Slope of Magnetic Field-Density Relation as An Indicator of Magnetic Dominance

The electromagnetic field is a fundamental force in nature that regulates the formation of stars in the universe. Despite decades of efforts, a reliable assessment of the importance of the magnetic fields in star formation relations remains missing. In star-formation research, our acknowledgment of the importance of magnetic field is best summarized by the Cruther+ 2010 B-rho relation. The relation is either interpreted as proof of the importance of a magnetic field in the collapse, or the result of self-similar collapse where the role of the magnetic is secondary to gravity. Using simulations, we find a fundamental relation, ${\cal M}_{\rm A}$-k$_{B-\rho}$(slope of $B-\rho$ relation) relation. This fundamental B-$\rho$-slope relation enables one to measure the Alfv\'enic Mach number, a direct indicator of the importance of the magnetic field, using the distribution of data in the B-$\rho$ plane. It allows us to drive the following empirical $B-\rho$ relation \begin{equation} \frac{B}{B_c} = {\rm exp}\left(\left(\frac{\gamma}{{\cal K}}\right)^{-1}\left( \frac{\rho}{\rho_c}\right)^\frac{\gamma}{{\cal K}}\right)\nonumber, \end{equation} which offers an excellent fit to the Cruther et al. data, where we assume ${\cal M}_{\rm A}-\rho$ relation. The foundational ${\cal M}_{\rm A}-{\rm k}_{B-\rho}$ relation provides an independent way to measure the importance of magnetic field against the kinematic motion using multiple magnetic field measurements. Our approach offers a new interpretation of Cruther+2010, where a gradual decrease in the importance of B at higher densities is implied.