On the evolution of the density pdf in strongly self-gravitating systems

The time evolution of the probability density function (PDF) of the mass density is formulated and solved for systems in free-fall using a simple appoximate function for the collapse of a sphere. We demonstrate that a pressure-free collapse results in a power-law tail on the high-density side of the PDF. The slope quickly asymptotes to the functional form $\mathrm{P}_v(\rho)\propto\rho^{-1.54}$ for the (volume-weighted) PDF and $\mathrm{P}_m(\rho)\propto\rho^{-0.54}$ for the corresponding mass-weighted distribution. From the simple approximation of the PDF we derive analytic descriptions for mass accretion, finding that dynamically quiet systems with narrow density PDFs lead to retarded star formation and low star formation rates. Conversely, strong turbulent motions that broaden the PDF accelerate the collapse causing a bursting mode of star formation. Finally, we compare our theoretical work with observations. The measured star formation rates are consistent with our model during the early phases of the collapse. Comparison of observed column density PDFs with those derived from our model suggests that observed star-forming cores are roughly in free-fall.