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 ΣTP between the lognormal and the power-law forms of the probability distribution function in a molecular cloud. Our turbulent magnetohydrodynamic simulations show that the value of ΣTP increases as the strength of both the initial magnetic field and turbulence increases. We develop an analytic expression for ΣTP based on the interplay of turbulence, a (strong) magnetic field, and gravity. The transition value ΣTP scales with {{ \mathcal M }}02, the square of the initial sonic Mach number, and β 0, the initial ratio of gas pressure to magnetic pressure. We fit the variation of ΣTP among different model clouds as a function of {{ \mathcal M }}02{β }0 or, equivalently, the square of the initial Alfvénic Mach number {{ \mathcal M }}{{A}0}2. This implies that the transition value ΣTP is an imprint of cloud initial conditions and is set by turbulent compression of a magnetic cloud. Physically, the value of ΣTP denotes the boundary above which the mass-to-flux ratio becomes supercritical and gravity drives the evolution.