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Approximate methods to estimate solutions to the radiative transfer equationare essential for the understanding of atmospheres of exoplanets and browndwarfs. The simplest and most popular choice is the "two-stream method" whichis often used to produce simple yet effective models for radiative transfer inscattering and absorbing media. Toon et al. (1989) (Toon89) outlined atwo-stream method for computing reflected light and thermal spectra and waslater implemented in the open-source radiative transfer model PICASO. In Part~Iof this series, we developed an analytical spherical harmonics method forsolving the radiative transfer equation for reflected solar radiation (Rooneyet al. 2023), which was implemented in PICASO to increase the accuracy of thecode by offering a higher-order approximation. This work is an extension ofthis spherical harmonics derivation to study thermal emission spectroscopy. Wehighlight the model differences in the approach for thermal emission andbenchmark the 4-term method (SH4) against Toon89 and a high-streamdiscrete-ordinates method, CDISORT. By comparing the spectra produced by eachmodel we demonstrate that the SH4 method provides a significant increase inaccuracy, compared to Toon89, which can be attributed to the increased order ofapproximation and to the choice of phase function. We also explore thetrade-off between computational time and model accuracy. We find that our4-term method is twice as slow as our 2-term method, but is up to five timesmore accurate, when compared with CDISORT. Therefore, SH4 provides excellentimprovement in model accuracy with minimal sacrifice in numerical expense.

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