Pressure \(P=\rho \sigma_v^2\) is computed at each level of each leaf of the dendrogram. \(\rho\) is not a direct observable, and is calculated as follows. Consider a set of pixels belonging to a given level in a given leaf, which we will call a contour. First, the CO intensity is summed over the contour and multiplied by the cube pixel size (in pc\(^2\) km s\(^{-1}\)) to obtain the ‘3-d integrated’ CO intensity \(I_{\rm 3d}\) (note that this is effectively a discrete integration). This is then converted to a total molecular gas mass \(M_{\rm mol}\) via
\[M_{\rm mol} = 6.7\, \mu \, m_{\rm H} \, I_{\rm 3d} \, X_{\rm CO},\]
where \(\mu\) is the mean molecular weight (taken to be 2.715 to account for H\(_2\) and He) and m\(_{\rm H}\) is the hydrogen atomic mass. The factor of 6.7 corrects for the relative underabundance of \(^{13}\)CO with respect to \(^{12}\)CO. We use the ‘standard’ X-factor of \(2.0~\times~10^{20}\) cm\(^{-2}\) (K km s\(^{-1}\))\(^{-1}\).
\(M_{\rm mol}\) is the total mass within the contour. To convert to a density, we estimate the spatial extent of the contour using the second moments of the set’s pixel distribution and then assume that the (unseen) third dimension is identical for simplicity. In particular, we define the ‘size’ as \(R=1.91 \, \sqrt{\sigma_x \, \sigma_y}\), where \(\sigma_i\) is the intensity-weighted standard deviation of the pixel distribution in direction \(i\), converted to physical units. \(R\) is essentially a characteristic spread in the \(xy\) pixel distribution for the set of pixels in a given contour. \(\rho\) is then simply \(M_{\rm mol} / V\), where the volume \(V = \pi^{3/2} \, R^3\), and we have assumed quasi-spherical geometry.
The 3-d velocity dispersion that enters into the pressure is calculated using the line-of-sight velocity and the assumption of an isotropic velocity field. For a given contour, \(\sigma_v\) is the intensity-weighted standard deviation of the (line-of-sight) velocity distribution of the pixels in that contour. are we missing a factor of sqrt(3)??.
Pressure is then put into the ‘standard’ units of \(P/k_{\rm B}\) in K cm\(^{-3}\).