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 , \operatorname { \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 \operatorname { \mathcal { M } } from the 2D column density , \Sigma , by analysing the fractal dimension , \operatorname { \mathcal { D } } .
We do this by computing \operatorname { \mathcal { D } } for six simulations , ranging between 1 and 100 in \operatorname { \mathcal { M } } .
From this data we are able to construct an empirical relation , \log \operatorname { \mathcal { M } } ( \operatorname { \mathcal { D } } ) = \xi _ { 1 } ( % \operatorname { erfc } ^ { -1 } [ ( \mathcal { D } - \operatorname { \mathcal { D } _ { \text { min } } } ) % / \Omega ] + \xi _ { 2 } ) , where \operatorname { erfc } ^ { -1 } is the inverse complimentary error function , \operatorname { \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 \operatorname { \mathcal { M } } \sim 10 and \operatorname { \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 .