# Grüneisen parameter formalism in the study of the Earth's core formation: a sensitivity study

Vincent Clesi, Renaud Deguen

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Vincent Clesi, Renaud Deguen. Grüneisen parameter formalism in the study of the Earth's core formation: a sensitivity study. Geophysical Journal International, 2024, 237, pp.1275-1284. 10.1093/gji/ggae117. insu-04841360

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# Grüneisen parameter formalism in the study of the Earth's core formation: a sensitivity study

Vincent Clesi<sup>1,2</sup> and Renaud Deguen<sup>1,3</sup>

Accepted 2024 March 22. Received 2024 March 19; in original form 2023 June 29

## SUMMARY

The Grüneisen parameter is an important parameter for the thermal state and evolution of the core, but its uncertainties and their implications are sometimes overlooked. Several formalisms using different parameters values have been used in different studies, making comparison between studies difficult. In this paper, we use previously published data sets to test the sensitivity of modelling the thermal state of the early core to the different formalisms and parameter values used to describe the evolution of the Grüneisen parameter with density. The temperature of the core obtained in our models is less sensitive to the uncertainties of the parameters used in Al'Tshuler *et al.* formalism than the uncertainties of the parameters used in Anderson formalism.

**Key words:** Composition and structure of the core; Equations of state; High-pressure behaviour; Numerical modelling; Planetary interiors.

## 1 INTRODUCTION

Constraining the core heat content, whether at the present time (Lin *et al.* 2003; Labrosse 2015) or for the primitive core (Clesi & Deguen 2023), implies making assumptions on the thermal expansion and compressibility of the core components. For this, the Grüneisen parameter (first defined in Grüneisen 1912) is often used. This parameter has the advantage of being comprised between 0.9 and 2 for metallic materials, which is a narrower range than the thermal expansion coefficient or bulk modulus, and its value has been determined by different methods: thermodynamic modelling (Anderson 1967; Al'Tshuler *et al.* 1987), *ab initio* (Dubrovinsky *et al.* 2000; Alfè *et al.* 2007), experiments (Jeanloz 1979; Umemoto & Hirose 2015). Using the Grüneisen parameter allows to simplify the models by getting rid of the thermal expansion parameter $\alpha$ which is more sensitive to the composition.

However, there are several approaches to model the variations of the Grüneisen parameter, especially with pressure. Some studies assume a constant value (Anderson & Ahrens 1994; Labrosse 2015), some use the Al'Tshulher formalism (Al'Tshuler *et al.* 1987; Dewaele *et al.* 2006; Umemoto & Hirose 2015), some use the power law first proposed by Anderson in 1967 (Anderson 1967; Dubrovinsky *et al.* 2000; Kuwayama *et al.* 2020), while others calculate it within the study: by *ab initio* in Alfè *et al.* (2007) and Ichikawa *et al.* (2014), by linear expansion in Badro *et al.* (2014). In this paper, we will use the results of our previous study (Clesi & Deguen 2023) to assess the model sensitivity to different approaches of Grüneisen parameter when modelling the initial heat

content of the core. We use previous results linking core composition and temperature, and test different ways of calculating the value of $\gamma$. By fitting the effect of each parameter on temperature, we can assess the variations introduced by using different formalisms, and the error introduced by variations of the parameters within the formalism chosen. We show that the formalism of Al'Tshuler (Al'Tshuler *et al.* 1987) is less prone to yield large errors in the calculations while being theoretically the most sound of all formalism studied.

## 2 GRÜNEISEN PARAMETER MODELLING AND THERMAL MODEL

### 2.1 Accretion scenario and thermal modelling

We use the accretion and core/mantle differentiation models that have been previously determined in Clesi & Deguen (2023). These models yield mantle compositions close to the Bulk Silicate Earth (BSE) given in McDonough & Sun (1995), while yielding compositions for the core compatible with a ~10 per cent wt. of light elements (Si and O) in the core. To determine the heat content and temperature of the core we consider the following steps:

(i) The initial temperature of each addition of metal is set at the bottom of the magma ocean, where the metal is assumed to equilibrate with the silicates. The initial temperature is therefore given by the liquidus of silicate at the pressure of the bottom of the magma ocean, as given by Andrault *et al.* (2011).

<footnote n="1">¹ENSL, UCBL, UJM, CNRS, LGL-TPE, Université de Lyon, Campus de la Doua, bâtiment Géode 2, rue Raphaël Dubois 69622 Villeurbanne Cedex, France.
E-mail: vclesi@uni-muenster.de</footnote>

<footnote n="2">²Department of Earth Science, Keith-Wiess Geological Laboratory, Rice University, 6100 Main Street Houston, TX 77005, USA</footnote>

<footnote n="3">³ISTerre, CNRS, IRD, Université Grenoble Alpes, Université Gustave Eiffel, Université Savoie Mont Blanc, 1381 Rue de la Piscine, 38610 Gières, France</footnote>
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Summary of the results from Clesi & Deguen (2023), showing the correlation between the isentropic temperature at the CMB at the end of accretion (y-axis), light elements concentration in the core (here Si and O, x-axis) and mean value of equilibrium pressure ($P_{eq}$, colour scale). The temperatures obtained here are obtained with the Grüneisen parameter calculated with Al'Tshuler formalism, with $\gamma_0 = 1.875$ and $\gamma_\infty = 1.305$. The points plotted here are the models matching the compositional constraints defined in Clesi & Deguen (2023).</figcaption></figure>

(ii) The metal is then heated by compression while migrating from the bottom of the magma ocean to the growing core. At each step of accretion its composition is different, and we do not consider any mixing, thus resulting in the formation of a stratified core [as in Jacobson et al. (2017)].

(iii) The initial temperature profile is then set by the additional compression of the metal up to the final core pressures. We use this initial temperature and density profile to calculate the heat content.

(iv) We assume that the core is then mixed from the stratified state to an isentropic state, and use the previously calculated heat content to get the corresponding temperature at the CMB ($T_{CMB}^{is}$). This is a strong assumption, since the core is often found to be stably stratified at the end of accretion (Clesi & Deguen 2023). Whether the core would be efficiently mixed depends on the radial variations in composition, ratio of temperature gradient to isentropic gradient (which depends on $\gamma$), and of the nature and intensity of the possible stirring processes (Jacobson et al. 2017; Bouffard et al. 2020). However, the main purpose of this assumption is to provide a single measure of the temperature of the core – the temperature $T_{CMB}^{is}$ of the CMB after mixing to an isentropic state – which can be seen as a convenient measure of the amount of heat stored into the core.

