On the relation between magnetic field strength and gas density in the interstellar medium: A multiscale analysis

The magnetic field strength to gas density relation in the interstellar medium is of fundamental importance. We present and compare Bayesian analyses of the B-n relation for two comprehensive observational data sets: a Zeeman data set and 700 observations using the Davis-Chandrasekhar-Fermi (DCF) method. Using a hierarchical Bayesian analysis we present a general, multi-scale broken power-law relation, $B=B_0(n/n_0)^{\alpha}$ , with $\alpha=\alpha_1$ for $nn_0$, and with $B_0$ the field strength at $n_0$. For the Zeeman data we find: $\alpha_1={0.15^{+0.06}_{-0.09}}$ for diffuse gas and $\alpha_2 = {0.53^{+0.09}_{-0.07}}$ for dense gas with $n_0 = 4.00^{+12.7}_{-2.90} \times 10^3$ cm$^{-3}$. For the DCF data we find: $\alpha_1={0.26^{+0.15}_{-0.15}}$ and $\alpha_2={0.77_{-0.15}^{+0.14}}$, with $n_0=13.9^{+10.1}_{-7.30} \times 10^4$ cm$^{-3}$, where the uncertainties give 68\% credible intervals. We perform a similar analysis on nineteen numerical magnetohydrodynamic simulations covering a wide range of physical conditions from protostellar disks to dwarf and Milky Way-like galaxies, completed with the AREPO, Flash, Pencil, and Ramses codes. The resulting exponents depend on several physical factors such as dynamo effects and their time scales, turbulence, and initial seed field strength. \textcolor{red}{We find that the dwarf and Milky Way-like galaxy simulations produce results closest to the observations.