deletions | additions       diff --git a/The_stellar_clusters_are_well__.tex b/The_stellar_clusters_are_well__.tex  index 1476525..34bb14a 100644  --- a/The_stellar_clusters_are_well__.tex  +++ b/The_stellar_clusters_are_well__.tex 
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M_T = \pi \lambda_T^2 \Sigma = \frac{64 \pi^3 G^2 \Sigma^3}{\varkappa^2}.  \end{equation}  We must infer $\Sigma$, $\sigma_v$ and $\varkappa$ from the ALMA data and supplementary information. We include the {\sc Hi} 21-cm map of the galaxy that is part of the THINGS \cite{Walter_2008} survey in our analysis. The low-resolution studies of \cite{Lundgren_2004}  In particular, we use the average line-of-sight velocity measurements of the 21-cm data to provide a constraint on the velocities at which the neutral ISM will be found in the spectrum. We then shift those assumed velocities to average spectra in radial bins, thereby determining an average line profile, even when the line cannot be readily discerned or characterized \citep[see][for details]{Schruba_2011}. To measure the surface density profiles, we shift and average the profiles in radial bins and convert the intergrated integrated  spectra to surface densities. The profiles of the curves are shown in Figure \ref{fig:surfdens}. \ref{fig:profiles}. There is reasonably good agreement between the ALMA data and the L04 work.  To calculate the surface density of the ISM, we first