Earth and Planetary Science Letters

Transcription

1 Earth and Planetary Science Letters 365 (2013) Contents lists available at SciVerse ScienceDirect Earth and Planetary Science Letters journal homepage: Assessing uncertainty in geochemical models for core formation in Earth Michael J. Walter a,n, Elizabeth Cottrell b a Department of Earth Sciences, University of Bristol, Queen s Road, Bristol BS8 1RJ, United Kingdom b Department of Mineral Sciences, National Museum of Natural History, Smithsonian Institution, 10th & Constitution Ave., Washington DC, 20560, USA article info Article history: Received 29 September 2012 Accepted 16 January 2013 Editor: L. Stixrude Keywords: siderophile elements regression models core formation model uncertainty abstract Unraveling the conditions at which Earth s metallic iron core formed yields important information about Earth s early accretion and differentiation history. Multi-variable statistical modeling of siderophile element partitioning between core-forming metallic liquids and silicate melts form the basis for physical models of core formation. While it seems clear that core segregation in a deep peridotitic magma ocean is generally consistent with many mantle siderophile element abundances, there is considerable disparity among extant physical models in terms of the key parameters of pressure, temperature and oxygen fugacity at which the core formed. Moreover, there is ongoing debate over whether simple single-stage equilibrium or more complex multi-stage accretion models are required by the partitioning data. Here we consider how variations in the statistical regression of partitioning data affect the outcomes of physical models for core formation. Taking extant experimental data sets for four well-studied siderophile elements (Ni, Co, W and V) as examples, we find that the regression model exerts a fundamental control on physical model outcomes. Further, the experimental data are currently too imprecise to discriminate among various single-stage and continuous core formation scenarios. Progress in the development of physical models requires better isolation of the independent variables that affect partition coefficients and verification of activity models at high pressure and temperature in order to reduce the global uncertainty in multi-variable statistical models. & 2013 Elsevier B.V. All rights reserved. 1. Introduction Models of core formation in Earth have historically been designed around the concept of segregation of a core-forming metal from a silicate mantle (Ringwood, 1966). The modern view of planetary accretion by energetic impacts among proto-planets implies that segregation and equilibration occurred when both phases were molten, probably within a deep magma ocean (Benz and Cameron, 1990; Canup, 2004; Chambers, 2004; Melosh, 1990; Tonks and Melosh, 1993). In principle, if the partitioning behavior of siderophile elements between metal and silicate are known, the conditions of equilibrium core formation can be deduced and important constraints placed on early Earth processes. The objective is to find a unique or progressive set of conditions of pressure, temperature, oxygen fugacity, and silicate melt and metal composition that can account for the observed mantle abundance of a suite of siderophile elements (see reviews by Righter (2003), Walter et al. (2000), Wood et al. (2006)). In practice, this requires high pressure and temperature experiments that equilibrate liquid metal with molten silicate, the accurate and precise analysis of siderophile elements in both phases, and a statistical n Corresponding author. Tel.: þ address: m.j.walter@bristol.ac.uk (M.J. Walter). model that parameterizes the partitioning behavior and provides a measure of model uncertainty. Currently there are many competing physical models for the origin of the siderophile element signature of Earth s mantle (Chabot et al., 2005; Corgne et al., 2008, 2009; Cottrell et al., 2009; Hillgren et al., 1994; Kegler et al., 2008; Li and Agee, 1996, 2001; Mann et al., 2009; Righter, 2011a; Righter and Drake, 1999; Righter et al., 1997; Rubie et al., 2011; Siebert et al., 2012, 2011; Thibault and Walter, 1995; Wade and Wood, 2005; Walter and Tronnes, 2004; Wood et al., 2008). Here we will limit our discussion to the two dominant paradigms, both of which require metal silicate equilibration in a deep magma ocean. In the simplest single-stage core formation scenario, the metal and silicate are assumed to have equilibrated at some average high pressure and temperature, possibly at the base of a deep magma ocean, (e.g. Corgne et al., 2009; Cottrell et al., 2009; Li and Agee, 1996, 2001; Mann et al., 2009; Righter, 2011a; Righter and Drake, 1999; Righter et al., 1997; Walter and Tronnes, 2004). In multistage or continuous accretion models, the magma ocean conditions evolve as the planet accretes with progressively changing conditions of variables such as pressure, temperature and oxygen fugacity (e.g. Corgne et al., 2008; O Neill, 1991; Siebert et al., 2011; Wade and Wood, 2005; Wood et al., 2008, 2006). Even though the number of elements included in models and the availability of experiments has grown considerably, there is X/$ - see front matter & 2013 Elsevier B.V. All rights reserved.

2 166 M.J. Walter, E. Cottrell / Earth and Planetary Science Letters 365 (2013) currently little consensus among studies regarding the conditions or mode of core formation required by the experimental data, except that high pressure and temperature equilibration is strongly implicated. A good example is the current debate over whether or not single-stage core formation models are viable, or whether continuous accretion with progressively changing pressure, temperature and oxygen fugacity are a requirement of the partitioning data (Corgne et al., 2008, 2009; Cottrell et al., 2009; Righter, 2011a; Siebert et al., 2011; Wade and Wood, 2005; Wood et al., 2008, 2006). In several studies it is argued that single-stage models are not viable because the mantle abundances of specific elements, and most notably vanadium (V), cannot be reconciled with available partitioning data for equilibration along the liquidus or solidus in a deep magma ocean at an oxygen fugacity corresponding to a mantle with 6.3 wt% Fe (Corgne et al., 2008; Siebert et al., 2011; Wade and Wood, 2005; Wood et al., 2008, 2006). In contrast, Righter (2011a) has argued using effectively the same data sets and elements that continuous accretion with progressive oxidation is not a unique requirement of the data, and that single-stage models are viable and perhaps even the preferred solution. Some authors have estimated the uncertainty in physical models for core formation (Chabot et al., 2005; Corgne et al., 2008, 2009; Cottrell et al., 2009; Li and Agee, 2001; Righter, 2011a; Righter and Drake, 1999; Righter et al., 1997; Rudge et al., 2010; Siebert et al., 2011), but the debate lacks a systematic study of how assumptions and uncertainties in the statistical models that describe element partitioning affect model outcomes. This makes it difficult to discern where the differences among physical model outcomes originate. Here we address two key questions relevant to evaluating equilibrium core formation models: (1) How do choices made during statistical modeling of the data, such as the form of the independent variables and the regression method, affect physical model outcomes? (2) Are the available experimental data precise enough to develop a unique physical model or discriminate between competing physical models? We investigate these questions by close inspection of the statistical regression models of experimental data for four siderophile elements: Ni, Co, W, and V. We chose these elements because their mantle abundances are generally well-constrained, data sets of 20 or more experiments in the pressure range of 5 25 GPa are available for each element from a single laboratory, and as a group they provide good leverage on the variables of pressure, temperature, oxygen fugacity, and silicate melt composition. We only draw on a subset of the published data in order to avoid systematic biases that arise when combining data sets from multiple labs. We assess the statistical regression models that describe the experimental partitioning data to determine how tightly the regression models can constrain the variables of core formation, and to elucidate how those constraints depend on the chosen regression model. 2. Methodology Partitioning experiments equilibrate metal and silicate over a range of pressures (P), temperatures (T), oxygen fugacities (fo 2 ), and metal and silicate compositions. In order to capture the variation of individual parameters, multivariable regression has been used extensively, both on individual data sets and in data compilations. To assess the impact of regression model and uncertainty on core formation modeling, we cull raw data from the literature, process it in a consistent manner, and regress the data using a variety of regression models. We then apply the regressions in models of core formation and take account of the regression uncertainties using a Monte Carlo randomsampling technique Data Selection of experimental data sets We focus on four elements from three high-pressure data sets. Each data set was generated within a single laboratory ensuring a standard experimental and analytical approach, and thereby avoiding intra-element uncertainty derived from inter-laboratory bias. We use the Ni and Co data of Kegler et al. (2008), thevdataof Wade and Wood (2001, 2005) and Wood et al. (2008), and the W data of Cottrell et al. (2009, 2010) (see Table A1 in Appendix A). Each data set has: (1) at least 22 experiments at Z5 GPa (data at pressures less than 5 GPa are excluded to avoid non-linear effects of pressure indicated in the data sets of Kegler et al. (2008) and Cottrell et al. (2009) in the lower pressure regime), (2) a relatively large range in the variables P, T, composition, and oxygen fugacity, and (3) metal compositions to which we can apply the activity model of Wade and Wood (2005). The last point precludes the large and influential Ni and Co data set of Li and Agee (1996, 2001) because the metals in these experiments contain 30 wt% S Activity models The iron-rich metal solvents in partitioning experiments may contain a variety of solutes. Interactions among the solutes and solvent can lead to deviations from ideal solution behavior, such that the abundance of an element is not proportional to its activity. This is a potentially large source of inter-experiment bias if a variety of metal compositions are used or if the concentration of a solute changes as a function of P and T. Corrections to the measured abundances to values in pure iron metal at infinite dilution are possible if interaction parameters are known, in which case partition coefficients among different experiments can then be directly compared. Here, we employ the algorithm of Wade and Wood (2005) ( uk/expet/metalact/) to correct for solute solute interactions in the iron alloy (see also Corgne et al. (2008)). This algorithm is based on interaction parameters and temperature extrapolations presented in the Steelmaking Data Sourcebook (Steelmaking, 1988) and the thermodynamic approach of Ma (2001). The inputs are the experimentally measured element mol fractions and run temperature, and the outputs are element activities at a given temperature. Carbon is a solute in many of the experimental metals and can have large affects on partitioning (Cottrell et al., 2009). Due to analytical difficulties in measuring C accurately, the algorithm estimates the high temperature carbon content of all carbonbearing metals on the basis of the free energy of dissolution at carbon saturation which is constrained by the experimentally determined 1 atm meltþgraphite liquidus (Shunk, 1969). This is appropriate as all C-bearing experiments were made in graphite capsules and so metals should be C saturated The dependent variable The dependent variable describes how each element (M) is distributed between the metal and silicate phases. For each experiment in this study, we calculated partition and distribution coefficients for each activity-corrected experiment as follows: D M ¼ ametal M X silicate M K D ¼ ðxsilicate FeO Þ n=2 ða metalþ M ða metal MFe Þðametal MO n=2 Þ ð1þ ð2þ

