A Method for a Pseudo-local Measurement of the Galactic Magnetic Field

It has been known for decades that the interstellar medium is threaded by a magnetic field. Many studies show that that the Galactic field consists of a Galaxy-wide, large scale field and a superposed, spatially-random component that is plausibly interpreted as magnetohydrodynamic (MHD) turbulence (Ferrière 2011; Han 2017; Ferrière 2020). It is probably safe to say that all observational investigations have concluded that the turbulent component is comparable in magnitude to the large scale component. Values for the ratio $\tfrac{\delta b}{{B}_{0}}$ where δb is the rms value of the turbulent component and B₀ is the magnitude of the large scale component, range from a few tens of percent to a number in excess of unity. Further discussion of results from the literature are given in Spangler (2021).

Some of the main diagnostic techniques such as Faraday rotation consist of path integrals over long lines of sight through the Galaxy. These provide an average of the Galactic field over a long line of sight (LOS) through the Galaxy, rather than the value at a set of points along that LOS. In this paper, I discuss a type of observation which utilizes the Faraday rotation technique to obtain an estimate of the Galactic field (strictly speaking the LOS component) in a localized region, rather than an entire LOS through the Galaxy. The localized region is comprised of the volume of an H ii region. In this sense, the technique provides a "local" measurement.

The observational impetus for this paper is a set of studies of the H ii regions Rosette Nebula and W4 by A. Costa and coworkers (Savage et al. 2013; Costa et al. 2016; Costa & Spangler 2018). Costa and co-workers carried out Faraday rotation measurements with the Very Large Array of background radio sources viewed through these two H ii regions. The main results of these investigations were: (1) rotation measures (RM) for sources viewed through the H ii regions were typically much larger than for nearby sources whose lines of sight probed the Galaxy, but not the H ii regions, (2) although there were large variations in the RM from source-to-source when the LOS passed through the H ii region, there was a well-defined average RM (magnitude and sign) for the H ii region, (3) the sign of the H ii region-induced RM (which dominated the total RM) was the same as the large scale Galactic field in that part of the sky. The inclusion of 4 additional H ii regions in the outer Galaxy studied by Harvey-Smith et al. (2011) extend and confirm this result (Spangler 2021). The important point is that the magnetic field in the "RM anomaly" associated with the H ii region is confined to the volume of the H ii region rather than being an average over a LOS of many kiloparsecs in the Galactic plane. The radii of the Rosette and W4 are approximately 20–25 parsecs, so the magnetic field is "local" in this sense.

I model the observed Faraday rotation measure as the sum of a background term due to the entire LOS through the Galaxy, and a local term due to the plasma in the H ii region. The RM due to the H ii region is determined by the LOS component (B_z) of the Galactic field at the location of the H ii region. I define a coordinate system which is convenient for describing the Galactic field. The ∥ direction is in the direction of the mean field, the p direction is perpendicular to the mean field, and in the Galactic plane, and the ⊥ direction completes the right-hand coordinate system and is perpendicular to the Galactic plane. The mean field is assumed to be in the Galactic plane and inclined at an angle Θ to the LOS.

Given this latter coordinate system, I express the magnetic field (mean field plus turbulence) by

$$\begin{eqnarray}&&{\boldsymbol{B}}=({B}_{0}+\delta {b}_{\parallel }){\hat{e}}_{\parallel }+\delta {b}_{p}{\hat{e}}_{p}+\delta {b}_{\perp }{\hat{e}}_{\perp }\end{eqnarray} \tag{ 1 }$$

In analogy with the Alfvènic turbulence in the solar wind, one expects $\langle {\left(\delta {b}_{p}\right)}^{2}\gt =\lt {(\delta {b}_{\perp })}^{2}\rangle$ and $\langle {(\delta {b}_{p})}^{2}\gt \gg \lt {(\delta {b}_{\parallel })}^{2}\rangle$. I assume the turbulent components are random, 0-mean quantities with an amplitude dominated by fluctuations on the outer scale l, l ≥ 2R, where R is the radius of the H ii region. If the turbulence is assumed to have the Alfvènic character described above, and the LOS is in the Galactic plane, only the δb_p contributes to the H ii region RM.

The quantity of interest to this study is the total probability that the observed RM_obs has the opposite sign to the Galactic average RM_bck. I refer to this as a "polarity reversal." I note this probability by P₋, which is a function of the argument X,

$$\begin{eqnarray}&&X\equiv \displaystyle \frac{({\rm{\Delta }}+\cos {\rm{\Theta }})}{\sqrt{2}\sigma \sin {\rm{\Theta }}}.\end{eqnarray} \tag{ 2 }$$

The quantity Δ is the ratio of the background to nebular RM. The larger the nebular RM is relative to the background RM_bck, the easier it is for the turbulence to produce a polarity reversal. The quantity σ is the rms of the normalized magnetic fluctuations in the p direction, ${\sigma }^{2}\equiv \langle {\left(\tfrac{\delta {b}_{p}}{{B}_{0}}\right)}^{2}\rangle$. The resulting expression for P₋ is (Spangler 2021)

$$\begin{eqnarray}&&{P}_{-}=\displaystyle \frac{1}{2}\left[1-\mathrm{erf}(X)\right]=\displaystyle \frac{1}{2}\mathrm{erfc}(X)\end{eqnarray} \tag{ 3 }$$

where erfc(X) is the complement of the error function. Equation (3) is plotted in Figure 1.

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It is of interest to calculate the quantity X for some observationally-relevant cases. I adopt values of σ in the range 0.5–0.7. Savage et al. (2013), Costa et al. (2016), Costa & Spangler (2018) found values of Θ for the Rosette Nebula and W4 to be in the range 50°–70°. Finally, a value of Δ = 0.1 is used. A value of σ = 0.70 and Θ = 70° gives X = 0.48; reference to Figure 1 shows that the probability of reversal P₋ is about 24 %. In such a case it is mildly surprising that all 6 H ii regions discussed in the Introduction produce "Faraday rotation anomalies" with the same polarity as the large scale field. On the other hand, a slightly smaller value of σ = 0.50 and Θ = 50°, also consistent with Costa's results for the Rosette Nebula and W4, would have X = 1.37. In this case, Figure 1 shows that the probability of a polarity reversal for a single source is less than 5%, and the results for the 6 H ii regions are unremarkable.

• 1.  
  In the case of the Galactic-anticenter-direction H ii regions Rosette Nebula and W4, the mean magnetic field in the H ii region has the same polarity as the general Galactic field in that part of space, despite the presence of strong MHD turbulence which would be expected to randomize the polarity. An additional 4 H ii regions studied by Harvey-Smith et al. (2011) share this property.
• 2.  
  I present formulas for the probability that the "point" magnetic field has the same polarity as the general field, which depends on properties of the turbulent magnetic fluctuations as well as the details of the LOS. For reasonable estimates, this probability could be non-negligible, being as high as 25% for plausible characteristics. As a result, the observation of 6 anticenter-direction H ii regions, all having the same magnetic polarity of the general field, becomes mildly curious.