Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 315 / 2 / b / Solution

Potential surface habitability requires a stable liquid-water interface, a temperature and pressure outside runaway-greenhouse and high-pressure-ice regimes, and an ${\mathrm{H}}_2$ inventory large enough for collision-induced greenhouse warming but not so deep that pressure and temperature sterilize the ocean. Long-term atmospheric retention, tolerable stellar ultraviolet and particle flux, nutrient cycling, and chemical disequilibrium compatible with metabolism are also needed. A location in the circumstellar habitable zone alone is insufficient.

For $M_p=5M_{\oplus }$ and $R_p=2R_{\oplus }$,Taking $T=300\mathrm{K}$ and hydrogen-rich mean molecular weight $\mu =2.3$ gives the atmospheric scale heightIf a strong feature spans $N_H=5$ scale heights and $R_s=0.5R_{\odot }$, the atmospheric spectral-feature amplitude isClouds, hazes, refraction, and smaller molecular abundance reduce this estimate.