3 M.J. Walter, E. Cottrell / Earth and Planetary Science Letters 365 (2013) where X is mol fraction, a is activity, and n is the valence of the cation. Note that partition coefficients, D, of an element, M, are dependent on oxygen fugacity according to the relation: logd M ¼ n=2logf O 2 þconstant ð3þ Whereas the distribution coefficient, K D, is independent of oxygen fugacity as the partition coefficient is normalized to the FeO/Fe ratio. The K D approach, however, assumes that the valence state, n, of the element is known and constant in interpolation and extrapolation Oxygen fugacity We define the oxygen fugacity relative to the iron wüstite equilibrium (DIW) for experiments containing liquid Fe-alloy as: DIW ¼ 2log asilicate FeO ð4þ a metal Fe where a is the activity of FeO in the silicate and Fe in the metal (Hillgren et al., 1994). The activity of Fe in the metal is calculated as described above. Activity coefficients for FeO are not expected to be unity and are likely greater than one in the experimental melts used here (e.g. 1.2 for basaltic melts) (see O Neill and Eggins (2002)). However, due to a lack of systematic data on the melt compositional dependence of the activity coefficient, we assume ideal mixing and approximate the activity of FeO in silicate melt by the measured mole fraction, X FeO Silicate composition The partitioning behavior of some elements, most notably W, exhibit a strong dependence on silicate melt composition. A popular method for quantifying such dependence is to parameterize the degree of melt polymerization using the ratio nbo/t (number of non-bridging oxygens to tetrahedrally coordinated silicon) as defined by Mysen (1988, 1991). While this simple, single-valued normalization may be incapable of totally capturing complex melt compositional and structural effects, especially at high pressures where higher coordination states occur in melts (O Neill et al., 2008; O Neill and Eggins, 2002), the form of this parameter will not significantly impact our conclusions Elemental abundances in Earth s mantle Modeling core formation in Earth requires knowledge of the primitive mantle and core abundances of the elements so that the core/mantle concentration ratios can be calculated, which here we call the bulk Earth partition coefficient, D Earth. Ni, Co and V are relatively refractory and compatible during partial melting, and their primitive mantle abundances are more precise than most siderophile elements (Palme and O Neill, 2003; Walter et al., 2000). W is also refractory, but its lithophile behavior during partial melting renders its primitive mantle abundance more uncertain (Arevalo and McDonough, 2008; Newsom et al., 1996). To establish the ranges of D Earth values we have used the recent estimates for the abundances of Ni, Co and V of Palme and O Neill (2003) and McDonough (2003) in the primitive mantle and core, respectively. We use the estimated one standard deviation uncertainties on mantle abundances provided by Palme and O Neill (2003) and assume a nominal 10% uncertainty on the core abundances provided by McDonough (2003). For W, we use the recent estimates of Arevalo and McDonough (2008) for primitive mantle and core abundances, and assign one standard deviation uncertainties as provided by those authors. The resulting range of estimates for log D Earth values are provided in Table 1, and they are generally consistent with the depletion factors calculated by Walter et al. (2000). Table 1 log D Earth values and associated uncertainties Data regression log D molar log D weight Ni Co W V The primary aim of this study is to quantify how the parameterization of experimental partitioning data impacts the outcomes of physical models for core formation. We restrict ourselves to investigating the impact of (1) statistical regression model, and (2) fixing coefficients in multivariable regressions Regression model In keeping with a standard approach in core formation modeling, partitioning data for an element, M, are regressed to a general equation of the form: logd M ¼ aþbup=t þwu1=t þdudiw þeunbo=t ð5þ When parameterizing in terms of the distribution coefficient, K D, the valence of M is assumed and regressions are made using Eq. (5) but the DIW coefficient is fixed such that: logd M ¼ aþbup=t þwu1=t n=2logðx FeO =a Fe ÞþeUnbo=t ð6þ Thus, log D is the dependent variable in all regression models and we refer to Eq. (6) as a fixed valence model, which is essentially identical to the K D approach, and denote this in tables and figures with FV. We perform least-squares multiple-linear regressions where all uncertainty is relegated to the dependent variable, and no error is assigned to independent variables. Further assumptions embedded in this approach are that independent variables are linearly independent and have constant variance (homoscedastic). While neither of these is likely to be true in detail for the data sets considered, we recognize but do not explore these assumptions. We use two kinds of multiple regression model, both of which are employed in the literature. In the first, we retain all coefficients and their associated uncertainties, even if some coefficients are statistically insignificant; we refer to this as a normal regression. In the second model, we perform a step-wise regression whereby insignificant coefficients are progressively eliminated, beginning with the least significant, until all remaining coefficients are significant. Here we follow convention and define significant terms as those having a P-value less than In all regressions the data are un-weighted Fixing variables Wade and Wood (2005) argue that uncertainties inherent in fitting and extrapolation of temperature effects from laboratory experiments at K to magma ocean temperatures exceeding 3000 K can be avoided by prescribing the effect of temperature based on free energy data obtained on pure substances at 1 atm (Barin et al., 1989; Chase, 1998). In contrast, Righter (2011a) argued that application of the 1 atm free energy data to conditions of high pressures and temperatures introduces different, and still un-quantified, errors. Here we use both approaches, and use the abbreviation FE when the temperature effect is assigned from free energy data. When assigning the temperature effect from free energy we use the values from Wood et al. (2008).