The details on the model and the different calculations are described in Clesi & Deguen (2023). The main result of the paper is that mean pressure of metal-silicate equilibrium, light elements concentration in the core, and core temperature are positively correlated. A summary of the results are presented in Fig. 1.

The initial temperature of the metal is set to be the liquidus temperature of the silicate (Andrault et al. 2011) at the bottom of the magma ocean where chemical equilibrium happens. It is given by:

$$T_{\text{eq}} = 1940 \left( \frac{P_{\text{eq}}}{29} + 1 \right)^{1/1.9}, \qquad (1)$$

where $P_{eq}$ is the pressure at the bottom of the magma ocean (in GPa) which is the pressure where metal and silicate are equilibrated. The temperature changes calculated in steps (ii) and (iii) are obtained from:

$$\frac{dT}{dP} = \frac{\gamma T}{K_s}, \qquad (2)$$

where $\gamma$ is the Grüneisen parameter of the metal, $T$ its temperature and $K_s$ its isentropic bulk modulus. We use the Murnaghan approximation for the bulk modulus,

$$K_s = K_0 + K' P, \qquad (3)$$

with $K_0 = 128.49$ GPa the bulk modulus for $P = 0$, and $K' = 3.67$ the first derivative of the bulk modulus, which yields the following equation of state for the metal:

$$\frac{\rho(P)}{\rho_0} = \left(1 + \frac{K'}{K_0} P\right)^{1/K'} \qquad (4)$$

with $\rho(P)$ is the density of the metal at pressure $P$. The value of $\rho_0$, that is the density of the metal at the reference pressure, is varying throughout accretion, depending on the composition of the metal, which is set by chemical equilibration with the silicates at the bottom of the magma ocean (see Clesi & Deguen (2023) for the details).

The heat content of the core is then calculated as

$$Q = 4\pi \int_{0}^{R_c} \rho(r) C_p T(r) r^2 dr, \qquad (5)$$

with $R_c = 3470$ km the total radius of the core, $C_p = 1000$ J kg$^{-1}$ K$^{-1}$ the specific heat of the metal and $T(r)$ the temperature in the core at the radius $r$; where the distance $r$ from the center and the pressure are linked by

$$P_{\text{core}}(r) = P_{\text{centre}} + \left( \frac{P_{\text{CMB}} - P_{\text{centre}}}{R_c^2} \right) r^2 \qquad (6)$$

with $P_{CMB}$ and $P_{centre}$ the pressure at the CMB and at the centre of the core, respectively. The isentropic temperature profile can then be obtained from

$$\left( \frac{\partial \ln T^{\text{is}}}{\partial \ln \rho^{\text{is}}} \right)_s = \gamma. \qquad (7)$$

where $\rho^{is}$ is the density profile of the core after isentropic mixing. As seen in eq. (7), the final isentropic temperature is a function of the Grüneisen parameter, $\gamma$. Depending on the $\gamma$ formalism (constant value, Al'Tshuler et al. (1987) or Anderson (1967) power laws), integration of eq. (7) will yield different results. We then consider that the core is fully mixed with a constant heat content. The isentropic mixed core temperature profile is determined by integration of eq. (7) for a mixed density profile $\rho^{is}(r)$, and combined with the heat content calculated by eq. (5), we can calculate the temperature at the CMB after mixing as

$$T_{\text{CMB}}^{\text{is}} = \frac{Q}{4\pi \int_{0}^{R_c} \rho^{\text{is}}(r) C_p T^{\text{is}}(r) r^2 dr}, \qquad (8)$$

which is evaluated numerically. In the following sections, we calculate $T_{CMB}^{is}$ from eq. (8) for the three different formalisms tested in this study, and we vary parameters values within each of the formalism to determine the sensitivity of $T_{CMB}^{is}$ to the formalisms and parameter values. Before varying the parameters, Fig. 2 shows how, for three solutions in the data set presented in Fig. 1, changing the formalism affects each step of the calculation. As can be seen on this figure, changing the value of the Grüneisen parameter tends
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Description of three steps of the model for different formalisms of the Grüneisen parameter tested in this study. Left-hand column: constant $\gamma = 1.7$. Middle column: Anderson's power law, with $\gamma_0 = 2.05$ and $b = 0.6$. Right-hand column: Al'Tshuler formalism with $\gamma_0 = 1.305$ and $\gamma_\infty = 1.875$. Each row represents one of the steps of the scenario described in Section 2.1. Top row: initial compression between the bottom of the magma ocean and the growing core-mantle boundary (step (i) and part of step (ii) in the text), with the liquidus curve in black and the adiabat of the first and last steps of the model shown by the arrows. The cross symbols represents the P-T conditions of the metal reaching the CMB at each step of accretion, before the core is fully formed. The arrows show the changes in P-T conditions undergone by the metal through the crystallized part of the mantle in the first step of accretion (round markers) and last step of accretion (square markers). Middle row: temperature profile of the core when stratified and after compression of the metal due to the growth of the core (step (ii) and step (iii) in the text). Bottom row: temperature profile in the core after mixing to an isentropic state (step (iv) in the text). The black dots mark the temperature at the CMB. A similar figure with more details on the model steps can be found at Clesi & Deguen (2023). Red curves: Model 1, obtained for the $f_c = 0.6$, $a_p = 0.4$ and $\lambda = 5$, with $P_{eq} = 19.7$ GPa and $\chi_{Si+O}^{core} = 2.47$ per cent. Blue curves: Model 2, obtained for the $f_c = 0.85$, $a_p = 0.6$ and $\lambda = 1$, with $P_{eq} = 34.1$ GPa and $\chi_{Si+O}^{core} = 5.24$ per cent. Green curves: Model 3, obtained for the $f_c = 1$, $a_p = 0.65$ and $\lambda = 0.4$, with $P_{eq} = 43.9$ GPa and $\chi_{Si+O}^{core} = 7.86$ per cent.</figcaption></figure>

to shift the temperature profiles up and down, irrespectively of the stratification.

## 2.2 Constant Grüneisen parameter

The first assumption that can be made is assuming $\gamma$ to be constant. Integration of eq. (2) then yields:

$$T(P) = T_{eq} \left( \frac{K' P + K_0}{K' P_{eq} + K_0} \right)^{\frac{\gamma}{K'}} \quad (9)$$

This equation is used as input to calculate the heat content (eq. 5) and then the value of $T_{CMB}^{is}$ (eq. 8).

