  Context: The analytical results of Chandrasekhar's semi-infinite diffuse
reflection problem is crucial in the context of stellar or planetary
atmosphere. However, the atmospheric emission effect was not taken into account
in this model, and the solutions are applicable only for diffusely scattering
atmosphere in absence of emission.
  Aim: We extend the model of semi-infinite diffuse reflection problem by
including the effects of thermal emission B(T ), and present how this affects
Chandrasekhar's analytical end results. Hence, we aim to generalize
Chandrasekhar's model to provide a complete picture of this problem.
  Method: We use Invariance Principle Method to find the radiative transfer
equation accurate for diffuse reflection in presence of B(T ). Then we derive
the modified scattering function S(${\mu},{\phi}; {\mu}_0 , {\phi}_0$ ) for
different kind of phase functions.
  Results: We find that, the scattering function S(${\mu}, {\phi}; {\mu}_0 ,
{\phi}_0$ ) as well as diffusely reflected specific intensity $I(0, {\mu};
{\mu}_0 )$ for different phase functions are modified due to the emission
$B(T)$ from layer ${\tau} = 0$. In both cases, B(T) is added to the results of
only scattering case derived by Chandrasekhar, with some multiplicative
factors. Thus the diffusely reflected spectra will be enriched and carries the
temperature information of ${\tau} = 0$ layer. As the effects are additive in
nature, hence our model reduces to the sub-case of Chandrasekhar's scattering
model in 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
Chandrasekhar's semi-infinite atmosphere problem.