4 168 M.J. Walter, E. Cottrell / Earth and Planetary Science Letters 365 (2013) The oxygen fugacity coefficient can also be fixed by choosing the valence state of the element. If the valence state is not chosen then the fugacity coefficient, DIW, is independent, and the fitted coefficients can have non-integer values (Cottrell et al., 2009; Li and Agee, 1996, 2001; Righter, 2011a; Righter and Drake, 1999; Righter et al., 1997; Walter and Thibault, 1995). However, this approach may be viewed as erroneous since elemental valence states should take integer values, especially for those elements that likely occur in only one valence state (e.g. Ni 2 þ,co 2 þ ). Mixed valence species (e.g. W 4 þ,w 6 þ ) may in principle yield meaningful non-integer valence states over specific ranges in oxygen fugacity (O Neill et al., 2008). To avoid non-integer valence states some authors choose to fix the elemental valence state during regression (Corgne et al., 2008, 2009; Righter, 2011a; Siebert et al., 2011; Wade and Wood, 2005; Wood et al., 2008), and we also test that approach here. We fix the Ni and Co valences at 2þ, the V valence at 3þ, and due to uncertainty in the appropriate W valence (Cottrell et al., 2009; Wade et al., 2012), we test both 4þ and 6þ valence states for this element. Thus, for each of the data sets we perform four types of multivariable linear regressions: log D (four free parameters), log D FV (three free parameters), log D FE (three free parameters), and log D FVFE (two free parameters). In combination with the two regression models, normal and stepwise, we perform eight regressions for each element. An exception is W, for which 12 regressions are made due to using both 4þ and 6þ valence states when regressing using the log D FV approach. Regressions were performed using both Excel s and Matlab s linear least-squares algorithms and outputs are independent of the choice of software Core formation models Single-stage equilibrium model We explore single-stage model solutions by considering all four elements simultaneously. By moving the intercept to the lefthand side of Eq. (5) we define a new parameter, D * M, where: logd n M ¼ logdearth M a ð7þ Thus, for Ni, Co, W and V we can write four equations in four unknowns: logd n Ni ¼ b Ni UP=T þw Ni U1=T þd NiUDIW þe Ni Unbo=t logd n Co ¼ b Co UP=T þw Co U1=T þd CoUDIW þe Co Unbo=t logd n W ¼ b WUP=T þw W U1=T þd W UDIW þe W Unbo=t logd n V ¼ b V UP=T þ w V U1=T þd VUDIW þ e V Unbo=t In matrix form, b¼ax, we have: 2 logd n b Ni Ni w 2 3 Ni d Ni e Ni P=T logd n Co b Co w Co d Co e Co 6 4 logd n 7 W 5 ¼ 1=T d 6 b W w W d W e 4 W 7 6 DIW logd n b V V w V d V e V nbo=t The coefficients in matrix A are provided by the regressions. The matrix equation is solved for the vector x (i.e. P, 1/T, DIW, nbo/t) using singular value decomposition with back substitution (Press et al., 1992). We define the unique P T DIW nbo/t solution that occurs when no errors are assigned to the terms in Eq. (5) (i.e. the coefficients are infinitely precise) as the nominal solution. Each element in matrices A and b has an associated regression uncertainty, and we assess the magnitude of the effects of uncertainty on core formation model outcomes using a Monte- Carlo-type random-sampling approach. We solve for x in the matrix equation and vary the coefficients in matrices A and b randomly (normal distribution) and independently in 10 5 iterations; increasing the number of iterations to 10 6 has no significant ð8þ effect on the results. In this way we establish the extent of the equilibrium model solution space. There is a unique solution for each iteration, although a large number of the solutions may be unphysical (e.g. negative P and T, subsolidus, etc.). Therefore, we further interrogate the solution space by specifying a P T DIW nbo/t range of interest that is credible for core formation in Earth. Within the prescribed limits we search for all solutions permissible given the parameter uncertainties Continuous accretion model We present results for two types of continuous accretion that closely emulate models presented in Wood et al. (2008) so that a fair comparison can be made with previous results. The primary features of the model are: (1) the planet accretes in 1% mass increments; (2) at each increment liquid metal and liquid silicate equilibrate at the base of a magma ocean, the depth of which is constant relative to the depth of the mantle core boundary (x¼0.35); (3) the equilibration is defined by a prescribed liquidus; (4) accreting material has a metal/silicate fraction equal to the bulk planet fraction (0.32 metal by weight), and the newly accreted metal and the entire silicate mantle equilibrate at each step; (5) either a constant mantle oxygen concentration where Fe mantle ¼6.3% (IW-2.2), or the mantle oxygen concentration is varied in two steps, one at 20% and another at 89% accretion, and at these steps the mantle Fe content increases from 0.6% to 1.6% and 1.6% to 6.3%, respectively. To assess uncertainty we again adopt the approach of random sampling of regression coefficients. The continuous accretion model is repeated 10 4 times, which we found is sufficient to establish the characteristics of the solution space, with each iteration having a new set of randomly selected regression coefficients for each element. We prescribe a peridotite magma ocean with nbo/t equal to 2.7, and our final oxygen fugacity in all models is set to be equal to IW-2.2 (Fe mantle 6.3 wt%); note that we do not assign uncertainty in these variables and so our accretion models are somewhat more restrictive than the single-stage models. 3. Results 3.1. Regressions Tabulated regression results are presented in Appendix A. Regression coefficients for Ni, Co and V do not match exactly those published in the original studies. In the case of Ni and Co, the regression coefficients we calculate diverge from those published in the original study because Kegler et al. (2008) did not apply the same solute solute interaction corrections we apply, and did not include the nbo/t term. In the case of V, our regressions using log D FVFE for experiments cannot reproduce published coefficients in Wood et al. (2008) because those authors include data from other studies (Chabot and Agee, 2003; Gessmann and Rubie, 1998) as well as unpublished data at o3 GPa. We find that coefficients derived from experiments at high P T are often inconsistent with theoretical expectations for both temperature and oxygen fugacity effects. For example, sizable temperature effects are expected from the free energy data for Ni, Co, and W, but the data sets selected here show no statistically significant temperature effects for these elements. The temperature effect measured for V is statistically significant but about half the size predicted from free energy data. In terms of oxygen fugacity effects, we find that the valence states of Ni and Co predicted from the Kegler et al. (2008) data are closer to þ3 than the expected þ2. For V a valence of about þ3.5 is predicted, and for W about þ4.5.

