The Effects of Mg/Si on the Exoplanetary Refractory Oxygen Budget

The most abundant element in the Earth is a nominally volatile one: oxygen. Constituting up to ∼50% of the planet's total atoms (McDonough 2003), within the Earth oxygen plays an important role at all scales: controlling the bulk structure of the Earth, the speciation of minerals within the mantle, and its surface habitability. For exoplanets, however, the structure and composition of these planets is currently determined through inference from planetary mass and radius (e.g., Valencia et al. 2006; Fortney et al. 2007; Seager et al. 2007; Sotin et al. 2007; Rogers & Seager 2010; Wagner et al. 2011; Zeng & Sasselov 2013; Lopez & Fortney 2014; Unterborn et al. 2016), with surface habitability limited to potential atmospheric observations (Seager & Bains 2015). Inverse models of terrestrial planets (e.g., Dorn et al. 2015) find these observables are sufficient only to constrain central core size and only where an atmosphere is assumed to be a negligible component of the planet's mass. Without this fully terrestrial assumption, mass–radius models cannot a priori determine whether a planet is a super-Earth or a mini-Neptune, that is, whether a planet is a massive terrestrial planet or contains a significant gaseous envelope. This is due to the first-order tradeoff in the size of the planet's metallic core with the thickness of its atmosphere.

Dorn et al. (2015) pointed out that there is considerable compositional and mineralogical degeneracy present in these models, that is, multiple planetary interior compositions are valid solutions given only the constraints of a planet's total mass and radius. They showed that adopting a host star's abundance of the refractory planet-building elements (Mg, Si, and Fe) as a proxy for planetary composition reduces this compositional degeneracy. Unterborn et al. (2016) expanded upon this approach by demonstrating that the Sun's refractory composition is a sufficient proxy for reproducing the Earth's bulk structure and mineralogy, but only by adopting a liquid iron core, upper mantle structure, and realistic light element budget for the core, which are all aspects that are absent in the "canonical" mass–radius model of Zeng & Sasselov (2013). These omissions cause the model to systematically overestimate the mass of Earth and "Earth-like" planets for a given radius. Furthermore, Unterborn et al. (2016) created a grid of benchmarked mass–radius models in order to quantitatively define "Earth-like" planets. The oxygen abundance of a planet in both models, however, was determined only as a consequence of the relative proportions of the core to mantle, rather than assumed to be the stellar value. As such, both models make broad assumptions on the bulk oxidation state of these exoplanets.

The size of a planetary core is a direct consequence of a planet's oxidation state. Core formation results from early differentiation in the planetary formation process as a consequence of the melting and immiscibility of metal and silicate, in which some fraction of the metal is not oxidized upon condensation. Therefore, to first-order, the fraction of free metal in the core is a function of the total oxygen abundance of a planet. The Sun contains ∼5 times more oxygen than the sum of the planet-building cations Mg, Si, and Fe (Asplund et al. 2005). Thus, if one were to erroneously assume that, like the refractory elements, a planet's oxygen abundance is mirrored between planet and star, one would predict the Earth is a planet that is entirely oxidized, with no core present. This discrepancy is due to the dual nature of oxygen: it is refractory, condensing as silicates and oxides, and volatile, condensing as ice. As such, the assumptions of stellar abundance being indicative of planetary abundance that apply for the refractory elements, as seen in Dorn et al. (2015) and Unterborn et al. (2016), cannot be assumed for oxygen. We therefore have no direct or indirect way to measure the abundance of the most dominant terrestrial planet-building element.

Equilibrium condensation models predict ∼23% of solar O entering into rocky phases (Lodders 2003). However, because oxygen behaves both as a refractory and volatile element, the fraction of oxygen condensing as a refractory, terrestrial-planet-building component will necessarily be a function of the bulk composition of the nebula, in particular the relative proportions of dominant rock-forming elements (Ca, Al, Ti, Mg, Si, and Fe) relative to oxygen, but also the oxidation state of these elements within the solid.

However, the existing condensation codes to calculate this speciation are either not open-source (e.g., Ebel & Grossman 2000; Lodders 2003) or proprietary (e.g., HSC Chemistry), hindering self-consistent comparisons across studies and testing of thermodynamic databases across a range of compositions not necessarily relevant to the solar system. As we seek to constrain the compositional and structural diversity of planetary systems outside of our own, we must understand how variations in stellar compositions affect the compositions of associated planets, and therefore their structures and mineralogies and thus the likelihood of them being "Earth-like." We present then, the Arbitrary Composition Condensation Sequence Calculator (ArCCoS), with a flexible thermodynamic database. ArCCoS calculates the stable assemblages of a gas and solid in equilibrium to determine the relative proportions and stoichiometry of the refractory condensing phases from which terrestrial planets are built.⁴

The chemical composition of the Earth is constrained from chondritic models, source material from the upper mantle (McDonough & Sun 1995; McDonough 2003; Javoy et al. 2010), and geophysical constraints such as the total mass of the planet, moment of inertia measurements, and seismic wave speeds. For this Earth composition, the dominant minerals by volume are the lower mantle minerals bridgmanite, with a perovskite structure (Mg,Fe)SiO₃, and ferropericlase, (Mg,Fe)O. The relative proportions of these two minerals is a function of the mantle's Mg/Si ratio. We highlight one potential utility of the ArCCoS code by examining the effects of variable stellar Mg/Si on the percentage of condensed oxygen in the refractory solid phases and the subsequent changes expected in the bulk structure of the resulting planet.