## 2.3 Power law formalism of Anderson

The second assumption is that the variation of $\gamma$ follows a power law of the form

$$\gamma = \gamma_0 \left( \frac{\rho_0}{\rho} \right)^b \quad (10)$$

where $\gamma_0$ is the Grüneisen parameter for $\rho = \rho_0$ and $b$ is the exponent of the power law. This formalism has been proposed by Anderson (1967), in order to simplify the calculation of the thermal expansion coefficient of some materials. This power law is practical for integration, and is especially fitted for Murnaghan equation of state, as it was one of the reasons for choosing this formalism in the original paper. eq. (2) with $\gamma$ given by eq. (10) has the following
analytical solution:

$$T(P) = T_{\text{eq}} \exp \left[ \frac{\gamma_0}{b} \left( \left( 1 + \frac{K' P_{\text{eq}}}{K_0} \right)^{-b/K'} - \left( 1 + \frac{K' P}{K_0} \right)^{-b/K'} \right) \right]. \quad (11)$$

## 2.4 Formalism of Al'Tshuler

The last model investigated in this study is given by

$$\gamma = \gamma_{\infty} + (\gamma_{0,j} - \gamma_{\infty}) \left(\frac{\rho_0}{\rho}\right)^{\beta}, \quad (12)$$

where $\beta = \gamma_{0,j}/(\gamma_{0,j} - \gamma_{\infty})$, and $\gamma_{0,j}$ the Grüneisen parameter for $\rho = \rho_0$ and $\gamma_{\infty}$ the asymptotic value of the Grüneisen parameter when $P \to \infty$. This equation also takes the form of a power law, used by multiple studies (Dewaele et al. 2006; Clesi & Deguen 2023). It is however different than the one from Anderson (1967), because it is derived from the theoretical isothermal density evolution of metal (Al'Tshuler et al. 1987). Though eq. (12) might be seen as an extended version of Anderson's power law (it is the sum of a power law in $\rho$ and a constant), note that the power exponent and the pre-factor of the power law part are here linked theoretically, which is not the case in Anderson's power law. One implication is that the number of parameters involved in Al'Tshuler's formalism is not higher than in Anderson's formalism, in spite of its seemingly greater complexity. With this formalism, integrating eq. (2) gives

$$T(P) = T_{\text{eq}} \left( \frac{K_0 + K' P}{K_0 + K' P_{\text{eq}}} \right)^{\frac{\gamma_\infty}{K'}} \times \exp \left[ \frac{\gamma_{0,j} - \gamma_\infty}{\beta} \left( \left( 1 + \frac{K' P_{\text{eq}}}{K_0} \right)^{-\frac{\beta}{K'}} - \left( 1 + \frac{K' P}{K_0} \right)^{-\frac{\beta}{K'}} \right) \right]. \quad (13)$$

## 2.5 Sensitivities of $T_{\text{CMB}}^{\text{is}}$ to the different parameters

In order to compare the three formalisms presented above, we fitted eqs (10) and (12) to a data set of Grüneisen parameter measurements for pure iron. This allows to obtain in a self-consistent way the values and uncertainties of the parameters appearing in Al'Tshuler et al. (1987) and Anderson (1967)'s formalisms. We use the data set provided by Murphy et al. (2011), to which we fitted the values of $\gamma_0$, $b$, $\gamma_\infty$ and $\gamma_{0,j}$ (Fig. 3). Given the limited number of points (10) in the data set, we created for each value of $\frac{\rho_0}{\rho}$ a random data set of 20 $\gamma$ values distributed following a normal law centred on the mean value with a standard deviation equal to the uncertainty given in Murphy et al. (2011). We then fitted the parameters of each formalism to the data set, thus allowing us to define a range of value for each parameter, given in Table 1, that will be tested throughout the study.

We use the subset of core formation models ($n=382$) from Clesi & Deguen (2023). Each solution represents a different evolution of the core composition while yielding an acceptable fit on the Bulk Silicate Earth (McDonough & Sun 1995). We then calculate for each solution the CMB temperature after core mixing with eqs (5) and (8) with the three different formalisms: eq. (9) for constant $\gamma$, eq. (10) for Anderson's power law, and eq. (13) for Al'Tshuler's formalism. We vary each parameter independently to assess their effect on the

mean and variance of $T_{\text{CMB}}^{\text{is}}$ value on the 382 models tested. In the following sections we discuss the strengths and weaknesses of each formalism in the same style.

For each formalism, we estimate the effect of varying parameter values on the mean value of $T_{\text{CMB}}^{\text{is}}$ in the data set. In order to estimate the sensitivity of the model results to the parameters, we fitted a linear equation to explain the mean $T_{\text{CMB}}^{\text{is}}$ value of our data set:

$$T_{\text{CMB}}^{\text{is}} = a_0 + a_1 x \quad (14)$$

where $x$ is one of the parameters ($\gamma, \gamma_0, b$ or $\gamma_\infty$).

The size of the data set we chose [382 different accretion scenarios, spanning different core compositions, see Clesi & Deguen (2023)] also allows us to assess the sensitivity of the actual values within a formalism to the others parameters in the models. Indeed, the variations of $T_{\text{CMB}}^{\text{is}}$, as shown in Clesi & Deguen (2023), depend also on the composition of the core (light element concentrations), the pressure of equilibrium and its variation throughout accretion as well as the oxygen fugacity of the impactors. A small dispersion of the $T_{\text{CMB}}^{\text{is}}$ values means that the Grüneisen parameter obtained for the set of parameter tends to mask the sensitivity of the model to other parameters (composition, pressure, etc.). A large dispersion of $T_{\text{CMB}}^{\text{is}}$ on the other hand means that the Grüneisen parameter tends to exacerbate the effect of the other parameters of the model.

As shown in the following sections, varying the values of the Grüneisen parameter can change the output of the same models by several hundreds of Kelvin. When modelling the compression of liquid metal, especially at high pressure, the data on compressibility is mostly derived from solid iron experiments and ab initio calculations. This situation presents several problems when modelling the Earth's core: the fact that liquid metal compressibility is not well constrained, and the fact that several light elements can affect this compressibility compared to pure iron. The Grüneisen parameter presents the advantage of having a limited range of value, thus extrapolating from solid iron value to the metal alloy forming the core limits the risk in terms of temperature calculation.

