P Faure

M A Bouhifd

M Boyet

G Manthilake

V Clesi

J.-L Devidal

Uranium and thorium partitioning in the bulk silicate Earth and the oxygen content of Earth's core

This study investigates the partitioning of U and Th between molten metal and silicate liquid (DU and DTh) during Earth's core-mantle differentiation. We report new Th partition coefficients between chondritic silicate melt and various Fe-rich alloys in the system Fe-Ni-C-S-Si as determined by experiments in a multi-anvil apparatus at 3-8 GPa, 2073-2373 K, and oxygen fugacities from 1.5 to 5 log units below the iron-wüstite (IW) buffer. By compiling all existing data on molten metal-silicate liquid partitioning of U and Th, we develop global models of U and Th partitioning between the mantle and core throughout Earth's accretion. The calculated concentrations in the Bulk Silicate Earth (BSE) are in agreement with previous studies (UBSE = 11.42 ± 0.45 ppb and ThBSE = 43.20 ± 1.73 ppb), whereas the contents of these radioactive elements in the Earth's core remain negligible. Compared to recent geochemical estimations, the calculated (Th/U)BSE supports EL rather than EH enstatite chondrites as the reduced building blocks of the Earth. Furthermore, we demonstrate that Th is much more sensitive than U to the oxygen content of the metallic phase. To reproduce the Th/U ratio of the BSE within its uncertainties, the oxygen content of the Earth's core must be lower than 4.0 wt%. By combining other existing constraints, this suggests that the core contains 2.0-4.0 wt% O.

The calculations of U and Th concentrations and Th/U in the BSE developed herein can be used as new constraints for determining the concentrations of other refractory lithophile elements in the BSE as soon as their metal-silicate partition coefficients are well constrained over the conditions of core segregation.

Introduction

Uranium and thorium are refractory elements that condensed from the solar nebula at high temperatures (>1400 K). One would thus assume that the Th/U ratio in the bulk silicate Earth (BSE) should be equal (to the first order) to that of chondrites if these elements were not fractionated during Earth's core formation. However, existing geochemical models show some discrepancies between the U and Th concentrations and Th/U ratio of the BSE compared to those of chondrites, as well as large uncertainties. For instance, [START_REF] Lyubetskaya | Chemical composition of Earth's primitive mantle and its variance[END_REF] show a relative depletion of refractory lithophile elements compared to previous estimates (e.g., [START_REF] Mcdonough | The composition of the Earth[END_REF][START_REF] Palme | Cosmochemical estimates of mantle composition[END_REF]. The BSE composition can be constructed using a geochemical approach based on mantle peridotite samples, which originate at depths shallower than the transition zone. In addition, model of the primitive mantle based on petrological data assumes global-scale homogeneity of the Earth's mantle. Moreover, the Th/U ratio of the upper mantle may have evolved with time due to changes in surface redox conditions.

The recycling of U from the surface into Earth's interior differs from that of Th because U exists in two valence states (4+ and 6+), and highly soluble U 6+ dominates under oxidized surface conditions. The terrestrial upper mantle sampled by mid-ocean ridge basalts shows a low Th/U ratio relative to that estimated from chondrites [START_REF] Jenner | Analysis of 60 elements in 616 ocean floor basaltic glasses[END_REF][START_REF] Gale | The mean composition of ocean ridge basalts: MEAN MORB[END_REF]. Indeed, the significant addition of recycled uranium into the upper mantle has been confirmed by high-precision 238 U/ 235 U measurements [START_REF] Andersen | The terrestrial uranium isotope cycle[END_REF].

The partitioning of radioactive elements between different Earth reservoirs is a topic of much debate since they play a major role in the thermal evolution of the Earth and its mantle dynamics. Indeed, some amount of radioactive heating is necessary to drive the early geodynamo. Moreover, several studies have recently proposed that U and Th become more siderophile and/or chalcophile under reducing conditions, and consequently could have been sequestered in non-negligible amounts (up to 10 ppb U and 21 ppb Th) in the Earth's core (e.g., [START_REF] Mccubbin | Is Mercury a volatile-rich planet?[END_REF][START_REF] Blanchard | The solubility of heatproducing elements in Earth's core[END_REF][START_REF] Chidester | Metal-silicate partitioning of U: Implications for the heat budget of the core and evidence for reduced U in the mantle[END_REF]Wohlers andWood, 2015, 2017, andreferences therein). It is thus essential to determine the budget of heat-producing elements in different planetary reservoirs.

In the present study, we present experimentally determined Th metal-silicate partitioning data at up to 8 GPa, 2373 K, and over a wide range of oxygen fugacity conditions and chemical compositions of the silicate and metallic phases, based on new chemical analyses of the experimental products reported in [START_REF] Bouhifd | Metal-silicate partitioning of Pb and U: Effects of metal composition and oxygen fugacity[END_REF]. The metal-silicate partitioning of Th has been the subject of very few experimental studies, and the only existing data were obtained at 1.5 GPa using mainly sulfide liquids [START_REF] Murrell | Partitioning of K, U, and Th between sulfide and silicate liquids: implications for radioactive heating of planetary cores[END_REF]Wohlers andWood, 2015, 2017). Then, we compiled all published experimental results on metal-silicate partitioning of these elements 𝐷 M = " M met " M sil (where 𝑋 M met and 𝑋 M sil are the wt% concentrations of element M in the metal and silicate phases, respectively). It is then possible to regress these data to better constrain the influence of physical parameters (pressure, temperature, oxygen fugacity) and the chemical compositions of the studied phases on DU and DTh. These models were subsequently used to derive the U and Th concentrations in the BSE and the core as well as the Th/U ratio of the BSE, and our results are consistent with previous results (e.g., [START_REF] Allègre | Chemical composition of the Earth and the volatility control on planetary genetics[END_REF][START_REF] Kargel | The composition and early evolution of earth[END_REF][START_REF] Lyubetskaya | Chemical composition of Earth's primitive mantle and its variance[END_REF]and references therein). We combined our results with recent lead isotopic analyses [START_REF] Wipperfurth | Earth's chondritic Th/U: Negligible fractionation during accretion, core formation, and crust-mantle differentiation[END_REF] to discriminate between different scenarios regarding Earth's building blocks. Furthermore, our models show also that the concentrations of U and Th in the Earth's core are less than 1.5 ppb and 5 ppb, respectively. Finally, the calculations performed herein provide new constraints on the oxygen content of the core, which cannot exceed 4 wt%.

Experimental methods

The run-products used in the present study are those described and analysed in [START_REF] Bouhifd | Metal-silicate partitioning of Pb and U: Effects of metal composition and oxygen fugacity[END_REF]. Here, we report new chemical analyses of thorium in the silicate and metallic phases and of oxygen in the Fe-rich alloys. In the following, we briefly summarize the experimental procedure and fully describe the analytical methods used to derive our new results.

Starting materials

Silicate starting materials consisted of a C1-chondrite model composition containing 49.6 wt% SiO2, 35.1 wt% MgO, 8.5 wt% FeO, 3.5 wt% Al2O3, and 3.3% CaO [START_REF] Wasson | Compositions of Chondrites[END_REF]. This silicate composition is similar to that used to determine the effect of water on the behavior of siderophile elements during core formation (e.g., [START_REF] Clesi | Effect of H2O on metal-silicate partitioning of Ni, Co, V, Cr, Mn and Fe: Implications for the oxidation state of the Earth and Mars[END_REF]. Doping was achieved with mixtures of U and Th dissolved in 0.5 M nitric acid (AAS standard solutions), which was denitrified in air at 1100 K for 4 h. The doping mixtures were added to the silicate starting material to achieve concentrations of 1000 to about 4000 ppm per element. Fe-rich alloy powders in the Fe-Ni-Si-S system were added to the silicate starting material in 50:50 proportions by weight to obtain several starting compositions. Oxygen fugacity (fO2) was varied between IW -2 and IW -5 by adding metallic Si to the starting material. Each starting composition was then mixed and ground under ethanol, and dried at ~400 K for at least 24 h.

High-pressure and high-temperature syntheses

The high-pressure and high-temperature experiments were performed by [START_REF] Bouhifd | Metal-silicate partitioning of Pb and U: Effects of metal composition and oxygen fugacity[END_REF] using the French national 1000-ton and 1500-ton multi-anvil facilities in Clermont-Ferrand, France. Briefly, the pressure assembly consisted of Cr-doped MgO octahedra (18 or 14 mm edge length) containing a LaCrO3 heater. The octahedra were compressed using 32 mm tungsten carbide anvils with truncation edge lengths of 11 and 8 mm for the 18 and 14 mm octahedra, respectively, and pyrophyllite gaskets. Samples were contained in either graphite or MgO single-crystal capsules in order to test the impact of the capsule material on the measured metal-silicate partition coefficients. The samples were first compressed to the desired pressure at room temperature (for instance for 5 GPa experiments, we use about 100 tons on a 14/8 assembly) and then heated to the target temperature for run durations of 1 to 2 min. These durations are sufficient to achieve equilibrium between silicate melt and metallic liquid as found for many other studies (e.g., [START_REF] Corgne | Metal-silicate partitioning and constraints on core composition and oxygen fugacity during Earth accretion[END_REF][START_REF] Cartier | Redox control of the fractionation of niobium and tantalum during planetary accretion and core formation[END_REF][START_REF] Bouhifd | Superchondritic Sm/Nd ratio of the Earth: Impact of Earth's core formation[END_REF][START_REF] Clesi | Low hydrogen contents in the cores of terrestrial planets[END_REF]; and references therein) and short enough to prevent excessive reaction between the capsule and the silicate or metallic phases. Temperatures were monitored with a thin axial W5Re/W26Re thermocouple whose junction was in direct contact with the top of the capsule. The thermocouples presented no difference of behavior when using either graphite or MgO single crystal capsule, showing the non-contamination of thermocouple during graphite experiments. The pressure and temperature uncertainties are estimated to be ±1 GPa and ±100 K, respectively. The samples were quenched by shutting off the power, which resulted in an initial quench rate of approximately 800 K/s. The samples were then decompressed over about 12 h. After the completion of each experiment, the run product was mounted in epoxy, polished, and carbon coated for chemical analyses.

