Model overview

CALLIOPE is a 0-D equilibrium outgassing solver for the magma-ocean atmosphere coupling. It treats the silicate mantle and the overlying gas-phase atmosphere as a single thermodynamic system in equilibrium at the surface, and asks: for a given total elemental inventory and a given magma-ocean state, what surface partial pressures and dissolved-volatile masses simultaneously satisfy (i) gas-phase chemical equilibrium, (ii) gas-melt solubility equilibrium, and (iii) elemental mass conservation?

This page summarises the model assumptions, the variables it solves for, and how it relates to the upstream papers (Bower et al. 2019 ¹, 2022 ²; Nicholls et al. 2024 ¹¹). Each component has its own dedicated page.

What is in the model

• Eleven gas-phase species (calliope.constants.volatile_species): H\(_2\)O, CO\(_2\), O\(_2\), H\(_2\), CH\(_4\), CO, N\(_2\), S\(_2\), SO\(_2\), H\(_2\)S, NH\(_3\).
• Five elements: H, C, N, S always solved. O is either a derived quantity set by the oxygen-fugacity buffer (buffered mode, equilibrium_atmosphere) or a fifth budgeted element with \(\Delta\mathrm{IW}\) as the additional unknown (authoritative-O mode, equilibrium_atmosphere_authoritative_O). The two modes share all physics functions and differ only in their unknown set; see Authoritative-oxygen mode.

• Six equilibrium reactions (gas-phase, surface temperature):

  Reaction	Source
  \(\mathrm{H_2O} \rightleftharpoons \mathrm{H_2} + \tfrac{1}{2}\,\mathrm{O_2}\)	JANAF (janaf_H2) and Schaefer & Fegley 2017 16 (schaefer_H)
  \(\mathrm{CO_2} \rightleftharpoons \mathrm{CO} + \tfrac{1}{2}\,\mathrm{O_2}\)	JANAF (janaf_CO) and Schaefer & Fegley 2017 16 (schaefer_C)
  \(\mathrm{CO_2} + 2\,\mathrm{H_2} \rightleftharpoons \mathrm{CH_4} + \mathrm{O_2}\)	Schaefer & Fegley 2017 16 (schaefer_CH4)
  \(\tfrac{1}{2}\,\mathrm{S_2} + \mathrm{O_2} \rightleftharpoons \mathrm{SO_2}\)	JANAF, doubled form (janaf_SO2)
  \(\tfrac{1}{2}\,\mathrm{S_2} + \mathrm{H_2} \rightleftharpoons \mathrm{H_2S}\)	JANAF, doubled form (janaf_H2S)
  \(\tfrac{1}{2}\,\mathrm{N_2} + \tfrac{3}{2}\,\mathrm{H_2} \rightleftharpoons \mathrm{NH_3}\)	JANAF, doubled form (janaf_NH3)

• One oxygen-fugacity buffer: Fischer et al. (2011) ⁷ iron-wüstite (default; close to atmodeller's Hirschmann composite across the magma-ocean range), or the legacy O'Neill & Eggins (2002) ¹⁴ IW. The shift \(\Delta\mathrm{IW}\) sets \(\log_{10} f_{\mathrm{O}_2}\) relative to the buffer; under the buffered mode it is a user-prescribed input, under the authoritative-O mode it is a solver unknown.

• One solubility law per species with multiple alternative compositions (peridotite, basalt, lunar glass, anorthite-diopside) selectable via constructor argument.

What is not in the model

• No interior structure: gravity, radius, mantle mass come in as scalar inputs. Use Zalmoxis for these.
• No interior thermal evolution: \(T_\mathrm{magma}\) and \(\Phi_\mathrm{global}\) come in as scalars. Use Aragog or SPIDER.
• No radiative transfer: surface partial pressures come out, optical depths and surface temperature come from AGNI or JANUS.
• No atmospheric escape: per-iteration mass loss is computed by the PROTEUS escape module (ZEPHYRUS).
• No solid-phase partitioning: dissolved-mass fields are written into _kg_solid slots that always read 0.0; CALLIOPE only resolves melt and gas reservoirs. The PROTEUS atmosphere modules handle solid-phase trapping if any.
• No real-gas EOS: all species are treated as ideal gases, so partial pressure \(\equiv\) fugacity. For non-ideal real-gas effects use atmodeller (Bower et al. 2025 ³).
• No condensation: every species is in the gas phase. Condensation chemistry happens in AGNI / JANUS.

