deletions | additions       diff --git a/methods.tex b/methods.tex  index d49322b..561e004 100644  --- a/methods.tex  +++ b/methods.tex 
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For spectral line emission, the emission and absorption coefficients are related to the Einstein $A$ and $B$ values of the transition: $j_\nu = h \nu_0 n_u A_{ul}\phi(\nu)$ and $\alpha_\nu = h\nu_0 \left( n_l B_{lu} - n_u B_{ul} \right)\phi(\nu)/4\pi$, respectively. Here, $\phi(\nu)$ is a normalized Doppler profile, corresponding to the ambient temperature, and $\nu_0$ is the central frequency. The radiative transfer equation then has a form of,  \begin{equation}  \label{eq:rad_trans} \label{eq:rad_trans_specific}  \frac{dI_{\nu}}{ds} = -\alpha_{\nu}I_{\nu} + j_{\nu}  = \frac{h \nu_0}{4\pi} \left[ \Delta n g_u B_{ul} I_{\nu} + n_u A_{ul} \right] \phi(\nu)  \end{equation}  where I have used the relation of the Einstein $B$ coefficients shown before to convert to $\Delta n$ in $\alpha_{\nu}$. Integrating over frequency, and substituting in Equation \ref{eq:pop_inverse}, yields  \begin{equation}  \label{eq:rad_trans}  \frac{dI}{ds} = \frac{h\nu_0}{4\pi \Delta \nu} \left[ \frac{\Delta n^{\circ} g_u B_{ul}}{1 + \bar{J}/\bar{J}_s} I + n_u A_{ul} \right]  \end{equation}  where $I \equiv \int I_{\nu}\phi(\nu)d\nu$, and $\Delta \nu \equiv \int I_{\nu}d\nu/I$ is the effective bandwidth. To solve this equation, the forms of $\bar{J}$ and $n_u$ must be specified. Masers are tightly beamed (see \S XXX REF SUBSECTION XXX), so $\bar{J} \approx I\Delta\Omega/4\pi$, assuming $I$ is constant across $\Delta\Omega$.