Time evolution of the galactic $B- \rho$ relation: the impact of the magnetic field morphology

One of the most frequently used indicators to characterize the magnetic field's influence on star formation is the relation between magnetic field strength and gas density ($B-\rho$ relation), usually expressed as $B \propto \rho^{\kappa}$. The value of $\kappa$ is an indication of the dynamical importance of the magnetic field during gas compression. Investigating the global magnetic field's impact on this relation and its evolution, we conduct MHD simulations of Milky-Way-like galaxies including gravity, star formation, and supernova feedback along with non-equilibrium chemistry up to $H_2$ formation fueling star formation. Two initial magnetic field morphologies are studied: one completely ordered (toroidal) and the other completely random. In these models, we study the dynamical importance of the magnetic field through the plasma $\beta$ and the $B-\rho$ relation. For both magnetic morphologies, low-density regions are thermally supported, while high-density regions are magnetically dominated. Equipartition is reached earlier and at lower densities in the toroidal model. However, the $B-\rho$ relation is not unique even within the same galaxy, as it consistently includes two different branches for a given density, with $\kappa$ ranging from about 0.2 to 0.8. The mean value of $\kappa$ for each model also displays significant variations over time, which supersede the differences between the two models. While our findings suggest that the magnetic field morphology does influence the galactic $B-\rho$ relation, its impact is transient, since time-averaged differences between the models fall within the large temporal scatter. The context and time-dependent nature of the $B-\rho$ relation underscore the need for comprehensive research and observations to understand the intricate role of magnetic fields in star formation processes across diverse galactic environments.