5 M.J. Walter, E. Cottrell / Earth and Planetary Science Letters 365 (2013) Adjusted R Ni Co W4+ W6+ V logd logd FE logd FV logd FVFE Fig. 1. The adjusted coefficient of determination, R 2, for stepwise regression models. Regressions are made to the activity-corrected experimental data at pressures of Z5 GPa from the studies of Kegler et al. (2008) for Ni and Co, Wade and Wood (2001, 2005) and Wood et al. (2008) for V, and Cottrell et al. (2009) for W. FV valence is fixed in regressions; FE temperature effect calculated from free energy data. Table 2 Nominal solutions to single-stage core formation. P T DIW nbo/t Normal a log D log D FE b log D FV4 c log D FVFE log D FV log D FVFE Stepwise log D log D FE log D FV log D FVFE log D FV log D FVFE a Normal and stepwise refer to regression style. b FE temperature effect from free energy data. c FV fixed valence state in regressions; 4 and 6 refer to fixed valence of 4þ or 6þ for W. The failure of the regressed temperature and oxygen fugacity coefficients to comply with values predicted from theory motivates fixing these coefficients during regression (Wade and Wood, 2005). Fig. 1 illustrates the effect of imposing the theoretical values during step-wise regression, as quantified by the adjusted coefficient of determination, R 2. The log D (stepwise) regression model provides a good fit to the data, with R 2 values for all elements. Assigning the temperature (w) and/or oxygen fugacity (d) coefficients drops the R 2 values for all elements. In the extreme case where both valence and temperature coefficients are assigned (e.g. log D FVFE ), the R 2 values for Ni and V drop below 0.4, and for Co the R 2 value drops to less than 0.2. We can generalize that the experimental data are not well modeled by the theoretical coefficients in these low R 2 regressions, and that the coefficients may be inaccurate under the experimental conditions. Conversely, regression of the W data set results in coefficients more similar to the theoretical values, so the R 2 does not fall dramatically. Another consequence of fixing the temperature and oxygen fugacity coefficients to theoretical values is that the terms are made infinitely precise in core formation models. Fig. 1 shows that this increase in model precision comes at the expense of the goodness of fit of the experimental data to the remaining variables. Thus, while the log D FVFE -based core formation models described below might appear more precise because there are fewer free terms, they may be less accurate. Other noteworthy observations from the regression results are: (1) there is debate over the statistical significance of the nbo/t term in the Kegler data set (Palme et al., 2011; Righter, 2011b). We find that Ni and Co have small but statistically significant nbo/ t terms in all regressions. (2) The P/T term for Co becomes insignificant in cases where the temperature effect is imposed by free energy. (3) The P/T and nbo/t effects are large, significant and effectively constant in all W regressions. (4) For W, a 4þ valence state is marginally better supported in terms of R 2 by the high P T experimental data than a 6þ valence state. (5) The P/T effect is not statistically significant in any V regression, whereas the nbo/t coefficient is significant in some cases but not others. (6) Most of the error for V is held in the intercept, which is not the case for Ni, Co and W. We recognize that these observations are unique to these data sets and activity models, and reiterate that our main purpose is not to adjudicate on the quality of the data sets or to exhaustively Temperature K logd logd FE logd FV logd FVFE4 logd FV6 logd FVFE6 Earth liquidus core-mantle boundary nbo/t Earth Pressure GPa IW Fig. 2. Nominal four-element (Ni, Co, W and V) solutions are shown in terms of (a) P versus T, and (b) nbo/t versus DIW, for single-stage equilibrium core formation. Solid symbols denote normal regressions, and open symbols stepwise regressions. The shaded liquidus region in (a) defines a Earth-credible region; in (b) this region occurs at the intersection of the shaded bands. The plots illustrate the wide range of solutions that occur depending on the choice of independent variables (spread in solid or open points) and on the choice of regression model (compare solid with open symbols of the same shape). None of the solutions satisfy Earth-credible values simultaneously for all four independent variables.

6 170 M.J. Walter, E. Cottrell / Earth and Planetary Science Letters 365 (2013) n = 100,000 compare these regressions with others in the published literature. Rather, it is to assess to what extent uncertainty in the regressions and choices made in how data are regressed can affect core formation model outcomes Single stage core formation models T (K) x P (GPa) x10 5 Fig. 3. A representative P T DIW projection showing the set of unique solutions in four-element, single-stage core formation models based on the regression coefficients for the log D (normal) regression, and when uncertainties in all model parameters are varied randomly and independently in a normal distribution about nominal coefficients in 10 5 iterations. Temperature (K) Pressure (GPa) logd regression non-liquidus solution liquidus solution mean liquidus solution Fig. 4. A P T projection showing a sub-region of the model log D (normal) solution space plotted in Fig. 3. Shown are unique four-element single-stage solutions in a P T region applicable to core formation in a deep peridotitic magma ocean. The open round symbols show model solutions along the peridotite liquidus that simultaneously satisfy Earth-credible values for DIW (IW ) and nbo/t ( ). Pluses are nonliquidus solutions that also satisfy these DIW and nbo/t constraints Nominal single-stage solutions The nominal solutions to Eq. (8), in which no uncertainty is assigned to regression coefficients, are listed in Table 2 for all regression models and are shown on P T and DIW nbo/t projections in Fig. 2. Physical model outcomes vary tremendously only as a function of the regression model. The variation in nominal solutions is extremely large depending on the choice of independent variables and on the choice of regression model. Five of the 12 nominal solutions occur at unphysical pressures for Earth, and the remaining seven yield solutions between about 60 and 125 GPa. Shown in Fig. 2 are shaded regions that we define as Earthcredible for single-stage core formation. Wade and Wood (2005) argued that single-stage metal silicate equilibration at the base of a magma ocean should occur at or near the liquidus temperature of mantle peridotite, and for simplicity we adopt their liquidus parameterization that is based on extrapolation of experimental data (Tronnes and Frost, 2002; Zerr et al., 1998). We define a liquidus region in Fig. 2a with a prescribed uncertainty of 710 GPa; at 130 GPa this liquidus is close to the recent liquidus determination of Fiquet et al. (2010) but above that of Andrault et al. (2011). On this basis we find that three of the 12 nominal solutions, or perhaps five if we relax our liquidus criterion somewhat, give credible pressures and temperatures for core formation in Earth. In Fig. 2b we define an Earth-credible region based on the assumption that the oxygen fugacity during core formation was DIW based on a mantle with an Fe content of 6.3 wt%, and assuming an ultramafic magma ocean with a nbo/t value of With these criteria, we find that none of the nominal solutions lie within an Earth-credible region for single stage core formation, with solutions tending to be too reducing (DIWo 2.5) or with nbo/t values inappropriate for an ultramafic magma ocean. Next we address how consideration of uncertainty on the terms in the model affects these conclusions. That is, how much must the coefficients change to move from a nominal model solution that lies outside our Earth-credible core formation region, to a solution that is credible? Table 3 Model solutions to single-stage core formation. n 1s a 2s 3s P 7 b T 7 DIW 7 nbo/t 7 Normal log D log D FE log D FV log D FVFE log D FV log D FVFE Stepwise log D log D FE log D FV log D FVFE log D FV log D FVFE Other labels as in Table 2. a 1s (2s,3s)¼number of solutions with all coefficients within one (two, three) standard errors from the nominal coefficients. b 7 ¼one standard error.

