deletions | additions       diff --git a/The_stellar_clusters_are_well__.tex b/The_stellar_clusters_are_well__.tex  index 3a8b718..2c3ac80 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. 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 calculate the surface density of the ISM, we first  Measuring $\sigma_v$ and $\varkappa$ for the gas disk requires measuring the rotation curve. We adopt the rotation curve analysis of \citet[][L04]{Lundgren_2004}, who used low resolution ($27''$) CO mapping to derive a rotation curve. They find the rotation curve is well modeled by an exponential disk. We confirm this by using their kinematic parameters (i.e., inclination and position angle) to estimate the amplitudes of the rotational motion for the {\sc Hi} 21-cm map of the galaxy that is part of the THINGS \cite{Walter_2008} survey. for.  In Figure \ref{fig:profiles}, we show the L04 rotation curve and the median absolute deviation of inferred rotational velocities for the 21-cm data around the rotation curve (grey region). There is good agreement between the two approaches, we adopt the functional form of the L04 curve rescaling from an adopted distance of 4.5 Mpc to 4.8 Mpc : \begin{equation}  V^2 = \frac{2 G M_d}{R_d} y^2 [I_0(y)K_0(y)-I_1(y)K_1(y)]  \end{equation}