Constant source transfer through a thin layer ($I_{\mathrm{out}}=I_{\mathrm{in}}{e}^{-\tau }+S(1-e^{-\tau })$)

Along a radial ray through a layer with constant source function, the incident specific intensity is attenuated by $e^{-\tau }$ and the layer adds $S(1-e^{-\tau })$. At small optical depth, $I_{\mathrm{out}}=I_{\mathrm{in}}+(S-I_{\mathrm{in}})\tau +O({\tau }^2)$. Whether the source exceeds the incident specific intensity determines the first-order emission or absorption contrast.