7 M.J. Walter, E. Cottrell / Earth and Planetary Science Letters 365 (2013) The single-stage model solution space We now consider the solution space that arises when all the error terms are included in a Monte Carlo analysis. As an example, Fig. 3 shows the entire solution space in a P T DIW projection for the log D Temperature (K) logd logd FE logd FV logd FVFE logd FV6 logd FVFE6 logd (s) logd FE (s) logd FV (s) logd FVFE (s) logd FV6 (s) logd FVFE6 (s) (normal) regression model; each regression type exhibits it own peculiarities in terms of spatial distribution. Fig. 3 shows that the solution space is expansive, with extreme variation around the nominal solution, and this is true for all regression models. Indeed, the majority of the solution space is non-physical (e.g. negative P and T, nbo/t o0 and 44) or unrealistic for core formation in Earth (e.g. P4130 GPa, T46000 K, DIWb 75, nbo/to2). We find that much of the extreme variation occurs when the terms are randomly varied within only one standard variance of the mean (1s), and so Fig. 3 illustrates the flexibility inherent in the model solution space. We now interrogate a region of interest within the overall solution space by again adopting the Earth-credible core formation conditions defined for the nominal solutions (Fig. 2). Fig. 4 is a P T section illustrating this case for the log D regression model. The circles show the liquidus solutions that fall within our restricted P T DIW nbo/t region of interest, and these are part of a larger cloud of solutions (crosses) that do not fall along our prescribed liquidus but do fit the DIW nbo/t criteria. We find for the log D regression case with an extremely unphysical nominal solution of about 170 GPa and 10,500 K, that when uncertainty is considered there are a range of single-stage solutions along the peridotite liquidus with an average of about 40 GPa and 3150 K. In Table 3 we list the single-stage model results for all our regression models in terms of our Earth-credible region of interest, and we find that liquidus solutions exist in all models when uncertainty is considered Pressure (GPa) Fig. 5. A P T projection showing mean values of four-element, single-stage liquidus solutions in a deep peridotitic magma ocean as a function of regression model. Solid symbols denote ordinary regressions and open symbols stepwise regressions. Error bars are 1s deviations about the mean and reflect the range of model solutions that fall within the prescribed Earth-credible region. Large symbols depict single stage solutions with all coefficients within 1s (large symbols), 2s (medium symbols) and 3s (small symbols), of their nominal values Assessing single-stage model consistency Here we provide a statistical basis for determining if a singlestage model is consistent with the regression coefficients by quantifying how much each coefficient must deviate from its regressed value to produce an Earth-credible core formation solution (see also Supplementary Material). Ultimately, there are no rigorous theoretical criteria for defining how much deviation is acceptable, but we adopt the following: (1) if all coefficients for each element in a given model solution are within one standard error (1s) Table 4 Model solutions to continuous core formation. Constant fo 2 Ni 1s a Co 1s W 1s V 1s Normal log D log D FE log D FV log D FVFE log D FV log D FVFE Stepwise log D log D FE log D FV log D FVFE log D FV log D FVFE Changing fo 2 Ni 1s Co 1s W 1s V 1s Normal log D log D FE log D FV log D FVFE log D FV log D FVFE Stepwise log D log D FE log D FV log D FVFE log D FV log D FVFE Other labels as in Table 2. a 71s is the range of solutions when coefficient errors are restricted to one standard error.

8 172 M.J. Walter, E. Cottrell / Earth and Planetary Science Letters 365 (2013) of their nominal coefficients, then a solution is highly consistent, (2) if all coefficients are within 2s then a solution is moderately consistent, and (3) if one or more coefficients are at 3s away from the nominal values then a solution is inconsistent. We recognize that this definition is arbitrary, but it provides a quantitative measure by which single-stage solutions can be compared among regression models. In Table 3 we itemize the number of single-stage liquidus solutions for each regression model that occur when all coefficients are within 1s, 2s or 3s of their nominal values. For example, in the log D (normal) regression illustrated in Fig. 4, none of the liquidus solutions occur when all coefficients are within 1s, 19 occur within 2s, and 37 occur within 3s. Thus, more than half of the solutions occur when one or more coefficients are stretched to 3s, and would be considered inconsistent with the regression coefficients, but about half the solutions are consistent at the 2s level. Table 3 shows that nine of the 11 regression models yield solutions to our prescribed singlestage core formation conditions while maintaining all coefficients within 2s, and two of those nine models have solutions that occur when all variables are within 1s. However, these two regression models, log D FVFE 6þ and log D FVFE 4þ (stepwise), have low goodness of fit for Ni, Co and V partitioning data. Thus, even though single stage core formation is highly consistent with these regression models, the regression models themselves may not be accurate representations of the experimental data. Fig. 5 plots the pressures and temperatures most consistent with each regression model, and the symbol size scales with the deviation of the coefficients from their nominal values. Variation in regression model alone produces a wide range of single-stage model solutions along the liquidus with mean solutions from about 30 to 70 GPa, reinforcing our earlier conclusion based on the nominal solutions. However, unlike the nominal cases that yielded no Earth-credible solutions, when uncertainty is included, the single-stage peridotite liquidus core formation model is generally consistent with the experimental partitioning data, at least for the chosen data sets and statistical criteria Continuous accretion core formation models We now present how regression models and their associated uncertainties affect the outcomes of continuous accretion models, 3.0 normal stepwise 2.0 normal stepwise 2.5 logd logd logdfe logdfv logdfvfe logds logdfes logdfvs logdfvfes 0.4 logd logdfe logdfv logdfvfe logds logdfes logdfvs logdfvfes normal stepwise normal stepwise logd logd logdfe logdfv logdfvfe logdfv6 logds logdfes logdfvs logdfvfes logdfvfe6 logdfv6s -5.0 logd logdfe logdfv logdfvfe logds logdfes logdfvs logdfvfes Fig. 6. Mean values and ranges of log D (weight) for (a) Ni, (b) Co, (c) W and (d) V, in continuous accretion models as a function of regression model (all weight-based). Mean values are calculated from 10 4 model iterations in which all coefficients are constrained to vary within 1s of nominal values. Error bars show the entire range of solutions from which the mean is determined when all coefficients are less than 1s. Closed symbols represent models in which relative oxygen fugacity is held constant at a value of IW-2.25, whereas open symbols represent models in which relative oxygen fugacity undergoes stepwise increases as in the models of Wood et al. (2008). Grey bars show the ranges for log D Earth. Vertical dashed lines separate normal from stepwise regression models.

