Table 1 free parameters in our model, the nominal parameter value for illustrative calculations (Figs. 2–4), and the full Monte Carlo range shown in later calculations (Fig. 5)

Initial conditions

Initial carbon, C (kg)

4 × 10²⁰

10²⁰–10^21.5

Approximate bulk silicate Earth carbon inventory^111,112,113

Initial free oxygen, O (kg)

6 × 10²¹

10²¹–10²²

Approximately Earth-like (ensures post-solidification mantle redox around quartz-fayalite-magnetite buffer if no nebular atmosphere).

Initial hydrogen, H (kg)

4 × 10²⁰

10²⁰–10²³

Point estimate is approximate bulk silicate Earth inventory, range encompasses large primary atmosphere.

Initial radionuclide U, Th, and K inventory (relative Earth)

1.0

0.33–30.0

Scalar multiplication of Earth’s radionuclide inventories in Lebrun, et al.²⁷. Allows for modest tidal heating.

Initial mantle FeO (mole fraction)

0.06

0.06

Earth-like value assumed for nominal calculations, but supplementary material investigates broad range of FeO (0.02 -0.2)

Stellar evolution and escape parameters

TRAPPIST-1 XUV saturation time,\({t}_{sat}\)

3.14

\({3.14}_{-1.46}^{+2.22}\) Gyr

XUV evolution parameters drawn randomly from joint distribution¹⁰¹.

Post saturation phase XUV decay exponent, \({\beta }_{decay}\)

−1.17

\(-{1.17}_{-0.28}^{+0.27}\)

XUV evolution parameters drawn randomly from joint distribution¹⁰¹.

Saturated log₁₀(F_XUV/F_BOLOMETRIC) flux ratio

-3.03

\(-{3.03}_{-0.23}^{+0.25}\)

XUV evolution parameters drawn randomly from joint distribution¹⁰¹.

Escape efficiency at low XUV flux, \({\varepsilon }_{low}\)

0.2

0.01–0.3

See escape section in Krissansen-Totton, et al.²⁸.

Transition parameter for diffusion limited to XUV-limited escape, \({\lambda }_{tra}\)

1.0

10⁻⁶–10¹*

See escape section in Krissansen-Totton, et al.²⁸.

XUV energy that contributes to XUV escape above hydrodynamic threshold, \({\zeta }_{high}\)

50%

0–100%

See escape section in Krissansen-Totton, et al.²⁸.

Cold trap temperature variation, \(\varDelta {T}_{cold-trap}\)

0 K

−30 to +30 K

Cold trap temperature, \({T}_{cold-trap}\), equals planetary skin temperature plus a fixed, uniformly sampled variation, \({T}_{cold-trap}={T}_{eq}{(1/2)}^{0.25}+\varDelta {T}_{cold-trap}\). Here, \({T}_{eq}\) is the planetary equilibrium temperature given assumed albedo.

Thermosphere temperature, \({T}_{thermo}\)

1000 K

200–5000 K*

^114,115,116

Interior parameter

Mantle viscosity coefficient

10 Pa

10¹–10³Pa s*

Solid mantle kinematic viscosity, \({\nu }_{rock}\), (m²/s) is given by the following equation:\({\nu }_{rock}={V}_{coef}3.8\times {10}^{7}\exp (\frac{350000}{8.314{T}_{p}})/{\rho }_{m}\)Here \({T}_{p}\) is mantle potential temperature (K) and \({\rho }_{m}\) is mantle density (kg/m³). See Krissansen-Totton, et al.²⁸.

Albedo

Bond albedo during magma ocean solidification

0.2

0.0–0.2

^85,86