A non-grey analytical model for irradiated atmospheres^⋆

I. Derivation

¹ Laboratoire Lagrange, UMR7293, Université de Nice Sophia-Antipolis, CNRS, Observatoire de la Côte d’Azur, 06300 Nice, France
e-mail: vivien.parmentier@oca.eu
² Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA

Received: 22 July 2013
Accepted: 25 November 2013

Abstract

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 (κ_v), 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 T_irr and at an angle μ_∗, 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 τ_lim = (κ_R/κ₁κ₂)(κ_Rκ_P/3)^1/2, it is mainly transported by the channel of lowest opacity whereas if it is absorbed at τ ≳ τ_lim it is mainly transported by the channel of highest opacity, independently of the spectral width of those channels. For low values of β (expected when lines are dominant), we find that the non-grey effects significantly cool the upper atmosphere. However, for β ≳ 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.