Conservation of mass and the law of mass action determine the condensation temperature of a solid in equilibrium with nebular gas. The governing equations are functions of the total pressure of the system (P_tot), the number of moles of each element in the system, and the distribution of these elements between each species in equilibrium. Mass balance is the sum of the number density, n in mol L⁻¹, of element or compound, X, in each gas phase, i, and solid phase, j:

$$\begin{eqnarray}&&{N}_{X}=\displaystyle \sum _{i}{\nu }_{i,X}{n}_{i}+\displaystyle \sum _{j}{\nu }_{j,X}{n}_{j},\end{eqnarray} \tag{ 1 }$$

where ν is the stoichiometric coefficient of compound i or j in the individual phases. The distribution of an element X between the gas and solid phases can then be calculated according to the law of mass action via the equilibrium constant, ${K}_{i,j}$:

$$\begin{eqnarray}&&\mathrm{ln}({K}_{i,j})=\mathrm{ln}\left(\displaystyle \frac{{n}_{i,j}}{{\displaystyle \prod }_{X}^{}{({n}_{X})}^{{\nu }_{(i,j),X}}}\ast {{RT}}^{\left(\displaystyle \sum _{X}{\nu }_{(i,j),X}\right)-1}\right)=\displaystyle \frac{-{\rm{\Delta }}{G}_{r}}{{RT}},\end{eqnarray} \tag{ 2 }$$

where n is the number density of a gas/solid/element i, j or X, R is the gas constant, T is the temperature (in K), and ${\rm{\Delta }}{G}_{r}$ is the Gibbs free energy of the reaction from the elements as defined by

$$\begin{eqnarray}&&{\rm{\Delta }}{G}_{r}={\rm{\Delta }}{G}_{i,j}-{\rm{\Delta }}{G}_{\mathrm{elements}}.\end{eqnarray} \tag{ 3 }$$

Assuming each gas phase is ideal and at constant volume, N_X is proportional to the partial pressure of element X in the system via

$$\begin{eqnarray}&&{N}_{X}=\displaystyle \frac{a(X)}{{\displaystyle \sum }_{i}{a}_{i}+{\displaystyle \sum }_{j}{a}_{j}}\ast \displaystyle \frac{{P}_{\mathrm{tot}}}{{RT}},\end{eqnarray} \tag{ 4 }$$

where a is the number of moles of element X from the input solar model or the number of moles of gas/solid, $i,j$ in the system. We simplify Equation (4) by only considering the dominant species by mole in the system: H, H₂, and the noble gases (He, Ne, Ar):

$$\begin{eqnarray}&&{N}_{X}=\displaystyle \frac{a(X)}{a({\rm{H}}+{{\rm{H}}}_{2}+\mathrm{noble}\,\mathrm{gasses})}\ast \displaystyle \frac{{P}_{\mathrm{tot}}}{{RT}}.\end{eqnarray} \tag{ 5 }$$

The temperature-dependent distributions of a(H) and a(H₂) are determined from their equilibrium coefficients taken from the JANAF tables (Chase 1998). At present, ArCCoS determines the speciation between ∼400 gas species, and 23 elements with ∼110 potential solid condensates (Tables 1 and 2). The ArCCoS database includes 23 elements: H, He, C, N, O, F, Ne, Na, Mg, Al, Si, P, S, Cl, Ar, K, Ca, Ti, Cr, Mn, Fe, Co, and Ni. Of these, we treat Si, Mg, Ca, Al, Ti, Ni, and Fe as the refractory elements.

Table 1.  Gas Species Included in ArCCoS

AlS	CH4	C2K2N2	H3N	F2S	S2	ClO	Cl3OP	F2O	MgF
AlF2	CNO	C2HF	K2	PS	CCl3F	ClTi	CaH2O2	F2Ti	FN
CCl2F2	CH2	Na2	H2MgO2	F3NO	F0S2	ClH	CrO2	TiF	FNO
AlF2O	CN2-trans	C2H	MgN	O3S	CHCl	Cl2Co	HNO2-trans	F3OP	Al
AlClF	CNNa	C2H2	KO	P2	CClF3	ClP	Cl2O	F2P	Ar
AlHO-cis	CS	CF2	NO3	PO2	CaCl	HS	CaCl2	F5P	C
AlClO	COS	C2F4	MgS	TiO2	CHF	Cl2K2	HNa	F3PS	Ca
CClN	CH2O	C2O	H2Na2O2	ClS	CHClF2	ClMg	Cl5P	F2N	Cl
AlHO2	C2	CF3	NS	FNO2	F2S2-cis	HO2	CaF2	HK	Co
AlN	C2Cl2	CF4	NSi	TiO	F2S2-trans	HO	CaHO	HKO	Cr
AlO	C2Cl4	CF4O	N2	SiO	FHO3S	HNaO	CaO	HMg	F
AlH	CP	CHNO	NO2	O2S	CClFO	Cl2Na2	C5	F4S	Fe
CCl3	CHO	N3	HSi	F7H7	S4	ClF3	FH	FP	H
CCl2O	CHP	NaO	H2	H2K2O2	S3	ClF5	CrO3	FPS	He
AlCl2	CN2-cis	C2HCl	K2O4S	O6P4	CF8S	Cl2	HNO2-cis	F3N	K
Al2O	CH3F	C2N	H2S	FS	SSi	ClNa	Cl3P	F2Na2	Mg
AlF	CO2	C2F2	NO	O2Si	CH2Cl2	Cl2Mg	C4N2	F4N2	Mn
CCl2	CH2ClF	PO	H2N	NiS	ClS2	ClHO	CrO	F2	N
Al2O2	CH3Cl	C2N2	H2O4S	Cl2S	P4S3	ClNO2	Cl3PS	F2N2-trans	Na
CCl	CH3	C2N2Na2	H2O	ClF5S	P4	ClNO	Cl4Co2	F2N2-cis	Ne
AlHO-trans	CS2	CF2O	NP	O2	CrN	PH	CaF	F6S	Ni
CClO	CH2F2	C3	H2N2	FeO	ClFO2S	ClK	CoF2	F2K2	O
AlO2	CKN	C2H4O	H3P	F3S	S8	ClOTi	Cl3Co	F2OS	P
Al2	CN	C2H4	H4N2	Cl2O2S	AlFO	ClO2	Cl2Ti	F2O2S	S
CCl4	CHN	N2O5	C4	F6H6	S5	ClF2OP	FHO	FO2	Si
AlCl	CO	C2F6	MgO	O3	CHCl2F	Cl2FOP	HNO3	F3P	Ti
CF	CHF3	Na2O4S	C3O2	F5H5	S6	ClFO3	FK	FOTi	Cl2${\mathrm{Cr}}^{\dagger }$
CFN	CHFO	N2O4	H2P	F4H4	S7	ClF	HNO	FNa	Cl2CrO2${}^{\dagger }$
CFO	CHCl3	N2O3	OS	F3H3	C2H4O	ClCo	HN	FNO3	${\mathrm{CrS}}^{\dagger }$
C2Cl6	CH	N2O	OS2	F2H2	P4O10	CaS	HMgO	NiO${}^{\dagger }$	F2${\mathrm{Mn}}^{\dagger }$
Cl2${\mathrm{Mn}}^{\dagger }$	${\mathrm{TiS}}^{\dagger }$	${\mathrm{CoO}}^{\ddagger }$	${\mathrm{MnO}}^{\ddagger }$

