Skip to content

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) 3 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) 14 524 2
basalt_dixon \(965\, p^{0.5}\) Dixon et al. (1995) 5, refit by Sossi 965 2
basalt_wilson \(215\, p^{0.7}\) Hamilton (1964) 7; Wilson & Head (1981) 15 215 1/0.7
anorthite_diopside \(727\, p^{0.5}\) Newcombe et al. (2017) 11 727 2
lunar_glass \(683\, p^{0.5}\) Newcombe et al. (2017) 11 683 2

About the peridotite constant 524

Sossi et al. (2023) 14 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) 12 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) 3 Table 1 compares all five compositions across the relevant pressure range; Nicholls et al. (2024) 12 uses peridotite as the fiducial, consistent with CALLIOPE's default.

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

Dixon et al. (1995) 5 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) 3 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 3 §2.2.1 discussion).

CO - SolubilityCO(composition='mafic_armstrong')

Armstrong et al. (2015) 1 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) 16.

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

Ardia et al. (2013) 2 Fe-free haplobasaltic-melt fit (their Eq. (8) with \(\ln K_0 = 4.93\) and \(\Delta V = 26.85\,\mathrm{cm}^3\)/mol at \(T_0 = 1400\,^\circ\)C, \(P_0 = 1\) bar, plotted as Fig. 11; calibrated over 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) 10
dasgupta (default) physical-state-dependent (see below) Dasgupta et al. (2022) 4

The Dasgupta et al. (2022) 4 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) 6 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) 3 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 8; Li et al. 2015 9; Yoshioka et al. 2019 16; Ardia et al. 2013 2); 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 8), CO (Yoshioka et al. 2019 16), and CH\(_4\) (Ardia et al. 2013 2), with non-ideal real-gas activity coefficients.

Validity envelope

Species (composition) Source Calibration \(T\) Calibration \(p\) Calibration \(f_{\mathrm{O}_2}\)
H\(_2\)O peridotite (default) Sossi et al. (2023) 14 2173 K (1900 \(^\circ\)C) 1 atm total (\(f_\mathrm{H_2O}\le 0.027\) bar) IW-1.9 to IW+6.0
H\(_2\)O basalt_dixon Dixon et al. (1995) 5 1473 K (1200 \(^\circ\)C) 176-717 bar \(p_\mathrm{H_2O}\) \(\sim\)QFM (\(\approx\) IW+3.5 to IW+5)
H\(_2\)O basalt_wilson Hamilton et al. (1964) 7 1373 K (1100 \(^\circ\)C) 1000-6000 bar \(p_\mathrm{H_2O}\) unbuffered for the pressure series (separate 1000-bar buffer series used MH, FMQ, MW buffers)
H\(_2\)O anorthite_diopside Newcombe et al. (2017) 11 1623 K (1350 \(^\circ\)C) 1 atm IW-2.3 to IW+4.8
H\(_2\)O lunar_glass Newcombe et al. (2017) 11 1623 K (1350 \(^\circ\)C) 1 atm IW-3.0 to IW+4.8
CO\(_2\) basalt_dixon (default) Dixon et al. (1995) 5 1473 K (1200 \(^\circ\)C) \(\le\)815 bar \(p_\mathrm{CO_2}\) \(\sim\)QFM (\(\approx\) IW+3.5 to IW+5)
CO mafic_armstrong (default) Armstrong et al. (2015) 1 1673 K (1400 \(^\circ\)C) 1.2 GPa \(p_\mathrm{tot}\) (1.0-1.2 GPa including Stanley et al. 2014 data co-fit) IW-3.65 to IW+1.46
CH\(_4\) basalt_ardia (default) Ardia et al. (2013) 2 1673-1723 K (1400-1450 \(^\circ\)C) 0.7-3 GPa \(p_\mathrm{tot}\) IW-9.5 to IW-1.4 (IW/Si and IWC/C buffers)
N\(_2\) libourel Libourel et al. (2003) 10 1673-1698 K (1400-1425 \(^\circ\)C) 1 atm linear regime \(\log_{10}f_{\mathrm{O}_2}\in[-10.7, -0.7]\) (\(\approx\) IW-1.3 to IW+9)
N\(_2\) dasgupta (default in dissolved_mass) Dasgupta et al. (2022) 4 1323-2600 K (1050-2327 \(^\circ\)C) 1 bar to 8.2 GPa \(p_\mathrm{tot}\) IW-8.3 to IW+8.7
S\(_2\) gaillard (default) Gaillard et al. (2022) 6 not stated by paper (1 atm calibration data) 1 atm IW-1 to FMQ+0.1 (\(\approx\) IW+3.5)

