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The solution of the radiative transfer equation for spherical ionized blob : \begin{equation} \frac{dI_{\nu}}{ds} = j_{\nu}-\alpha I_{\nu} \end{equation}

and solution is (Ref:} \begin{equation} I_{\nu}(s) = I_{\nu}(s_0)e^{-\tau_{\nu}(s_0, s)}+ \int_{0}^{s} j_{\nu}(s')e^{-\tau_{\nu}(s', s)} ds' \end{equation}

This is solved using $I(0)=0$. where $\tau_{\nu}(s', s)=\tau_{\nu}(s)-\tau_{\nu}(s')$. It is not clear to me why $\tau_{\nu}(s', s)$ is written this way.

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See here – Vijay Murthy Jun 27 '12 at 8:34

It is just an integrated variable to avoid confusion, nothing more. It will disappear after the integration.

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This is a simplified version of radiative transfer equation, in which the scattering source term is treated as a source function $j_{\nu}$. Thus, in the solution, the optical distance from the scattering source $j_{\nu}(s')$ to the detecting position $s$ is written as a function of the two position/path variables to emphasize the idea that the specific intensity is the result of incident intensity $I_{\nu}(s_0)$ and scattering intensity $j_{\nu}(s')$ all over the other positions, undergoing a series of extinction processes.

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