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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?

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Remembering my lessons...

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.

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