2025MNRAS.540.2762W

The relationship between magnetic field strength B and gas density n 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, multiscale broken power-law relation, $B=B_0(n/n_0)^{\alpha }$, with $\alpha =\alpha _1$ for $n< n_0$ and $\alpha _2$ for $n>n_0$, 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 $n_0 = 0.40^{+1.30}_{-0.30}\times 10^4$ cm$^{-3}$. For the DCF data, we find: $\alpha _1={0.26^{+0.01}_{-0.01}}$ and $\alpha _2={0.77_{-0.15}^{+0.14}}$, with $n_0=14.00^{+10.00}_{-7.00}\times 10^4$ cm$^{-3}$, where the uncertainties give 68 per cent credible intervals. We perform a similar analysis on nineteen numerical magnetohydrodynamic simulations covering a wide range of physical conditions from protostellar discs to dwarf and Milky Way-like galaxies, computed 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. We find that the dwarf and Milky Way-like galaxy simulations produce results closest to the observations.