On the other hand, this advantage of having less chance to be far away from the results can become a disadvantage when trying to be more precise (for instance investigating the effect of small variations in composition on the temperature).

In the following sections we discuss the effect and robustness of the different formalisms presented above by using the sensitivity results obtained by fitting eq. (14) to the mean values of $T_{\text{CMB}}^{\text{is}}$. To do so we will estimate the variation of temperature induced by a deviation from the following values:

(i) In Section 3, a constant $\gamma$ of 1.7, as used by Labrosse (2015).

(ii) In Section 4, Anderson (1967) formalism with $\gamma_0 = 2.05$ and $b = 0.63$ from Kuwayama et al. (2020).

(iii) In Section 5, Al'Tshuler et al. (1987) formalism with $\gamma_0 = 1.875$ and $\gamma_\infty = 1.305$ from Clesi & Deguen (2023).

## 3 CONSTANT GAMMA

Fig. 4 shows the effect of varying the value of a constant $\gamma$ on the isentropic temperature at the CMB. The range of $\gamma$ tested is between 0.9 and 2.1, which is representative of the range of values of $\gamma$ for iron given by several authors (see Dubrovinsky et al. (2000) or Wagle & Steinle-Neumann (2019) and references therein). $T_{\text{CMB}}^{\text{is}}$ is increasing with increasing $\gamma$: the mean value goes from 3424 K to 4615 K. The effect of $\gamma$ on the mean CMB temperature can be fitted by a polynomial function, for which the parameters are given in Table 2.
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Results of the fit from the data of Murphy *et al.* (2011) with the formalism of Anderson (1967) (left-hand panel) and Al'Tshuler *et al.* (1987) (right-hand panel). The data from Murphy *et al.* (2011) have been bootstrapped to account for the error bar by randomly distributing 20 values for each point following a Gaussian distribution, thus allowing the fit to be made on ~200 points instead of 10.</figcaption></figure>

<table><caption>**Table 1.** Values obtained from fitting the measurements of $\gamma$ from Murphy *et al.* (2011) to the Anderson's and Al'Tshuler's formalisms.</caption><thead><tr><th>Parameters</th><th>γ<sub>0</sub></th><th>b</th><th>γ<sub>0,j</sub></th><th>γ<sub>∞</sub></th></tr></thead><tbody><tr><td>Mean value</td><td>1.875</td><td>0.752</td><td>1.933</td><td>0.916</td></tr><tr><td>Minimum value</td><td>1.555</td><td>0.432</td><td>1.608</td><td>0.591</td></tr><tr><td>Maximum value</td><td>2.195</td><td>1.07</td><td>2.258</td><td>1.241</td></tr><tr><td>1σ</td><td>0.08</td><td>0.08</td><td>0.0812</td><td>0.0812</td></tr></tbody></table>

<figure><img src="image_5.png" /><figcaption>**Figure 4.** Variation of the mean $T_{\text{CMB}}^{\text{is}}$ determined by eqs (8) and (9) as a function of $\gamma$. The error bar are the $2\sigma$ variation on the subset of core formation models used in this study. For comparison the estimates of current CMB temperature from Nomura *et al.* (2014) (dotted blue line) and Davies *et al.* (2015) (dashed green line) are also shown.</figcaption></figure>

<table><caption>**Table 2.** Values of parameters fitting the trend of mean values in Fig. 4. The equation fitted is eq. (14) with $x$ replaced by $\gamma$.</caption><thead><tr><th>a<sub>0</sub></th><th>a<sub>1</sub></th><th>χ<sup>2</sup></th></tr></thead><tbody><tr><td>2116</td><td>1202</td><td>1.6676</td></tr></tbody></table>

The $2\sigma$ values are higher as $\gamma$ decreases: from 440 K at $\gamma = 0.9$ to 60 K at $\gamma = 2.05$. This indicates that the variability coming for the different accretion histories is buffered by high values of $\gamma$.

From Table 2, it is possible to calculate the $\Delta T$ (error on the final temperature at the CMB) induced by changing the value of $\gamma$ from a reference value chosen to be $\gamma = 1.7$. With the value of $a_1$ from Table 2, changing $\gamma$ by $\pm 0.1$ induces a variation $\Delta T = 120$ K, which is higher than the dispersion of values observed in the other formalisms tested in this study. This trend shows that the results are highly sensitive to the value of a constant $\gamma$.

## 4 ANDERSON'S POWER LAW

The formalism of Anderson (1967) has two parameters that can affect the sensitivity of $T_{\text{CMB}}^{\text{is}}$ to $\gamma$. The parameter $b$ values are empirically fitted for each composition of the metal, with values spanning from 0.63 (Kuwayama *et al.* 2020) and 0.69 (Dubrovinsky *et al.* 2000) to 1.0 (McQueen *et al.* 1970) and 1.69 (Jeanloz 1979). In the case of liquid metal, most of the values converges toward $b \le 1$. Here we tested values of $b$ between 0.4 and 1.1 so as to cover the entire range of $b$ values obtained from the fitting of Murphy *et al.* (2011)'s data set (see Table 1). As for the $\gamma_0$ value, it is the value for liquid metal at 1 bar, and since $\gamma$ decreases with pressure it has to be higher than the value of $\gamma$ at high pressures. The published range for $\gamma_0$ is between 1.59 (Brown & McQueen 1986) and 2.05 (Kuwayama *et al.* 2020), with 1.713 (Anderson & Ahrens 1994) and 1.8 (Dubrovinsky *et al.* 2000) having also been proposed and used. As for the value of $b$, we tested the range presented in Table 1, which is 1.5 to 2.2.

The top panel of Fig. 5 shows that increasing the value of $\gamma_0$ tends to increase the mean value of $T_{\text{CMB}}^{\text{is}}$, irrespectively of the value of $b$. For $\gamma_0 = 1.2$, the mean value of $T_{\text{CMB}}^{\text{is}}$ is between 3227 and 3492 K, and for $\gamma_0 = 2.1$ the mean value is between 3869 and 4474 K, depending on the value of $b$; for $b = 1.1$, the temperature varies between 4131 and 3500 K over the same $\gamma_0$ range. The sensitivity of $T_{\text{CMB}}^{\text{is}}$ to $\gamma_0$ is stronger at the lower values of $b$, though this effect is not very strong. The bottom panel of Fig. 5 shows the effect of increasing the exponent of the power law. The higher $b$ is, the lower the temperature is: for $b = 0.4$ the temperature is between 4535 and 3750 K depending on $\gamma_0$ values; for $b = 1.1$ the mean value is between 3527 and 4131 K. The higher the value of $\gamma_0$ and the lower the value of $b$, the lower the dispersion of values: for $b = 0.4$, the $2\sigma$ variation is 22 K at $\gamma_0 = 2.2$ and 150 K at $\gamma_0 = 1.5$; for $b = 1.1$ the $2\sigma$ is 48 and 160 K for $\gamma_0 = 2.2$ and $\gamma_0 = 1.5$, respectively.