Note. All thermochemical data are taken from Chase (1998), unless noted.

References. ${}^{\dagger }$: Knacke et al. (1991); ${}^{\ddagger }$: Pedley & Marshall (1983).

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All reactions are considered to occur at a given P and T from the reference elements, with the exception of H, N, O, F, and Cl where the molecular form (e.g., O₂) is chosen as the reference form. Data for these reference phases for enthalpy and entropy were adopted from the JANAF tables (Chase 1998), with linear interpolation in T. The Gibbs free energy of the reactant elements is therefore:

$$\begin{eqnarray}&&{\rm{\Delta }}{G}_{\mathrm{elements}}=\displaystyle \sum _{X}{\nu }_{X}{[H(T)-T\ast S(T)]}_{X},\end{eqnarray} \tag{ 6 }$$

where H(T) is the enthalpy of formation at temperature T, S is the entropy, and ${\nu }_{X}$ is the stoichiometric constant of element X in the reaction.

2.1. Gases

Above 2000 K, no solids are stable and only gas-phase equilibria need to be considered. For example: consider the gas reaction of water from its constituent elements in their reference states:

$$\begin{eqnarray}&&{{\rm{H}}}_{2}({\rm{g}})+\displaystyle \frac{1}{2}{{\rm{O}}}_{2}({\rm{g}})\leftrightarrow {{\rm{H}}}_{2}{\rm{O}}({\rm{g}}).\end{eqnarray} \tag{ 7 }$$

At a given T, the free energies of each species are known and their equilibrium coefficients can be written in terms of their partial pressures (p_X):

$$\begin{eqnarray}&&{K}_{{{\rm{H}}}_{2}{\rm{O}}}=\displaystyle \frac{{p}_{{{\rm{H}}}_{2}{\rm{O}}}}{{p}_{{{\rm{H}}}_{2}}{p}_{{{\rm{O}}}_{2}}^{0.5}}.\end{eqnarray} \tag{ 8 }$$

The number density of gaseous water (${n}_{{{\rm{H}}}_{2}{\rm{O}}}$) is then taken from the ideal gas law

$$\begin{eqnarray}&&{n}_{{{\rm{H}}}_{2}{\rm{O}}}=\displaystyle \frac{{p}_{{{\rm{H}}}_{2}{\rm{O}}}}{{RT}}={n}_{{{\rm{H}}}_{2}}{n}_{{{\rm{O}}}_{2}}^{0.5}{K}_{{{\rm{H}}}_{2}{\rm{O}}}{({RT})}^{0.5},\end{eqnarray} \tag{ 9 }$$

where ${n}_{{{\rm{H}}}_{2}}$ and ${n}_{{{\rm{O}}}_{2}}$ are the number densities of molecular hydrogen and oxygen, respectively. Similar equations are written for each gaseous species. For example, the mass balance equation for oxygen is written as,

$$\begin{eqnarray}&&{N}_{{\rm{O}}}={n}_{{{\rm{H}}}_{2}{\rm{O}}}+{n}_{\mathrm{CO}}+2{n}_{{\mathrm{CO}}_{2}}+\cdots ,\end{eqnarray} \tag{ 10 }$$

which in turn can be written in terms of the concentrations component reference elements:

$$\begin{eqnarray}{N}_{{\rm{O}}} & = & {n}_{{{\rm{H}}}_{2}}{n}_{{{\rm{O}}}_{2}}^{0.5}{K}_{{{\rm{H}}}_{2}{\rm{O}}}{({RT})}^{0.5}+{n}_{{\rm{C}}}{n}_{{\rm{O}}}^{0.5}{K}_{{\mathrm{CO}}_{2}}{({RT})}^{0.5}\\ & & +2{n}_{{\rm{C}}}{n}_{{{\rm{O}}}_{2}}{K}_{{\mathrm{CO}}_{2}}({RT})+\cdots .\end{eqnarray} \tag{ 11 }$$

Combining Equations (5) and (11) for each of the 23 elements in the system solved in ArCCoS represents a system of 23 nonlinear equations.

