Revised mass-radius relationships for water-rich rocky planets more irradiated than the runaway greenhouse limit

Mass-radius relationships for water-rich rocky planets are usually calculated assuming most water is present in condensed (either liquid or solid) form. Planet density estimates are then compared to these mass-radius relationships, even when these planets are more irradiated than the runaway greenhouse irradiation limit (around 1.1 times the insolation at Earth for planets orbiting a Sun-like star), for which water has been shown to be unstable in condensed form and would instead form a thick H2O-dominated atmosphere. Here we use a 1-D radiative-convective inverse version of the LMD generic numerical climate model to derive new theoretical mass-radius relationships appropriate for water-rich rocky planets that are more irradiated than the runaway greenhouse irradiation limit, meaning planets endowed with a steam, water-dominated atmosphere. As a result of the runaway greenhouse radius inflation effect introduced in previous work, these new mass-radius relationships significantly differ from those traditionally used in the literature. For a given water-to-rock mass ratio, these new mass-radius relationships lead to planet bulk densities much lower than calculated when water is assumed to be in condensed form. In other words, using traditional mass-radius relationships for planets that are more irradiated than the runaway greenhouse irradiation limit tends to dramatically overestimate -possibly by several orders of magnitude- their bulk water content. In particular, this result applies to TRAPPIST-1 b, c, and d, which can accommodate a water mass fraction of at most 2, 0.3 and 0.08%, respectively, assuming planetary core with a terrestrial composition. In addition, we show that significant changes of mass-radius relationships (between planets less and more irradiated than the runaway greenhouse limit) can be used to remove bulk composition degeneracies in multiplanetary systems such as TRAPPIST-1. Broadly speaking, our results demonstrate that non-H2/He-dominated atmospheres can have a first-order effect on the mass-radius relationships, even for rocky planets receiving moderate irradiation. Finally, we provide an empirical formula for the H2O steam atmosphere thickness as a function of planet core gravity and radius, water content, and irradiation. This formula can easily be used to construct mass-radius relationships for any water-rich, rocky planet (i.e., with any kind of interior composition ranging from pure iron to pure silicate) more irradiated than the runaway greenhouse irradiation threshold.