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

Abstract

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 Alfv\'enic 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.