melting process is assumed, the P-T conditions to obtain the same melt composition from a primitive mantle are different: for near-fractional polybaric melting the potential temperature is about 40°C lower (1410-1425°C) and the melting zone extends to a shallower depth (0.8-1.0 GPa, 70-85 km) than for batch melting (1450-1470°C, 2.0 GPa, 170 km). For a previously depleted mantle, the degree of melting required to maintain the melt composition is somewhat lower (5-7 wt.% vs. 20-25 wt.%), and thus T p is even lower (1375-1395°C), with negligible differences between the melting processes at such a low melt fraction. In detail, melting a depleted mantle reproduces more accurately the composition of the Adirondack basalts and their K 2 O/Na 2 O ratio in particular. Therefore, the T p of mantle source was likely 100-150°C lower than the previous estimate of 1520-1550°C (Filiberto, 2017). The mantle sources of other potential Martian primary melts and their implications for the evolution of partial melting in the Martian mantle will be discussed in an upcoming manuscript. Finally, in a next step, MAGMARS will be coupled to convection models to account more accurately for the feed-back effects between mantle convection, partial melting and crust formation, and create new models of the thermo-chemical evolution of Mars.

Data Availability Statement
The experimental data and the MAGMARS model (Matlab scripts) are available in Supporting Information S1 and on GitHub (https://github.com/mxcllnt/MAGMARS), where the code will be maintained and frequently updated. The version used for this study is MAGMARSv1.0 (Collinet et al., 2021). The experimental data has also been added to the Library of Experimental Phase Relations (LEPR).