Abstract
ABSTRACT We derive an analytic expression for the transitional column density value ( ) between the lognormal and power-law form of the probability distribution function (PDF) in star-forming molecular clouds. Our expression for 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 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 and the measured values from the numerical simulations and observations (to within 1.2 AV). 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.