# Why the source function is equal to Planck function when we have a local thermodynamic equilibrium?

I understand that the source function $$S_λ$$ for the special case of blackbody radiation is equal to the Planck function $$B_λ$$. However, in the broader case of a local thermodynamic equilibrium (and not the special case of a blackbody) I would expect that $$S_λ=εB_λ$$ where $$ε$$ the emissivity

and the equation of radiative transfer to be:

$$\frac{dI_λ}{k_λρds}=-I_λ+εB_λ$$

and not

$$\frac{dI_λ}{k_λρds}=-I_λ+B_λ$$

Where do I make a mistake?

In LTE, the collisions dominate over the radiative transitions, then the probability of an emitted photon to be "destroyed" by a collision is much higher than to be scattered (abosrbed and re-emitted) by the atom. The $$\epsilon$$ parameter defines this concept ($$\epsilon=\frac{C_{ul}}{C_{ul}+A_{ul}}$$). For this reason, in the LTE regime $$\epsilon=1$$. In the case of NLTE (non-LTE), the source function is defined as $$S_{\lambda}=\epsilon B_{\lambda} + (1-\epsilon)J_{\lambda}$$, where the $$J_{\lambda}$$ term considers the scattering role of the radiation field.