For the gas phases (Table 1), the thermodynamic data are from Chase (1998), Knacke et al. (1991), and Pedley & Marshall (1983). At temperatures within the range reported for a given species in Chase (1998) and Pedley & Marshall (1983), ${\rm{\Delta }}{G}_{\mathrm{gas}}$ as a function of temperature is determined by linear interpolation. For those species taken from Knacke et al. (1991), we follow their methodology for determining the ${\rm{\Delta }}{G}_{\mathrm{gas}}$ for an individual gas phase.

2.2. Solid Condensation

Condensation of solids from the gas phase occurs when the partial pressure of the component elements exceeds the equilibrium constant and thus the relationship

$$\begin{eqnarray}&&{K}_{j}-\displaystyle \prod _{X}{({n}_{X}{RT})}^{{\nu }_{X}}=0\end{eqnarray} \tag{ 12 }$$

is achieved, where ${\nu }_{X}$ is the stoichiometric constant for each constituent element. For example, the first solid to condense for a system of solar composition at P_tot = 10⁻³ bar is corundum at 1770 K when ${P}_{{\mathrm{Al}}_{2}{{\rm{O}}}_{3}}={K}_{{\mathrm{Al}}_{2}{{\rm{O}}}_{3}}$. For corundum, Equation (12) is then written as

$$\begin{eqnarray}&&{K}_{{\mathrm{Al}}_{2}{{\rm{O}}}_{3}}-{n}_{\mathrm{Al}}^{2}{n}_{{{\rm{O}}}_{2}}^{1.5}{{RT}}^{3.5}=0.\end{eqnarray} \tag{ 13 }$$

The temperature at which the constraint in Equation (12) is met is considered the initial or "appearance" condensation temperature. Once corundum begins to condense, some fraction, ${n}_{{\mathrm{Al}}_{2}{{\rm{O}}}_{3}}$, of the solid is now in equilibrium with the gas phase and each elemental number density in Equation (1) must adjust to this new constraint. As gas chemistry changes upon cooling, many of the first condensates become unstable, and rereact with the gas to form new phases. A solid is considered to be removed from the system when the solid's number density, n_j, falls below 1 × 10⁻¹⁰ mol per mol of atoms in the system, following Sharp & Wasserburg (1995). For corundum, this occurs at 1731 K (Table 3), with hibonite (CaAl₂O₁₉) becoming the new stable host of Al. This temperature is considered the solid's "disappearance" temperature. Its constituent element abundances are returned to the gas phase and the calculation is repeated.

Gibbs free energy values of solid phases are either linearly interpolated when ${\rm{\Delta }}{G}_{\mathrm{solid}}$ is directly available or derived from reported specific heat functions (Table 2). In order to self-consistently calculate ${\rm{\Delta }}{G}_{\mathrm{solid}}$ from specific heat data, we begin with the definition of ${\rm{\Delta }}{G}_{\mathrm{solid}}$,

$$\begin{eqnarray}&&{\rm{\Delta }}{G}_{\mathrm{solid}}={\rm{\Delta }}H(P,T)-T\ast S(P,T),\end{eqnarray} \tag{ 14 }$$

where

$$\begin{eqnarray}{\rm{\Delta }}H(P,T) & = & {\rm{\Delta }}H({P}_{r},{T}_{r})+{\displaystyle \int }_{{T}_{r}}^{T}{C}_{p}(T){dT}\\ & & +{\displaystyle \int }_{{P}_{r}}^{P}\left\{V({P}_{r},{T}_{r})-T{\left(\displaystyle \frac{\partial V}{\partial T}\right)}_{P}\right\}{dP},\end{eqnarray} \tag{ 15 }$$

and

$$\begin{eqnarray}&&S(P,T)=S({P}_{r},{T}_{r})+{\int }_{{T}_{r}}^{T}\displaystyle \frac{{C}_{p}(T)}{T}{dT}+{\int }_{{P}_{r}}^{P}{\left(\displaystyle \frac{\partial V}{\partial T}\right)}_{P}{dP},\end{eqnarray} \tag{ 16 }$$

where ${\rm{\Delta }}H(P,T)$ and $S(P,T)$ are the enthalpy of formation from the elements and third law entropy at P, and T, ${\rm{\Delta }}H({P}_{r},{T}_{r})$, and $S({P}_{r},{T}_{r})$ are the same values at reference P and T (1 bar, 298.15 K) and V is the molar volume. The ambient pressures in these calculations are small (P_tot ∼ 10⁻³ bar); we therefore ignore the volume integrals in our calculations of ${\rm{\Delta }}{G}_{\mathrm{solid}}$. As an exploratory study, we omit any formulation of solid solutions and consider only those condensates composed of pure, end-member phases. The inclusion of solid-solution models increases the complexity of the ArCCoS code and will be included in future updates and studies.