A linear fit (eq. 14) of the effect of both parameters on the mean values can be performed with good $\chi^2$ values. The fitted parameter
<figure><img src="image_6.png" /><figcaption>**Figure 5.** Top panel: effect of $\gamma_0$ on $T_{\text{CMB}}^{\text{is}}$ for different values of b. Red squares: $b = 0.4$. Dark blue squares: $b = 0.6$. Dark green squares: $b = 1.1$. Bottom panel: effect of b on $T_{\text{CMB}}^{\text{is}}$ for different values of $\gamma_0$. Red squares: $\gamma_0 = 2.2$. Dark green squares: $\gamma_0 = 1.875$. Dark blue squares: $\gamma_0 = 1.5$. $T_{\text{CMB}}^{\text{is}}$ is calculated from eqs (8) and (11). For comparison, we also show the estimates of present-day CMB temperature from Nomura et al. (2014) (dotted blue line, lower estimate) and the review of Davies et al. (2015) (dashed green line).</figcaption></figure>

<table><caption>**Table 3.** Values of parameter fitting the trend of mean values in Fig. 5. The equations fitted are: $\bar{T}_{\text{CMB}}^{\text{is}} = a_0 + a_0\gamma_0$, and $\bar{T}_{\text{CMB}}^{\text{is}} = a_0 + a_1b$. The first part of the table shows the variation for fixed values of b (Fig. 5, left-hand panel), and the second part shows the variation for fixed values of $\gamma_0$ (Fig. 5, right-hand panel).</caption><thead><tr><th>Conditions</th><th>a<sub>0</sub></th><th>a<sub>1</sub></th><th>&chi;<sup>2</sup></th></tr></thead><tbody><tr><td>b = 0.4</td><td>1984</td><td>1158</td><td>0.1855</td></tr><tr><td>b = 0.6</td><td>2059</td><td>1065</td><td>0.1151</td></tr><tr><td>b = 1.1</td><td>2197</td><td>877</td><td>0.0424</td></tr><tr><td>&gamma;<sub>0</sub> = 2.2</td><td>4764</td><td>-587</td><td>0.4068</td></tr><tr><td>&gamma;<sub>0</sub> = 1.875</td><td>4319</td><td>-445</td><td>0.0244</td></tr><tr><td>&gamma;<sub>0</sub> = 1.5</td><td>3872</td><td>-317</td><td>0.0041</td></tr></tbody></table>

are shown in Table 3, showing a positive effect of $\gamma_0$, and a negative effect of b.

Let us now assume that we used the values of $\gamma_0 = 2.05$ and $b = 0.6$ from Kuwayama et al. (2020) to make the calculation on the models of Clesi & Deguen (2023). It is then possible to use the parameters from Table 3 to calculate what is the induced error if the 'true' values are different. For $\gamma_0 = 2.2$, changing b by only

0.1 (corresponding to 15 per cent of the range of values given in Table 1) changes $T_{\text{CMB}}^{\text{is}}$ by 58 K. If we rather consider $b = 0.6$, then a variation of 0.1 (also corresponding to 15 per cent of the range of values given in Table 1) for the parameter $\gamma_0$ yields $\Delta T \sim 100$ K.

The error induced by getting a wrong value for the exponent is therefore less important than getting the value of $\gamma_0$ wrong, but the variations are not negligible, especially if both parameters estimations are wrong: if for example $\gamma_0 = 1.875$ and $b = 0.4$, then the final temperature calculated with the reference values ($\gamma_0 = 2.05$ and $b = 0.6$) is overestimated by $\sim 300$ K.

## 5 AL'TSHULER POWER LAW

The formalism of Al'Tshuler et al. (1987) depends on two parameters, $\gamma_0$ and $\gamma_\infty$, with $\gamma_\infty < \gamma_0$. The values of $\gamma_\infty$ represent the minimum value of the Grüneisen parameter for infinite pressure (i.e. when the compressibility reaches a minimum asymptotic value) due to the quantum-statistical Grüneisen coefficient under extreme pressure (Gilvarry 1956; Burakovsky & Preston 2004). For liquid iron this value is between 1 and 1.4 (Dewaele et al. (2006), and references therein), and we tested values between 0.6 and 1.25 as given in Table 1. As for $\gamma_0$, it is the value of the Grüneisen parameter at the pressure of the reference state ($\rho/\rho_0 = 1$). Therefore it is higher than $\gamma_\infty$ and close to the values of the parameter $\gamma_0$ from Anderson (1967), studied in the previous section. Here we tested values between 1.6 and 2.25, as given by the results of the fit in Table 1. Fig. 6 shows that the temperature is positively correlated with both $\gamma_0$ and $\gamma_\infty$. Higher values of $\gamma_0$ lead to less dispersion of the results: for instance, at $\gamma_\infty = 0.6$, the $2\sigma$ value for the data set is 35 K for $\gamma_0 = 2.25$ and 144 K for $\gamma_0 = 1.6$. When $\gamma_0$ is fixed, varying the value of $\gamma_\infty$ has less impact on the dispersion of the results: for instance at $\gamma_0 = 1.6$, the $2\sigma$ for the data set is 141 K for $\gamma_\infty = 0.9$ and 128 K for $\gamma_\infty = 1.25$.

The variation of the mean temperature of our data set is more affected by varying $\gamma_0$ than $\gamma_\infty$. For instance, the mean temperature goes from 4319 to 3762 K with $\gamma_\infty = 1.25$ and for $\gamma_0$ varying from 2.25 to 1.6, respectively. On the other hand the temperature decreases from 4319 to 4254 K with $\gamma_0 = 2.25$ for $\gamma_\infty$ varying from 1.25 to 0.6, respectively.

The linear fits of the mean values of $T_{\text{CMB}}^{\text{is}}$ yield good $\chi^2$ values, with the parameters values given in Table 4. $T_{\text{CMB}}^{\text{is}}$ correlates positively with both $\gamma_0$ and $\gamma_\infty$, and the strongest effect of $\gamma_0$ is due to the higher value of $a_1$ (Table 4).