9 M.J. Walter, E. Cottrell / Earth and Planetary Science Letters 365 (2013) and whether they can be statistically distinguished from singlestage models. Continuous accretion model results are listed in Table 4 in terms of the log D value acquired by the model Earth at the end of accretion as a function both of the regression model and the accretion scenario. Fig. 6 displays the range of solutions obtained when applying a conservative cut-off of 1s to the uncertainty in the coefficients. That is, the quoted errors do not represent 1s uncertainty about the mean solutions, but rather represent the range of solutions when all the coefficients are constrained to vary by no more than 1s during the random sampling. Like the single-stage models, the continuous accretion model solution space expands considerably when regression error is considered. Regression models with fewer independent variables yield narrower 71s ranges. However, we again must emphasize that the apparent increase in precision is primarily a consequence of zero associated uncertainty on the fixed variables. The regression models themselves vary in how well they represent the experimental partitioning data (Fig. 1). The ranges of model solutions that are obtained when using regressions with more independent variables are generally larger than in cases where variables are fixed with no associated error. We draw several conclusions from the continuous accretion modeling: (1) Ni and Co are not useful for discriminating among accretion models (Wade and Wood, 2005; Wood et al., 2008). All models produce outcomes that can reproduce the observed log D Earth values well within 1s uncertainty on the regression coefficients, and there is only a marginal difference between constant and increasing oxygen fugacity scenarios. (2) W and V, as a consequence of their greater sensitivity to oxygen fugacity (W and V) and temperature (V), are more sensitive to the choice of regression model and show considerable variation in physical model outcomes. (3) The constant oxygen fugacity scenario does not produce any mean solutions that overlap observed log D Earth values for W or V, consistent with the results of Wade and Wood (2005) and Wood et al. (2008). However, when only 1s uncertainty on regressions coefficients is considered, seven of the W models and two of the V models provide solutions that satisfy log D Earth values; at the 2s level (not shown) all models produce consistent solutions. (4) The progressive oxidation scenario generally yields mean model solutions for all elements that are closer to log D Earth values than in the constant oxidation scenario. At the 1s level of uncertainty on regression coefficients progressive oxidation provides more consistent solutions to the log D Earth constraints for W and V. At the 2s level, however, the progressive oxidation and fixed oxidation models are effectively indistinguishable. (5) At the 1s level, all stepwise regression models for V provide solutions for continuousaccretionwithvariableoxygen fugacity whereas no solutions are achieved for accretion at constant oxygen fugacity. Thus, regression model choices (e.g., normal versus stepwise, log D versus log D FVFE ) directly inform model outcomes, and Fig. 6 illustrates how divergent views about core formation found in the literature may hinge on the regression model employed. As with single-stage models, our results indicate that the flexibility inherited from the regression models is too large to effectively discriminate among competing accretion scenarios. Both accretion scenarios considered here are arguably consistent with the experimental data sets. Certainly a range of accretion models between these two can also provide consistent solutions, and the specific characteristics of progressive accretion as modeled here and by Wood et al. (2008) are not uniquely required by the data. 4. Discussion and conclusions Equilibration of metal and silicate in a deep peridotitic magma ocean is the modern paradigm for core formation, and within this framework debate has emerged over the style of core formation that is most consistent with the experimental partitioning data. This debate is important because an accurate model for the conditions at which Earth s core formed can provide information about early Earth conditions and processes that can be compared to knowledge obtained from theoretical and isotopic models. Quantifying the constraints imposed by the metal silicate partitioning data is one means by which we may discriminate among competing models. Our results show that the regression models for the four elements and the data sub-sets we considered are currently too imprecise to permit a statistically compelling discrimination among various single-stage and continuous core formation scenarios. We find that at the 2s level of uncertainty on regression coefficients, both single-stage and continuous accretion scenarios provide consistent solutions for nearly all the regression models tested. In contrast, at the more strict 1s level of uncertainty several, but not all, regression models permit either single-stage solutions or continuous accretion solutions at constant oxygen fugacity of IW-2. There are also other kinds of uncertainty we have not addressed that will increase the global uncertainty in multi-element core formation models. For example, all of the mean model solutions within our region prescribed for single-stage core formation occur at pressures and temperatures far in excess of those in the experimental data sets requiring considerable extrapolation; the statistically most acceptable single-stage solutions occur at pressures of GPa. Pressure induced changes in coordination and element valence state in silicate melts can lead to unpredictable changes in partitioning behavior not captured by the regressions of the lower P T data. Obtaining partitioning data at pressures approaching 100 GPa and at temperatures in excess of 3000 K, like those reported in recent studies (Bouhifd and Jephcoat, 2011; Siebert et al., 2012), are key for validating the pressure Table A1 Experiments included in regressions. Ni, Co a W b V c BKHP1 W01 No 6 BKHP5 W09 No 5 BKHP6 W19 HT FeC BKHP15 W28 HT60 BKHP17 W29 HT8020c PHP1 W32 HT70 BKHP2 W33 LT11 BKHP7 W38 LT21 BKHP11 W41 R3 BKHP20 W42 s2266 BKHP21 W49 s2265 BKHP22 W91 A1a BKHP23 W45 A2a BKHP13 W46 A3 BKHP14 W50 MK81 BKHP24 W51 MK84 BKHP25 W81 B1 BKHP26 W92 B2 BKHP29 W64 b3 BKHP39 W65 c1a BKHP40 W88 c5 BKHP38 W BKHP47 W94 BKHP12 W95 S3575 W62 W68 W87 W90 a Data from Kegler et al. (2008). b Data from Cottrell et al. (2009). c Data from Wade and Wood (2001, 2005) and Wood et al. (2008).

10 174 M.J. Walter, E. Cottrell / Earth and Planetary Science Letters 365 (2013) effects predicted from the available data, and can increase both the precision and accuracy of core formation models. The independent variable that is the least well-constrained experimentally for the four elements is the effect of temperature, which is ironic because early predictions about the importance of this effect stimulated much of the subsequent partitioning work (Murthy, 1991). Working to isolate this effect, as done for Ni and Co in the study of Chabot et al. (2005), is of prime importance to increase the precision in multi-element core formation models. If free energy data are used to fix this variable, uncertainty in the thermodynamic data should be assessed and propagated into models; here for example we assumed no uncertainty when fixing the temperature effect from free energy and so underestimated the uncertainty in those models. Model accuracy depends also on assumptions about the valence-state of elements. Further studies constraining the oxygen fugacity ranges over which valences states may change for multi-valent elements such as W are needed, perhaps in combination with synchrotron-based XANES analyses of runproducts to obtain direct determinations of valence state. The presence of light elements in core-forming metals can also have an important influence on partitioning (Corgne et al., 2009), and models incorporating these effects require independent constraints on the light elements in the core with uncertainties propagated into core formation models. A final source of uncertainty not considered here arises from inter-laboratory biases in experimental data sets. In contrast to our approach of using only a restricted sub-set of the available data, most studies maximize experimental data points in regressions by mixing data from multiple laboratories, and the uncertainties associated with these biases have yet to be systematically quantified. Table A2 Coefficients for normal and stepwise regressions of experimental data. Regression a n k 1s b 1/T 1s P/T 1s DIW 1s nbo/t 1s R 2 Valence c Normal Ni log D log D FE log D FV log D FVFE Stepwise log D log D FE log D FV log D FVFE All coefficients significant in normal regression Normal Co log D log D FE log D FV log D FVFE Stepwise log D log D FE log D FV log D FVFE Normal W log D log D FE log D FV log D FVFE log D FV log D FVFE Stepwise log D log D FE All coefficients significant in normal regression log D FV log D FVFE4 All coefficients significant in normal regression log D FV log D FVFE6 All coefficients significant in normal regression Normal V log D log D FE log D FV log D FVFE Stepwise log D log D FE log D FV log D FVFE FE temperature effect from free energy data; FV fixed valence state; 4 and 6 refer to fixed valence of 4þ or 6þ for W. a Normal regressions maintain all coefficients, stepwise regression progressively eliminate insignificant coefficients. b One standard error on the coefficients as calculated in the least squares regression. c Valence state of the element as calculated from the regression or as fixed in the regression.

