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