Bayesian Revisit of the Relationship between the Total Field Strength and the Volume Density of Interstellar Clouds

Abstract

The Zeeman effect has been the only method to directly probe the magnetic field strength in molecular clouds. The Bayesian analysis of Zeeman measurements carried out by Crutcher et al. is the only reference for cloud magnetic field strength. Here we extended their model and Bayesian analysis of the relation between field strength (B) and volume density (n) in the following three directions based on the recent observational and theoretical development. First, we take R, the observational uncertainty of n, as a parameter to be estimated from data. Second, the restriction of α, the index of the B–n relationship, is relieved from [0, 0.75] to [0, 1]. Third, we allow f, the minimum-to-maximum B ratio, to vary with n. Our results show that taking R as a parameter provides a better fitting to the B–n relationship and much more reliable estimates on R, f, and the changing point of α. Arguably our most important finding is that α cannot be reliably estimated by any of the models studied here, either from us or Crutcher et al., if R > 2, which is indeed the case from our estimate. This is the so-called errors-in-variables bias, a well known problem for statisticians.

Investigating the role of magnetic fields in star formation using molecular line profiles

“…values for the mass-to-flux ratio for pre-stellar objects are often close to the critical value and have large uncertainties (Soam et al 2018;Beltrán et al 2019). Concerns have also been raised about the reliability of magnetic field strengths measured via the Zeeman effect and their relation to the gas density, due to assumptions about the abundance of the OH molecule (Tassis et al 2014) and the statistical methods employed (Tritsis et al 2015;Jiang, Li & Fan 2020).…”

Magnetic Fields in Molecular Clouds—Observation and Interpretation

“…Besides that, the ρ 0 derived by [3] is much lower than the value from simulations, as discussed in Section 2.2. The authors in [5] found that some parameters in the Bayesian analysis performed by [3] were overly restricted. The most important is the uncertainty of ρ, which was fixed within a factor of two (R = 2) by [3].…”

“…A smaller BIC value is preferred. Our simple modifications [5] reduced the BIC of the model of [3] by more than 100.…”

“…Galaxies 2021, 9, x FOR PEER REVIEW 6 of 22 highest posterior density interval (HPDI) of each parameter in the model should not be significantly lower than 95% if the algorithm is effective [24]. The results are shown in Figure 2, adopted from [5], where Model C is almost identical to that of [3]. We [5] released the restriction of R = 2 in Model C to create Model B.…”

“…Model C is almost identical to Crutcher's [3], but with no limit on the slope index n (Crutcher et al limited n to be under 0.75). Model B further releases the restriction on the density uncertainty (factor of 2) used by [3]; see [5] for Model A. Note that all models assume a slope-changing density of ρ 0 .…”