Table 2.  List of Solid Species Included in ArCCoS Including References

Misc. Solids				Chase (1998)
Åkermanite	Ca2MgSi2O7${}^{\dagger }$	Grossite	CaAl4O7§	Al	FNa	Mg2Si
Albite	NaAlSi3O8${}^{\dagger }$	Grossular	Ca3Al2Si3O12${}^{\dagger }$	AlN	F2Fe	Mn
Almandine	Fe3Al2Si3O2${}^{\dagger }$	Hibonite	CaAl2O19${}^{\ddagger }$	Al2S3	F2Mg	N4Si3
Andalusite	Al2SiO5${}^{\dagger }$	Ilmenite	FeTiO3${}^{\dagger }$	Al6Si2O13	Fe	Na
Anorthite	CaAl2Si2O8${}^{\dagger }$	Jadeite	NaAlSi2O6${}^{\dagger }$	C	FeH2O2	NaO2
Anthophyllite	Mg7Si8O22(OH)2${}^{\dagger }$	Lime	CaO${}^{\dagger }$	C2Mg	FeH3O3	Na2O3Si
Brucite	Mg(OH)2${}^{\dagger }$	Magnesite	MgCO3${}^{\dagger }$	C3Al4	FeO	Na2SO4-δ
Ca-aluminate	CaAl2O4§	Magnetite	Fe3O4${}^{\dagger }$	C3Mg2	FeO4S	Na2SO4-III
Calcite	CaCO3${}^{\dagger }$	Merwinite	Ca3MgSi2O8${}^{\dagger }$	Ca-α	FeS2-I	Ni
Clinoenstatite	MgSiO3${}^{\dagger }$	Monticellite	CaMgSiO4${}^{\dagger }$	Ca-β	FeS2-II	P-I
Corderite	Mg2Al4Si5O18${}^{\dagger }$	Periclase	MgO${}^{\dagger }$	CaCl2	Fe2O2S3	S
Corundum	Al2O3${}^{\dagger }$	Perovskite	CaTiO3${}^{\sharp }$	CaH2O2	HK	S2Si
Cristobalite	SiO2${}^{\dagger }$	Pyrope	Mg3Al2Si3O12${}^{\dagger }$	CaS	HNa	SiC-α
Diopside	CaMgSi2O6${}^{\dagger }$	Quartz-α	SiO2${}^{\dagger }$	ClK	H2Mg	SiC-β
Dolomite	CaMg(CO3)2${}^{\dagger }$	Sanidine	KAlSi3O8${}^{\dagger }$	ClNa	K	Ti-α
Enstatite	MgSiO3${}^{\dagger }$	Sillimanite	Al2SiO5${}^{\dagger }$	Cl2Fe	K2O	Ti-β
Fayalite	Fe2SiO4${}^{\dagger }$	Sphene	CaTiSiO5${}^{\dagger }$	Cl2Mg	K2O3Si	TiH2
Ferrosilite	FeSiO3${}^{\dagger }$	Spinel	MgAl2O4${}^{\dagger }$	Co	Mg	TiO-α
Forsterite	Mg2SiO4${}^{\dagger }$	Talc	Mg3Si4(OH)2${}^{\dagger }$	CoO	MgO4S	TiO-β
Gehlenite	Ca2Al2SiO7${}^{\dagger }$			FK	MgS	Ti3O5-α
Wollastonite	CaSiO3${}^{\dagger }$			Ti3O5-β

References. ${}^{\dagger }$: Berman (1988); § : C_p from Berman & Brown (1985); 298 K data from Berman (1983); ${}^{\ddagger }$: Berman (1983); ${}^{\sharp }$: Robie et al. (1978).

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2.3. Algorithm

At each temperature, this system is solved using the scipy root finding package with a least-squares method using a modified Levenberg–Marquardt algorithm as implemented in MINPACK1 (Moré et al. 1980). Models are run beginning at 2500 K, where only monoatomic gasses are present in the nebula and a solution can be easily found. ArCCoS requires an initial guess in order to begin solving Equation (1). For the initial temperature calculation this is the concentration calculated in Equation (5), with subsequent initial guesses being the solution from the previous temperature step. Equilibrium is assumed when the sum of the least-squares difference in the mass balance (Equation (1)) for all elements is less than 10⁻¹⁵ mol. Once a solution is found, the temperature is lowered by 2 K and the calculation is repeated at the new temperature. ArCCoS continues solving Equation (1) until 100% of each refractory element is condensed (Si, Mg, Fe, Ca, Al, Ni and Ti). The total percentage of oxygen in the refractory phases, %RO, is then

$$\begin{eqnarray} &&\% \mathrm{RO}=\displaystyle \frac{{\displaystyle \sum }_{j}^{}{\nu }_{j,O}{n}_{j}}{{N}_{{\rm{O}}}}.\end{eqnarray} \tag{ 17 }$$

2.4. Model Benchmark

Ebel & Grossman (2000, hereafter EG00) is the most similar model for benchmarking ArCCoS, as we adopt the same input thermodynamic database and a similar computational approach. For a gas at 10⁻³ bar of the solar composition reported in EG00 (Table 3). our model predicts the same condensing solids, with the exception of cordierite and Cr-spinel (a Cr-rich solid-solution of Mg and Cr-spinel) at 1330 and 1230 K, respectively (Table 3). Furthermore, our calculated appearance and disappearance are within ∼15 K. These discrepancies are likely a consequence of our non-incorporation of solid solutions in this model.

