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

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 $B/B_0 \propto (n/n_0)^{α_1}$ for $n \le n_0$ and $(n/n_0)^{α_2}$ 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 $α_1 = 0.18^{+0.02}_{-0.02}$, $α_2 = 0.63^{+0.08}_{-0.05}$, $n_0 = 1630^{+2560}_{-1430}\,\text{cm}^{-3}$, and $B_0 = 7.60^{+2.00}_{-3.47}\,μ\text{G}$.

Figures (12)

• Figure 1: Full data set including Zeeman data from Crutcher2010 and Hwang2024 and pulsar data from Seta2025. Lines show Crutcher2010 upper-limit power law (solid line) and the Whitworth2025 MAP result (dashed line), which, as discussed in Appendix \ref{['Append:code']}, had too high a value of $B_0$. (Note that Fig. 1 of Seta2025 used the preliminary result from an early preprint version of Whitworth2025.) The outlier in the Zeeman data is circled for identification, see main text for further details Crutcher2010.
• Figure 2: The inferred upper limit of magnetic field strength as a function of number density, based on hierarchical Bayesian analysis of Model A for each dataset (DS1–DS4: see subfigure captions). The solid black line in all plots is the upper envelope proposed by C10. The red line shows the MAP relationship between $B_z$ and $n$ based on our analysis, and the blue lines are 100 random posterior draws. The scaling envelope of $f_B$, shown as the pink band, represents how far $B$ could plausibly lie below the MAP. The dashed line shows the result published in Whitworth2025, which had an incorrectly high value of $B_0$, as is further discussed in Appendix \ref{['Append:code']}. Error bars for $n$ are omitted for readability.
• Figure 3: The inferred relationship of magnetic field strength as a function of number density, based on hierarchical Bayesian analysis for each dataset for Model B, with fixed linear density error parameter $(f_n + 1)$. Here we plot the results for the different values of $(f_n + 1)$ on the same plot for each dataset for ease of comparison between the error treatments. Otherwise the same as Fig. \ref{['fig:model_A_grid']}.
• Figure 4: MAP results for model C with separate values of $f_B$ above and below $n_0$. The error ranges for $f_{B,1}$ (green) and $f_{B,2}$ (orange) are shown. Otherwise the same as Fig. \ref{['fig:model_A_grid']}.
• Figure 5: MAP results for model D, including explicit pulsar density errors where available. All notations are the same as Fig. \ref{['fig:model_C_grid']}.