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

Abstract

Approximate methods for radiative transfer equations that are fast, reliable, and accurate 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. (hereafter, Toon89) outlined a two-stream method for computing reflected light and thermal spectra that 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 that 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 four-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 four-term method is twice as slow as our two-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.

Testing Approximate Infrared Scattering Radiative-transfer Methods for Hot Jupiter Atmospheres

“…Due to the generalized nature of the RT methods used here, our findings are likely applicable to atmospheres across the exoplanet spectrum, from ultrahot Jupiters to warm Neptunes and rocky planets. However, we caution conditions of high gravity and strong scattering on cloudy brown dwarfs, such as those tested in Rooney et al (2024), will need to be tested further before AAE and VIM can be recommended for brown dwarf modeling efforts. Software: DISORT and DISORT-twostr: http://www.rtatmocn.…”

“…In Rooney et al (2024), the Toon89 method was compared to a spherical harmonic (SH) method with the source function technique. They found that the Toon89 method produced spectral OLR differences of ∼20% or more at certain wavelengths compared to their DISORT reference calculations for their examples of cloudy brown dwarfs.…”

“…1. The larger gravity used in the brown dwarf models of Rooney et al (2024), making the OLR more sensitive to strong scattering components deeper in the atmosphere. 2.…”

“…We suspect these two factors, as well as the assumption of a hemispheric-mean phase function in the Toon89 method compared to the Henyey-Greenstein function in the SH method, are responsible for the much larger errors seen in Rooney et al (2024). Owing to this, the EAA and VIM methods will need to be separately tested for brown dwarf conditions similar to Rooney et al (2024), though we speculate that, since a lot of the errors in the Toon89 method stem from the use of the hemispheric-mean phase approximation (Rooney et al 2024), VIM may be more accurate than Toon89 in some cases, but probably not as accurate as the SH method itself.…”