deletions | additions       diff --git a/methods.tex b/methods.tex  index 05ec9be..dec27da 100644  --- a/methods.tex  +++ b/methods.tex 
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The Doppler width $\Delta\nu_D$ is a function of the molecular mass and kinetic temperature \citep{stahler_palla_2004}. Using the ratio of the intensity difference between the central frequency $\nu_0$ and where the intensity falls to $e^{-1}$, the change in the linewidths is  \begin{equation}  \label{eq:line_narrow}  \Delta\nu^\star = \frac{\Delta\nu_D}{\sqrt{\alpha_0 s \right}} s}}  \end{equation}  This result, as shown in \citet{stahler_palla_2004}, shows that the maser linewidth decreases with increasing path length. However, this result was derived using an unsaturated maser. A maser saturates from the inside out of it's line profile, since the greatest intensity is at the center. Thus, the center saturates before the wings of the profile do. The profile must then broaden, since the unsaturated wings are growing exponentially. This effect sets a limit on the narrowing of the line \citep{Elitzur_1992, stahler_palla_2004}. When the maser is completely saturated, it maintains a Doppler linewidth.