The Self-gravitating Gas Fraction and The Critical Density for Star Formation

We analytically calculate the star formation efficiency and dense self-gravitating gas fraction in the presence of magneto-gravo-turbulence using the model of Burkhart (2018), which employs a piecewise lognormal and powerlaw density Probability Distribution Function (PDF). We show that the PDF transition density from lognormal to powerlaw forms is a mathematically motivated critical density for star formation 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 instantaneous star formation efficiency ($\epsilon_{\rm inst}$) and depletion time ($\tau_{\rm depl}$) can be calculated from the dense self-gravitating gas fraction represented as the fraction of gas in the PDF powerlaw tail. We minimize the number of free parameters in the analytic expressions for $\epsilon_{\rm inst}$ and $\tau_{\rm depl}$ by using the PDF transition density instead of a parameterized critical density for collapse and thus provide a more direct pathway for comparison with observations. We test the analytic predictions for the transition density and self-gravitating 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 should be weakly anti-correlated with the sonic Mach number while the depletion time should increase with increasing sonic Mach number. The star formation efficiency and depletion time depend primarily on the dense self-gravitating gas fraction, which in turn depends on the interplay of gravity, turbulence and stellar feedback. Our model prediction is in agreement with recent observations, such as the M51 PdBI Arcsecond Whirlpool Survey (PAWS).