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

We analytically calculate the star formationefficiency and dense self-gravitatinggas 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 PDFtransition density from lognormal to powerlaw forms is a mathematically motivated critical density for star formationand 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 PDFtransition density is taken as the critical density, the instantaneous star formationefficiency ($\epsilon_{\rm inst}$) and depletion time ($\tau_{\rm depl}$) can be calculated from the dense self-gravitatinggas fraction represented as the fraction of gas in the PDFpowerlaw tail. We minimize the number of free parameters in the analytic expressions for $\epsilon_{\rm inst}$ and $\tau_{\rm depl}$ by using the PDFtransition 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-gravitatinggas 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 formationefficiency should be weakly anti-correlated with the sonic Mach number while the depletion time should increase with increasing sonic Mach number. The star formationefficiency and depletion time depend primarily on the dense self-gravitatinggas 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 WhirlpoolSurvey (PAWS).