The relation between the turbulent Mach number and observed fractal dimensions of turbulent clouds

Abstract

Supersonic turbulence is a key player in controlling the structure and star formation potential of molecular clouds (MCs). The three-dimensional (3D) turbulent Mach number, $\mathcal{M}$, allows us to predict the rate of star formation. However, determining Mach numbers in observations is challenging because it requires accurate measurements of the velocity dispersion. Moreover, observations are limited to two-dimensional (2D) projections of the MCs and velocity information can usually only be obtained for the line-of-sight component. Here we present a new method that allows us to estimate $\mathcal{M}$ from the 2D column density, $\Sigma$, by analysing the fractal dimension, $\mathcal{D}$. We do this by computing $\mathcal{D}$ for six simulations, ranging between $1$ and $100$ in $\mathcal{M}$. From this data we are able to construct an empirical relation, $\log\mathcal{M}(\mathcal{D}) = \xi_1(\text{erfc}^{-1} [(\mathcal{D}-\mathcal{D}_{\text{min}})/\Omega] + \xi_2),$ where $\text{erfc}^{-1}$ is the inverse complimentary error function, $\mathcal{D}_{\text{min}} = 1.55 \pm 0.13$ is the minimum fractal dimension of $\Sigma$, $\Omega = 0.22 \pm 0.07$, $\xi_1 = 0.9 \pm 0.1$ and $\xi_2 = 0.2 \pm 0.2$. We test the accuracy of this new relation on column density maps from $Herschel$ observations of two quiescent subregions in the Polaris Flare MC, `saxophone' and `quiet'. We measure $\mathcal{M} \sim 10$ and $\mathcal{M} \sim 2$ for the subregions, respectively, which is similar to previous estimates based on measuring the velocity dispersion from molecular line data. These results show that this new empirical relation can provide useful estimates of the cloud kinematics, solely based upon the geometry from the column density of the cloud.