Effect of Thermal Emission in Isotropic Scattering Atmospheres: An Invariant-embedding Extension of Chandrasekhar’s H(μ) Function

Chandrasekhar’s H ( μ ) function forms the foundation of radiative transfer theory for semi-infinite, isotropically scattering atmospheres under external illumination. However, the classical formulation does not account for thermal emission from internal heat sources, which is essential in many astrophysical environments, including hot Jupiters, brown dwarfs, and strongly irradiated exoplanets, where reradiated stellar energy significantly alters the emergent intensity. To address this limitation, we extend Chandrasekhar’s diffuse reflection framework by incorporating intrinsic thermal emission within the invariant-embedding formalism developed by R. Bellman et al. In this approach, thermal emission enters as an embedded invariant contribution to the source function, leading to a generalized angular redistribution function M ( μ ), following Sengupta. Building on the formulation of S. Chandrasekhar and its recent extension, we derive the governing nonlinear integral equations for M ( μ ) and express them in terms of the direction cosine μ , the thermal emission coefficient U ( T ) = B ( T )/ F , and the single-scattering albedo ${\tilde{\omega }}_{0}$ . High-precision numerical values of $M(\mu ,U,{\tilde{\omega }}_{0})$ are computed for μ ∈ [0, 1], U < 0.7, and ${\tilde{\omega }}_{0}\lt 1$ using a stable iterative scheme based on Gaussian quadrature. In the limit of vanishing thermal emission, the formulation reduces to Chandrasekhar’s classical H ( μ ) function, validating the approach. As an application, we consider the ultrashort-period exoplanet K2-137b and identify the wavelength range 0.85–2.5 μ m, where the model is most applicable, corresponding to the capabilities of JWST, the Hubble Space Telescope, and ARIEL.