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

TL;DR

This study refines the relationship between magnetic field strength and gas density in the interstellar medium by incorporating diffuse gas data and density uncertainties, confirming a non-zero power-law slope and a broad transition density.

Contribution

It extends previous models by integrating pulsar observations and explicitly modeling density uncertainties, providing stronger constraints on the magnetic field-density relation in diffuse gas.

Findings

Confirmed a non-zero exponent in diffuse gas ($eta_1$).

Estimated transition density around 1630 cm$^{-3}$.

Derived magnetic field strength near 7.6 μG at the transition.

Abstract

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 $B_0 = 7.60^{+2.00}_{-3.47}\,\mu\text{G}$.