bksw_2010_eqn_12

\begin{matrix}
\frac{e_{\gamma}}{e_{th}} = \frac{Q_{Br} T L_{\textup{diff}}}{\beta c n_p \varepsilon} = \frac{\alpha n_p n_e \sigma_T c k T \frac{1}{\beta n_e \sigma_T} \sqrt{\frac{m_e c^2}{k T}} \Lambda g}{\beta c n_p \varepsilon} =
\\
= \frac{\alpha}{\beta^2} \sqrt{\frac{m_e c^2}{k T}} \Lambda g \approx \alpha \frac{m_p c^2}{k T} \sqrt{\frac{m_e c^2}{k T}} \approx \alpha \frac{m_p}{m_e} \left( \frac{m_e c^2}{k T} \right )^{3/2}

\end{matrix}