The Star Formation Rate in the Gravoturbulent Interstellar Medium

Stars form in supersonic turbulent molecular clouds that are self-gravitating. We present an analytic determination of the star formation rate (SFR) in a gravoturbulent medium based on the density probability distribution function of molecular clouds having a piecewise lognormal and power law form. This is in contrast to previous analytic SFR models that are governed primarily by interstellar turbulence which sets purely lognormal density PDFs. In the gravoturbulent SFR model described herein, low density gas resides in the lognormal portion of the PDF. Gas becomes gravitationally unstable past a critical density ($\rho_{crit}$), and the PDF begins to forms a power law. As the collapse of the cloud proceeds, the transitional density ($\rho_t$) between the lognormal and power law portions of the PDF moves towards lower-density while the slope of the power law ($\alpha$) becomes increasingly shallow. The star formation rate per free-fall time is calculated via an integral over the lognormal from $\rho_{crit}$ to $\rho_t$ and an integral over the power law from $\rho_t$ to the maximum density. As $\alpha$ becomes shallower the SFR accelerates beyond the expected values calculated from a lognormal density PDF. We show that the star formation efficiency per free fall time in observations of local molecular cloud increases with shallower PDF power law slopes, in agreement with our model. Our model can explain why star formation is spatially and temporally variable within a cloud and why the depletion times observed in local and extragalactic giant molecular clouds vary. Both star-bursting and quiescent star-forming systems can be explained without the need to invoke extreme variations of turbulence in the local interstellar environment.