On the radiative equilibrium of irradiated planetary atmospheres

Context. The evolution of stars and planets is mostly controlled by the properties of their atmosphere. This is particularly true in the case of exoplanets close to their stars, for which one has to account both for an (often intense) irradiation flux, and from an intrinsic flux responsible for the progressive loss of the inner planetary heat. Aims. The goals of the present work are to help understanding the coupling between radiative transfer and advection in exoplanetary atmospheres and to provide constraints on the temperatures of the deep atmospheres. This is crucial in assessing whether modifying assumed opacity sources and/or heat transport may explain the inflated sizes of a significant number of giant exoplanets found so far. Methods. I use a simple analytical approach inspired by Eddington’s approximation for stellar atmospheres to derive a relation between temperature and optical depth valid for plane-parallel static grey atmospheres which are both transporting an intrinsic heat flux and receiving an outer radiation flux. The model is parameterized as a function of mean visible and thermal opacities, respectively. Results. The model is shown to reproduce relatively well temperature profiles obtained from more sophisticated radiative transfer calculations of exoplanetary atmospheres. It naturally explains why a temperature inversion (stratosphere) appears when the opacity in the optical becomes significant compared to that in the infrared. I further show that the mean equivalent flux (proportional to T 4) is conserved in the presence of horizontal advection on constant optical depth levels. This implies with these hypotheses that the deep atmospheric temperature used as outer boundary for the evolution models should be calculated from models pertaining to the entire planetary atmosphere, not from ones that are relevant to the day side or to the substellar point. In these conditions, present-day models yield deep temperatures that are ∼ 1000 K too cold to explain the present size of planet HD 209458b. An tenfold increase in the infrared to visible opacity ratio would be required to slow the planetary cooling and contraction sufficiently to explain its size. However, the mean equivalent flux is not conserved anymore in the presence of opacity variations, or in the case of non-radiative vertical transport of energy: The presence of clouds on the night side or a downward transport of kinetic energy and its dissipation at deep levels would help making the deep atmosphere hotter and may explain the inflated sizes of giant exoplanets.