The \(f_{\mathrm{O}_2}\) column reports the calibration footprint (the range of \(f_{\mathrm{O}_2}\) over which the experiments that produced the fit were performed), not the law's functional dependence: only the Gaillard S\(_2\) and Dasgupta N\(_2\) laws carry an explicit \(f_{\mathrm{O}_2}\) term, while every other law in the table is an \(f_{\mathrm{O}_2}\)-independent expression fitted to data from experiments performed at the listed \(f_{\mathrm{O}_2}\) range. The choice between H\(_2\)O laws (peridotite vs basalt) is one of the larger uncertainties in early magma-ocean modelling (the line-26 note expands on the Sossi peridotite vs Dixon basalt prefactor difference); the \(f_{\mathrm{O}_2}\) footprint here describes where the data lived, not how robust the fit is across compositions.

The Dasgupta N\(_2\) and Gaillard S\(_2\) rows quote the ranges that Dasgupta et al. (2022) 4 Equation 10 (n=137 compiled data) and Gaillard et al. (2022) 6 Equation 10 (refit of O'Neill & Mavrogenes (2002) 13 plus 8 other experimental sources, n=369) report directly. The Gaillard refit data are all at 1 atm; the formula is applied at magma-ocean pressures in CALLIOPE without an explicit pressure correction. The Libourel linear regime stops at IW-1.3, below which the same paper documents a sharp transition to chemical (network-bound N\(^{3-}\)) dissolution with \(\sim\)5 orders of magnitude higher solubility; the libourel law in CALLIOPE captures only the oxidising-end linear regime and underestimates dissolved N below IW-1.3. The CO, CH\(_4\) and Dasgupta-N\(_2\) rows give the total-pressure range, since the Poynting and reduced-N branches depend on \(p_\mathrm{tot}\) as well as the species partial pressure.

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

  • Equilibrium chemistry: how the gas-phase speciation feeds into the partial pressures the solubility laws then consume.
  • Mass balance & solver: how dissolved + atmospheric masses are summed to close the elemental conservation constraints.
  • Oxygen fugacity: the IW-buffer parameterisations that drive the \(f_{\mathrm{O}_2}\) dependence in the Gaillard S\(_2\) and Dasgupta N\(_2\) laws.
  • Authoritative-oxygen mode: how the Gaillard \(\ln(p_\mathrm{S_2}/f_{\mathrm{O}_2})\) and Dasgupta \(-1.6\,\Delta\mathrm{IW}\) terms enter the 5-residual mass balance once \(\Delta\mathrm{IW}\) becomes a solver unknown.
  • API reference for calliope.solubility.

  1. L. S. Armstrong, M. M. Hirschmann, B. D. Stanley, E. G. Falksen, S. D. Jacobsen, Speciation and solubility of reduced C-O-H-N volatiles in mafic melt: implications for volcanism, atmospheric evolution, and deep volatile cycles in the terrestrial planets, Geochimica et Cosmochimica Acta, 171, 283–302, 2015. SciX. ↩↩

  2. P. Ardia, M. M. Hirschmann, A. C. Withers, B. D. Stanley, Solubility of CH\(_4\) in a synthetic basaltic melt, with applications to atmosphere-magma ocean-core partitioning of volatiles and to the evolution of the Martian atmosphere, Geochimica et Cosmochimica Acta, 114, 52–71, 2013. SciX. ↩↩↩↩