Electron Microprobe

The major and minor element compositions of the metal and silicate phases were analyzed with a CAMECA SX100 electron microprobe with a 20 kV accelerating voltage at the Laboratoire Magmas et Volcans (LMV, Clermont-Ferrand, France). Major element analyses in silicate phases were performed at 50 nA with a 10-20 μm defocused beam, with on-peak counting times ranging from 5 to 80 s depending on element concentrations and detection limit requirements (e.g. [START_REF] Bouhifd | Metal-silicate partitioning of Pb and U: Effects of metal composition and oxygen fugacity[END_REF]. Wollastonite was used as a standard for Si and Ca, and MgO, Al2O3, and Fe2O3 for Mg, Al, and Fe, respectively. To analyze K and S, we used natural KAlSi3O8 and ZnS as standards, respectively. For metallic phase analyses, we used a defocused beam (10-20 μm), beam currents of 15 nA for major elements (Fe, Ni, and Si) or 150 nA for trace elements, and on-peak counting times ranging from 5 to 60 s. Pure Ni, Si, and Fe metals were used as standards. For samples embedded in graphite capsules, we estimated the carbon content of the Fe-alloys from the deviation of the analytical totals from 100%.

In the present study, we essentially focused our chemical analyses to determine the concentrations of Th in the silicate phase and oxygen contents in the metal phases. To analyze Th in the silicate, thorianite (ThO2) was used as standard. For metal analyses, pure Ni, Si, and Fe (NIST 665) metals were used as standards. To analyze oxygen in metallic phases, NIST 665 and Fe2O3 were used as standards. Prior to analyses, the samples and standards were polished and carbon coated in the same way, then kept under vacuum until analysis. Analyses were performed using 20 kV accelerating voltage, a beam current of 15 nA, and a defocused beam of 30 μm, and the detection limit for oxygen in the metals was 550 ppm.

During the oxygen analyses, we found that the nominally oxygen-free standards contained about 0.21 ± 0.02 wt% O at 20 kV accelerating voltage; this value was consistent among the standards. We additionally observed that the measured oxygen contents of the standards decreased with increasing voltage from 7 to 15 kV before reaching a plateau between 15 and 25 kV (the maximum voltage tested). We concluded that the measured oxygen contents in the O-free standards were the result of rapid adsorption of oxygen onto the surface of the standards. Therefore, we subtracted the value obtained from the standards at 20 kV from every analysis of our run-products. The corrected concentrations are presented in Tables A1 andA2. The reported oxygen analyses are in good agreement with the recent parameterization for metal-silicate oxygen partitioning reported by [START_REF] Fischer | High pressure metal-silicate partitioning of Ni, Co, V, Cr, Si, and O. Geochim[END_REF].

LA-ICP-MS

U and Th concentrations in both metal and silicate phases were analyzed by laser ablation inductively coupled plasma mass spectrometry (LA-ICP-MS) at LMV. We used a quadrupole ICP-MS (Agilent 7500-cs, Agilent Technologies) coupled with a 193 nm ArF excimer-type laser (Resonetics M-50). The laser was fired on selected sample regions at a repetition rate of 3 Hz with 8 mJ output and a spot size of 15-44 μm. Pit depths were roughly equivalent to the spot size.

During LA-ICP-MS analyses, the Ca signal was monitored during metal analyses and the Fe and Ni signals during silicate analyses. Any sudden variation of their contents was indicative of phase contamination (i.e., metal contamination during silicate phase analysis or vice versa). Ablation was performed under a pure He flow for durations of about 80 s (blank count rates were measured for 20 s prior to 60 s of ablation). The analyte was carried to the ICP torch by a gas mixture in which N2 + Ar gases were added to the pure He gas. Reference glasses NIST SRM 610, NIST 612, and BCR2-G were used for external calibration [START_REF] Gagnon | Quantitative analysis of silicate certified reference materials by LA-ICPMS with and without an internal standard[END_REF], and 29 Si, 57 Fe, and 60 Ni were used for internal calibration. All data were processed using the GLITTER software package [START_REF] Achterberg | Data reduction software for LA-ICP-MS[END_REF]. Detection limits calculated for U and Th were below 1 ppm.

Results

The experimental conditions and calculated U and Th partition coefficients (DU and DTh, respectively) are summarized in Table 1, and the chemical compositions of the quenched silicate and metallic melts are reported in Tables A1 andA2. Typical run products contained two immiscible phases: silicate glass and Fe-rich alloy (Fig. 1).

Microstructures of the run products showed that both phases were fully molten during the experiments. In some cases, we observed dendritic microstructures in the silicate. When dendritic microstructures were present within the region of interest, LA-ICP-MS analyses were performed using a 44 µm spot size to average the heterogeneous quenched texture.

Several criteria indicate the attainment of thermodynamic equilibrium in these experimental samples: (i) multiple analyses in the silicate phase show that it is homogeneous with no evidence of chemical zoning, and (ii) all metal blobs embedded in the silicate matrix are of similar composition.

A critical parameter during metal-silicate partitioning using the multi-anvil technique is the nature of the capsule, because no material can be completely inert to both silicate and metal liquids. Here we used two different capsules: MgO single-crystal and graphite capsules. Using graphite capsules, we usually observed low analytical totals for the metallic phase, i.e., between 93 and 98.9 wt% (Table A1), indicating significant amounts of carbon dissolved in the Fe-rich alloy. However, we did not observe any effect of the capsule material on DU or DTh considering analytical errors over the range of P-T conditions explored in this study (Fig. 2). Similar behavior has been observed for the metal-silicate partition coefficients of U and other lithophile elements [START_REF] Bouhifd | Metal-silicate partitioning of Pb and U: Effects of metal composition and oxygen fugacity[END_REF]2015;and references therein). Finally, our experimental results showed that varying the concentrations of U and Th in the starting materials had no effect on the resulting metal-silicate partition coefficients (Tables A1 andA2).

Oxygen fugacity

The prevailing oxygen fugacities in our experiments were calculated based on the Fe and FeO contents of the quenched metal and silicate phases, respectively, by considering the equilibrium reaction between Fe metal and Fe oxide:

Fe metal + ½ O2 ↔ FeO silicate (1)
The fO2 can thus be estimated relative to the Iron-Wüstite buffer (IW) from the activities of Fe in the metal and FeO in the silicate following the equation:

log fO 2 (DIW) = 2×log ) ) FeO sil ) Fe met *, (2)
where 𝑎 FeO sil = 𝛾 FeO sil × 𝑋 FeO sil and 𝑎 Fe met = 𝛾 Fe met × 𝑋 Fe met represent the activities of FeO in the silicate liquid and Fe in the metal, respectively, 𝛾 FeO sil and 𝛾 Fe met are their respective activity coefficients, and 𝑋 FeO sil and 𝑋 Fe met their respective mole fractions.

We calculated the activity coefficient of Fe in the Fe-rich alloys based on the formula of [START_REF] Ma | Thermodynamic description for concentrated metallic solutions using interaction parameters[END_REF] using the interaction parameters ε defined in the Steelmaking Data Sourcebook (1988) (see also [START_REF] Wade | Core formation and the oxidation state of the Earth[END_REF][START_REF] Siebert | Systematics of metal-silicate partitioning for many siderophile elements applied to Earth's core formation[END_REF]

𝜒 . 6 𝜒 4 6 ) 1 1 -𝜒 . + 1 1 -𝜒 4 + 𝜒 . 2(1 -𝜒 4 ) 6 -1* (3) /01
.21

This treatment is available using the activity calculator (http://www.earth.ox.ac.uk/~expet/metalact/) provided by the university of Oxford [START_REF] Wade | Core formation and the oxidation state of the Earth[END_REF]. No pressure correction is made in our calculations due to the lack of reliable data for all elements.

To determine 𝛾 FeO sil for all silicate melt compositions, we used the average value (𝛾 FeO sil = 1.5 ± 0.5 ) derived from studies at 1 bar by O' Neill and Eggins (2002) for several melt compositions in the CaO-MgO-Al2O3-SiO2 system, and by [START_REF] Holzheid | The activities of NiO, CoO and FeO in silicate melts[END_REF] for melts in the anorthite-diopside system and komatiitic basalt, assuming no pressure effect [START_REF] Toplis | The thermodynamics of iron and magnesium partitioning between olivine and liquid: criteria for assessing and predicting equilibrium in natural and experimental systems[END_REF]. The calculated fO 2 of our experiments ranged from IW -5

to IW -1. Values determined for each experiment are reported in Tables A1 andA2. For comparison, we also report the value of fO 2 calculated assuming ideality (𝛾 FeO sil = 𝛾 Fe met = 1), and the difference between the two values is within the uncertainties of the calculations. Thus, in the following, we use the fO 2 values calculated for the non-ideal system.

Parameterizations of the result

As reported in previous studies (e.g., [START_REF] Righter | Prediction of siderophile element metalsilicate partition coefficients to 20 GPa and 2800 °C: the effect of pressure, temperature, oxygen fugacity, and silicate and metallic melt compositions[END_REF], the metal-silicate partition coefficient of element M (𝐷 M met/sil ) is linked to the free energy of the reaction, and can be related to T, P, and fO2 by the following equation, to which we have added three terms to account for the metal composition (here, the silicate composition was kept constant, reflecting the early mantle during core formation):

log@𝐷 M <=>/?@A A = 𝑎 log(𝑓𝑂 6 ) + 𝑏 1 𝑇 + 𝑐 𝑃 𝑇 + 𝑑 log(1 -𝑋 S met ) + 𝑒 log(1 -𝑋 Si met ) + 𝑓 log(1 -𝑋 O met ) + 𝑐𝑠𝑡, ( 4
)
where a is related to the valence of element M in the silicate melt, and b and c are related to the enthalpy @ ∆D°F G A and volume @ H∆I°F G A of the system, respectively. Parameters d, e, and f are constants related to the chemical composition of the metallic phase. The term cst includes the effect of the entropy of the system @ ∆J°F A.

