From CO2- to H2O-dominated atmospheres and back - How mixed outgassing changes the volatile distribution in magma oceans around M dwarf stars

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Volume 693 (January 2025)

A&A, 693 (2025) A303

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Table A.1

Placeholder for differentiation of mass balance equations.

mass balance H₂O	mass balance CO₂
(volatile 1)	(volatile 2)
$ɡ = F_{H_{2} O}$ $={{F}_{{{\text{H}}_{2}}\text{O}}}$	$l = F_{{CO}_{2}}$ $l\text{ }\!\!~\!\!\text{ = }\!\!~\!\!\text{ }{{F}_{\text{C}{{\text{O}}_{2}}}}$
$a_{1} = - M_{H_{2} O}^{moa} (1)$ ${{a}_{1}}=-M_{{{\text{H}}_{2}}\text{O}}^{\text{moa }\!\!~\!\!\text{ }}\text{ }\!\!~\!\!\text{ (1)}$	$a_{2} = - M_{{CO}_{2}}^{moa} (1)$ ${{a}_{2}}=-M_{\text{C}{{\text{O}}_{2}}}^{\text{moa }\!\!~\!\!\text{ }}\text{ }\!\!~\!\!\text{ (1)}$
$b_{1} = k_{H_{2} O} \cdot M^{crystal}$ ${{b}_{1}}={{k}_{{{\text{H}}_{2}}\text{O}}}\cdot {{M}^{\text{crystal }\!\!~\!\!\text{ }}}$	$b_{2} = k_{{CO}_{2}} \cdot M^{crystal}$ ${{b}_{2}}={{k}_{\text{C}{{\text{O}}_{\text{2}}}}}\cdot {{M}^{\text{crystal }\!\!~\!\!\text{ }}}$
f = 3.44 × 10⁻⁸ (Schaefer) or 2.08 × 10⁻⁶ (E-T)	v = 2.08 × 10⁻⁶ (E-T)
e =0 (Schaefer) or −3 × 10⁻³ (E-T)	u = −5 × 10⁻⁴ (E-T)
exp₁ = 1/0.74 (Schaefer) or 1/0.52 (E-T)	exp₂ = 1/0.45 (E-T)
$w_{1} = μ_{{CO}_{2}} / μ_{H_{2} O}$ ${{w}_{1}}={{\mu }_{\text{C}{{\text{O}}_{2}}}}/{{\mu }_{{{\text{H}}_{2}}\text{O}}}$	$w_{2} = μ_{H_{2} O} / μ_{{CO}_{2}}$ ${{w}_{2}}={{\mu }_{{{\text{H}}_{2}}\text{O}}}/{{\mu }_{\text{C}{{\text{O}}_{2}}}}$

Both volatile systems

h = M^liq
$h = M^{liq}$ $h={{M}^{\text{liq}}}$
$d = \frac{4 π R_{P l}^{2}}{ɡ}$ $d=\frac{4\pi R_{Pl}^{2}}{}$

(1) Due to numerical reasons, we add in the code implementation mass budget correction terms to a₁ and a₂. See Sect. D for a description and discussion of code stability and performance.

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