Table 3.  Comparison of Appearance (in) and Disappearance (out) Temperatures (in K) of Solid Phases between This Study and the Model of Ebel & Grossman (2000)

	Ebel & Grossman (2000)		This study
Solid	In	Out	In	Out
Corundum	1770	1726	1771	1731
Hibonite	1728	1686	1735	1699
Grossite	1698	1594	1712	1592
Perovskite	1680	1458	1681	1396
CaAl2O4	1624	1568	1623	1557
Melilite	1580	1434	1575	1456
(Gehlenite)
Grossite	1568	1502	1558	1486
Hibonite	1502	1488	⋯	⋯
Spinel	1488	1400	1487	1463
Melliilite	⋯	⋯	1457	1447
(Åkermanite)
Fe	1462	⋯	1453	⋯
Clinopyroxene	1458	⋯	1449	⋯
Olivine	1444	⋯	1446	⋯
Plagioclase	1406	1318	1466	⋯
Ti3O5	1368	1242	1397	1214
Ni	⋯	⋯	1382	⋯
Orthopyroxene	1366	⋯	1372	⋯
Co	⋯	⋯	1272	⋯
Corderite	1330	⋯	⋯	⋯
Cr-Spinel	1230	⋯	⋯	⋯
TiO2	⋯	⋯	1215	⋯

Note. Those solids with no reported disappearance temperature remain stable at the termination of the calculation. All calculations are run at P_tot = 10⁻³ bar.

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Table 4.  Stellar Abundances Adopted in ArCCoS (Abundances Are Normalized Such That log(N_H) = 12.0)

Element	Anders & Grevesse (1989)	Lodders (2003)	Asplund et al. (2005)
H	12.0	12.0	12.0
He	10.99 ± 0.035	10.899 ± 0.01	10.93 ± 0.01
C	8.56 ± 0.04	8.39 ± 0.04	8.39 ± 0.05
N	8.05 ± 0.04	7.83 ± 0.11	8.39 ± 0.06
O	8.93 ± 0.035	8.69 ± 0.05	8.66 ± 0.05
Ne	8.09 ± 0.10	7.87 ± 0.10	7.84 ± 0.06
Mg	7.59 ± 0.05	7.55 ± 0.02	7.53 ± 0.09
Al	6.48 ± 0.07	6.46 ± 0.02	6.37 ± 0.06
Si	7.55 ± 0.05	7.54 ± 0.02	7.51 ± 0.04
Fe	7.51 ± 0.03	7.47 ± 0.03	7.45 ± 0.05
Ca	6.34 ± 0.02	6.34 ± 0.03	6.31 ± 0.04
Ti	4.93 ± 0.02	4.92 ± 0.03	4.90 ± 0.06
F	4.48 ± 0.30	4.46 ± 0.06	4.56 ± 0.30
Cl	5.27 ± 0.30	5.26 ± 0.06	5.50 ± 0.30
S	7.27 ± 0.06	7.19 ± 0.04	7.14 ± 0.05
Na	6.31 ± 0.03	6.30 ± 0.03	6.17 ± 0.04
Ar	6.56 ± 0.10	6.55 ± 0.08	6.18 ± 0.08
Cr	5.68 ± 0.03	5.65 ± 0.05	5.64 ± 0.10
Ni	6.25 ± 0.04	6.22 ± 0.03	6.23 ± 0.04
P	5.57 ± 0.04	5.46 ± 0.04	5.36 ± 0.04
K	5.13 ± 0.13	5.11 ± 0.05	5.08 ± 0.07
Co	4.91 ± 0.04	4.91 ± 0.03	4.92 ± 0.08
Mn	5.53 ± 0.03	5.50 ± 0.03	5.39 ± 0.03

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Adopting the solar abundances of Lodders (2003) (Table 4), we calculate values similar to their reported appearance, 50% condensation temperatures, which is the temperature at which 50% of the element is condensed, and %RO (Table 5), with any difference being due to either the same solid-solution details discussed above or differences in the adopted input thermodynamic database, which Lodders (2003) does not report.

Table 5.  Comparison of the Initial Condensation Temperature and 50% Condensation Temperature for the Major Terrestrial-planet-building Elements between This Work and Lodders (2003)

	Tappearance (K)		${T}_{50 \% }$ (K)
Element	This Study	Lodders (2003)	This Study	Lodders (2003)
Al	1665	1677	1643	1653
Ca	1609	1659	1508	1517
Ti	1583	1593	1569	1582
Mg	1397	1387	1336	1336
Si	1477	1529	1318	1310
Fe	1356	1357	1329	1310
Avg. Diff.	22.5 K		9.8 K

Note. Calculations were performed using the input solar abundances of Lodders (2003, Table 4) and at P_tot = 10⁻⁴ bar.

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The solar model adopted by EG00 is that of Anders & Grevesse (1989) (Table 4), which has been revised by Grevesse & Noels (1993), Palme & Beer (1993), Grevesse et al. (1996), Grevesse & Sauval (1998, 2002), Lodders (2003), Asplund et al. (2005) and Asplund et al. (2009), the most recent of which includes a three-dimensional, time-dependent hydrodynamical model of the solar atmosphere. This latter model, however, does not agree with that of the CI chondrites, which are often assumed to be indicative of the composition of the solar photosphere, particularly with respect to Mg. We therefore adopt the second most recent solar abundances of Asplund et al. (2005) as our preferred solar compositional model (Table 4).