Using all existing data (presented here along with published data) for molten metal-silicate liquid partitioning of U (63 data points) and Th (32 data points) for which the T, P, fO2, and silicate and metal compositions are all well characterized, we refined the coefficients in equation ( 4) by multiple linear regression following a previously validated approach [START_REF] Bouhifd | Metal-silicate partitioning of Pb and U: Effects of metal composition and oxygen fugacity[END_REF][START_REF] Righter | Partition coefficients at high pressure and temperature[END_REF][START_REF] Righter | Partitioning of Mo, P and other siderophile elements (Cu, Ga, Sn, Ni, Co, Cr, Mn, V, and W) between metal and silicate melt as a function of temperature and silicate melt composition[END_REF].

Data available in the literature come from a variety of sources (detailed in Appendix B) using a variety of synthesis techniques presenting consistent results: piston-cylinder, multi-anvil cell as well as diamond anvil cell. In order to obtain the best-fit parametrizations, we excluded experiments with high uncertainties (>75% of the reported partition coefficients), and data involving liquid metal-solid silicate equilibrium considered as irrelevant to magma ocean conditions. For consistency, the literature values for fO2 and partition exchange coefficients were recalculated using the reported concentrations of each element.

To determine the best-fit solutions for U and Th using equation ( 4), we performed several tests to establish the robustness of these parameterizations (Appendix C). The resulting parametrizations succeed in reproducing the experimental partition coefficients for a given set P, T, fO2 and chemical composition (Fig. 3a,b), and will thus be used to discuss the behavior of U and Th as a function of P, T, fO2, and the chemical composition of the core.

Effects of pressure and temperature on DU and DTh

The first test of our parameterizations of the metal-silicate partitioning of U and Th

show that, within the range of experimental conditions investigated, the pressure has only negligible effects on DU and DTh. For instance, at ΔIW -2 and constant chemical composition of the metallic phase (79.5 wt% Fe, 10 wt% Ni, 6 wt% Si, 2.5 wt% O, 2 wt% S), the effect of pressure on DU is relatively small. Varying the pressure from 10 to 50 GPa (at 3500 K) lead to a small variation of DU from about 5×10 -3 to 8×10 -3 . However, data for Th are only available up to 8 GPa. By plotting the DTh/DU ratio versus pressure (Fig. 4), we found that this ratio is constant within uncertainties. So we set the parameter c (related to the pressure in equation ( 4)) to zero. By doing so, we found that the quality of subsequent parameterizations with c = 0 is unchanged (Appendix C).

At constant pressure (between 5 and 50 GPa), a temperature increase from 2500 to 3500 K induces an increase of DU and DTh of about one order of magnitude. This suggests that, of the two parameters, temperature is the factor that most strongly influences the molten metal-silicate liquid partitioning of U and Th, consistent with results for U reported by [START_REF] Chidester | Metal-silicate partitioning of U: Implications for the heat budget of the core and evidence for reduced U in the mantle[END_REF].

Oxidation state of U and Th at reducing conditions

Metal-silicate partitioning experiments can be used to estimate the valence of any metallic element M in a silicate liquid as function of fO2. The partitioning of an element M between silicate and metal involves an exchange reaction between its oxidized form (MOn/2) in the silicate melt and its reduced form in the metal (M), which depend on fO2

and the valence n of the element in the silicate as reported in the following reaction:

MO K/6 ↔ K L O 6 + M (5)
It is therefore possible to interpret the parameter a of our multi parameter linear regression as equal to -K L for each element. From equation ( 4), we determined that parameter a for thorium is -0.95 ± 0.25, consistent with its valence of 4+ in silicate melts at reducing conditions from ΔIW -5 to ΔIW -1. However, for uranium, we determined parameter a to be -0.73 ± 0.09, indicating a more complicated exchange reaction than for Th. Previous studies have proposed that the valence of U in silicate melts under reducing conditions can be 4+, as suggested by low pressure (<20 GPa) metal-silicate partitioning experiments (e.g. [START_REF] Malavergne | New high-pressure and high-temperature metal/silicate partitioning of U and Pb: Implications for the cores of the Earth and Mars[END_REF][START_REF] Bouhifd | Metal-silicate partitioning of Pb and U: Effects of metal composition and oxygen fugacity[END_REF][START_REF] Wohlers | Uranium, thorium and REE partitioning into sulfide liquids: Implications for reduced S-rich bodies[END_REF], or 2+ (e.g. [START_REF] Chidester | Metal-silicate partitioning of U: Implications for the heat budget of the core and evidence for reduced U in the mantle[END_REF][START_REF] Blanchard | The solubility of heatproducing elements in Earth's core[END_REF][START_REF] Boujibar | U, Th, and K partitioning between metal, silicate, and sulfide and implications for Mercury's structure, volatile content, and radioactive heat production[END_REF].

The parameterization performed herein shows that the parameter a for U does not correspond to any single valence state (4+ or 2+). By fixing the parameter a to -1 or to -0.5, reflecting a valence of 4+ or 2+ for uranium in silicate melts at reducing conditions, the quality of our parameterizations remains the same as for a = -0.73. So we decided to keep this parameter equals to -0.73 ± 0.09 avoiding any supplementary hypothesis on the parameterizations. In fact, when we report the uranium partition coefficients as a function of oxygen fugacity for experiments where the S content is low (XS < 2 wt%), we could see a change of the valence of U in silicate melt from 4+ to 2+ around ΔIW -2 (Fig. 5).

This behavior is consistent with the change of the oxidation state of Nb and Ta in silicate melts at reducing conditions [START_REF] Cartier | Redox control of the fractionation of niobium and tantalum during planetary accretion and core formation[END_REF]. However, further experiments and X-ray absorption near edge structure analyses are required to confirm and better characterize the valence of uranium in silicate melts at the conditions of Earth's core formation.

Effects of silicate and metal compositions on DU and DTh

The existing data on molten metal-silicate liquid partitioning of uranium show disagreement about the effect of silicate composition on DU. [START_REF] Chidester | Metal-silicate partitioning of U: Implications for the heat budget of the core and evidence for reduced U in the mantle[END_REF] proposed that the SiO2 content of the silicate has a measurable effect on DU. However, [START_REF] Blanchard | The solubility of heatproducing elements in Earth's core[END_REF] found that there is no effect of the SiO2 content nor the degree of polymerization of the silicate melt on DU. By combining all existing data on molten metal -silicate liquid partitioning of U and Th we found that the effect of SiO2 has no significant impact on the partitioning of Th and U (see Appendix C).

The effect of carbon dissolved in the metallic phase on DU and DTh is estimated by comparing C-bearing and C-free experiments. As reported in Tables A1 andA2, the presence of carbon in the metallic phase has only a negligible impact on DU and DTh (Fig.

2

). This is in good agreement with the C-U interaction parameter in a C-saturated iron alloy, listed as 0 at temperatures around 1600-1700 K in the Steelmaking Data Sourcebook (1988). Thus, since we do not observe any significant impact of carbon (up to 4.8 wt%) on DU and DTh (Fig. 2), we do thus not include a parameter for C in our parameterizations for U and Th.

We also found that varying the Si content of the metallic phase did not lead to significant variations of the Th/U ratio of the BSE nor concentrations of U and Th in the different reservoirs. For example, an accretion scenario with a Si-free core and a scenario using the parametrization taken from [START_REF] Fischer | High pressure metal-silicate partitioning of Ni, Co, V, Cr, Si, and O. Geochim[END_REF] (which propose a core with 8.5 wt% Si) lead to a 1% variation of the Th/U ratio of the silicate reservoir when all other parameters are kept constant. So, we do not include a parameter for Si in our parameterizations.

The effect of sulfur in the metallic phase on DU and DTh seems to depend on S concentration and experimental pressure. For instance, we noticed a small dependence of DU and DTh on S content for concentrations <10 ± 2 wt%; DU and DTh increase by less than half an order of magnitude when the S content is increased from 2 to 10 wt%, consistent with low-pressure (Fig. 6) and high-pressure experimental data (e.g. [START_REF] Bouhifd | Metal-silicate partitioning of Pb and U: Effects of metal composition and oxygen fugacity[END_REF][START_REF] Blanchard | The solubility of heatproducing elements in Earth's core[END_REF][START_REF] Chidester | Metal-silicate partitioning of U: Implications for the heat budget of the core and evidence for reduced U in the mantle[END_REF]. However, we observed a strong dependence of DU and DTh on S concentrations at >20 wt% S. At low pressures (<20 GPa) and ΔIW -2, increasing the sulfur content from 2 to 35 wt% (corresponding to the FeS composition) results in an increase of DU from 8.4×10 -4 to 4.4×10 -1 (roughly three orders of magnitude), and DTh shows a smaller variation from 8.4×10 -4 to 8.3×10 - 2 . At higher pressures (around 50 GPa), 3000 K, and ΔIW -2, increasing the S content from 2 to 35 wt% led to an increase of DU from about 4.6×10 -3 to 2.4, making U a slightly siderophile element at these extreme conditions. At the same conditions, the same increase in S content increases DTh from 6.2×10 -3 to 0.6; Th thus remains a lithophile element at all investigated P, T, and fO2 conditions and chemical compositions. We note that for sulfide-silicate partitioning at ΔIW -2, DU is ~10 times higher than DTh. Similar conclusions were made by [START_REF] Wohlers | Uranium, thorium and REE partitioning into sulfide liquids: Implications for reduced S-rich bodies[END_REF] from low-pressure and lowtemperature data. However, at lower oxygen fugacities around IW -5, they differ by only a factor of 2.

