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, MA-kB-ρ(slope of B-ρ relation) relation. This fundamental B-ρ-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-ρ plane. It allows us to drive the following empirical B-ρ relation {equation} B/B_c = { exp}(({γ}{{ K}})⁻¹( ρ/ρ_c)γ{{ K}}), {equation} which offers an excellent fit to the Cruther et al. data, where we assume MA-ρ relation. The foundational MA- kB-ρ 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.