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

Context. Semi-grey atmospheric models (with one opacity for the visible and one opacity for the infrared) are useful to understand the global structure of irradiated atmospheres, their dynamics and the interior structure and evolution of planets, brown dwarfs and stars. But when compared to direct numerical radiative transfer calculations for irradiated exoplanets, these models systematically overestimate the temperatures at low optical depth, independently of the opacity parameters. Aims. We wish to understand 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 to link the energy of the beam to its radiation pressure, 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 κv, two infrared opacities, κ1 and κ2, and β the fraction of the energy in the beam with opacities κ1. We consider that the atmosphere receives an incident irradiation in the visible with an effective temperature Tirr and at an angle µ∗, and that it is heated from below with an effective temperature Tint. Results. Our irradiated non-grey model is found to provide a range of temperatures that is consistent with that obtained by numerical calculations. We find that because of the variable opacities in the infrared, a smaller fraction of the source function is contributing to the heating of the upper layers leading to smaller temperatures than in the semi-grey models. For small values of β (expected whe