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The Magnetic Field versus Density relation in Star-Forming Molecular Clouds

We study the magnetic field to density ($B-ρ$) relation in turbulent molecular clouds with dynamically important magnetic fields using nonideal three-dimensional magnetohydrodynamic simulations. Our simulations show that there is a distinguishable break density $ρ_{\rm T}$ between the relatively flat low density regime and a power-law regime at higher densities. We present an analytic theory for $ρ_{\rm T}$ based on the interplay of the magnetic field, turbulence, and gravity. The break density $ρ_{\rm T}$ scales with the strength of the initial Alfvén Mach number $\mathcal{M}_{\rm A0}$ for sub-Alfvénic ( $\mathcal{M}_{\rm A0}<1$) and trans-Alfvénic ($\mathcal{M}_{\rm A0} \sim 1$) clouds. We fit the variation of $ρ_{\rm T}$ for model clouds as a function of $\mathcal{M}_{\rm A0}$, set by different values of initial sonic Mach number $\mathcal{M_{\rm 0}}$ and the initial ratio of gas pressure to magnetic pressure $β_{\rm 0}$. This implies that $ρ_{\rm T}$, which denotes the transition in mass-to-flux ratio from the subcritical to supercritical regime, is set by the initial turbulent compression of the molecular cloud.