Analytical Models of Exoplanetary Atmospheres. II. Radiative Transfer via the Two-stream Approximation

Abstract

We present a comprehensive analytical study of radiative transfer using the method of moments and include the effects of non-isotropic scattering in the coherent limit. Within this unified formalism, we derive the governing equations and solutions describing two-stream radiative transfer (which approximates the passage of radiation as a pair of outgoing and incoming fluxes), flux-limited diffusion (which describes radiative transfer in the deep interior) and solutions for the temperature-pressure profiles. Generally, the problem is mathematically under-determined unless a set of closures (Eddington coefficients) is specified. We demonstrate that the hemispheric (or hemi-isotropic) closure naturally derives from the radiative transfer equation if energy conservation is obeyed, while the Eddington closure produces spurious enhancements of both reflected light and thermal emission. We concoct recipes for implementing two-stream radiative transfer in stand-alone numerical calculations and general circulation models. We use our two-stream solutions to construct toy models of the runaway greenhouse effect. We present a new solution for temperature-pressure profiles with a non-constant optical opacity and elucidate the effects of non-isotropic scattering in the optical and infrared. We derive generalized expressions for the spherical and Bond albedos and the photon deposition depth. We demonstrate that the value of the optical depth corresponding to the photosphere is not always 2/3 (Milne's solution) and depends on a combination of stellar irradiation, internal heat and the properties of scattering both in optical and infrared. Finally, we derive generalized expressions for the total, net, outgoing and incoming fluxes in the convective regime.

Accepted by ApJS. 23 pages, 11 figures, 3 tables, 158 equations. No change from previous version except for title (to match ApJS convention)

References in corpus (5)

• On the radiative equilibrium of irradiated planetary atmospheres
• Atmospheric Retrieval for Super-Earths: Uniquely Constraining the Atmospheric Composition with Transmission Spectroscopy
• Theoretical Spectra and Light Curves of Close-in Extrasolar Giant Planets and Comparison with Data
• On the Absorption and Redistribution of Energy in Irradiated Planets
• An Analytic Radiative-Convective Model for Planetary Atmospheres

Cited by in corpus (31)

• The imprint of exoplanet formation history on observable present-day spectra of hot Jupiters
• VULCAN: an Open-Source, Validated Chemical Kinetics Python Code for Exoplanetary Atmospheres
• Atmospheric Dynamics of Hot Exoplanets
• Self-luminous and irradiated exoplanetary atmospheres explored with HELIOS
• GENESIS: New Self-Consistent Models of Exoplanetary Spectra
• Atmospheric Regimes and Trends on Exoplanets and Brown Dwarfs
• Deciphering thermal phase curves of dry, tidally locked terrestrial planets
• Simulating gas giant exoplanet atmospheres with Exo-FMS: Comparing semi-grey, picket fence and correlated-k radiative-transfer schemes
• The GAPS Programme at TNG XXXVIII. Five molecules in the atmosphere of the warm giant planet WASP-69b detected at high spectral resolution
• Evidence for disequilibrium chemistry from vertical mixing in hot Jupiter atmospheres. A comprehensive survey of transiting close-in gas giant exoplanets with warm-Spitzer/IRAC
• On the Chemical Abundance of HR 8799 and the Planet c
• Retrieval Study of Brown Dwarfs Across the L-T Sequence
• The GAPS Programme at TNG XXXIX -- Multiple molecular species in the atmosphere of the warm giant planet WASP-80 b unveiled at high resolution with GIANO-B
• Super-Earths and Earth-like Exoplanets
• Retrieval study of cool, directly imaged exoplanet 51 Eri b
• Is the atmosphere of the ultra-hot Jupiter WASP-121b variable?
• The GAPS Programme at TNG LV. Multiple molecular species in the atmosphere of HAT-P-11 b and review of the HAT-P-11 planetary system
• Retrieval Analysis of the Emission Spectrum of WASP-12b: Sensitivity of Outcomes to Prior Assumptions and Implications for Formation History
• Auto-Differentiable Spectrum Model for High-Dispersion Characterization of Exoplanets and Brown Dwarfs
• Models of Warm Jupiter Atmospheres: Observable Signatures of Obliquity
• The THOR+HELIOS general circulation model: multi-wavelength radiative transfer with accurate scattering by clouds/hazes
• Analytical Models of Exoplanetary Atmospheres. IV. Improved Two-stream Radiative Transfer for the Treatment of Aerosols
• Atmospheric stability and collapse on tidally locked rocky planets
• EOS: Atmospheric Radiative Transfer in Habitable Worlds with HELIOS
• Modeling the Thermal Bulge of A Hot Jupiter with the Two-Stream Approximation
• The Effect of Metallicity on the Non-Equilibrium Abundance of Hydrogen Dominated Exoplanet Atmosphere
• The effect of spectroscopic binning on atmospheric retrievals
• Using a neural network approach to accelerate disequilibrium chemistry calculations in exoplanet atmospheres
• The SPACE Program. I. The featureless spectrum of HD 86226 c challenges sub-Neptune atmosphere trends
• Spherical Harmonics for the 1D Radiative Transfer Equation I: Reflected Light
• How does the shape of the stellar spectrum affect the Raman scattering features in the albedo of exoplanets?