Mathematical statement

CALLIOPE assembles one mass-conservation equation per solved element. Each equation has the structure

Under the buffered mode the equation set spans \(e \in \{\mathrm{H}, \mathrm{C}, \mathrm{N}, \mathrm{S}\}\), the four primary partial pressures are the unknowns, and the \(4\times 4\) system is solved with scipy.optimize.fsolve (Powell hybrid). Under the authoritative-O mode the equation set spans \(e \in \{\mathrm{H}, \mathrm{C}, \mathrm{N}, \mathrm{S}, \mathrm{O}\}\), the unknown vector is extended with \(\Delta\mathrm{IW}\), and the \(5\times 5\) system is solved with the same outer loop (Authoritative-oxygen mode).

The seven secondary partial pressures are not independent in either mode: they are algebraic functions of the primaries via the six equilibrium constants, evaluated at \(T = T_\mathrm{magma}\) and \(\log_{10} f_{\mathrm{O}_2} = \log_{10} f_{\mathrm{O}_2}^\mathrm{IW}(T) + \Delta\mathrm{IW}\).

The four pieces of physics decompose cleanly:

Lineage

• Bower et al. (2019) ¹ introduced the H\(_2\)O + CO\(_2\) mass-balance + Henry's-law treatment that CALLIOPE inherits, including the molar-mass correction \(\mu_v / \bar\mu\) in the column-mass relation that earlier studies (Elkins-Tanton 2008 ⁶; Lebrun et al. 2013 ⁹; Salvador et al. 2017 ¹⁵; Nikolaou et al. 2019 ¹³) had omitted.
• Bower et al. (2022) ² added the H\(_2\), CO, CH\(_4\) extensions and the explicit Schaefer & Fegley (2017) ¹⁶ IVTHANTHERMO / Chase (1998) ⁴ JANAF equilibrium constants for the H\(_2\)O/H\(_2\), CO\(_2\)/CO, and CO\(_2\)+H\(_2\)/CH\(_4\) couples; also adopted the O'Neill & Eggins (2002) ¹⁴ IW buffer (their Eq. 7) as the parameterisation of mantle redox state.
• Nicholls et al. (2024) ¹¹ introduced N\(_2\) via the Libourel et al. (2003) ¹⁰ and Dasgupta et al. (2022) ⁵ solubility laws, which is the species set in calliope.solve.equilibrium_atmosphere today.
• Nicholls et al. (2026) ¹² demonstrated the sulfur extension (S\(_2\), SO\(_2\), H\(_2\)S) on L 98-59 d, validating the equilibrium constants and the Gaillard et al. (2022) ⁸ S\(_2\) solubility law against in-situ photochemical inferences.

Validity range

CALLIOPE is calibrated for surface temperatures of roughly \(1000 \le T_\mathrm{magma} \le 4000\) K and surface pressures of roughly \(0.1 \le p_\mathrm{surf} \le 5000\) bar. The lower end of the pressure range is set by numerical stability of the speciation walk; the upper end is the loose envelope above which one or more solubility laws extrapolate. Individual solubility laws have tighter calibration windows than the envelope (Dixon CO\(_2\): \(\le\)815 bar; Sossi H\(_2\)O peridotite: 1 atm; Hamilton H\(_2\)O basalt: 1-6 kbar; Ardia CH\(_4\): 0.7-3 GPa total pressure), see the per-law table in Solubility laws. Outside the envelope above:

• Below \(T \sim 1000\) K the JANAF fits used for the equilibrium constants extrapolate beyond their validation range. The PROTEUS wrapper enforces a configurable T_floor (default 700 K), which clips temperatures below T_floor to this value, since thermochemical equilibrium does not necessarily hold at cooler temperatures.
• Above \(T \sim 4000\) K the mantle-atmosphere partitioning approximation breaks down; switch to atmodeller (Bower et al. 2025 ³).
• The Sossi (2023) peridotite and Newcombe (2017) anorthite-diopside / lunar-glass H\(_2\)O laws are calibrated at 1 atm. The Dixon (1995) basalt fit is calibrated to 717 bar \(p_\mathrm{H_2O}\). For surface pressures above \(\sim\)1 kbar use the Hamilton (1964) basalt fit (1-6 kbar calibration range) or atmodeller for higher-pressure non-ideal behaviour. Applying the Sossi or Newcombe fits at kbar pressures extrapolates the partial-pressure input by 3 orders of magnitude beyond calibration; the resulting dissolved-mass error scales as \(p^{0.5}\) for the power-law fits, so the error is a factor of \(\sim\)30 at 1 kbar.
• Solid-phase partitioning is ignored; CALLIOPE strictly handles melt + gas. Use it only when \(\Phi_\mathrm{global} > 0\), or accept that all dissolved masses will be zero.