Tomographic Imaging of the Sagittarius Spiral Arm's Magnetic Field Structure

We selected the Sagittarius arm, the nearest major spiral arm in the inner Galactic plane with abundant observable stars in optical polarimetry, as our first target to create a tomographic image of the magnetic field in a spiral arm. We observed a target field within +10° < l < +20° to avoid the Aquila Rift and to have a good sky position from the Higashi-Hiroshima Observatory (see Section 2.2).

To define the target region, in addition to the above constraints, we imposed the following conditions, referring to the Gaia Data Release 2 (DR2; Gaia Collaboration et al. 2018) catalog, which was the latest Gaia release when the observation was being planned:

• 1.  
  A sufficient number of stars (≳100) with Gaia distances are distributed across all distances up to ∼3 kpc.
• 2.  
  The interstellar extinction increases gradually with distance along the LOS, rather than experiencing concentrated increases at specific distances.

These conditions impose a continuous sampling of the magnetic field across the Sagittarius arm along the LOS. Consequently, we selected a $17^{\prime} \times 10^{\prime}$ field centered at l = +1415, b = −147.

We obtained linear polarimetry in the Cousins R band (R_C band: λ = 0.65 μm) using the Hiroshima Optical and Near-infrared Camera (HONIR; Akitaya et al. 2014) on the 1.5 m Kanata Telescope, Higashi-Hiroshima Observatory, on 2021 August 5. The optics of the HONIR instrument consists of a rotating half-wave plate, a focal mask of five equally spaced slits with a 50% opening ratio, and a Wollaston prism that splits the incident light into two orthogonally polarized images next to each other on the detector focal plane (see Section 5 of Akitaya et al. 2014). As a result, five pairs of images with orthogonal polarizations are exposed across the entire surface of the detector. To cover the $7\buildrel{\,\prime}\over{.} 0\times 9\buildrel{\,\prime}\over{.} 6$ detector field of view (FOV) with multiple exposures, we made 3 × 3 spatial dithers with a 312 step in the east–west direction and a 200 step in the north–south direction.

To measure the polarization parameters q ≡ Q/I and u ≡ U/I of each star, we acquired photometry with four PAs of the half-wave plate at 0°, 45°, 225, and 675 (Kawabata et al. 1999). As a result, we obtained a total of 36 exposures, with each exposure lasting 75 s.

We covered the target field using two adjacent FOVs centered at l = +1411, b = −141 and l = +1418, b = −153. The combined FOV size was $17\buildrel{\,\prime}\over{.} 0\times 10\buildrel{\,\prime}\over{.} 5$ as shown in Figure 1.

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We measured stellar intensities by aperture photometry using SExtractor (Bertin & Arnouts 1996). The typical size of the point-spread function is ∼18, and we fixed the aperture diameter to 65 (24 pixels).

We calibrated the instrumental polarization by observing the unpolarized standard star G191-B2B on 2021 July 27. The measured instrumental polarization, an offset vector to the origin in the q–u parameter space, is q_inst = 0.01% ± 0.02% and u_inst = −0.04% ± 0.02%, which are negligible for our measurements. The stability of the instrumental polarization, measured over a period of 10 months including the observational period, is consistently better than 0.1%, and is thus considered negligible for our measurements. The variation of the instrumental polarization across the detector is better than 0.1% and can also be considered negligible (Akitaya et al. 2014).

We calibrated the polarization PA by observing the strongly polarized standard stars BD+64 106, BD+59 389, and HD 204827 (Schmidt et al. 1992) on 2021 July 27 and August 30. The achieved calibration accuracy is better than 04 and the stability during the observational period was estimated to be better than 03.

We calibrated the polarization efficiency of the instrument by observing an artificially polarized star through a wire-grid polarizer inserted before the half-wave plate. The measured efficiency is 99.1% ± 0.01%, by which we scaled the observed polarization fractions.

