deletions | additions       diff --git a/section_Analysis_subsection_GMC_properties__.tex b/section_Analysis_subsection_GMC_properties__.tex  index 73193ca..d977c9e 100644  --- a/section_Analysis_subsection_GMC_properties__.tex  +++ b/section_Analysis_subsection_GMC_properties__.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 Rosolowsky \& Leroy (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.