The Transition from a Lognormal to a Power-Law Column Density Distribution in Molecular Clouds: An Imprint of the Initial Magnetic Field and Turbulence

We introduce a theory for the development of a transitional column density $Sigma_{rm TP}$ between the lognormal and the power-law forms of the probability distribution function (PDF) in a molecular cloud. Our turbulent magnetohydrodynamic simulations show that the value of $Sigma_{rm TP}$ increases as the strength of both the initial magnetic field and turbulence increases. We develop an analytic expression for $Sigma_{rm TP}$ based on the interplay of turbulence, a (strong) magnetic field, and gravity. The transition value $Sigma_{rm TP}$ scales with $mathcal{M}^2_{rm 0}$, the square of the initial sonic Mach number, and $beta_{0}$, the initial ratio of gas pressure to magnetic pressure. We fit the variation of $Sigma_{rm TP}$ among different model clouds as a function of $mathcal{M}^2_{rm 0} beta_{0}$, or equivalently the square of the initial Alfvenic Mach number $mathcal{M}^2_{rm A0}$. This implies that the transition value $Sigma_{rm TP}$ is an imprint of cloud initial conditions and is set by turbulent compression of a magnetic cloud. Physically, the value of $Sigma_{rm TP}$ denotes the boundary above which the mass-to-flux ratio becomes supercritical and gravity drives the evolution.