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).
