The Star Formation Rate in the Gravoturbulent Interstellar Medium

We analytically determine the star formation rate (SFR) based on a piecewise power law and lognormal probability distribution function (PDF) of density in giant molecular clouds. This is in contrast to previous analytic SFR models that use purely lognormal density PDFs. 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 infinity. As $\alpha$ becomes shallower the SFR increases beyond the expected values calculated from a lognormal density PDF. We show that the star formation efficiency per free fall time in local molecular clouds increases with shallower PDF power law slopes, in agreement with our model. The approach presented here yields star formation rates in good agreement with both local GMCs and extragalactic observations. Our model can explain why star formation is spatially and temporally variable within a cloud and can accelerate and why the depletion times observed in local and extragalactic giant molecular clouds vary. Our model can explain both star-bursting and quiescent star-forming systems without the need to invoke extreme variations in the local interstellar environment.