Spherical Harmonics for the 1D Radiative Transfer Equation. I. Reflected Light

Abstract

A significant challenge in radiative transfer theory for atmospheres of exoplanets and brown dwarfs is the derivation of computationally efficient methods that have adequate fidelity to more precise, numerically demanding solutions. In this work, we extend the capability of the first open-source radiative transfer model for computing the reflected light of exoplanets at any phase geometry, PICASO (Planetary Intensity Code for Atmospheric Spectroscopy Observations). Until now, PICASO has implemented two-stream approaches to the solving the radiative transfer equation for reflected light, in particular following the derivations of Toon et al. In order to improve the model accuracy, we have considered higher-order approximations of the phase functions; namely, we have increased the order of approximation from two to four, using spherical harmonics. The spherical harmonics approximation decouples spatial and directional dependencies by expanding the intensity and phase function into a series of spherical harmonics, or Legendre polynomials, allowing for analytical solutions for low-order approximations to optimize computational efficiency. We rigorously derive the spherical harmonics method for reflected light and benchmark the four-term method (SH4) against Toon et al. and two independent and higher-fidelity methods (CDISORT and doubling method). On average, the SH4 method provides an order-of-magnitude increase in accuracy, compared to Toon et al. Finally, we implement SH4 within PICASO and observe only a modest increase in computational time, compared to two-stream methods (20% increase).

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

“…where the location within the atmosphere is specified by τ ä [0, τ N ], (where τ N is the cumulative optical depth), I is the azimuthally averaged intensity and w 0 is the single-scattering albedo, B(T) is the Planck function at temperature T, and m m¢ P , ( )is the azimuthally averaged scattering phase function. We note the similarities between the radiative transfer equation for thermal emission (Equation ( 1)) and that for reflected light, outlined in Rooney et al (2023). The difference lies in the final term on the right-hand side, the source term S (T), defined as…”

“…We emphasize that all other terms in the azimuthally averaged, one-dimensional radiative transfer Equation (1) are identical for reflected light and thermal emission. This allows us to largely follow the spherical harmonics model derivation outlined in Rooney et al (2023) for reflected light, with a few modifications to allow for the different source term. We will highlight these differences throughout this work.…”

“…We will highlight these differences throughout this work. However, we refer the reader to Rooney et al (2023) for a more in-depth discussion of the general model derivation.…”

“…Closely following the methodology outlined in Rooney et al (2023), we arrive at the layerwise solution:…”

“…Though these corrections often help to improve accuracy, it has also been shown that improved accuracy can be achieved by considering four-stream approximations (Liou 1974;Cuzzi et al 1982;Liou et al 1988;Rooney et al 2023). Four-stream methods are also still able to leverage the δ-approximation to allow for strong forward scattering.…”