The variations in temperature are minimized if $\gamma_0 \ge 2$ and $\gamma_\infty \le 1$, as shown by the corresponding values of the fit in Table 4. Therefore, choosing high values of $\gamma_0$ [like 1.837 as in Dewaele et al. (2006) and Clesi & Deguen (2023) or 2.05 as in Kuwayama et al. (2020)] combined with relatively low values of $\gamma_\infty$ can minimize the error in the output. The $\gamma_\infty$ value of 1.3 used in Dewaele et al. (2006) and Clesi & Deguen (2023) or 1.2 for Dubrovinsky et al. (2000) are a bit high in terms of minimizing the dispersion and error in the output.

Let us now assume that the values of $\gamma_0 = 1.875$ and $\gamma_\infty = 1.305$ in the original study of Clesi & Deguen (2023) [taken from Dewaele et al. (2006)] are wrong. It is then possible to estimate the error induced in the final mean temperature by calculating the variation in temperature induced by a variation in the values of $\gamma_0$ and $\gamma_\infty$ using the parameters in Table 4. For $\gamma_0 = 1.875$, a variation in $\gamma_\infty$ value by 0.1 induces a variation of the temperature of $\sim 15$ K. The main parameter that can induce error is $\gamma_0$: for $\gamma_\infty = 1.25$, a deviation of $\pm 0.1$ in the value of $\gamma_0$ leads to a deviation of
<figure><img src="image_7.png" /><figcaption>**Figure 6.** Evolution of the mean value of $T_{\text{CMB}}^{\text{is}}$ as a function of $\gamma_0$ (top) and $\gamma_{\infty}$ (bottom) in the formalism of Al'Tshuler et al. (1987), given by eqs (13) and (8). On the top panel $\gamma_{\infty}$ values are fixed at 1, 1.25, 1.305 and 1.5. On the bottom panel, $\gamma_0$ values are fixed at 2.05, 1.875, 1.75 and 1.5. For comparison we also show the estimates of current CMB temperature from Nomura et al. (2014) (dotted blue line) and Davies et al. (2015) (dashed green line).</figcaption></figure>

<table><caption>**Table 4.** Values of parameters fitting the trend of mean values in Fig. 6. The equation fitted is: $T_{\text{CMB}}^{\text{is}} = a_0 + a_1\gamma_x$, with $\gamma_x$ being $\gamma_0$ or $\gamma_{\infty}$. The first part of the table are for fixed values of $\gamma_{\infty}$, the second one for fixed values of $\gamma_0$. The value of $\chi^2$ for each fit is given in the last column.</caption><thead><tr><th>Conditions</th><th>a<sub>0</sub></th><th>a<sub>1</sub> &chi;<sup>2</sup></th><th></th></tr></thead><tbody><tr><td>&gamma;<sub>0</sub> = 2.25</td><td>4191</td><td>100</td><td>0.0047</td></tr><tr><td>&gamma;<sub>0</sub> = 1.9</td><td>3845</td><td>115</td><td>0.0013</td></tr><tr><td>&gamma;<sub>0</sub> = 1.6</td><td>3560</td><td>150</td><td>0.0020</td></tr><tr><td>&gamma;<sub>&infin;</sub> = 0.6</td><td>2151</td><td>933</td><td>0.0538</td></tr><tr><td>&gamma;<sub>&infin;</sub> = 0.9</td><td>2188</td><td>927</td><td>0.0629</td></tr><tr><td>&gamma;<sub>&infin;</sub> = 1.25</td><td>2312</td><td>890</td><td>0.1284</td></tr></tbody></table>

~90 K. Then if the value of $\gamma_0$ is not the one proposed by Dewaele et al. (2006), but the one proposed by Kuwayama et al. (2020) ($\gamma_0$ = 2.05), even if the value of $\gamma_{\infty}$ is correct, then $\Delta T$ would be a positive 200K, meaning that the initial value from Clesi & Deguen (2023) underestimates the temperature. Getting the value wrong would place the conclusions of the study towards a conservative estimate of the temperature.

# 6 DISCUSSION

We propose in the previous sections an overview of the effect of each parameter on the output of a given model. All three formalisms have their own merits and none of them should be discarded a priori. In this section we will provide an estimate of the error induced by the type and values used in each formalism, and argue that one can choose the formalism that suits the best the purpose of the study.

## 6.1 Thermodynamic theory compliance versus practicality

In terms of theoretical merits, Al'Tshuler et al. (1987) is the more correct formalism. It is derived from the study of variations in isotherms using the original definition of the parameter of Grüneisen (1912), and Mie-Grüneisen equation of state. The whole original paper of Al'Tshuler et al. (1987), is very strong in terms of theoretical compliance, since the relationship between density and Grüneisen parameter is derived by calculus alone. It also takes into account the asymptotic behaviour of $\gamma$ at high pressure (Burakovsky & Preston 2004). On the other hand, the formalism of Anderson (1967) is justified only by the sentence:

> "Assuming the power law $\gamma = \gamma_0(V/V_0)^q$

followed by an integration of the $\gamma$ function. In terms of theoretical soundness, it is less sound than Al'Tshuler et al. (1987) study. But in terms of integration and data fitting, it is more convenient. Indeed, this formalism combined with a Murnaghan equation of state yield an easy-to-integrate formula, while still fitting the experimental data.

The same kind of reasoning applies for studies using constant $\gamma$, despite its limitations. Table 2 shows that the variations in temperature induced by a choice of constant value are much larger. Furthermore, there is extensive experimental (Boehler & Ramakrishnan 1980; Dubrovinsky et al. 2000) and theoretical (Gilvarry 1956; Al'Tshuler et al. 1987) evidences that the Grüneisen parameter is not independent of pressure. However, the integration of a constant parameter within a much more complicated model tends to simplify readability and interpretations. One example is the study of the energetics of the core (e.g. Labrosse 2015): $\gamma$ is not expected to vary strongly within the core, and the effect of these variations is likely secondary compared to the effect of thermal conductivity variations.

