The Dense Gas Fraction and The Critical Density Required for Star Formation
We analytically calculate the star formation efficiency and dense gas fraction in the presence of self-gravitating super-Alfv\'enic turbulence using the model of Burkhart (2018) which employs a piecewise lognormal and power law density PDF. We show that the PDF transition density between lognormal and power law forms is a mathematically motivated critical density and can be physically related to the density where the Jeans length is comparable to the sonic length, i.e. the post-shock critical density for collapse. When the PDF transition density is taken as the critical density, the star formation efficiency ($\epsilon$) and depletion time ($t_{\rm depl}$) can be calculated from the dense self-gravitating gas faction represented as the fraction of gas in the PDF power law tail. We minimize the number of free parameters in the expressions for $\epsilon$ and $t_{\rm depl}$ by removing the parameterized critical density criterion for collapse and thus provide a more direct pathway for comparison with observations. We test the analytic predictions for the transition density and dense gas fraction against AREPO moving mesh gravoturbulent simulations and find good agreement. We predict that, when gravity dominates the density distribution in the star forming gas, the star formation efficiency and depletion time should be weakly anti-correlated with the sonic Mach number. The star formation efficiency and depletion time depend primarily on the slope of the power law tail, which directly quantifies the fraction of dense self-gravitating gas and the feedback efficiency. 
Our model prediction is in agreement with recent observations, such as the M51 PdBI Arcsecond Whirlpool Survey (PAWS).