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Solubility laws

CALLIOPE uses gas-melt solubility laws to relate the dissolved concentration of each volatile species in the magma ocean to its surface partial pressure. The general form is the modified Henry's law,

\[ X_i^\mathrm{melt} = \alpha_i\, p_i^{\,1/\beta_i}, \]

with \(X_i^\mathrm{melt}\) in ppmw, \(p_i\) in bar, and species-specific empirical constants \(\alpha_i, \beta_i\). Bower et al. (2022) Equation (1) writes the same relation in terms of fugacity; CALLIOPE assumes ideal-gas behaviour so \(f \equiv p\).

This page lists the implemented laws, the experimental sources behind each, and the alternative compositions a user can select. The corresponding code lives in solubility.py; the speciation-time call paths are in solve.dissolved_mass.

Per-species inventory

H\(_2\)O - SolubilityH2O(composition=...)

Composition Law Source \(\alpha\) (ppmw bar\(^{-1/\beta}\)) \(\beta\)
peridotite (default) \(524\, p^{0.5}\) Sossi et al. (2023) 524 2
basalt_dixon \(965\, p^{0.5}\) Dixon et al. (1995), refit by Sossi 965 2
basalt_wilson \(215\, p^{0.7}\) Hamilton (1964); Wilson & Head (1981) 215 1/0.7
anorthite_diopside \(727\, p^{0.5}\) Newcombe et al. (2017) 727 2
lunar_glass \(683\, p^{0.5}\) Newcombe et al. (2017) 683 2

About the peridotite constant 524

Sossi et al. (2023) report two values for the prefactor depending on the FTIR absorption coefficient used: \(\alpha = 524 \pm 16\) ppmw bar\(^{-0.5}\) from the basaltic-glass calibration (\(\epsilon_{3550} = 6.3\) m\(^2\) mol\(^{-1}\)), and \(\alpha = 647\) ppmw bar\(^{-0.5}\) from the peridotite-glass calibration. CALLIOPE uses 524 to match the value adopted in Nicholls et al. (2024) and the PROTEUS-side fiducial; the label peridotite refers to the experimental melt composition (Sossi+2023 used a peridotitic starting composition), not to the spectroscopic-calibration choice. If you want the full peridotite-glass calibration, instantiate SolubilityH2O('peridotite') and override the constant manually, or wait for an upstream change.

The choice between peridotite (default) and basalt is one of the larger uncertainties in early magma-ocean modelling. Bower et al. (2022) Table 1 compares all five compositions across the relevant pressure range; Nicholls et al. (2024) uses peridotite as the fiducial, consistent with CALLIOPE's default.

CO\(_2\) - SolubilityCO2(composition='basalt_dixon')

Dixon et al. (1995) MORB fit, with an explicit Poynting-like temperature/pressure correction:

\[ X_{\mathrm{CO_2}}^\mathrm{melt}\,[\text{mol fr.}] = 3.8 \times 10^{-7} \cdot p_{\mathrm{CO_2}} \cdot \exp\left(-\frac{23 (p_{\mathrm{CO_2}} - 1)}{83.15\, T_\mathrm{magma}}\right) \]

then converted from molar to ppmw via the algebraic conversion in Bower et al. (2022) Equation (3):

\[ X_{\mathrm{CO_2}}^\mathrm{melt}\,[\text{ppmw}] = 10^4 \cdot \frac{4400 X_{\mathrm{CO_2}}^\mathrm{melt}}{36.6 - 44 X_{\mathrm{CO_2}}^\mathrm{melt}}. \]

This is the only solubility law in CALLIOPE that depends on \(T_\mathrm{magma}\); the others ignore the temperature term that experimental data only weakly constrains (Bower et al. 2022 §2.2.1 discussion).

CO - SolubilityCO(composition='mafic_armstrong')

Armstrong et al. (2015) mafic-melt fit:

\[ \log_{10} X_\mathrm{CO}^\mathrm{melt}\,[\text{ppmw}] = -0.738 + 0.876\, \log_{10} p_\mathrm{CO} - 5.44 \times 10^{-5} \cdot p_\mathrm{tot} \]

The \(-5.44 \times 10^{-5} \cdot p_\mathrm{tot}\) term is a total-pressure (Poynting) correction that reduces solubility at high pressures. CO solubility is generally an order of magnitude or more lower than CO\(_2\), consistent with the experimental constraints summarised in Yoshioka et al. (2019).

CH\(_4\) - SolubilityCH4(composition='basalt_ardia')

Ardia et al. (2013) basalt fit (their Fig. 6 best-fit, 0.7-3 GPa):

\[ X_\mathrm{CH_4}^\mathrm{melt}\,[\text{ppmw}] = p_\mathrm{CH_4} \cdot \exp\left(4.93 - 1.93\,p_\mathrm{tot}^{[\mathrm{GPa}]}\right) \]

with both pressures converted from bar to GPa internally before evaluation. The exponential term encodes the Poynting correction: at high total pressures, the molar volume of dissolved CH\(_4\) in the melt becomes significantly smaller than that of gas-phase CH\(_4\), so solubility decreases with increasing \(p_\mathrm{tot}\). The factor of \(\exp(1.93) \approx 6.9\) between \(p_\mathrm{tot} = 1\) bar and \(p_\mathrm{tot} = 1\) GPa is verified in tests/test_stoichiometry.py::TestCH4Solubility.

