deletions | additions       diff --git a/The_stellar_clusters_are_well__.tex b/The_stellar_clusters_are_well__.tex  index 34bb14a..4c2dac1 100644  --- a/The_stellar_clusters_are_well__.tex  +++ b/The_stellar_clusters_are_well__.tex 
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\caption{The power law indices $\alpha$ and truncation masses $M_c$ for giant molecular clouds and stellar clusters (from Adamo et al. 2015) in bins of equal area. The properties were derived for clouds more massive than $3\times 10^5 M_\odot$, and clusters more massive than 5000 M$_\odot$.}   \end{table}  We compare the characteristic masses and slopes derived from the empirical distributions to the characteristic masses produced by the Jeans instability and the Toomre instability. The Jeans instability is the characteristic fragementation fragmentation  mass for a thin sheet of mass with support from a velocity distribution $\sigma_v$ and surface mass density $\Sigma$. Such a sheet will fragment into the characteristic (2D) Jeans mass for the system: \begin{equation}  M_{J} = \frac{\sigma_v^4}{\pi G^2 \Sigma}.  \end{equation} 
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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 \citet[][L04a]{Lundgren_2004}  and average \cite[][L04b]{Lundgren_2004b} mapped  the profiles galaxy  in radial bins CO $(1-0)$  and convert the integrated spectra to surface densities. The profiles $(2-1)$ emission using SEST achieving a resolution  of the curves are shown in Figure \ref{fig:profiles}. There is reasonably $27''$ at their best. However, this work provides a  good agreement between the ALMA data and measure of  the L04 work. overall gas masses.  We measure data from the original cubes, using 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 any given 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 measure  the surface density profiles, we shift and average the profiles in radial bins and convert the integrated spectra to surface densities. The profiles  of the ISM, we first surface density curves are shown in Figure \ref{fig:profiles}. There is reasonably good agreement between the ALMA data and the L04 work, after scaling to our adopted distance of 4.8 Mpc and CO-to-H$_2$ conversion factor of $2\times 10^{20}~\mathrm{H_2/(K~km~s^{-1})}$.  Measuring$\sigma_v$ and  $\varkappa$ and $\sigma_v$  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. 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}  where $y\equiv R/(2R_d)$, $M_d$, the disk mass, is $6.4\times 10^{10}~M_{\odot}$ and $R_d$, the scale length, is 2.9 kpc. For this curve, the epicyclic frequency is   \begin{equation}  \varkappa^2 = \frac{4 G M_d}{R_d^3} \left\{ \left[2 I_0(y)+yI_1(y)\right]K_0(y)-\left[y I_0(y)+I_1(y)\right]K_1(y)\right\} I_0(y)+I_1(y)\right]K_1(y)\right\}.  \end{equation}  Finally, we measure the ISM velocity dispersion using the {\sc Hi} and CO data. For each radial bin, we shift the spectra to a common centre velocity and measure the intensity-weighted second moment of the resulting profile. The resulting velocity dispersion is shown in the bottom panel of Figure \ref{fig:profiles}.