The abundances of the major planet-building elements (Mg, Fe, Si) from the Asplund et al. (2005) solar model are within 10% of the chondritic Earth model of McDonough (2003, Table 7). We find that when 100% of refractory elements are stable in condensed solid phases, 22.8% of solar oxygen is condensed in refractory phases or 50.9% of the total moles in the system, which is ∼5.5% greater than McDonough (2003, Table 7). For this composition, enstatite (MgSiO₃) is the dominant host of O regardless of the solar model (Figure 1). For Asplund et al. (2005) (Figure 1(a)), enstatite is responsible for 68% of the total %RO, followed by forsterite (18%), anorthite (9%), and diopside (5%). These phases are also the dominant hosts of both Mg and Si (Figure 2). While there is significant oxygen variability in solar models, the condensation sequence for each of the models of Anders & Grevesse (1989), Lodders (2003), and Asplund et al. (2005) do not vary with respect to the relative proportions of the refractory elements or moles of oxygen condensed, but only in the fraction of oxygen condensed (13.7%, 22.8%, and 22.8%, respectively, Figure 1).

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To demonstrate the impact of variable stellar composition, we calculate the condensation sequences when varying solar Mg/Si between 0.63 $\leqslant$ Mg/Si $\leqslant$ 1.67. We accomplish this in two ways: (1) varying Si between 159% and 60% of solar (Mg/Si = 1.05), and (2) varying Mg between 63% and 167% of solar, each while holding all other abundances constant. While neglecting associated variability in the other major elements is unrealistic, varying only Mg/Si provides a simplified case to illustrate the importance of stellar composition's effect on exoplanet interior compositions and resulting core-mantle structures, as well as its importance for providing predictions of a terrestrial exoplanet's mass.

We find that between 0.63 $\leqslant$ Mg/Si $\leqslant$ 1.67, in case (1), changes in Si result in a 14% difference in the percent of refractory oxygen (18–32 %RO) compared to case (2) in which %RO varies by 8% (20–28 %RO) by exclusively changing Mg (Figure 3). Furthermore, for constant Mg/Si, cases (1) and (2) result in different %RO. As with the calculations of solar composition, enstatite, forsterite, anorthite, and diopside are the dominant host phases for O, with quartz becoming a major host at lower Mg/Si (Figures 4(a), (b)). For a given change in Mg/Si though, we find that Si abundances more drastically affect %RO compared to those in Mg. For example, when the solar Si abundance is decreased by 60%, %RO increases by ∼10% compared to solar, whereas a similar increase in Mg only increases %RO by ∼5%. A similar result is found when Mg or Si is decreased. This behavior is due to the different stable oxidation states of each cation, Si⁴⁺ and Mg²⁺, thereby binding two (SiO₂) and one (MgO) condensed oxygen atoms. Therefore, changes in the total silicon abundance will have a greater effect on %RO.

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3.1. Discussion

The Earth's composition and differentiation into metal core and silicate crust and mantle are a reflection of the protoplanetary disk of solar composition from which the Earth formed, as well as any fractionation that occurred during planetary formation. As refractory elements, the dominant planetary cations Mg, Si, Fe, Al, and Ca are not expected to considerably fractionate relative to each other during planet formation. Combining the abundances of these elements with the oxygen budget predicted by ArCCoS allows us to estimate the relative mass proportion of oxidized mantle phases to core. For example, given the Asplund et al. (2005) solar composition and stoichiometric oxidation first of Mg, then Si, Al, Ca, Ti, and finally Fe, 99% of the total oxygen budget is consumed by Mg, Si, Ca, and Ti, of which 95% is due to Mg and Si alone (Table 6). The remaining oxygen oxidizes ∼14% of the Fe, leaving the remainder in the reduced, metallic form. If all of this metallic iron and nickel segregate into the Earth's core, it accounts for 31.9 wt% of the planet, or ∼99% of the core's actual mass.

Table 6.  Stoichiometric Oxidation Results for the Calculated Condensation Sequence of the Solar Composition of Asplund et al. (2005) for Both a Pure Fe/Ni Core and an Fe/Ni Core with 4.1 wt% Si and 0.77 wt% ${{\rm{O}}}^{\ddagger }$ (%RO = 22.8)

	Abundance	%O	Resulting Planet
Oxide	(mol)	Remaining	Properties
Fe/Ni Core
MgO	32.5	67.5	Core	32.0 mass%
SiO2	31.0	5.5
Al2O3	1.1	2.2	Mantle
CaO	2.0	0.2	MgSiO3	95.3 mol%
Ti3O5	0.02	0.1	MgO	4.5 mol%
FeO	0.1	0.0	FeO	0.2 mol%
Core${}^{\dagger }$	29.6	⋯	Mg/Si	1.05
Fe/Ni/Si/O Core
MgO	32.7∗	67.3	Core	29.4 mass%
SiO2	29.1	9.2
Al2O3	1.13	5.8	Mantle
CaO	2.0	3.8	MgSiO3	80.0 mol%
Ti3O5	0.02	3.7	MgO	10.0 mol%
FeO	3.7	0.0	FeO	10.0 mol%
Core${}^{\dagger }$	28.0	⋯	Mg/Si	1.12

Note. Molar abundances are normalized so that there are only 100 moles of O available to oxidize material (%RO*O = 100).

References. ${}^{\dagger }$: remaining after all O exhausted by formation of oxides. ${}^{\ddagger }$: core contains 7.7 mol% Si, 5.8 mol% Ni, and 2.6 mol% O. ∗: normalized such that %RO*O—O_core = 100.

