Revisiting rocky planets cooling phase and magma ocean occurrence with a consistent atmospheric model

Rocky planets can host two distinct types of surface magma oceans. First, post-accretion magma oceans, sustained by the residual heat from accretional impacts [1]. The planet’s surface can remain molten for extended periods depending on the planet’s volatile content and the atmospheric properties. Second, permanent magma oceans, maintained by intense stellar irradiation, possibly combined with a strong greenhouse effect. For Earth and Venus, early evolution likely involved both mechanisms, with their large water inventories (at least 0.01 % of the planetary mass) making a dense steam atmosphere unavoidable during these phases.

In both cases, a classic approach to modelling these hot, dense atmospheres has been to assume that they are entirely convective from the surface to the photosphere, where most of the planet’s thermal emission is radiated to space. This assumption, first formalized by Kasting [3], greatly simplifies the problem, the atmospheric profile follows an adiabatic lapse rate, and the outgoing longwave radiation becomes independent of the surface temperature over a wide range [350-2000 K] triggered by the runaway greenhouse threshold near 270 W/m^2. If the combined internal and stellar heat flux exceeds this limit, complete vaporization of the water reservoir occurs, and the surface temperature rises above 2000 K, leading to the formation of a magma ocean.

This approximation has been systematically applied to hot, thick terrestrial atmospheres. For example, to estimate magma ocean lifetimes [4, 6], to describe Venus's past [2], Earth's future, or to model the mass-radius relationships of highly irradiated rocky planets  [8, 9] and to predict or interpret JWST observations of hot terrestrial exoplanets like TRAPPIST-1b, c, and d.

However, recent work [3] has shown that this assumption is generally invalid for hot and dense atmospheres, where radiative zones may develop and break the fully convective structure. For thick atmospheres the assumption of full convection breaks down, radiative processes dominate over convection in large portions of the atmosphere. This invalidates the widely used inverse climate modeling method, where a prescribed atmospheric profile is coupled to radiative transfer in order to determine equilibrium fluxes. Instead, a consistent radiative-convective equilibrium must be solved for each atmospheric layers.

In this study, we present an improved coupling model between the magma ocean evolution and a consistent 1D radiative-convective atmospheric model using the Exo_k framework [5]. This code allows efficient computation of radiative transfer and convective adjustment, including effects such as dry and moist convection, condensation, precipitation and turbulent diffusion. Importantly, it can handle the wide range of radiative timescales-spanning seconds to tens of thousands of years-that arise in thick steam atmospheres.

Focusing on Venus, we reassess its early thermal evolution. Classical models predict that its steam atmosphere, maintained by insolation, kept the surface in a molten state for hundreds of millions of years-long enough for hydrodynamic escape to remove the planet’s water [2]. In summary, our work demonstrates the critical importance of using consistent radiative-convective atmospheric models to accurately capture magma ocean lifetimes and volatile evolution. These insights have many implications not only for the early histories of Venus and Earth but also for exoplanet characterization and the interpretation of future observations with the James Webb Space Telescope (JWST) and other missions.

References
[1] L. Elkins-Tanton. Linked magma ocean solidification and atmospheric growth for Earth and Mars.Earth and Planetary Science Letters, 2008.  
[2] K. Hamano, Y. Abe, and H. Genda. Emergence of two types of terrestrial planet on solidification  of magma ocean. Nature, 2013.
[3] J. F. Kasting. Runaway and moist greenhouse atmospheres and the evolution of Earth and Venus. Icarus, 1988.  
[4] T. Lebrun, H. Massol, E. Chassefi.re, A. Davaille, E. Marcq, P. Sarda, F. Leblanc, and  G. Brandeis. Thermal evolution of an early magma ocean in interaction with the atmosphere. Journal of Geophysical Research: Planets, 2013
[5] J. Leconte. Spectral binning of precomputed correlated-k coefficients. Astronomy & Astrophysics, 2021.
[6] H. Massol, A. Davaille, and P. Sarda. Early Formation of a Water Ocean as a  Function of Initial CO2 and H2O Contents in a Solidifying Rocky Planet. Journal  of Geophysical Research: Planets, 2023
[7] F. Selsis, J. Leconte, M. Turbet, G. Chaverot, and E. Bolmont. A cool runaway greenhouse without  surface magma ocean. Nature, 2023
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[9] M. Turbet, D. Ehrenreich, C. Lovis, E. Bolmont, and T. Fauchez. The runaway greenhouse radius  inflation effect - An observational diagnostic to probe water on Earth-sized planets and test the  habitable zone concept. Astronomy & Astrophysics, 2019