Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 315 / 3 / c / Solution

The location between pure-silicate and pure-water planetary mass-radius curves does not determine a unique composition. Possibilities include a rocky iron-silicate interior with a deep water layer, a rock-water mixture, or a mostly rocky planet whose small hydrogen-helium envelope inflates its radius. The atmosphere could accordingly be hydrogen-helium, steam-rich, carbon-dioxide-rich, or secondary gas released from the interior. Residence in the circumstellar habitable zone constrains stellar flux but does not by itself distinguish these models or guarantee surface liquid water.

For an atmosphere spanning fixed base and photospheric pressures,Thus two compositions haveAt equal temperature, a hydrogen-helium atmosphere with $\mu \simeq 2.3$ is abouttimes thicker than a steam atmosphere with $\mu \simeq 18$. This large difference is one reason atmospheric composition contributes strongly to the exoplanet interior-composition degeneracy.