By performing the parameterizations presented herein, we found the oxygen content of the metallic phase to exert the strongest influence on metal-silicate U and Th partitioning. DTh is more sensitive than DU to the oxygen content of the metal. For instance, at similar P-T conditions and ΔIW -2, an increase in the oxygen content of the metal from 2 to 5 wt% increases DU by less than one order of magnitude (from ~5×10 -4 to 3×10 -3 ) but DTh by about two orders of magnitude (from ~2×10 -4 to 1.7×10 -2 ). This difference will provide strong constraint on the composition of the Earth's core, as discussed in section 4.2.

Finally, we note that we were able to separate the effects of oxygen and sulfur on DU and DTh by compiling all existing data from S-rich systems at low pressures to S-free systems at high pressures, as S-rich and S-free metals are known to incorporate oxygen at low and high pressure, respectively. For example, whereas [START_REF] Blanchard | The solubility of heatproducing elements in Earth's core[END_REF] used only high-pressure experiments to constrain the impact of O on metal-silicate U partitioning, our parameterization incorporates data from S-rich experiments at low pressure (Wohlers andWood, 2015, 2017) and S-free to S-rich experiments at high pressure [START_REF] Blanchard | The solubility of heatproducing elements in Earth's core[END_REF][START_REF] Chidester | Metal-silicate partitioning of U: Implications for the heat budget of the core and evidence for reduced U in the mantle[END_REF].

By taking into account all the features discussed above, the best-fit solutions of our parameterizations for molten metal -silicate liquid partitioning of uranium and thorium are reported in equations ( 6) and ( 7). These equations will be used to discuss the behavior of U and Th during the Earth's core segregation. Where 𝐷 U <=>/?@A and 𝐷 Th <=>/?@A are molten metal -silicate liquid partition coefficients of uranium and thorium (on weight basis), respectively; fO2 is the oxygen fugacity relative to the iron-wüstite buffer; T is the temperature in K; 𝑋 S met and 𝑋 O met are the wt% contents of S and O in the metallic phase, respectively.

Discussion

U and Th concentrations and the Th/U ratio of the bulk silicate Earth

The U and Th contents of the BSE depend on the total amounts of both elements on Earth, which are directly connected to the nature of Earth's building blocks and their partitioning between silicate and metal reservoirs. Geochemical data from chondrites show that absolute U and Th concentrations both vary by about 23% among the different chondrite groups, whereas the Th/U ratio is relatively constant, varying by only 7% [START_REF] Wasson | Compositions of Chondrites[END_REF]. To reproduce the present-day mantle FeO content of about 8 wt%, the Earth must have been accreted from a large fraction (80-90%) of reduced bodies (EH or EL enstatite chondrites) with 10 to 20% of Earth's mass likely accreted from more oxidized bodies (e.g., [START_REF] Clesi | Effect of H2O on metal-silicate partitioning of Ni, Co, V, Cr, Mn and Fe: Implications for the oxidation state of the Earth and Mars[END_REF][START_REF] Rubie | Accretion and differentiation of the terrestrial planets with implications for the compositions of early-formed Solar System bodies and accretion of water[END_REF].

Furthermore, [START_REF] Dauphas | Calcium-48 isotopic anomalies in bulk chondrites and achondrites: Evidence for a uniform isotopic reservoir in the inner protoplanetary disk[END_REF] showed that the terrestrial 17 O, 48 Ca, 50 Ti, 62 Ni, and 92 Mo isotopic compositions can be reproduced by a mixture of 91% enstatite, 7% ordinary, and 2% carbonaceous chondrites. Indeed, this chondritic composition can explain the subchondritic Nb/Ta ratio of the BSE [START_REF] Cartier | Redox control of the fractionation of niobium and tantalum during planetary accretion and core formation[END_REF].

Similarly, we present here models of planetary accretion focusing on the evolution of the U and Th contents and the Th/U ratio of the BSE. We adopted continuous core formation models similar to those reported by [START_REF] Rubie | Accretion and differentiation of the terrestrial planets with implications for the compositions of early-formed Solar System bodies and accretion of water[END_REF] and [START_REF] Clesi | Effect of H2O on metal-silicate partitioning of Ni, Co, V, Cr, Mn and Fe: Implications for the oxidation state of the Earth and Mars[END_REF].

In our models, the Earth accreted from impactors of a given chemical composition, which then equilibrated (fully or partially) with the magma ocean of the proto-Earth until the planet obtained its final mass (ME). We assume that impactors were already differentiated into silicate mantles and metallic cores [START_REF] Taylor | Accretion of differentiated planetesimals to the Earth[END_REF]. This differentiation is defined by a pressure and temperature of equilibrium, leading to the partitioning of their bulk compositions coherent with their masses MI (from 0.01 ME to 0.1 ME). This assumption only affects the results when the cores of impactors fail to completely equilibrate with the terrestrial magma ocean, which is predicted to have occurred only during the final few relatively large impacts (e.g. [START_REF] Deguen | Turbulent metal-silicate mixing, fragmentation, and equilibration in magma oceans[END_REF] (see below).

At each step of accretion (1% ME), the impactor's metal and silicate equilibrate within the proto-mantle, which accounts for 68% of the planet's mass. We tested various increments of accretion (0.1%, 1%, and 10% ME) and observed no impact on the final result as long as the ratio MI/ME is maintained. We set the equilibration pressure to 50% of the core-mantle boundary pressure at each step. This fraction is derived from previous studies on the partitioning of siderophile elements sensitive to pressure (e.g., [START_REF] Boujibar | Metal-silicate partitioning of sulfur, new experimental and thermodynamical constraints on planetary accretion[END_REF][START_REF] Fischer | High pressure metal-silicate partitioning of Ni, Co, V, Cr, Si, and O. Geochim[END_REF], leading to a final pressure of equilibration of 60 GPa [START_REF] Siebert | Metal-silicate partitioning of Ni and Co in a deep magma ocean[END_REF]. Temperature is fixed at the liquidus of chondritic melts [START_REF] Andrault | Solidus and liquidus profiles of chondritic mantle: Implication for melting of the Earth across its history[END_REF], which evolved from 1900 K at 1% accretion to 3500 K during the final stage of accretion. Oxygen fugacity is constrained by the building block composition and the amount of reduced / oxidized building block along accretion. Since a large amount of reduced bodies is required to meet Earth's isotopic composition [START_REF] Dauphas | Calcium-48 isotopic anomalies in bulk chondrites and achondrites: Evidence for a uniform isotopic reservoir in the inner protoplanetary disk[END_REF][START_REF] Dauphas | The isotopic nature of the Earth's accreting material through time[END_REF][START_REF] Javoy | The chemical composition of the Earth: Enstatite chondrite models[END_REF][START_REF] Boyet | Enstatite chondrites EL3 as building blocks for the Earth: The debate over 146 Sm-142 Nd systematics[END_REF], this model follows the fO 2 paths defined in [START_REF] Clesi | Effect of H2O on metal-silicate partitioning of Ni, Co, V, Cr, Mn and Fe: Implications for the oxidation state of the Earth and Mars[END_REF], i.e., ranging from IW -5 to IW -2. However, The fO2 path of Earth accretion has a negligible effect on the resulting (Th/U)BSE which vary by less than 0.7% when considering an accretion at constant fO2 (IW-2 throughout accretion) instead of a reduced to oxidized accretion scenario. The fO2 path mainly affects the content of the metallic phase in light elements (see below). Partitioning is determined using our parameterizations for U and Th (Table 2). Following the approach of Deguen et al.

(2014), we take into account the effect of partial metal-silicate equilibration on elemental partitioning, which mainly affects the last 40% of Earth's accretion. We consider that equilibration between the proto-Earth's mantle and the metal of the impactors becomes less efficient as accretion progresses (e.g., [START_REF] Deguen | Turbulent metal-silicate mixing, fragmentation, and equilibration in magma oceans[END_REF][START_REF] Rubie | Accretion and differentiation of the terrestrial planets with implications for the compositions of early-formed Solar System bodies and accretion of water[END_REF]. For the last giant impact that formed the Moon (MI = 0.1 ME), we find that the impactor's core merges almost completely with the Earth's proto-core, with less than 5% of its mass equilibrating with the silicate mantle. However, we find that the impact of this equilibration on our parameterization is very limited, in agreement with Rudge et al.

(2010). Since we consider two mostly lithophile elements, disequilibrium models are as compatible with the geochemical observations as equilibrium models, and the elemental concentrations of the BSE remain within uncertainties in either case.

At each step of accretion, the O, Si, and Ni content of the metallic core is calculated using the parameterization of [START_REF] Fischer | High pressure metal-silicate partitioning of Ni, Co, V, Cr, Si, and O. Geochim[END_REF]. The C content of the core remains constant at 0.2 wt% during accretion [START_REF] Wood | Accretion of the Earth and segregation of its core[END_REF], and a late addition of sulfurrich material increases the S content of the core to 2 wt% to fit geophysical, geochemical, and experimental estimates (e.g., [START_REF] Allègre | The chemical composition of Earth[END_REF][START_REF] Boujibar | Metal-silicate partitioning of sulfur, new experimental and thermodynamical constraints on planetary accretion[END_REF][START_REF] Dreibus | Cosmochemical constraints on the sulfur content of the Earth's core[END_REF][START_REF] Palme | Cosmochemical estimates of mantle composition[END_REF] A major contribution of enstatite chondrites during Earth's accretion is supported by the strong similarities between mass-independent isotopic ratios measured in enstatite chondrite and terrestrial samples (e.g., [START_REF] Javoy | The integral enstatite chondrite model of the Earth[END_REF][START_REF] Dauphas | The isotopic nature of the Earth's accreting material through time[END_REF][START_REF] Boyet | Enstatite chondrites EL3 as building blocks for the Earth: The debate over 146 Sm-142 Nd systematics[END_REF].