N\(_2\) - SolubilityN2(composition='dasgupta') (in dissolved_mass)

CALLIOPE provides two N\(_2\) solubility laws but dissolved_mass hard-codes dasgupta:

Composition Law Source
libourel \(0.0611\, p_\mathrm{N_2}\) (linear Henry's law) Libourel et al. (2003)
dasgupta (default) physical-state-dependent (see below) Dasgupta et al. (2022)

The Dasgupta et al. (2022) law adds an \(f_{\mathrm{O}_2}\)-dependent reduced-N contribution on top of the molecular dissolution:

\[ X_\mathrm{N_2}^\mathrm{melt}\,[\text{ppmw}] = \sqrt{p_\mathrm{N_2}^{[\mathrm{GPa}]}} \cdot \exp\left(\frac{5908\sqrt{p_\mathrm{tot}^{[\mathrm{GPa}]}}}{T_\mathrm{magma}} - 1.6\,\Delta\mathrm{IW}\right) + p_\mathrm{N_2}^{[\mathrm{GPa}]} \cdot c_\mathrm{melt} \]

where the prefactor \(c_\mathrm{melt}\) depends on the silicate composition. The mole fractions \(x_\mathrm{SiO_2}\), \(x_\mathrm{Al_2O_3}\), \(x_\mathrm{TiO_2}\) are exposed as constructor kwargs SolubilityN2(x_SiO2=..., x_Al2O3=..., x_TiO2=...) and default to the Earth-mantle estimates \(x_\mathrm{SiO_2} = 0.56\), \(x_\mathrm{Al_2O_3} = 0.11\), \(x_\mathrm{TiO_2} = 0.01\), giving \(c_\mathrm{melt} = \exp(4.67 + 7.11 \cdot 0.56 - 13.06 \cdot 0.11 - 120.67 \cdot 0.01) \approx 407\). The two terms compete: at low \(p_\mathrm{N_2}\) and reducing conditions, the first term (reduced N dissolved as nitride / amide complexes) dominates; at high \(p_\mathrm{N_2}\) and oxidising conditions, the second term (molecular N\(_2\) Henry's law) dominates. The breakpoint occurs roughly at \(p_\mathrm{N_2}^{[\mathrm{GPa}]} \approx 10^{-5}\) for \(\Delta\mathrm{IW} = 0\).

S\(_2\) - SolubilityS2(composition='gaillard')

Gaillard et al. (2022) sulfide-saturated mafic-melt law:

\[ \log_{e} X_\mathrm{S_2}^\mathrm{melt}\,[\text{ppmw}] = 13.8426 - \frac{26476}{T_\mathrm{magma}} + 0.124\,x_\mathrm{FeO}^{[\text{wt\%}]} + 0.5\,\ln\frac{p_\mathrm{S_2}}{f_{\mathrm{O}_2}} \]

where \(x_\mathrm{FeO}\) is the melt FeO content in wt%, exposed as the constructor kwarg SolubilityS2(x_FeO=...) and defaulting to \(10\) wt% (Earth-mantle reference). The \(0.5\,\ln(p_\mathrm{S_2}/f_{\mathrm{O}_2})\) term ties the solubility to redox state: at fixed \(p_\mathrm{S_2}\), more reducing conditions (\(f_{\mathrm{O}_2}\) smaller) increase the dissolved sulfur (sulfide regime); more oxidising conditions decrease it.

The implementation refuses to evaluate at \(p_\mathrm{S_2} < 10^{-20}\) bar (returns 0.0) to avoid log(0) in the rare case where the solver bottoms out at exactly zero S inventory.

What about the missing laws?

CALLIOPE does not include explicit solubility laws for H\(_2\), NH\(_3\), SO\(_2\), H\(_2\)S, or O\(_2\). Their dissolved masses are computed from the primary-species solubilities (H\(_2\)O for H-bearing species, CO\(_2\) for C-bearing species, S\(_2\) for S-bearing species, N\(_2\) for N-bearing species) via stoichiometric atom-counting in dissolved_mass(). This is consistent with Bower et al. (2022) Section 2.2.3 which sets \(\alpha_\mathrm{H_2} = \alpha_\mathrm{CO} = \alpha_\mathrm{CH_4} = 0\) on the grounds that experimentally constrained solubilities are 1-2 dex smaller than those of H\(_2\)O / CO\(_2\) at equivalent fugacities (Hirschmann et al. 2012; Li et al. 2015; Yoshioka et al. 2019; Ardia et al. 2013); within the CALLIOPE framework the same logic justifies omitting solubility for the three S species and ammonia.

If you need explicit dissolution of reduced species into the melt, the atmodeller project provides full per-species solubility laws including H\(_2\) (Hirschmann et al. 2012), CO (Yoshioka et al. 2019), and CH\(_4\) (Ardia et al. 2013), with non-ideal real-gas activity coefficients.

Validity envelope

Species Calibration \(T\) range Calibration \(p\) range
H\(_2\)O peridotite 2173 K \(\le\)1 bar to several kbar
H\(_2\)O basalt (Dixon) 1473 K 176-2021 bar
CO\(_2\) (Dixon) \(\le\)2000 K \(\le\)815 bar
CO (Armstrong) \(\sim\)1700 K \(\le\)3 GPa
CH\(_4\) (Ardia) 1573-1873 K 0.7-3 GPa
N\(_2\) (Dasgupta) 1500-2200 K 0-3 GPa
S\(_2\) (Gaillard) 1473-1773 K, \(f_{\mathrm{O}_2} <\) IW+1 sulfide-saturated regime

CALLIOPE deliberately makes no attempt to flag extrapolation: the laws are evaluated formally outside their calibration ranges to keep the solver well-posed. For applications outside the bracket, treat the dissolved masses as upper bounds and check sensitivity by switching solubility laws via the constructor argument.

See also