Grain opacity and the bulk composition of extrasolar planets. II. An analytical model for the grain opacity in protoplanetary atmospheres
Abstract
Context. We investigate the grain opacity kgr in the atmosphere of protoplanets. This is important for the planetary mass-radius relation since kgr affects the H/He envelope mass of low-mass planets and the critical core mass of giant planets. Aims. The goal of this study is to derive an analytical model for kgr. Methods. Our model is based on the comparison of the timescales of microphysical processes like grain settling in the Stokes and Epstein regime, growth by Brownian motion coagulation and differential settling, grain evaporation, and grain advection due to envelope contraction. With these timescales we derive the grain size, abundance, and opacity. Results. We find that the main growth process is differential settling. In this regime, kgr has a simple functional form and is given as 27 Q/8 H rho in the Epstein regime and as 2 Q/H rho for Stokes drag. Grain dynamics lead to a typical radial structure of kgr with high ISM-like values in the top layers but a strong decrease in the deeper parts where the grain-free molecular opacities take over. Conclusions. In agreement with earlier results we find that kgr is typically much lower than in the ISM. The equations also show that a higher dust input in the top layer does not strongly increase kgr with two important implications. First, for a formation of giant planet cores via pebbles, there could be the issue that pebbles increase the grain input high in the atmosphere due to ablation. This could potentially increase kgr hindering giant planet formation. Our study shows that this adverse effect should not occur. Second, it means that a higher stellar [Fe/H] which presumably leads to a higher surface density of planetesimals only favors giant planet formation without being detrimental to it due to an increased kgr. This corroborates the result that core accretion explains the increase of the giant planet frequency with [Fe/H].
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