Effect of Thermal Emission in Isotropic Scattering Atmospheres: An Invariant-Embedding Extension of Chandrasekhar's $H(\mu)$-Function

Abstract

Chandrasekhar's H(mu)-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 re-radiated 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. In this approach, thermal emission enters as an embedded invariant contribution to the source function, leading to a generalized angular redistribution function M(mu). We derive the governing non-linear integral equations for M(mu) and express them in terms of the direction cosine mu, the thermal emission coefficient U(T)=B(T)/F, and the single-scattering albedo omega_0. High-precision numerical values of M(mu,U,omega_0) are computed for mu in [0,1], U<0.7, and omega_0<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(mu)-function, validating the approach. As an application, we consider the ultra-short-period exoplanet K2-137b and identify the wavelength range 0.85--2.5 micron where the model is most applicable, corresponding to the capabilities of JWST, HST, and ARIEL.