Paper Review of ​Gravity or Turbulence? The velocity dispersion-size relation

Larson

​Larson's observations of molecular clouds revealed the following relations between molecular cloud densities \rho, velocity dispersions \sigma, and size R

begin{equation} \sigma \propto R^\alpha \mbox{ with } \alpha=.5 \end{equation}

begin{equation} \alpha_{vir} = \frac{5\sigma^2 R}{GM} \sim 1 \end{equation}

These two also imply:

begin{equation}\tag{3} \rho \propto \frac{\sigma^2}{R^2} \rightarrow \rho \propto R^{\beta} \mbox { with } \beta = -1 \end{equation}

Equation 1 is not too surprising since it is the relation expected for highly compressible supersonic turbulence. The second equation is curious since it implies that not only are molecular clouds virialized, but together with equation 1 implies that molecular clouds all have the same column density (\Sigma \propto \rho R \propto R^0). Of course this can be explained if molecular clouds are pressure bound by the ISM which is at constant pressure.

In fact the density-size relation \rho \propto \frac{1}{R} follows from consideration of the Jeans length R = \lambda_J = c_s \left(\frac{\pi}{G \rho} \right)^{1/2} for a region initially in pressure equilibrium with a constant background ISM pressure P_0.

begin{equation} \rho c_s^2 = P_0 \Rightarrow c_s = \sqrt{\frac{P_0}{\rho}} \Rightarrow R = \sqrt{\frac{P_0 \pi}{G}} \frac{1}{\rho} \end{equation}

Solomon

​Solomon et al. followed up by examining 273 GMC's in the 12CO 1-0 transtiion and confirmed the scaling relations of Larson

Although the ratio of virial mass to CO emission was not completely linear M_{VT} \propto L_{CO}^{.81}

Heyer

​Heyer et al. recently re-analyzed the GMC's from the Solomon sample but using a lower opacity tracer (13CO 1-0) and with higher spatial and spectral resolution.

They found variation in the column densities, as well as a dependence of the velocity dispersion on column density:

begin{equation}\tag{4} \sigma \propto \Sigma^{1/2} R^{1/2} \end{equation}

which implies that

begin{equation}\tag{5} \alpha_{vir} \propto \frac{\sigma^2 R}{GM} \propto \frac{R^2 \Sigma}{GM} \mbox{ is constant} \end{equation}

If the column density is constant - then one recovers Larson's laws. However the observed variations in the column density conflict with the idea of a turbulent cascade and instead point to 'gravitational forcing' (ie denser cores have higher velocity dispersions because they are virialized.)

Which finally brings us to ​Vazquez et al.

They argue that the variation in column densities implies that molecular clouds are not in virial equilibrium and pressure bound by the ISM but instead are in a state of hierarchical gravitational collapse.

This is similar to the gravational fragmentation scenario proposed by Hoyle except that one does not start with a uniform medium and does not expect everything to collapse simultaneously.