However, the most recently published trace element measurements of enstatite chondrites (Barrat et al., 2014;Dauphas and Pourmand, 2015) reveal variable U and Th concentrations and significant differences relative to the compilation published by Wasson and Kallaymen (1988). As U and Th concentrations are low in enstatite chondrites, obtaining accurate measurements remains challenging. Thus, in our model, we consider the enstatite chondrite U and Th concentrations of Barrat et al. (2014) and Dauphas and Pourmand (2015) to represent the reduced end-members. We note here the significant effect of weathering on the Th/U ratios of samples reported in these studies.

Under oxidizing surface conditions, these elements show contrasting behavior, with U behaving as a water-soluble hexavalent species, whereas Th remains in a tetravalent state [START_REF] Kramers | Two terrestrial lead isotope paradoxes, forward transport modelling, core formation and the history of the continental crust[END_REF]. Prolonged exposure to atmospheric conditions would therefore raise the Th/U ratio of extraterrestrial samples. For instance, the average Th/U ratios of enstatite chondrites (falls and finds) are 4.23 ± 0.81 for EL chondrites and 3.59 ± 0.52 for EH chondrites (1σ uncertainties), but these ratios are lowered to (Th/U)EL = 3.85 ± 0.52 and (Th/U)EH = 3.24 ± 0.21 when only chondrite falls are included. Therefore, we only took into account concentrations measured in chondrites fall (Table 3). Results

published by Barrat et al. (2014) and Dauphas and Pourmand (2015) were obtained using classical ICPMS technique and isotope dilution by MC-ICPMS, respectively. Three samples (2 EL and 1 EH) are significantly different from the average Th/U values and have been removed from the dataset (see Appendix D). The average values are 4.39±0.52 and 3.24±0.17 for EL and EH groups, respectively. Identical values are obtained when only concentrations obtained by isotope dilution technique are considered.

We tested several scenarios involving different carbonaceous chondrites as the oxidized building blocks of the Earth. Scenarios 2, 6, 7, and 8 (Table 4) which involve different type of carbonaceous chondrites, show no significant differences in the final concentrations of the modeled reservoirs. Therefore, we henceforth use CI chondrite falls (Table 3) as a proxy for the oxidized building blocks as they have the most precisely analyzed trace element concentrations (Barrat et al., 2012). 4).

The same calculations show the Earth's core to contain about 1.28 ppb U and 4.80 ppb Th using EH chondrites (scenario 1) or 1.24 ppb U and 5.22 ppb Th using EL chondrites (scenario 2). Such U and Th concentrations would produce much less than 1 TW of energy due to radioactive decay, compared to the estimated 5-15 TW heat flux escaping the core [START_REF] Lay | Core-mantle boundary heat flow[END_REF]. Thus, even for the highest D U and D Th values, the U and Th contents of the Earth's core represent a negligible contribution to the heat flow from the core to the mantle. These results are marginally affected by increasing the amount of the reduced materials from 80 to 90% (scenarios 1, 2 and 9, Table 4).

Our results are also in agreement, within uncertainties, with the Th/U concentrations reported by several authors (e.g., [START_REF] Allègre | Chemical composition of the Earth and the volatility control on planetary genetics[END_REF][START_REF] Mcdonough | The composition of the Earth[END_REF][START_REF] Kargel | The composition and early evolution of earth[END_REF]Lyubetskaya and Korenaga, 2007, and references therein). However, given the large uncertainties on the published Th/U value of the BSE (up to 35% of the reported value), it is not possible to discriminate between the different scenarios proposed here. Nevertheless, [START_REF] Wipperfurth | Earth's chondritic Th/U: Negligible fractionation during accretion, core formation, and crust-mantle differentiation[END_REF]

recently proposed

Th/U = 3.776 0P.PRS 5P.166 for the BSE, derived from the 208 Pb*/ 206 Pb* ratios (where the asterisks denote the radiogenic component) of various terrestrial rocks compared to the chondritic 208 Pb*/ 206 Pb* value [START_REF] Blichert-Toft | The Solar System primordial lead[END_REF]. Considering the errors on their result, our results using EL chondrites match the BSE Th/U value, whereas models using EH chondrites produce Th/U ratios significantly different (at 95% confidence) from the estimated BSE value (Fig. 7). Although combinations of EL and EH chondrites are possible scenarios for Earth's accretion (scenarios 3, 4, and 5, Table 4), EL chondrites must represent the main portion of the mixture. This observation supports the notion that EL rather than EH chondrites represent the dominant fraction of Earth's building blocks [START_REF] Boyet | Enstatite chondrites EL3 as building blocks for the Earth: The debate over 146 Sm-142 Nd systematics[END_REF].

Th/U ratio of the BSE and the oxygen content of Earth's core

As discussed in section 3.2.3, our parameterizations of DU and DTh show that the oxygen content of the Earth's core had a major impact on the fractionation of Th and U during core segregation. Indeed, the most recent accretion models and high-pressure and high-temperature studies suggest that Earth's core contains at least some amount of oxygen (e.g., [START_REF] Bouhifd | Convergence of Ni and Co metal-silicate partition coefficients in the deep magma-ocean and coupled silicon-oxygen solubility in iron melts at high pressures[END_REF][START_REF] Tsuno | Simultaneous partitioning of silicon and oxygen into the Earth's core during early Earth differentiation: simultaneous partitioning of silicon and oxygen into earth's core[END_REF][START_REF] Siebert | Metal-silicate partitioning of Ni and Co in a deep magma ocean[END_REF][START_REF] Badro | A seismologically consistent compositional model of Earth's core[END_REF][START_REF] Fischer | High pressure metal-silicate partitioning of Ni, Co, V, Cr, Si, and O. Geochim[END_REF][START_REF] Rubie | Accretion and differentiation of the terrestrial planets with implications for the compositions of early-formed Solar System bodies and accretion of water[END_REF]and references therein). Existing models propose a large range of plausible oxygen contents in the Earth's core. Based on geochemical arguments, [START_REF] Allègre | Chemical composition of the Earth and the volatility control on planetary genetics[END_REF] concluded that the core contains 5.0 ± 0.5 wt% O, whereas [START_REF] Javoy | The chemical composition of the Earth: Enstatite chondrite models[END_REF] reported 2.55 ± 1.39 wt% O. By coupling geochemical and geophysical constraints, [START_REF] Badro | A seismologically consistent compositional model of Earth's core[END_REF]2015) concluded that the core contains 2.7 to 5 wt% O. [START_REF] Fischer | High pressure metal-silicate partitioning of Ni, Co, V, Cr, Si, and O. Geochim[END_REF] estimated the oxygen content of the core by following similar accretion scenarios and found that the Earth's core contains 2.1-2.6 wt% O. Finally, to fit the present-day mantle concentrations of some moderately siderophile elements (Cr, V, and Mn), [START_REF] Siebert | Terrestrial Accretion Under Oxidizing Conditions[END_REF] suggested that the core should contain 2 to 5.5 wt% O. A general consensus of the literature can be found on a minimum content of oxygen in the core of 2 wt% O.

By tracing the evolution of the Th/U ratio of the BSE versus the oxygen content of the core (Fig. 8) using our parameterizations of DU and DTh (Eq. 4, Table 2), at 4.0 wt% O in the core, our modeled Th/U ratio of the BSE is beyond the 95% confidence level (2σ) of the BSE value. In this model, we equilibrate a CI-like chondritic composition with a proto-Earth (Barrat et al., 2012), simulating the last stages of core segregation when the most oxygen was incorporated into the metallic phase. For these calculations, we use the BSE Th/U value (3.776 0P.PRS 5P.166 ) given by [START_REF] Wipperfurth | Earth's chondritic Th/U: Negligible fractionation during accretion, core formation, and crust-mantle differentiation[END_REF], which is in good agreement with several other studies (e.g., [START_REF] Allègre | The chemical composition of Earth[END_REF][START_REF] Kargel | The composition and early evolution of earth[END_REF].

By combining the geophysical and geochemical constraints reported above with the added constraint of the Th/U ratio of the BSE, the core must contain 2 to 4 wt% O.

Impact of a late sulfide addition on U and Th partitioning

Another process that could have fractionated the Th/U ratio of the BSE is the late addition of sulfide (FeS) to the segregating core (e.g., [START_REF] Wohlers | Uranium, thorium and REE partitioning into sulfide liquids: Implications for reduced S-rich bodies[END_REF]. The S content of the Earth's core is about 2 wt% (e.g., [START_REF] Allègre | The chemical composition of Earth[END_REF][START_REF] Badro | A seismologically consistent compositional model of Earth's core[END_REF][START_REF] Boujibar | Metal-silicate partitioning of sulfur, new experimental and thermodynamical constraints on planetary accretion[END_REF][START_REF] Dreibus | Cosmochemical constraints on the sulfur content of the Earth's core[END_REF]Palme and O'Neill, 2003, and references therein), and different scenarios exist to explain how the Earth acquired sulfur during core segregation. If S was added throughout Earth's accretion, we find that the U and Th contents of the core and the BSE would have barely been affected. If S was added exclusively in the last 10-20% of Earth's accretion (e.g. [START_REF] Wood | Cooling of the Earth and core formation after giant impact[END_REF], our continuous core formation calculations show that the U and Th concentrations of the BSE would be lowered by 3%, whereas the variation of the Th/U ratio of the BSE would be negligible. Furthermore, our parameterizations (Eq. 4, Table 2) suggest that the late addition of sulfur-rich bodies would have increased the U and Th concentrations in the core by less than 6%. Thus, the late addition of sulfide (10-20%) to the Earth should not have drastically affected the U and Th contents of the BSE and the core, which contribute no more than 1 TW of the energy required to sustain the geodynamo. Finally, the addition of sulfide to the core, at any time during Earth's accretion, would not be observable based on the Th/U ratio of the BSE.