  3. D. J. Bower, K. Hakim, P. A. Sossi, P. Sanan, Retention of water in terrestrial magma oceans and carbon-rich early atmospheres, The Planetary Science Journal, 3(4), 93, 2022. SciX. ↩↩↩↩↩

  4. R. Dasgupta, E. Falksen, A. Pal, C. Sun, The fate of nitrogen during parent body partial melting and accretion of the inner Solar System bodies at reducing conditions, Geochimica et Cosmochimica Acta, 336, 291–307, 2022. SciX. ↩↩↩↩

  5. J. E. Dixon, E. M. Stolper, J. R. Holloway, An experimental study of water and carbon dioxide solubilities in mid-ocean ridge basaltic liquids. Part I: Calibration and solubility models, Journal of Petrology, 36(6), 1607–1631, 1995. SciX. ↩↩↩↩

  6. F. Gaillard, F. Bernadou, M. Roskosz, M. A. Bouhifd, Y. Marrocchi, G. Iacono-Marziano, M. Moreira, B. Scaillet, G. Rogerie, Redox controls during magma ocean degassing, Earth and Planetary Science Letters, 577, 117255, 2022. SciX. ↩↩↩

  7. D. L. Hamilton, C. W. Burnham, E. F. Osborn, The solubility of water and effects of oxygen fugacity and water content on crystallization in mafic magmas, Journal of Petrology, 5(1), 21–39, 1964. ↩↩

  8. M. M. Hirschmann, A. C. Withers, P. Ardia, N. T. Foley, Solubility of molecular hydrogen in silicate melts and consequences for volatile evolution of terrestrial planets, Earth and Planetary Science Letters, 345–348, 38–48, 2012. SciX. ↩↩

  9. Y. Li, R. Dasgupta, K. Tsuno, The effects of sulfur, silicon, water, and oxygen fugacity on carbon solubility and partitioning in Fe-rich alloy and silicate melt systems at 3 GPa and 1600 °C: implications for core-mantle differentiation and degassing of magma oceans and reduced planetary mantles, Earth and Planetary Science Letters, 415, 54–66, 2015. SciX. ↩

  10. G. Libourel, B. Marty, F. Humbert, Nitrogen solubility in basaltic melt. Part I. Effect of oxygen fugacity, Geochimica et Cosmochimica Acta, 67(21), 4123–4135, 2003. SciX. ↩↩

  11. M. E. Newcombe, A. Brett, J. R. Beckett, M. B. Baker, S. Newman, Y. Guan, J. M. Eiler, E. M. Stolper, Solubility of water in lunar basalt at low pH\(_2\)O, Geochimica et Cosmochimica Acta, 200, 330–352, 2017. SciX. ↩↩↩↩

  12. H. Nicholls, T. Lichtenberg, D. J. Bower, R. Pierrehumbert, Magma ocean evolution at arbitrary redox state, Journal of Geophysical Research: Planets, 129, e2024JE008576, 2024. SciX. ↩↩

  13. H. St. C. O'Neill, J. A. Mavrogenes, The sulfide capacity and the sulfur content at sulfide saturation of silicate melts at 1400 °C and 1 bar, Journal of Petrology, 43(6), 1049–1087, 2002. SciX. ↩

  14. P. A. Sossi, P. M. E. Tollan, J. Badro, D. J. Bower, Solubility of water in peridotite liquids and the prevalence of steam atmospheres on rocky planets, Earth and Planetary Science Letters, 601, 117894, 2023. SciX. ↩↩↩

  15. L. Wilson, J. W. Head, Ascent and eruption of basaltic magma on the Earth and Moon, Journal of Geophysical Research, 86(B4), 2971–3001, 1981. SciX. ↩

  16. T. Yoshioka, D. Nakashima, T. Nakamura, S. Shcheka, H. Keppler, Carbon solubility in silicate melts in equilibrium with a CO-CO\(_2\) gas phase and graphite, Geochimica et Cosmochimica Acta, 259, 129–143, 2019. SciX. ↩↩↩