| Abstract : | The analytical results of Chandrasekharʼs semi-infinite diffuse reflection problem is crucial in the context of the stellar or planetary atmosphere. However, the atmospheric emission effect was not taken into account in this model, and the solutions are applicable only for a diffusely scattering atmosphere in the absence of emission. We extend the model of the semi-infinite diffuse reflection problem by including the effects of thermal emission B(T), and present how this affects Chandrasekharʼs analytical end results. Hence, we aim to generalize Chandrasekhar’s model to provide a complete picture of this problem. We use Invariance Principle Method to find the radiative transfer equation accurate for diffuse reflection in the presence of B(T). Then we derive the modified scattering function S(μ, f; μ_0 , f_0 ) for different kinds of phase functions. We find that the scattering function S(μ, f; μ_0 , f_0 ) as well as the diffusely reflected specific intensity I(0, μ; μ_0 ) for different phase functions are modified due to the emission B(T) from layer τ = 0. In both cases, B(T) is added to the results of the only scattering case derived by Chandrasekhar, with some multiplicative factors. Thus the diffusely reflected spectra will be enriched and carry the temperature information of the τ = 0 layer. As the effects are additive in nature, hence our model reduces to the
sub-case of Chandrasekharʼs scattering model in the case of B(T) = 0. We conclude that our generalized model provides more accurate results due to the inclusion of the thermal emission effect in Chandrasekharʼs semi-infinite atmosphere problem. |