11 M.J. Walter, E. Cottrell / Earth and Planetary Science Letters 365 (2013) In final analysis, our fundamental conclusion is that core formation models based on experimental partitioning data for the four siderophile elements Ni, Co, W, and V are too imprecise to discriminate among single-stage and continuous accretion core formation scenarios. Siderophile element partitioning data are currently flexible enough to fit a wide range of core formation scenarios once conceived. Practitioners who use experimental data to constrain models of core formation should be cognizant of the impact that the regression model has on physical model outcomes. The methods employed here can be expanded to include more elements, but we predict that this conclusion will hold because other extant data sets are likely to have comparable or worse precision than those chosen for this study. Further experimental and theoretical studies are required to increase the precision and accuracy of models to a point where their full predictive power can be achieved. Acknowledgments We thank Prof. Bernie Wood for supplying computer code for activity corrections. Nancy Chabot and an anonymous reviewer are thanked for their constructive comments. This work was supported by NSF grant EAR to EC, and NERC grant NE/f019084/1to MJW. Appendix A. Regression tables See Tables A1 and A2. Appendix B. Supporting information Supplementary data associated with this article can be found in the online version at References Andrault, D., Bolfan-Casanova, N., Lo Nigro, G., Bouhifd, M.A., Garbarino, G., Mezouar, M., Solidus and liquidus profiles of chondritic mantle: implication for melting of the Earth across its history. Earth Planet. Sci. Lett. 304, Arevalo Jr., R., McDonough, W.F., Tungsten geochemistry and implications for understanding the Earth s interior. Earth Planet. Sci. Lett. 272, Barin, I., Sauert, F., Schultze-Rhonhof, E., Sheng, W.S., Thermochemical Data of Pure Substances, Part I and Part II. Verlagsgesellschaft, Weinheim, Germany. Benz, W., Cameron, G.W., Terrestrial effects of the giant impact. In: Newsom, H.E., Jones, J.H. (Eds.), Origin of the Earth. Oxford University Press, New York, pp Bouhifd, M.A., Jephcoat, A.P., 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. Earth Planet. Sci. Lett. 307, Canup, R., Simulations of a late lunar-forming impact. Icarus 168, Chabot, N.L., Agee, C.B., Core formation in the Earth and Moon: new experimental constraints from V, Cr, and Mn. Geochim. Cosmochim. Acta 67, Chabot, N.L., Draper, D.S., Agee, C.B., Conditions of core formation in the Earth: constraints from Nickel and Cobalt partitioning. Geochim. Cosmochim. Acta 69, Chambers, J.E., Planetary accretion in the inner Solar System. Earth Planet. Sci. Lett. 223, Chase, M.W., NIST-JANAF Thermochemical Tables, Fourth edition. National Institute of Standards and Technology. Corgne, A., Keshav, S., Wood, B.J., McDonough, W.F., Fei, Y., Metal silicate partitioning and constraints on core composition and oxygen fugacity during Earth accretion. Geochim. Cosmochim. Acta 72, Corgne, A., Siebert, J., Badro, J., Oxygen as a light element: a solution to single-stage core formation. Earth Planet. Sci. Lett. 288, Cottrell, E., Walter, M.J., Walker, D., Metal silicate partitioning of tungsten at high pressure and temperature: implications for equilibrium core formation in Earth. Earth Planet. Sci. Lett. 281, Cottrell, E., Walter, M.J., Walker, D., Erratum to Metal silicate partitioning of tungsten at high pressure and temperature: implications for equilibrium core formation in Earth (vol. 281, pg 275, 2009). Earth Planet. Sci. Lett. 289, Fiquet, G., Auzende, A.L., Siebert, J., Corgne, A., Bureau, H., Ozawa, H., Garbarino, G., Melting of peridotite to 140 Gigapascals. Science 329, Gessmann, C.K., Rubie, D.C., The effect of temperature on the partitioning of nickel, cobalt, manganese, chromium, and vanadium at 9 GPa and constraints on formation of the Earth s core. Geochim. Cosmochim. Acta 62, Hillgren, V.J., Drake, M.J., Rubie, D.C., High-pressure and high-temperature experiments on core mantle segregation in the accreting earth. Science 264, Kegler, P., Holzheid, A., Frost, D.J., Rubie, D.C., Dohmen, R., Palme, H., New Ni and Co metal silicate partitioning data and their relevance for an early terrestrial magma ocean. Earth Planet. Sci. Lett. 268, Li, J., Agee, C.B., Geochemistry of mantle core differentiation at high pressure. Nature 381, Li, J., Agee, C.B., The effect of pressure, temperature, oxygen fugacity and composition on partitioning of nickel and cobalt between liquid Fe Ni S alloy and liquid silicate: implications for the Earth s core formation. Geochim. Cosmochim. Acta 65, Ma, Z.T., Thermodynamic description for concentrated metallic solutions using interaction parameters. Metallurgical and Materials Transactions B- Process Metallurgy and Materials Processing Science 32, Mann, U., Frost, D.J., Rubie, D.C., Evidence for high-pressure core mantle differentiation from the metal silicate partitioning of lithophile and weaklysiderophile elements. Geochim. Cosmochim. Acta 73, McDonough, W.F., Compositional model for the Earth s core. In: Carlson, R.W. (Ed.), The Mantle and Core, Treatise on Geochemistry. Elsevier-Pergamon, Oxford, pp Melosh, H.J., Giant impacts and the thermal state of the early Earth. In: Newsom, H.E., Jones, J.H. (Eds.), Origin of the Earth. Oxford University Press, New York, pp Murthy, V.R., Early differentiation of the Earth and the problem of mantle siderophile elements a new approach. Science 253, Mysen, B.O., Structure and Properties of Silicate Melts. Elsevier (Ed.), Amsterdam. Mysen, B.O., Properties of magmatic liquids. In: Kushiro, I. (Ed.), Physical Chemistry of Magmas. Springer-Verlag, New York, pp Newsom, H.E., Sims, K.W.W., Noll, P.D., Jaeger, W.L., Maehr, S.A., Beserra, T.B., The depletion of tungsten in the bulk silicate Earth: constraints on core formation. Geochim. Cosmochim. Acta 60, O Neill, H.S.C., The origin of the moon and the early history of the earth a chemical-model. 2. The Earth. Geochim. Cosmochim. Acta 55, O Neill, H.S.C., Berry, A.J., Eggins, S.M., The solubility and oxidation state of tungsten in silicate melts: implications for the comparative chemistry of W and Mo in planetary differentiation processes. Chem. Geol. 255, O Neill, H.S.C., Eggins, S.M., The effect of melt composition on trace element partitioning: an experimental investigation of the activity coefficients of FeO, NiO, CoO, MoO 2 and MoO 3 in silicate melts. Chem. Geol. 186, Palme, H., Kegler, P., Holzheid, A., Frost, D.J., Rubie, D.C., Comment on Prediction of metal silicate partition coefficients for siderophile elements: an update and assessment of PT conditions for metal silicate equilibrium during accretion of the Earth by K. Righter, EPSL 304 (2011) , Earth Planet. Sci. Lett. 312, Palme, H., O Neill, H.S.C., Cosmochemical estimate of mantle composition. In: Carlson, R.W. (Ed.), The Mantle and Core, Treatise on Geochemistry. Elsevier-Pergamon, Oxford, pp Press, W.H., Teukolsky, S.A., Vetterling, W.T., PFlannery, B.P., Numerical Recipes in Fortran. Cambridge University Press, New York. Righter, K., Metal silicate partitioning of siderophile elements and core formation in the early Earth. Annu. Rev. Earth Planet. Sci. 31, Righter, K., 2011a. Prediction of metal silicate partition coefficients for siderophile elements: an update and assessment of PT conditions for metal silicate equilibrium during accretion of the Earth. Earth Planet. Sci. Lett. 304, Righter, K., 2011b. Reply to the comment by Palme et al. on Prediction of metal silicate partition coefficients for siderophile elements: an update and assessment of PT conditions for metal silicate equilibrium during accretion of the Earth. Earth Planet. Sci. Lett. 312, Righter, K., Drake, M.J., Effect of water on metal silicate partitioning of siderophile elements: a high pressure and temperature terrestrial magma ocean and core formation. Earth Planet. Sci. Lett. 171, Righter, K., Drake, M.J., Yaxley, G., Prediction of siderophile element metal silicate partition coefficients to 20 GPa and C: the effects of pressure, temperature, oxygen fugacity, and silicate and metallic melt compositions. Phys. Earth Planet. Inter. 100, Ringwood, A.E., Chemical evolution of terrestrial planets. Geochim. Cosmochim. Acta 30, Rubie, D.C., Frost, D.J., Mann, U., Asahara, Y., Nimmo, F., Tsuno, K., Kegler, P., Holzheid, A., Palme, H., Heterogeneous accretion, composition and core mantle differentiation of the Earth. Earth Planet. Sci. Lett. 301, Rudge, J.F., Kleine, T., Bourdon, B., Broad bounds on Earth s accretion and core formation constrained by geochemical models. Nat. Geosci. 3, Shunk, F.A., Constitution of Binary Alloys, McGraw-Hill, New York, NY. Siebert, J., Badro, J., Antonangeli, D., Ryerson, F.J., Metal silicate partitioning of Ni and Co in a deep magma ocean. Earth Planet. Sci. Lett. 321,

