The razor's edge of collapse: the transition point from lognormal to power-law distributions in molecular clouds

We derive an analytic expression for the transitional column density value (${\eta }_t$) between the lognormal and power-law form of the probability distribution function (PDF) in star-forming molecular clouds. Our expression for ${\eta }_t$ depends on the mean column density, the variance of the lognormal portion of the PDF, and the slope of the power-law portion of the PDF. We show that ${\eta }_t$ can be related to physical quantities such as the sonic Mach number of the flow and the power-law index for a self-gravitating isothermal sphere. This implies that the transition point between the lognormal and power-law density/column density PDF represents the critical density where turbulent and thermal pressure balance, the so-called "post-shock density." We test our analytic prediction for the transition column density using dust PDF observations reported in the literature, as well as numerical MHD simulations of self-gravitating supersonic turbulence with the Enzo code. We find excellent agreement between the analytic ${\eta }_t$ and the measured values from the numerical simulations and observations (to within 1.2 A_V). We discuss the utility of our expression for determining the properties of the PDF from unresolved low-density material in dust observations, for estimating the post-shock density, and for determining the H i–H₂ transition in clouds.