Conclusion

In the present study, we presented new metal -silicate partition coefficients of Th in a fully liquid environment. Combined with literature data from previous studies, we developed new parametrizations of the behavior of U and Th in the context of a magma ocean, that demonstrate the dependence of DU and DTh on the metallic melt composition and the temperature.

The new parameterizations are then used to better constrain the conditions of core formation and the composition of the core, in a multi-stage core growth models, by comparing the resulting (Th/U)BSE to the latest result by [START_REF] Wipperfurth | Earth's chondritic Th/U: Negligible fractionation during accretion, core formation, and crust-mantle differentiation[END_REF]. When considering building block mixing that reproduce the major isotopic and elementary features of the BSE, the calculated concentrations (UBSE = 11.42 ± 0.45 ppb and ThBSE = 43.20 ± 1.73 ppb) best reproduce (Th/U)BSE when using reduced building blocks with an EL chondrites type composition rather than EH. This support the idea that EL chondrites (rather than EH) constitute a major part of Earth's reduced building blocks. In the meantime, the contents U and Th in the Earth's core remain negligible, and cannot be significantly affected by the addition of sulphur. Finally, we show that, in order to reproduce the latest Th/U ratio of the BSE within its uncertainties [START_REF] Wipperfurth | Earth's chondritic Th/U: Negligible fractionation during accretion, core formation, and crust-mantle differentiation[END_REF], the oxygen content of the Earth's core cannot exceed 4wt%. Runs 975 (4 GPa) and 1022 (6 GPa) were performed in graphite capsules, and runs 979 (4 GPa) and 986 (6 GPa) in MgO single-crystal capsules (Table 1). [START_REF] Wohlers | Uranium, thorium and REE partitioning into sulfide liquids: Implications for reduced S-rich bodies[END_REF]. These experimental data are corrected from the effect of T and oxygen content of the metallic phase using the equations ( 6) and ( 7). In the present plot we excluded data involving sulfide liquids, and only consider partitioning data between silicate melt (with S content < 5 wt%) and Fe-rich alloys. The solid lines of slope -1 or -0.5 represent uranium in valence 4+ or 2+, respectively. Th/U in BSE [START_REF] Wipperfurth | Earth's chondritic Th/U: Negligible fractionation during accretion, core formation, and crust-mantle differentiation[END_REF]. 95 % confidence limit on observed value Model Th/U in BSE 95 % confidence limit on model value 2003), [START_REF] Lyubetskaya | Chemical composition of Earth's primitive mantle and its variance[END_REF] and [START_REF] Wipperfurth | Earth's chondritic Th/U: Negligible fractionation during accretion, core formation, and crust-mantle differentiation[END_REF], respectively. Scenarios 9 and 10 are building block mixing proposed by [START_REF] Dauphas | Calcium-48 isotopic anomalies in bulk chondrites and achondrites: Evidence for a uniform isotopic reservoir in the inner protoplanetary disk[END_REF] and [START_REF] Dauphas | The isotopic nature of the Earth's accreting material through time[END_REF], respectively. For these scenarios, since no precision is given considering the EC type used in these scenarios, we used the average of the EH and EL values given in Table 3, CI chondrites were considered as representative of carbonaceous chondrites (CC) chondrites and ordinary chondrites (OC) concentrations is an average of L LL and H chondrites concentration taken from [START_REF] Wasson | Compositions of Chondrites[END_REF].

4.5 ×10 -5 1.6 ×10 -3 3.7 ×10 -3 2.3 ×10 -5 ±2.7 ×10 -5 ±2.8 ×10 -5 ±0.8 ×10 -3 ±0.9×10 -3 ±1.5 ×10 -5 DTh 3.9 ×10 -5 1.3 ×10 -5 1.1 ×10 -3 1.3 ×10 -3 ±1.1 ×10 -5 ±0.3 ×10 -5 ±0.4 ×10 -3 ±0.15×10 -3
DU 2.1 ×10 -5 1.8 ×10 -4 1.5 ×10 -3 1.7 ×10 -3 1.6 ×10 -2 ±1.3 ×10 -5 ±0.7 ×10 -4 ±0.3 ×10 -3 ±0.7 ×10 -3 ±0.5 ×10 -2
DTh 6.05 ×10 -5 8.2 ×10 -5 9.9 ×10 -5 1.9 ×10 -4 4.0 ×10 -3 ±2.00 ×10 -5 ±1.8 ×10 -5 ±1.9 ×10 -5 ±1.0 ×10 -4 ±0.5 ×10 -3 n.d. not analysed.

1 Pressure uncertainties are estimated to be ± 0.5 GPa of the reported values.

2 Temperature uncertainties are within ± 100 K.

3 Oxygene concentration in metal is the result of microprobe analysis in samples subtracted of the concentration analyzed in pure Fe sample Nist665 (XoNist665=0.21±0.01 wt%). 4 For the run #1029, total of about 1.5 wt% of trace elements in the silicate are not reported. 5 Oxygen fugacities relative to Iron-Wüstite buffer assuming that the activities of Fe and FeO were equal to their mole fraction. 6 Oxygen fugacities relative to Iron-Wüstite buffer assuming that γFeO = 1.5. and γFe is calculated using the formula of [START_REF] Ma | Thermodynamic description for concentrated metallic solutions using interaction parameters[END_REF] Prior to each regression, we ran a correlation analysis to avoid using redundant variables. We calculated the Pearson correlation coefficient (r) for each pair of variables described here. The Pearson correlation coefficients of all variables used in the final parameterizations are listed in Table C.1. We also calculated the determination coefficient (r̂), presented in Table C.2, which quantify the proportion of linear variance of one variable that can be explained by the other variable. In our dataset, only a few noticeable correlations can be found between the variables used in the parameterizations. The most important correlation was found between SiO2 in the silicate and Si in the metal (r = 0.82, r̂ = 0.67). We therefore decided to test these variables in separate regressions. For the rest of the variables, the fact that no correlations were found in our database can be explain by the diversity of the experimental conditions (P, T, fO2) and the variety of starting compositions used by the different authors, which renders variables that are usually correlated in natural systems uncorrelated here.

a) log(DU) log(fO2) 1/T log(1-XS) log(1-XO) log(fO2)

1.00 Regressions were performed for every non-redundant variable. For each regression, the overall model significance was determined with a Fisher-Snedecor test (F value and associated pvalue), and a Student's t-test was run for every single variable (t-stat and associated p-value).

We used a significance level of α = 5%, meaning that any value associated with a p-value < 0.05 and a t-stat > 1.96 is considered to be of statistical significance, at the 95% confidence level. Every variable found to be statistically insignificant was set to 0 and the remaining variables were regressed another time. For each new regression, potential correlations of the remaining variables were checked before calculation. In the end, we obtained a regression with six statistically significant and uncorrelated variables (other than cst) to constrain the evolution of DTh and DU.

We first ruled out of the parameterization terms related to SiO2 in the silicate and Ni in the metal, which were found to be of no statistical significance. We then removed the variables related to C and Si in the metal from the regression process: although they are found to be significant variable while having a weak effect, we realized parameterization with and without these parameters and the overall model robustness increased for both U and Th parameterizations when these parameters are not taking into account. The coefficients of other variables remained unchanged. Furthermore, considering the absence of any major effect on samples within graphite capsules and our qualitative analyses of metallic C in samples (obtained from the deviation of microprobe totals from 100%), we removed C in the metal from the final parameterizations.

The effect of pressure on DTh is poorly constrained by the available dataset (cTh = 276 ± 289, with a p-value of 0.54 and t-stat = 1.02 in the first parameterization), and, while the value of cu is shown to be of statistical significance (cU = 15.6 ± 45.2, with a p-value of 0.04 and t-stat = -2.52) it remains fairly negligible and overlaps with 0, within the error bar. By plotting the DTh/DU ratio versus pressure (Fig. 4), we found that this ratio is constant within uncertainties. We therefore decided to fix the parameter cTh and cU equal to 0, reducing the number of variables in the two parameterizations (other than cst) to four, without any effect on the other variables.

The final parameterizations of log DU and log DTh present fairly robust overall model significance. 85% and 82% (R 2 ) of the respective variations of log DU and log DTh observed within the data are accounted for by the variables included in the final parameterization. The Fisher-Snedecor test values of each model (96.76 for U and 35.65 for Th) associated to its pvalue (significance F) confirm the overall pertinence of these parameterizations. The significance of each variable used in these models is also confirmed by their calculated pvalues.

Final parameterization of logDU

Regression Statistics

Multiple R 0.92 Dauphas and Pourmand (2015) (plain symbol) and Barrat et al. (2014) (open symbol). U and Th concentrations are obtained using isotope dilution measurements in Dauphas and Pourmand (2015) and classical ICPMS technique measurements in Barrat et al., (2012), with a precision on U/Th ratio of 2.4% and 7.15% (2stdev), respectively. To determine these errors, we have considered the replicate measurements published for chondrites using the same analytical techniques. Seven measurements of the carbonaceous chondrite Allende were published in [START_REF] Pourmand | Distribution coefficients of 60 elements on TODGA resin: Application to Ca, Lu, Hf, U and Th isotope geochemistry[END_REF]. They obtain a precision of 2.4% (2stdev) on Th/U (3.77±0.09). Barrat et al. (2012) published 6 measurements on Orgueil (carbonaceous chondrite). They obtained precision of 7.15% (2stdev) on Th/U (3.57±0.25). These error bars are represented in the figure. They are always smaller than the symbol size for data published by Dauphas and Pourmand (2015). Among the data represented in this plot, 3 samples are significantly outside the error bar (dark blue and orange rectangle, for EH and EL chondrites, respectively), which have been removed from the dataset. The resulting Th/U average values are significantly different: (Th/U)EH=3.24±0.17 and (Th/U)EL=4.39±0.52 (light blue and orange fields, respectively).

