Spherical Harmonics for the 1D Radiative Transfer Equation II: Thermal Emission

Approximate methods to estimate solutions to the radiative transfer equation are essential for the understanding of atmospheres of exoplanets and brown dwarfs. The simplest and most popular choice is the "two-stream method" which is often used to produce simple yet effective models for radiative transfer in scattering and absorbing media. Toon et al. (1989) (Toon89) outlined a two-stream method for computing reflected light and thermal spectra and was later implemented in the open-source radiative transfer model PICASO. In Part~I of this series, we developed an analytical spherical harmonics method for solving the radiative transfer equation for reflected solar radiation (Rooney et al. 2023), which was implemented in PICASO to increase the accuracy of the code by offering a higher-order approximation. This work is an extension of this spherical harmonics derivation to study thermal emission spectroscopy. We highlight the model differences in the approach for thermal emission and benchmark the 4-term method (SH4) against Toon89 and a high-stream discrete-ordinates method, CDISORT. By comparing the spectra produced by each model we demonstrate that the SH4 method provides a significant increase in accuracy, compared to Toon89, which can be attributed to the increased order of approximation and to the choice of phase function. We also explore the trade-off between computational time and model accuracy. We find that our 4-term method is twice as slow as our 2-term method, but is up to five times more accurate, when compared with CDISORT. Therefore, SH4 provides excellent improvement in model accuracy with minimal sacrifice in numerical expense.