12 176 M.J. Walter, E. Cottrell / Earth and Planetary Science Letters 365 (2013) Siebert, J., Corgne, A., Ryerson, F.J., Systematics of metal silicate partitioning for many siderophile elements applied to Earth s core formation. Geochim. Cosmochim. Acta 75, Steelmaking, Steelmaking Data Sourcebook. The Japan Society for the Promotion of Science, the 19th Committee on Steelmaking. Gordon and Breach Science Publishers, Montreux. Thibault, Y., Walter, M.J., The influence of pressure and temperature on the metal silicate partition-coefficients of nickel and cobalt in a model-c1 chondrite and implications for metal segregation in a deep magma ocean. Geochim. Cosmochim. Acta 59, Tonks, W.B., Melosh, H.J., Magma ocean formation due to giant impacts. J. Geophys. Res. 98, Tronnes, R.G., Frost, D.J., Peridotite melting and mineral melt partitioning of major and minor elements at GPa. Earth Planet. Sci. Lett. 197, Wade, J., Wood, B.J., The Earth s missing niobium may be in the core. Nature 409, Wade, J., Wood, B.J., Core formation and the oxidation state of the Earth. Earth Planet. Sci. Lett. 236, Wade, J., Wood, B.J., Tuff, J., Metal silicate partitioning of Mo and W at high pressures and temperatures: evidence for late accretion of sulphur to the Earth. Geochim. Cosmochim. Acta 85, Walter, M.J., Newsom, H.E., Ertel, W., Holzheid, A., Siderophile elements in the Earth and Moon: metal/silicate partitioning and implications for core formation. In: Canup, R.M., Righter, K. (Eds.), Origin of the Earth and Moon. The University of Arizona Press, Tucson. Walter, M.J., Thibault, Y., Partitioning of tungsten and molybdenum between metallic liquid and silicate melt. Science 270, Walter, M.J., Tronnes, R.G., Early Earth differentiation. Earth Planet. Sci. Lett. 225, Wood, B.J., Wade, J., Kilburn, M.R., Core formation and the oxidation state of the Earth: additional constraints from Nb, V and Cr partitioning. Geochim. Cosmochim. Acta 72, Wood, B.J., Walter, M.J., Wade, J., Accretion of the Earth and segregation of its core. Nature 441, Zerr, A., Diegeler, A., Boehler, R., Solidus of Earth s deep mantle. Science 281,

13 Supplementary Material. Quantifying Differences Among Regression Models In order to gain a deeper understanding of the balance among variables required to reach a particular model solution, and how this balance changes depending on regression model, we quantify in P-T-ΔIW-nbo/t space how much change occurs on individual variables in terms of shifts in partition coefficients. We begin by writing an expression which in a 4-D Cartesian coordinate system describes the shift in all parameters in going from the nominal solution (P-T- ΔIW-nbo/t) n to the model solution (P-T-ΔIW-nbo/t) m in terms of logd for each element, M. First, the nominal solution is written as: log D nominal M = [! n A n +! n B n +! n C n +! n D n + I n ] (S.1) Where α, β, χ, δ are the regression coefficients, A, B, C, D stand for P/T, 1/T, ΔIW, and nbo/t, I is the intercept and the subscripts n stands for nominal. The position of the model solution, m, relative to the nominal solution, can be quantified as: [ ] + [! n B n +! n!b m +!" m B n!b m ] + [! n C n +! n!c m C n!c m ] + [ ] + log D M model =! n A n +! n!a m A n!a m! n D n +! n!d m D n!d m (S.2) I n +!I m and the difference in the nominal and model solution quantified as:! log D = log D M model " log D M nominal = [! n!a m A n!a m ] + [! n!b m +!" m B n!b m ] + [ ] +! n!c m C n!c m (S.3) [! n!d m D n!d m ] +!I m This treatment provides us with a quantitative measure of how each term contributes in moving from the nominal solution with no error, to a model solution where coefficient uncertainties are considered (e.g. Δα = α n ΔΑ m + Δα m Α n + Δα m ΔΑ m, etc.). ΔlogD is zero because all of the

14 changes in the terms (ΔP/T, Δ1/T, ΔIW, Δnbo/t, ΔlogD Earth ) in going from the nominal to model solution must sum to zero in terms of changes in logd; the target D Earth remains the same for both solutions. We can apply this approach on a point-to-point basis (e.g. each point in the model solution space), but by way of example we have chosen to consider the mean liquidus solution for two models, logd and logd FV FE 6+ (Table 3, main text). We have chosen these models because they provide very different results in terms of both the regressions and the model solutions, as well as their statistical consistency. The results are illustrated on Fig. S.1 using bar-diagrams that show for individual elements how much each variable contributes to maintain a ΔlogD of zero in moving from the nominal to the model solution. As an example of the information that can be gleaned from this analysis, consider W and V in the logd (normal) model (Fig. S.1a). Note the large negative shift in ΔlogD for W that occurs because in moving from the nominal to the model solution, nbo/t changes from ~ 0.7 to ~ 2.7, with a corresponding change in ΔlogD of ~ -1.5 log units. This shift is compensated for to maintain ΔlogD of zero by positive deviations in ΔlogD in all the other terms. In the case of V in the logd model, the 1/T term produces a large negative shift that, together with a smaller negative shift due to the nbo/t term, is compensated for by other terms, but primarily through a large shift in the intercept term. We can also see that the two regression models produce quite different results. Shifts in ΔlogD are smaller for the logd FV FE 6+ model because the nominal solution is closer to our prescribed core formation solution region. The sign of the shifts also changes for many variables, again as a consequence of the position of the nominal solution in P-T-ΔIW-nbo/t space relative to the model solution.

15 Figure S.1. Bar plots showing relative changes in logd of each independent variable and for each element when moving from the nominal single-stage solution to the mean Earth-credible liquidus model solution (Table 3 in main text) using the (a) logd (normal regression) and (b) logd FVFE 6+ regressions. ΔlogD must remain at zero in moving from the nominal conditions to the model conditions, and these diagrams show quantitatively how each variable changes in order to accommodate this constraint. A general observation that applies to all regression models and is illustrated in Figures S.1a and S.1b, is that in moving from nominal to model solutions there are much larger shifts in ΔlogD for W and V than for Ni and Co, which simply reflects the large dependencies of these elements on independent variables such as nbo/t (W), temperature (V) and oxygen fugacity (W 6+ ). If the deviations in these terms between the nominal and model solutions are large, then large shifts in these terms must be compensated for by deviations in other terms. This is where the uncertainty on the regression coefficients comes into play, providing the flexibility (e.g. < 1σ variation) or inflexibility (e.g. > 3σ variation) in the models to maintain ΔlogD at zero. While the information in Figure S.1 describes how much and in what direction each variable contributes to the solution, it is also informative to track how frequently a given variable must

16 be stretched outside of a particular range, say 1σ, in order to achieve a model solution. In general, we find that model likelihood is not particularly sensitive to any one element or variable, that is, to achieve model solutions all variables for all elements are perturbed. For example, Figure S.2 shows the number of times each variable for each element deviated by more than 1σ to achieve a model solution for the logd (normal) and logd FV FE 6+ cases, and variance on individual terms are, generally speaking, equally parsed among all variables that have an associated uncertainty. Figure S.2. Bar plots showing number of times each variable for each element deviated by more than 1σ to achieve a model solution for the logd (normal) and logd FV FE 6+ cases when moving from the nominal single-stage solution to the mean Earth-credible liquidus model solution (Table 3 in main text) using the (a) logd and (b) logd FVFE 6+ regressions.