R

  log $𝐷 U "#$/&'( & = (-0.72 ± 0.07) × log(𝑓𝑂 ) ) + (-8346 ± 1804) × * + + (-15.33 ± 1.13) × log(1 -𝑋 S met ) + (-51.55 ± 7.86) × log(1 -𝑋 O met ) + (-1.52 ± 0.52) (6) log $𝐷 Th "#$/&'( & = (-0.96 ± 0.10) × log(𝑓𝑂 ) ) + (-9967 ± 2618) × * + + (-13.76 ± 1.40) × log(1 -𝑋 S met ) + (-137.82 ± 26.63) × log(1 -𝑋 O met) + (0.77 ± 1.12) (7)

( 2015 )

 2015 and references therein) propose Earth's core concentrations up to 5.85 wt% O and 2.04 wt% Si. To determine the viability of any accretion scenarios and the resulting U and Th concentrations in the different reservoirs, we also studied the evolution of Ni, Co, Nb, and Ta, elements that have well-constrained metal-silicate partitioning behavior in the literature. The parameterizations used to constrain the behaviors of these elements are those of Fischer et al. (2015) for Ni and Co and Cartier et al. (2014) for Nb and Ta. As a first test of validity, any accretion scenarios studied herein must reproduce the Ni/Co and Nb/Ta elemental ratios in the present-day mantle.

Figure 1 .

 1 Figure 1. Backscattered electron image of sample #979 in its graphite capsule after

Figure 2 .

 2 Figure 2. Effect of the nature of the capsule (C or MgO) on U and Th partition

Figure 3 .

 3 Figure 3. Comparison between (a) U and (b) Th metal-silicate partition coefficients

Figure 4 .

 4 Figure 4. Evolution of DTh/DU as a function of pressure in S-free metal-silicate

Figure 5 .

 5 Figure 5. Molten metal -silicate liquid partition coefficients as a function of

Figure 6 .

 6 Figure 6. Evolution of (a) U and (b) Th partition coefficients with the concentration

Figure 7 .

 7 Figure 7. Compilation of Th/U ratios estimated for the BSE compared with the

Figure 8 .

 8 Figure 8. Evolution of the Th/U ratio of the BSE with the O content of the Earth's

  Figure 1.

  Figure 5.

  Figure 8.

Figure S1 :

 S1 Figure S1: Th/U measured in fall EH and EL chondrites. Data fromDauphas and Pourmand (2015) (plain symbol) andBarrat et al. (2014) (open symbol). U and Th concentrations are obtained using isotope dilution measurements inDauphas and Pourmand (2015) and classical ICPMS technique measurements inBarrat et al., (2012), with a precision on U/Th ratio of 2.4% and 7.15% (2stdev), respectively. To determine these errors, we have considered the replicate measurements published for chondrites using the same analytical techniques. Seven measurements of the carbonaceous chondrite Allende were published in[START_REF] Pourmand | Distribution coefficients of 60 elements on TODGA resin: Application to Ca, Lu, Hf, U and Th isotope geochemistry[END_REF]. They obtain a precision of 2.4% (2stdev) on Th/U (3.77±0.09).Barrat et al. (2012) published 6 measurements on Orgueil (carbonaceous chondrite). They obtained precision of 7.15% (2stdev) on Th/U (3.57±0.25). These error bars are represented in the figure. They are always smaller than the symbol size for data published byDauphas and Pourmand (2015). Among the data represented in this plot, 3 samples are significantly outside the error bar (dark blue and orange rectangle, for EH and EL chondrites, respectively), which have been removed from the dataset. The resulting Th/U average values are significantly different: (Th/U)EH=3.24±0.17 and (Th/U)EL=4.39±0.52 (light blue and orange fields, respectively).

  . Variations of the O and Si contents in the metal are related to the proportion of reduced and oxidized building blocks, which influence the fO2 path. At the end of accretion model in which Earth is built from a major proportion

(80%) of reduced materials, the Earth's core contains 0.2 wt% C, 2 wt% S, 6.75 ± 0.75 wt% Si, and 2.8 ± 0.7 wt% O. On the other hand, models in which Earth is built from more oxidized materials (following an accretion with constant fO2 at IW-2, Fischer et al.

Table 4

 4

	summarize the results of several scenarios of Earth accretion involving
	various building block mixing. All these scenarios reproduce the Nb/Ta and Ni/Co
	elemental ratios of the BSE, within uncertainties, as shown in Table 4. Our calculations
	produce a BSE with 11.77 ± 0.45 ppb U, 40.42 ± 1.62 ppb Th, and Th/U = 3.43 ± 0.13
	when using only EH chondrites as the reduced material (scenario 1, Table 4), and
	11.42±0.45 ppb U, 43.20±1.73 ppb Th, and Th/U = 3.78 ± 0.15 when using only EL
	chondrites as the reduced material (scenarios 2, Table

Table 1 .

 1 Experimental conditions (P, T, f O2 ) and metal -silicate partitioning coefficients of U and Th for the different samples. The partition coefficient values of U are taken from[START_REF] Bouhifd | Metal-silicate partitioning of Pb and U: Effects of metal composition and oxygen fugacity[END_REF]

	Sample	Capsule	P (Gpa)	T (K)	log fO2 (ΔIW)	DU *	DTh
	#969	Graphite	4	2128	-1.43	5.65×10 -5	3.86×10 -5
	#975	Graphite	4	2073	-1.56	4.51×10 -5	1.26×10 -5
	#1026	Graphite	5	2273	-0.82	1.64×10 -3	1.14×10 -3
	#1027	Graphite	5	2273	-4.42	3.73×10 -3	1.34×10 -3
	#1022	Graphite	6	2173	-1.74	2.30×10 -5
	#979	MgO	4	2075	-1.70	2.07×10 -5	2.02×10 -5
	#986	MgO	6	2173	-1.64	1.76×10 -4	8.22×10 -5
	#1021	MgO	6	2173	-1.44	1.54×10 -3	9.92×10 -5
	#1028	MgO	8	2373	-1.45	1.71×10 -3	1.89×10 -4
	#1029	MgO	8	2373	-4.67	1.64×10 -2	4.00×10 -3

*

Table 2 .

 2 Results of the multi-variable linear regression based on Equation 4 using our experimental data together with those listed in Appendix B.

	Elements N o .	R 2	a	b	d	f	cst
	U	74 0.85	-0.72 ± 0.07	-8346 ± 1103	-15.33 ± 1.13	-51.55 ± 7.86	-1.53 ± 0.52
	Th	44 0.82	-0.96 ± 0.10	-9967 ± 2618	-13.76 ± 1.40	-137.82 ± 26.63	0.77 ± 1.12
	N o .: number of experiments; R 2 : correlation coefficient.

Table 3 .

 3 U and Th abundances (in ppb) for CI and EC chondrites used in this work. Only samples which fall where observed have been selected here, taken from the most recent published trace element measurements on chondrites. The n value represents the number of samples the U and Th concentration are based on.

	n	U	1!	Th	1!	Th/U	1!	references
	CI chondrites 7	7.72	0.23	28.30	1.42	3.66	0.11	Barrat et al, 2012
	EL chondrites 8	6.60	1.00	28.70	3.11	4.39	0.52	Barrat et al, 2014;
								Dauphas and Pourmand,
	EH chondrites 7	8.33	0.69	26.95	1.51	3.24	0.17	2015

Table 4 .

 4 Different scenarios of Earth's accretion involving reducing chondrites (EH or EL type) and more oxidized ones (CI, CO, CV or CM) in various proportions. CI, EH and EL concentrations are presented in Table3. Other chondrite compositions are taken from[START_REF] Wasson | Compositions of Chondrites[END_REF], but scenarios 5,6 and 7 show that recent CI chondrites U and Th concentrations can be used as a proxy for carbonaceous chondrites. The uncertainties presented here are all 2!. Nb/Ta and Ni/Co ratios of the resulting BSE as well as the Si and O content of the associated core are calculated for each scenario. Each scenario presented here reproduce, within 2 ! uncertainties, the Nb/Ta and Ni/Co of the BSE, presented for comparison in the last row of this table. Nb/Ta, Ni/Co and Th/U of the BSE, are taken from Münker et al. (

Table A1 .

 A1 Chemical analysis of the run products using graphite capsules.

