A non-grey analytical model for irradiated atmospheres ? I. Derivation

Context. Semi-grey atmospheric models (with one opacity for the visible and one opacity for the infrared) are useful for understanding the global structure of irradiated atmospheres, their dynamics, and the interior structure and evolution of planets, brown dwarfs, and stars. When compared to direct numerical radiative transfer calculations for irradiated exoplanets, however, these models systematically overestimate the temperatures at low optical depths, independently of the opacity parameters. Aims. We investigate why semi-grey models fail at low optical depths and provide a more accurate approximation to the atmospheric structure by accounting for the variable opacity in the infrared. Methods. Using the Eddington approximation, we derive an analytical model to account for lines and/or bands in the infrared. Four parameters (instead of two for the semi-grey models) are used: a visible opacity (kappa(v)), two infrared opacities, (kappa(1) and kappa(2)), and beta (the fraction of the energy in the beam with opacities kappa(1)). We consider that the atmosphere receives an incident irradiation in the visible with an effective temperature T-irr and at an angle mu(*), and that it is heated from below with an effective temperature T-int. Results. Our non-grey, irradiated line model is found to provide a range of temperatures that is consistent with that obtained by numerical calculations. We find that if the stellar flux is absorbed at optical depth larger than tau(lim) = (kappa(R)/kappa(1)kappa(2))(kappa(R)kappa(P)/3)(1/2), it is mainly transported by the channel of lowest opacity whereas if it is absorbed at tau greater than or similar to tau(lim) it is mainly transported by the channel of highest opacity, independently of the spectral width of those channels. For low values of beta (expected when lines are dominant), we find that the non-grey effects significantly cool the upper atmosphere. However, for beta greater than or similar to 1/2 (appropriate in the presence of bands with a wavelength-dependence smaller than or comparable to the width of the Planck function), we find that the temperature structure is affected down to infrared optical depths unity and deeper as a result of the so-called blanketing effect. Conclusions. The expressions that we derive can be used to provide a proper functional form for algorithms that invert the atmospheric properties from spectral information. Because a full atmospheric structure can be calculated directly, these expressions should be useful for simulations of the dynamics of these atmospheres and of the thermal evolution of the planets. Finally, they should be used to test full radiative transfer models and to improve their convergence.