## 6.2 Assessing the uncertainties in the output of the model

Using the data set of Murphy et al. (2011), we fitted the range of plausible value for each parameter in the formalism of Anderson (1967) and Al'Tshuler et al. (1987). This allows us to assess the error induced by choosing one formalism over another, and the error induced by choosing a parameter value over another in a given formalism. When using the mean value presented in Table 1 in each formalism, we can compare the effect of choosing one formalism over another by calculating, for each model in the data set, the difference in $T_{\text{CMB}}^{\text{is}}$. In Table 5, we show the results of this comparison with $\Delta T = T_{\text{CMB}}^{\text{is}}(\text{Anderson}) - T_{\text{CMB}}^{\text{is}}(\text{Al'Tshuler})$. The formalism of Anderson tends to yield higher values of $T_{\text{CMB}}^{\text{is}}$ in any case, but the difference is small (8 K, 0.2 per cent of variation maximum). Therefore, as long as the parameters of each formalism are consistently fitted to the same data set, choosing a formalism does not induce much variations. On the other hand, the range of values chosen within a formalism is much more important than the variations induced by a chosen formalism. In Table 6, we show the
<table><caption>**Table 5.** Difference $\Delta T$ between the core temperature $T_{CMB}^{is}$ obtained using Anderson's and Al'Tshuler's power laws with the mean values of the parameters given in Table 1. The table gives the mean difference amongst our core-formation models (Clesi & Deguen 2023), as well as the minimum and maximum of $\Delta T$. The formalism of Al'Tshuler is the reference point for calculating the relative variation in temperature.</caption><thead><tr><th></th><th>Absolute<br>ΔT (K)</th><th>Relative ΔT<br>(per cent)</th></tr></thead><tbody><tr><td>Mean</td><td>3.21</td><td>0.08</td></tr><tr><td>Minimum</td><td>0.008</td><td>0.0001</td></tr><tr><td>Maximum</td><td>8.05</td><td>0.20</td></tr></tbody></table>

range of variation when using the maximum and minimum values of the parameters presented in Table 1. Variation in the values of $\gamma_0$ is the most important: the mean variation induced by a change in $\gamma_0$ value in both formalisms yield a mean $\Delta T$ of ~300 K, slightly lower in the Al'Tshuler formalism (Table 6). Varying the parameter $\gamma_\infty$ within the range given in Table 1 yield a low error range, between -26 and +50 K (Table 6). On the other hand, the range of b values given in Table 1 yield variations of temperature between -136 and +156 K. The Anderson formalism is more prone to yielding large error: taking into account both parameters value ranges, the final value of $T_{CMB}^{is}$ can vary by ~20 per cent, with a mean variation of 11-15 per cent. The range of $T_{CMB}^{is}$ is smaller when using Al'Tshuler formalism: given the small variation induced by an error on $\gamma_\infty$, the maximum error on the final value of $T_{CMB}^{is}$ is ~10 per cent, with a mean error of ~8 per cent. In summary, the Al'Tshuler formalism yield lower uncertainties than the Anderson formalism on this type of model. However, when fitting any of the formalism to the same data set, there is little to none variation in the output (Table 5). Since, as it is done in this study, it is possible to explain satisfactorily the same data with two different formalisms, the choice of formalism is not critical. For instance the data of Boehler & Ramakrishnan (1980) is fitted with Anderson (1967) formalism, but is used by Al'Tshuler et al. (1987) to test the formalism. The two formalisms are close in terms of mathematical writing (both of them are power laws) so it may be that for a given problem and a given data set of $\gamma$ values, either formalism can be used (with different values of parameters). In this instance it depends on the quality of the data available and the best fit available. This problem of uncertainties range and representativity of the data needs to be addressed when choosing a formalism and the values of parameters.

## 6.3 Choosing a formalism and its parameter value: a function of the study's goal

In the original study of Clesi & Deguen (2023) the choice has been made to use the formalism of Al'Tshuler et al. (1987) with the values of Dewaele et al. (2006). In the supplementary information of the same study are presented different results with the Grüneisen parameter formalism of Anderson (1967) with the values of Kuwayama et al. (2020). The results are sensibly different with everything else being the equal. In this section we argue that, for this particular type of model, it is indeed better to use Al'Tshuler et al. (1987) formalism, because it enhances the robustness and replicability of the results. Indeed, using Al'Tshuler formalism is limiting the variance and the risk of error, as shown previously in Section 5, and limits the overall uncertainty of the result as shown in Section 6.2. Furthermore, in the models presented in Clesi & Deguen (2023) and briefly re-explained in Section 2.1, there are several hypotheses that are made and are a source of possible error in the model;