	P (GPa) 1	4		4				5		5	6
	T (K) 2		2128		2073			2273		2273	2173
	Run			969		975				1026		1027	1022
	Silicate
	EPMA (wt%)
	SiO2		48.93±0.57	48.20±0.43 43.00±0.89 54.05±0.39 50.37±0.42
	Al2O3		4.33±0.31	3.72±0.22	3.59±0.42	3.83±0.10	2.60±0.12
	FeO		8.44±0.13	7.54±0.17 13.83±0.63 0.32±0.04	7.50±0.45
	MgO		33.29±0.77	36.39±0.55 33.93±0.82 35.76±0.49 34.86±0.53	BSE (ppb)	Core (ppb)
	Scenario Reduced material % Oxidized material % CaO 2.87±0.11 2.77±0.13 2.60±0.21 3.05±0.05	2.88±0.12
	K2O		0.47±0.02	0.44±0.03	2.25±0.22	Nb/Ta 2.47±0.03	Ni/Co 0.30±0.03	U	Th	Th/U	U (ppm)	Th (ppm)
	NiO	1	EH 0.13±0.01	80 0.19±0.02 CI	20 0.14±0.02	13.75 ±0.55 19.1 ±0.80 11.77 ±0.45 40.42 ±1.62 3.43 ±0.13 1.28 ±0.05 4.80 ±0.20 n.d. 0.13±0.02
	PbO UO2	2	EL 0.16±0.01 0.54±0.03	80 0.14±0.01 CI 0.52±0.03	20 0.21±0.02 n.d.	n.d. 14.68 ±0.59 18.7 ±0.76 11.42 ±0.48 43.20 ±1.73 3.78 ±0.15 1.24 ±0.07 5.22 ±0.21 0.09±0.01 0.17±0.01 0.48±0.03
	Total	3	EH-EL 99.16	40-40	99.91	CI		99.56	20	13.76 ±0.56 19.1 ±0.78 11.61 ±0.46 41.88 ±1.67 3.60 ±0.14 1.24 ±0.05 4.82 ±0.20 100.13 99.21
	4 LA-ICP-MS (ppm)	EH-EL	20-60			CI			20	14.03 ±0.57 19.1 ±0.77 11.51 ±0.46 42.52 ±1.70 3.69 ±0.15 1.25 ±0.06 5.05 ±0.20
	U			3896±100	3769±156		905±108	1126±22	4353±220
	Th		5	EH-EL 3881±72	60-20 3957±163 CI		20 879±71	13.75 ±0.56 19.1 ±0.79 11.68 ±0.47 41.16 ±1.64 3.52 ±0.14 1.28 ±0.05 4.83 ±0.20 1100±12 4120±140
	Metal EPMA (wt%)	6	EL	80			CM			20	13.08 ±0.60 19.45 ±0.81 11.49 ±0.46 43.20 ±2.20 3.76 ±0.25 1.24 ±0.12 5.23 ±0.42
	Fe		7	EL 58.73±0.59	80 61.03±1.10 54.01±1.20 83.30±0.32 74.61±0.68 CO 20 14.46 ±0.55 19.23 ±0.84 11.46 ±0.63 42.40 ±2.06 3.70 ±0.23 1.24 ±0.11 5.23 ±0.41
	Ni S		8	EL 36.55±0.76 --	80 35.42±0.99 19.60±1.10 0.07±0.04 20.25±0.69 CV 20 13.88 ±0.54 19.27 ±0.82 11.40 ±0.72 42.18 ±2.51 3.70 ±0.30 1.25 ±0.16 5.24 ±0.56 --23.28±0.82 ----
	Si		9	EC --	91	--	OC-CC	--	7-2 11.28 ±0.66 18.75 ±0.89 11.39 ±0.85 42.82 ±2.02 3.76 ±0.32 1.25 ±0.13 5.29 ±0.51 15.76±0.65 --
	Pb 3 O	10	EC 0.73±0.10 0.03±0.01	71 0.33±0.14 OC-CO 0.11±0.01 5-24 12.23 ±0.61 19.73 ±0.88 11.80 ±0.83 43.66 ±2.06 3.70 ±0.32 1.20 ±0.13 5.15 ±0.50 n.d. 0.10±0.02 0.05±0.04 0.75±0.13 0.16±0.02 0.03±0.02
	Total	BSE	96.32		97.06			97.86		14.00 ±0.60 18.90 ±0.32 99.51 95.20	3.78 ±0.24
	LA-ICP-MS (ppm)
	U			0.22±0.10	0.17±0.10	1.48±0.60	4.2±0.96	0.10±0.06
	Th			0.15±0.04	0.05±0.01	1.00±0.30	1.47±0.15	n.d.
	3 log fO2 (ΔIW)	-1.81		-1.95			-1.25		-4.95	-2.07
	4 log fO2 (ΔIW)	-1.43		-1.56			-0.82		-4.42	-1.74
	DU			5.6 ×10 -5

Table A2 .

 A2 Chemical analyses of run products using MgO-single crystal capsules (wt%).

	P (GPa) 1	4	6	6	8	8
	T (K) 2	2075	2173	2173	2373	2373
	Run	979	986	1021	1028	4 1029
	Silicate
	EPMA (wt%)
	SiO2	48.20±0.85	47.00±0.76	49.75±1.20	47.84±0.82	47.41±0.71
	Al2O3	4.14±0.46	5.80±0.42	2.60±0.25	4.06±0.39	4.31±0.24
	FeO	8.86±0.55	8.00±0.51	9.50±0.62	9.12±0.53	0.31±0.02
	MgO	35.41±0.77	34.45±0.85	33.86±0.85	34.41±0.78	37.73±0.95
	CaO	2.57±0.38	2.55±0.32	1.90±0.21	2.45±0.24	2.55±0.15
	K2O	0.35±0.02	0.37±0.03	0.18±0.04	1.25±0.46	3.15±0.17
	NiO	0.14±0.01	0.14±0.01	0.13±0.01	0.10±0.01	n.d.
	PbO	0.05±0.02	n.d.	n.d.	n.d.	0.10±0.01
	UO2	0.36±0.05	0.35±0.05	0.11±0.01	0.10±0.02	0.28±0.02
	Total	100.08	98.66	98.03	99.33	95.84
	LA-ICPMS (ppm)
	U	3858±478	3300±424	846±76	850±130	859±51
	Th	3969±491	3162±67	1008±95	899±190	851±41
	Metal
	EPMA (wt%)
	Fe	66.91±0.54	56.11±0.90	61.39±1.53	60.34±0.9	84.64±0.59
	Ni	30.02±0.22	40.52±0.58	17.97±0.81	18.24±0.57	0.26±0.07
	S	--	--	20.03±0.82	19.69±0.33	--
	Si	--	--	--	--	13.81±0.31
	Pb	0.38±0.06	0.23±0.03	0.12±0.01	0.19±0.02	0.27±0.10
	3 O	0.05±0.02	0.06±0.02	0.33±0.04	0.42±0.04	0.05±0.03
	Total	97.57	97.12	99.94	98.9	99.24
	LA-ICPMS (ppm)
	U	0.08±0.04	0.58±0.15	1.30±0.16	1.45±0.36	14.1±3.5
	Th	0.08±0.04	0.26±0.05	0.10±0.01	0.17±0.05	3.4±0.3
	5 log fO2 (ΔIW)	-2.05	-1.99	-1.76	-1.80	-5.02
	6 log fO2 (ΔIW)	-1.70	-1.64	-1.44	-1.45	-4.67

log(DTh) log(fO2) 1/T log(1-XS) log(1-XO)

	a)
	log(DU)	log(fO2)	1/T	log(1-XS)	log(1-XO)
	log(fO2) 1.00
	1/T	0.08	1.00
	log(1-XS) 0.14	0.08	1.00
	log(1-XO) 0.09	0.04	0.08	1.00
	b)
	log(DTh) log(fO2)	1/T	log(1-XS)	log(1-XO)
	log(fO2) 1.00
	1/T	0.08	1.00
	log(1-XS) 0.09	0.13	1.00
	log(1-XO) 0.10	0.00	0.09	1.00
	1/T	-0.27	1.00
	log(1-XS) 0.38	-0.29	1.00
	log(1-XO) -0.31	0.19	-0.29	1.00
	b)
	log(fO2)	1.00
	1/T	-0.29	1.00
	log(1-XS) 0.29	-0.35	1.00
	log(1-XO) -0.31	-0.04	0.30	1.00

Table C.1. Pearson correlation coefficients (r) for each pair of variables used in the final parameterization of a) DU and b) DTh.

Table C .

 C 2. Determination coefficients (r̂ ) for each pair of variables used in the final regressions for a) U and b) Th. r̂ is the square product of the Pearson correlation coefficient, r, which quantifies the proportion of linear variance of one variable that is explained by the other variable. The low r̂ values presented here suggest the absence of multi-collinearity in our model.

  Table C.3. Details of the regressions of equation (4) and associated significance tests.

	Final parameterization of logDTh
	Regression Statistics
	Multiple R	0.88
	R 2	0.82
	2 Adjusted R 2 0.76 0.85
	Adjusted R 2 0.83 Standard 0.71 error Standard error 0.67 Observations 44
	Observations 74
	Variance Analysis
	Variance Analysis degrees of freedom Square Sum degrees of freedom Square Sum Regression 4 72.87	Mean Sum Mean Sum 18.21	F-value F-value 35.65	Significance F Significance F 1.53 .10 -12
	Regression Residual	4 39	182.46 19.92	45.61 0.51	96.76	9.24 .10 -28
	Residual Total	70 43	32.99 92.79	0.47
	Total	74	215.45
	cst	Coefficients Coefficients 0.77	Standard errors Standard errors 1.10	t-Stat t-Stat 0.24	p-value p-value 0.80
	cst a	-1.52 -0.96	0.05 0.10	-2.94 -3.20	0.004 0.002
	a b	-0.72 -9967	0.10 2618	-9.69 -5.01	1.44 .10 -14 7.22 .10 -6
	b d	-8346 -13.7	1103 1.40	-8.29 -9.79	5.26 .10 -12 4.60 .10 -12
	d f	-15.3 -137.8	1.10 26.6	-13.50 3.61 .10 -21 -6.15 2.12 .10 -5
	f	-51.5	7.80	-6.55	7.85 .10 -9

Acknowledgments

We thank Denis Andrault, Nathalie Bolfan-Casanova, and Tahar Hammouda for fruitful discussions. We also thank xx for editorial handling. Official reviews by the associate editor as well as well as anonymous reviewers were helpful to improve the clarity of the paper.This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant Agreement No 682778 -ISOREE). The multi-anvil apparatus of Laboratoire Magmas et Volcans is financially supported by the CNRS (Instrument national de l'INSU). This work was also supported by the French Government Laboratory of Excellence Initiative N° ANR-10_LABX-006, the Région Auvergne, and the European Regional Development Fund. This is Laboratory of Excellence ClerVolc contribution N° 398.

Appendix B. Compilation of the experimental data used to derive the coefficients of Equation (4) for U and Th. The uncertainties on the reported data can be found in the original papers. To determine the best parameterization of the behavior of Uranium and Thorium considering the database available, we performed a stepwise regression to determine the most pertinent variables to include in our parameterization and their best-fit parameters.

We first listed all the variables with a potential influence on the U and Th partition coefficients.

In addition to the variables present in the final parameterization (Table 2), we also tested the metallic C, Ni and Si concentration and the SiO2 concentration in the silicate. These parameters are available for all experiments in Appendix B.