2024ApJ...976..209Z

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 fields is best summarized by the R. M. Crutcher et al. B-ρ relation,\begin{eqnarray*}\mathrm{log}B(\rho )/\mathrm{Gauss}=\left\{\begin{array}{l}-5,\,\mathrm{if}\,\rho \,\lesssim \,{10}^{-20}\,{\rm{g}}\,{\mathrm{cm}}^{-3}\\ \displaystyle \frac{2}{3}\cdot \mathrm{log}\rho +\mathrm{log}{\rho }_{0},\,\mathrm{if}\,\rho \,\gtrsim \,{10}^{-20}\,{\rm{g}}\,{\mathrm{cm}}^{-3},\end{array}\right.\end{eqnarray*}whose interpretation remains controversial. The relation is either interpreted as proof of the importance of a magnetic field in gravitational collapse or as the result of self-similar collapse where the role of the magnetic field is secondary to gravity. Using simulations, we find a fundamental relation, ${{ \mathcal M }}_{{\rm{A}}}$-k_B–ρ (the slope of the B-ρ relation):\begin{eqnarray*}\displaystyle \frac{{{ \mathcal M }}_{{\rm{A}}}}{{{ \mathcal M }}_{{\rm{A}},{\rm{c}}}}={{\rm{k}}}_{B-\rho }^{{ \mathcal K }}\approx \displaystyle \frac{{{ \mathcal M }}_{{\rm{A}}}}{7.5}\approx {{\rm{k}}}_{B-\rho }^{1.7\pm 0.15}.\end{eqnarray*}This fundamental B-ρ slope relation enables one to measure the Alfvénic 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 apply the following empirical B-ρ relation:\begin{eqnarray*}\displaystyle \frac{B}{{B}_{c}}=\exp \left({\left(\displaystyle \frac{\gamma }{{ \mathcal K }}\right)}^{-1}{\left(\displaystyle \frac{\rho }{{\rho }_{c}}\right)}^{\displaystyle \frac{\gamma }{{ \mathcal K }}}\right)\approx \displaystyle \frac{B}{{10}^{-6.3}{\rm{G}}}\approx \exp \left(9{\left(\displaystyle \frac{\rho }{{10}^{-16.1}{\rm{g}}\,{\mathrm{cm}}^{-3}}\right)}^{0.11}\right)\ ,\end{eqnarray*}which offers an excellent fit to the Crutcher et al. data, where we assume an ${{ \mathcal M }}_{{\rm{A}}}-\rho $ relation ($\tfrac{{{ \mathcal M }}_{{\rm{A}}}}{{{ \mathcal M }}_{{\rm{A}},{\rm{c}}}}={\left(\tfrac{\rho }{{\rho }_{c}}\right)}^{\gamma }\approx {{ \mathcal M }}_{{\rm{A}}}/7.5\approx {\left(\rho /{10}^{-16.1}\,{\rm{g}}\,{{\rm{cm}}}^{-3}\right)}^{0.19}$). The foundational ${{ \mathcal M }}_{{\rm{A}}}-{{\rm{k}}}_{B-\rho }$ relation provides an independent way to measure the importance of the magnetic field against the kinematic motion using multiple magnetic-field measurements. Our approach offers a new interpretation of the classical B-ρ relation, where a gradual decrease in the importance of B at higher densities is implied.