Volume 562, February 2014
|Number of page(s)||17|
|Section||Planets and planetary systems|
|Published online||21 February 2014|
A non-grey analytical model for irradiated atmospheres⋆
Laboratoire Lagrange, UMR7293, Université de Nice Sophia-Antipolis, CNRS,
Observatoire de la Côte d’Azur,
2 Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA
Accepted: 25 November 2013
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 (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 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 τlim = (κR/κ1κ2)(κRκP/3)1/2, it is mainly transported by the channel of lowest opacity whereas if it is absorbed at τ ≳ τlim 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.
Key words: radiative transfer / planets and satellites: atmospheres / stars: atmospheres / planetary systems
A FORTRAN implementation of the analytical model is available at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (184.108.40.206) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/562/A133
© ESO, 2014
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