The Razor's Edge of Collapse: The Transition Point from Lognormal to
  Powerlaw in Molecular Cloud PDFs
  We derive an analytic expression for the transitional column density value
($s_t$) between the lognormal and power-law form of the probability
distribution function (PDF) in star-forming molecular clouds. Our expression
for $s_t$ 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 $s_t$ 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 $s_t$ and the measured values from the numerical
simulations and observations (to within 1.5 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 HI-H$_2$ transition in clouds.