among others: the number of element in the compositional model, the equilibrium rate, the discretization of core/mantle segregation in 20 steps, the choice of equation of state, the parameters values of equation of state, the neglect of dissipation and diffusion, the thermal state of the solid mantle. All of these hypotheses are more accurately described and justified in the original publication. On top of these simplifying hypotheses, the values of $\gamma_0$ and $\gamma_\infty$ from Dewaele et al. (2006) are assumed to be independent of the composition of the core, which might be a source of error in the model. Choosing a robust formalism that limits the variation if those values are wrong is then a better option: for instance the main purpose of the publication of Clesi & Deguen (2023) is to show the existence of a correlation between core composition and its temperature, doing so by applying a number of hypotheses, which is a broader goal than getting a precise value for the core temperature. Thus, limiting the scattering of the results when many other process in the model might also be a source of scattering helps us to get a better view of the problem. After all, for models of this type, it may be hard to tell if the scattering of the results is an actual scattering or an artifact created by the hypotheses and calculations techniques. This kind of limitation in the scattering also facilitates comparison between studies. For instance, one topic that is highly debated is the amount of light elements such as N, H or C in the core (Malavergne et al. 2019; Grewal et al. 2019; Fischer et al. 2020; Blanchard et al. 2022; Suer et al. 2023). The presence of such elements in the core will affect the temperature of the core by affecting the density of the metal, which in turn affects the temperature (through the effect of density on $\gamma$; see eqs (2), (7) and (8). However, each of the aforementioned study use a different model of accretion with a different set of hypothesis than in the Clesi & Deguen (2023). If one were to calculate the effect of carbon on temperature using the data and accretion models of Fischer et al. (2020) or Blanchard et al. (2022) studies in combination with a thermal evolution model, and find a significant effect of the carbon concentration on the temperature, can this effect be attributed to carbon or to the type of accretion and thermal model used to calculate carbon concentration and temperature? Among the source of uncertainties is the Grüneisen parameter formalism and value. Using a less sensitive formalism such as Al'Tshuler will at least close one of the point of discussion about the validity of the results: whether or not the values of $\gamma_0$ and $\gamma_\infty$ are the 'true' values, at least the error is low and if differences arise between models, then they are probably not due to the Grüneisen parameter. On the other hand, if the end goal is to best describe the entirety of the phenomenon or get a precise estimate of the core temperature (Driscoll & Davies 2023; Dobrosavljevic et al. 2022), then the best formalism is the one that fits the best the data, or the values that are calculated directly within ab initio studies (Vočadlo 2007; Alfè 2009; Alfè et al. 2007). If one would use a core segregation model to calculate the actual temperature at the CMB instead of highlighting the correlations between parameters, or actually deriving a precise value on those correlations, then the choice of formalism and parameters value must be driven by the quality of the data, the quality of the fit, and the range of uncertainties on the parameters as highlighted in Section 6.2. As an example, let us assume one wants to calculate the effect of hydrogen incorporation in the core on temperature using for instance models from Clesi et al. (2018), Malavergne et al. (2019) or Suer et al. (2023). There are some data available on the hydrogen effect on the Grüneisen parameter (Umemoto & Hirose 2015). If Anderson's formalism fitted to the data from Umemoto & Hirose (2015) yields a narrower range of value for $\gamma_0$ and b, than the range of value for $\gamma_0$ and $\gamma_\infty$ obtained through fitting the data with Al'Tshuler formalism, then using the formalism of Anderson
<table><caption>**Table 6.** Variation of $\Delta T$ within a chosen formalism. The absolute value are in K, the number in parenthesis are the variation relatively to the mean value in per cent. Top part of the Table: Anderson's power law parameter. Bottom part: Al'Tshuler power law parameter. The variation on temperature is calculated using the temperature obtained by calculating with the mean value of the parameter presented in Table 1. The terms maximum and minimum refers to the absolute deviation from the initial value, not to the value itself (i.e. -205 K is a higher value than -313 K but the absolute variation is lower).</caption><thead><tr><th></th><th>Mean ΔT</th><th>Minimum ΔT</th><th>Maximum ΔT</th></tr></thead><tbody><tr><td>γ<sub>0</sub> = 1.555</td><td>-300 K (-7.56 per cent)</td><td>-226 K (-5.47 per cent)</td><td>-334 K (-8.52 per cent)</td></tr><tr><td>γ<sub>0</sub> = 2.195</td><td>331.5 K (8.34 per cent)</td><td>243 K (5.90 per cent)</td><td>372 K (9.49 per cent)</td></tr><tr><td>b = 0.432</td><td>152 K (3.8 per cent)</td><td>136 K (3.30 per cent)</td><td>156 K (3.98 per cent)</td></tr><tr><td>b = 1.07</td><td>-132 K (8.34 per cent)</td><td>-118 K (5.89 per cent)</td><td>-136 K (9.48 per cent)</td></tr><tr><td>γ<sub>0</sub> = 1.608</td><td>-279 K (-7.03 per cent)</td><td>-205 K (-4.98 per cent)</td><td>-313 K (-7.99 per cent)</td></tr><tr><td>γ<sub>0</sub> = 2.258</td><td>311 K (7.83 per cent)</td><td>223 K (5.43 per cent)</td><td>351 K (8.97 per cent)</td></tr><tr><td>γ<sub>∞</sub> = 0.591</td><td>-27.15 K (-0.68 per cent)</td><td>-26 K (-0.64 per cent)</td><td>-27 K (-0.70 per cent)</td></tr><tr><td>γ<sub>∞</sub> = 1.241</td><td>47 K (1.17 per cent)</td><td>45 K (1.09 per cent)</td><td>47 K (1.19 per cent)</td></tr></tbody></table>

would be better, especially if the range is narrow enough to yield smaller error than the one presented in Table 6.

## 6.4 Implications for the CMB temperature

As highlighted in the previous sections, there are some limitations to the inferences that can be made yet as to the relationship between the Grünesien parameter and the core temperature. The main goal of this study is to derive the sensitivity of $T_{CMB}^{is}$ to the variations of parameters controlling the Grüneisen parameter. However, from the sensitivity study some implications can be drawn about the initial temperature of the core.

All three formalisms applied to the model described in Section 2.1 can yield acceptable $T_{CMB}^{is}$ for the Earth when compared to current estimates of CMB temperature from Zhang et al. (2016) or Davies et al. (2015) (4000 ± 200 K), Nomura et al. (2014) (3570 ± 200 K) or Dobrosavljevic et al. (2022) (3500 ± 200 K). In all the formalisms presented, it is possible to find values of the parameters that yield initial $T_{CMB}^{is}$ higher than the current estimates listed above. However, none of the values tested and presented in Figs 4, 5 and 6 can yield initial temperature at the CMB compatible with the estimates of Andrault et al. (2017) (5400 ± 100 K) or Driscoll & Davies (2023) (5000–6000 K). This may be due to the own limitations of the model used in this study. Indeed, among other limitations described in the original paper (Clesi & Deguen 2023), the viscous dissipation that tends to increase temperature of the core (King & Olson 2011) is neglected, thus leading to lower temperatures. Furthermore the choice of the Murnaghan equation of state to simplify the calculations can also lead to an underestimation of temperature. This does show the importance of having constraints on this parameter when trying to constrain the core temperature precisely.

## 7 CONCLUSION

The Grüneisen parameter $\gamma$ is an important parameter when studying the thermal state of the core, yet its value is not very well known for different composition of iron alloys in the core. Different formalisms are used throughout the literature: constant values, *ad hoc* power law (Anderson 1967) and thermodynamically derived power law (Al'Tshuler et al. 1987). With this sensitivity study, we show that the thermodynamically derived power law of Al'Tshuler et al. (1987) is less likely to yield errors when the actual values of the parameters controlling $\gamma$ are not precisely known, and

is theoretically more sound than the *ad hoc* power law of Anderson (1967).

However, with the data at hand it is not yet possible to exclude any formalism or parameter values based on this study alone. Nonetheless, the sensitivity of temperature to the Grüneisen parameter can be high depending on the formalism adopted and need to be acknowledged when modelling temperature evolution. Further work in constraining the compositional dependencies of the parameters would greatly improve the thermal models of the core and their links to the light element concentrations.

## ACKNOWLEDGMENTS

This work was supported by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant number 716429). ISTERRE is part of Labex OSUG@2020 (ANR10 LABX56). The authors declare no other sources of funding and no competing interest.

## DATA AVAILABILITY

The code and data will be made available upon demand to the corresponding author.

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