In a pure scattering medium, the radiative transfer equations written in terms of the photon number intensity ni (i = O, X) take the form (see e.g. Alexander, Mészáros & Bussard 1989; Mészáros, Pavlov & Shibanov 1989)
(15)
\begin{eqnarray}
\mu _z\frac{\mathrm{d}n_i}{\mathrm{d}\tau }&amp;=&amp; \sum _{k=\mathrm{O,X}}\int \bigg \lbrace -\sigma _{ik}(\alpha \rightarrow \alpha ^{\prime })n_i(\alpha )\left [1+n_k(\alpha ^{\prime })\right] \nonumber \\
&amp;&amp;+ \,\sigma _{ki}(\alpha ^{\prime }\rightarrow \alpha )\bigg (\frac{\varepsilon ^{\prime }}{\varepsilon }\bigg )^2n_k(\alpha ^{\prime })\left[1+n_i(\alpha )\right ]\bigg \rbrace \mathrm{d}\varepsilon ^{\prime }\mathrm{d}\Omega ^{\prime }, 
\end{eqnarray}
where stimulated scattering is accounted for. Here, dτ = neσTds is the infinitesimal Thomson depth, with s being a parameter along the photon propagation direction. However, the previous expression can be considerably simplified. In fact, in LTE, the scattering cross-sections (1) must obey the detailed balance condition (see e.g. Alexander et al. 1989; Mészáros et al. 1989; Alexander & Mészáros 1991):
(16)
\begin{eqnarray}
\sigma _{ki}(\alpha ^{\prime }\rightarrow \alpha ) =\bigg (\frac{\varepsilon }{\varepsilon ^{\prime }}\bigg )^2\exp \left[-(\varepsilon -\varepsilon ^{\prime })/kT\right]\sigma _{ik}(\alpha \rightarrow \alpha ^{\prime })\,;
\end{eqnarray}
in this way, equation (15) becomes
(17)
\begin{eqnarray}
\mu _z\frac{\mathrm{d}n_i}{\mathrm{d}\tau } &amp;=&amp; \sum _{k=\mathrm{O,X}}\int \bigg \lbrace -\sigma _{ik}(\alpha \rightarrow \alpha ^{\prime })n_i(\alpha ) \nonumber \\
 &amp;&amp;+ \, \sigma _{ik}(\alpha \rightarrow \alpha ^{\prime })F_i(\alpha ,\varepsilon ^{\prime })n_k(\alpha ^{\prime })\bigg \rbrace \mathrm{d}\varepsilon ^{\prime }\mathrm{d}\Omega ^{\prime }, 
\end{eqnarray}
where
(18)
\begin{eqnarray}
F_i(\alpha ,\varepsilon ^{\prime })\equiv \exp \left [-(\varepsilon -\varepsilon ^{\prime })/kT\right]\left(1+n_i(\alpha )\right)-n_i(\alpha ).
\end{eqnarray}
Finally, under the assumption of conservative scattering (ε = ε΄) and substituting the expressions (1), one obtains
(19)
\begin{eqnarray}
\mu _z\frac{\mathrm{d}n_\mathrm{O}}{\mathrm{d}\tau } &amp;=&amp; -\bigg [1-\mu ^2_\mathrm{Bk}+\frac{3\mu ^2_\mathrm{Bk}}{4}\bigg (\frac{\varepsilon }{\varepsilon _\mathrm{B}}\bigg )^2\bigg ]n_\mathrm{O}(\alpha ) \nonumber \\
&amp;&amp;+\,\frac{3}{8\pi }\int _{4\pi }\bigg [\left(1-\mu _\mathrm{Bk}^2\right)\left(1-\mu _\mathrm{Bk}^{\prime 2}\right)n_\mathrm{O}(\alpha ^{\prime }) \nonumber \\
&amp;&amp;+\,\bigg (\frac{\varepsilon }{\varepsilon _\mathrm{B}}\bigg )^2\mu ^2_\mathrm{Bk}\cos ^2\left(\phi _\mathrm{Bk}-\phi ^{\prime }_\mathrm{Bk}\right)n_\mathrm{X}(\alpha ^{\prime })\bigg ]\mathrm{d}\Omega ^{\prime }, \nonumber \\
\ \nonumber \\
\mu _z\frac{\mathrm{d}n_\mathrm{X}}{\mathrm{d}\tau } &amp;=&amp; -\bigg (\frac{\varepsilon }{\varepsilon _\mathrm{B}}\bigg )^2n_\mathrm{X}(\alpha ) \nonumber \\
&amp;&amp;+ \,\frac{3}{8\pi }\bigg (\frac{\varepsilon }{\varepsilon _\mathrm{B}}\bigg )^2\int _{4\pi }\left[\sin ^2\left(\phi _\mathrm{Bk}-\phi ^{\prime }_\mathrm{Bk}\right)n_\mathrm{X}(\alpha ^{\prime })\right. \nonumber \\
&amp;&amp;\left.+\,\mu ^{\prime 2}_\mathrm{Bk}\cos ^2\left(\phi _\mathrm{Bk}-\phi ^{\prime }_\mathrm{Bk}\right)n_\mathrm{O}(\alpha ^{\prime })\right]\mathrm{d}\Omega ^{\prime }.
\end{eqnarray}
Citation: Mészáros et al. 1989
Citation function: Uses; Uses