On the relation between magnetic field strength and gas density in the interstellar medium. II. Density uncertainties and diffuse gas constraints

The relationship between magnetic field strength and gas density is essential to understand the interstellar medium and star formation. Zeeman measurements in dense atomic and molecular gas phases have traditionally been used to directly probe magnetic field strengths in the Milky Way. This allowed derivation of a relationship between magnetic field strength $B$ and gas number density $n$. We recently generalized this relation as a two-part power-law with non-zero slopes and a transition density given as $B/B_0 \propto (n/n_0)^{\alpha_1}$ for $n \le n_0$ and $(n/n_0)^{\alpha_2}$ for $n > n_0$. Here, we extend our previous hierarchical Bayesian framework by incorporating a large body of pulsar observations that probe the diffuse interstellar medium and explicitly modelling density uncertainties through a global log-density correction parameter $R$ applied to all densities. We also account for magnetic field geometry and measurement uncertainties through a magnetic hyperparameter to estimate $B$. This results in a stronger constraint on the diffuse gas part of the $B$--$n$ relation. Our results confirm a non-zero exponent in the diffuse gas and a broad transition density with our best model and data set yielding maximum a posteriori results of $\alpha_1 = 0.18^{+0.02}_{-0.02}$, $\alpha_2 = 0.63^{+0.08}_{-0.05}$, $n_0 = 1630^{+2560}_{-1430}\,\text{cm}^{-3}$, and $B_0 = 7.60^{+2.00}_{-3.47}\,\mu\text{G}$.