Analytical Models of Exoplanetary Atmospheres. VI. Full Solutions for Improved Two-stream Radiative Transfer Including Direct Stellar Beam

TL;DR

This paper derives full solutions for an improved two-stream radiative transfer model that includes direct stellar beam effects, enhancing accuracy for planetary atmosphere simulations especially with aerosols.

Contribution

It introduces an analytical correction factor and fitting function for the improved two-stream model, incorporating the direct stellar beam into radiative transfer calculations.

Findings

Derived an analytical expression for the correction factor.

Provided a simple fitting function for practical use.

Validated the model reproduces Eddington and quadrature closures.

Abstract

Two-stream radiative transfer is a workhorse in the Earth, planetary and exoplanetary sciences due to its simplicity and ease of implementation. However, a longstanding limitation of the two-stream approximation is its inaccuracy in the presence of medium-sized or large aerosols. This limitation was lifted by the discovery of the improved two-stream technique, where the accuracy of the scattering greenhouse effect is matched to that of multi-stream calculations by construction. In the present study, we derive the full solutions for improved two-stream radiative transfer, following its introduction by Heng & Kitzmann (2017), and include contributions from the direct stellar beam. The generalization of the original two-stream flux solutions comes in the form of a correction factor, traditionally set to unity, which is the ratio of a pair of first Eddington coefficients. We derive an analytical expression for this correction factor and also provide a simple fitting function for its ease of use by other workers. We prove that the direct stellar beam is associated with a second Eddington coefficient that is on the order of unity. Setting this second Eddington coefficient to 2/3 and $1/\sqrt{3}$ reproduces the Eddington and quadrature closures, respectively, associated with the direct beam. We use our improved two-stream solutions for the fluxes to derive two-stream source function solutions for both the intensity and fluxes.