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This, however, neglects the presence of light elements (e.g., S, Si, O) in Earth's core. The presence of these light elements within the Earth's core is necessary to account for the seismically observed density difference between the core's mass and that of a pure Fe–Ni alloy (Birch 1952; Jeanloz 1979), and is consistent with the observation of such light elements in some chondritic meteorites (Weisberg et al. 2006). The Earth's core density can be constrained by its internal structure as well as the major element ratios of the Earth, assuming a density that is ∼94% of pure Fe (Unterborn et al. 2016). Assuming Si and O are the dominant light elements present in the core and enter the core at a ratio of Si/O ∼3 (Fischer et al. 2015), ∼7 mol% of all Si must be present in the core to account for the 6% mass deficit. Conserving the mass of the system, this incorporation of Si and O into the core reduces the mass of the core to 29.4 wt%, or 91% of the Earth core mass.

As the balance of the oxidized material after core formation forms the silicate Earth, we can use this stoichiometric method to also estimate the relative proportions of mantle phases. Assuming the mantle mineralogy is that of iron-bearing bridgmanite ((Mg,Fe)SiO₃) as formed through the reaction of ferropericlase ((Mg,Fe)O) and silica (SiO₂) and all other elements (Ca, Al, Ti) that are fractionated into the melt-extracted crust, we calculate a depleted mantle composition of ∼80% bridgmanite and a remaining 10% each of both periclase and wüstite (Table 6). This mantle Mg/Si, then, is ∼1.12, which is lower than the 1.25 of the chondritic Earth model (McDonough & Sun 1995; McDonough 2003). This latter value is heavily weighted toward xenolith rock samples from Earth's upper mantle. There is considerable debate as to whether the mantle is compositionally homogeneous, in which the Mg/Si of the upper mantle may not be characteristic of the bulk mantle (Matas et al. 2007; Javoy et al. 2010). These studies predict lower average mantle Mg/Si compared to the upper mantle weighted, "pyrolitic" model.

While this simplistic stoichiometric model underpredicts the relative size of the Earth's core to mantle under realistic core compositional conditions, this underestimate is likely due to our predicted relative oxygen abundance being greater than the Earth model of McDonough (2003) (Table 7). This overabundance of O causes more Fe to oxidize to FeO, thereby lowering the mass of the core. When Si is included in the core, this mass deficit is exacerbated due to the creation of fewer moles of mantle SiO₂. The oxygen that would be oxidized by Si then goes on to oxidize Fe instead, reducing the size of the core further. To lower, %RO then, some fraction of the oxidized refractory elements must condense as reduced, rather than oxidized phases. A likely source of this reduction is the incorporation of light elements into metallic Fe during the condensation process. In the case of Si, any amount that is incorporated into Fe rather than oxidized phases such as enstatite or forsterite, will return 2 mol of O back to the gas phase, thus lowering %RO while still allowing for the condensation of refractory elements. We are currently working to address these discrepancies by including solid-solution models within ArCCoS for the incorporation of light elements (e.g., S, Si) into condensing Fe-alloy, as well as FeO in olivine. It should be noted, however, that for a 1 Earth-radius planet of solar composition and an Earth-like core composition, the model has a planetary mass only 4% smaller than the Earth's true mass and is well within the current observational error for planetary mass (Cottaar et al. 2014; Unterborn et al. 2016). Thus, we predict the bulk structure and mineralogy of the Earth within the current observational uncertainty, and these more detailed calculations are beyond the scope of this paper.

Table 7.  Comparison of the Abundances of the Major, Planet-forming Elements between McDonough (2003) and Our Calculation of Percent Refractory Oxygen, Adopting the Solar Model of Asplund et al. (2005)

	Asplund et al. (2005)	McDonough (2003)
Element	mol %	mol %	$\%$ Difference
Fe	13.8	15.2	−9.5
Ni	0.83	0.82	+3.4
Mg	16.5	16.8	−1.8
Si	15.8	15.2	+3.9
Al	1.1	1.5	−25.3
Ca	1.0	1.1	−11.2
O	50.9†	49.3	+3.4
Sum	99.93	99.92	⋯

Reference. ${}^{\dagger }$: %RO = 22.8.

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3.2. Effects of Varying Mg/Si

We present here the Python-based, open-source software package, Arbitrary Composition Condensation Sequence Calculator (ArCCoS). It is designed for calculating the stability of solid phases in equilibrium with gas, with a wide variety of applications including determining the composition of the first solids in an exoplanetary system during formation, dust condensation in molecular clouds and the interstellar medium, and any other application where solid/gas-phase equilibria are necessary. To date, though, all software to calculate these condensation sequences are commercially available or closed-source. Here, we show that ArCCoS, combined with simple stoichiometry, can reproduce the Earth's bulk structure and total mass to within observational uncertainty. Furthermore, we show here that changes in stellar Mg and Si abundances, and thus bulk Mg/Si, affect the relative core sizes of exoplanets and their mantle mineralogy. These changes provide observable differences in total planetary mass for a given stellar composition. This model assumes the composition at 100% refractory element condensation is representative of the "average" terrestrial planet in a system. We are working to address the more complicated question of planets forming from condensates within specific radial "feeding zones," where local compositional and oxygen fugacity gradients may exist. This more complex methodology may be preferable to adopting this average composition approach for studies systems with multiple terrestrial planets (e.g. TRAPPIST-1). This method, however, provides a broad approach to studying the potential geology of a terrestrial exoplanet-bearing system, and provide new data to gauge a systems potential to be "Earth-like." We are also addressing incorporating more sophisticated solid-solution models into ArCCoS. Such improved models are likely required to address the modeled discrepancy of Venus mass–radius models not being well fit by an Earth or solar composition (Ringwood & Anderson 1977; Unterborn et al. 2016).

This work is supported by NSF CAREER grant EAR-60023026 to W.R.P.