deletions | additions       diff --git a/section_Analysis_label_sec_cprops__.tex b/section_Analysis_label_sec_cprops__.tex  index ceb0b34..72b04ca 100644  --- a/section_Analysis_label_sec_cprops__.tex  +++ b/section_Analysis_label_sec_cprops__.tex 
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\end{equation}  where $\alpha_{\mathrm{CO}}=4.35 M_{\odot}\mbox{ pc}^{-2}~\mathrm{(K~km~s^{-1})^{-1}}$. The velocity dispersion is the linearly-extrapolated, emission-weighted second moment of the velocity axis, corrected for the channel width. Similarly radius of the cloud is the root-mean-square of the linearly-extrapolated, emission-weighted second moments of the major and minor spatial axis of the emission. The radius is also corrected for the instrumental response by assuming an elliptical beam and subtracting its width in quadrature. See \citet{Rosolowsky_2006} for details. The algorithm corrects for the case where the beam and cloud position angles are not aligned.   The virial mass of the cloud is calculated from the radius and line width of the molecular cloud: $M_{\mathrm{vir}} = 5 \sigma^2 R/G$. Comparing the virial and luminous masses gives insight to the dynamical nature of the molecular clouds identified in the data. The average surface density is calculated from the luminous mass $\Sigma = M_{\mathrm{lum}}/(\pi R^2)$. Typical uncertainties are 0.2 dex in the velocity dispersion and line width and 0.3 dex in the mass estimates (both virial and luminous), though these errors grow when the signal to noise approaches the $4\sigma$ threshold. We adopt a mass completeness limit of $M_{\mathrm{min}}=3\times 10^{5}~M_{\odot}$, corresponding formally to a $50\sigma$ aggregate detection of a molecular cloud. This is a factor of $\sim 8$ larger than the minimum mass that would be admitted by our masking procedure, but represents a conservative treatment of the spatial filtering artifacts in the center of the galaxy, which makes cloud identification less certain.