We converted the measured normalized Stokes parameters, q and u, defined in equatorial coordinates, into Galactic coordinates, q_Gal and u_Gal. This transformation allowed us to align the polarization measurements with the Galactic coordinate system for further analysis and interpretation. The details of the coordinate conversion process are described in Appendix A.

We referred to the Gaia Data Release 3 (DR3; Gaia Collaboration et al. 2023) catalog and cross-matched the observed stars with detections of polarization within a search radius of 1''. We referred to a Gaia-based catalog by Bailer-Jones et al. (2021) for the distance of each star. Among their distance estimations, we adopted "geometric" distances, including distance estimates for all our observed stars.

We limited our search by applying the conditions of a renormalized unit weight error ≤1.4 and a parallax_over_error ≥3 in the Gaia DR3 catalog, and a stellar distance uncertainty (a 68% confidence interval) ≤20%. In addition, we selected data with an estimated error δ P ≤ 0.3% for the fractional polarization, which was typically achieved by stars with R_C ≤ 15.5 mag. Following this procedure, we identified 184 stars within the observed field. In investigating interstellar extinction in the observed region, we referred to 259 stars meeting the criteria of distance uncertainty ≤20% and A_G values available in the DR3 catalog. There were 130 stars found in both data sets. We analyzed all available data for both polarization and extinction, regardless of their availability in the other data set. We summarize the identified stars in Table 6 in Appendix B.

Figure 1 shows the spatial distribution of the observed polarization pseudovectors (white segments), indicating the PAs and polarization fractions (P). The derivation of these values from the observed q and u values is detailed in Appendices A and B. Of the 184 stars employed in the following analyses, 105 stars with a polarization PA uncertainty δPA ≤ 10° are plotted in the figure. The distribution of B_POS traced by stellar polarimetry appears to be a perfect mix of various PAs in space. The histogram of PAs shown in Figure 2 shows a bimodal distribution centered around 30° and 140°. PA = 90°, which is the direction parallel to the Galactic plane, corresponds to the minimum of the distribution. Thus the observed B_POS is not parallel to but predominantly tilted from the Galactic plane. The PAs and their distribution do not show particular variations or trends with sky coordinates (Figure 1).

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Figures 1 and 2 also show Planck's observed magnetic field PA for the same region (orange segments). In the following, we refer to the polarimetry data observed by the Planck satellite at 353 GHz (data release 3; Planck Collaboration et al. 2020a), as provided by IRSA (Planck Team 2020), with a resolution set to $10^{\prime}$.

Given that the Stokes parameters in the Planck data products are provided in the HEALPix convention rather than the IAU convention, we estimate the polarization PA of the data using the following equation:

$$\begin{eqnarray*}&&{\mathrm{PA}}_{\mathrm{Planck}}=0.5\times \arctan 2(-{U}_{\mathrm{Planck}},{Q}_{\mathrm{Planck}}).\end{eqnarray*}$$

We estimate the PA of the magnetic field by rotating the polarization PA observed by Planck by 90°. We will use the term "Planck's observed magnetic field" or "the Planck magnetic field" for simplicity. Similar to our stellar polarimetry data, the Planck magnetic field shows PAs deviating from 90° (1400, 711, 1210, and 660 from north to south in Figure 1). However, the angle offset from 90° is generally larger for the stellar polarimetry magnetic field orientations.

The distance dependence of the optical polarimetry data is shown in Figure 3. Specifically, we show how PA and P vary as a function of the Gaia stellar distances estimated by Bailer-Jones et al. (2021). Note that PA is mostly nonparallel to the Galactic plane, as shown in Figure 2.

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Figure 3 also shows interstellar extinction (A_G) values taken from the Gaia DR3 catalog. We find an apparent increase of about 2 mag in A_G at distances beyond ∼1.2 kpc. It further becomes A_G ≥ 2.5 mag beyond ∼2 kpc. We can attribute this increase in interstellar extinction at distances of about 1.2–2 kpc to the dust in the Sagittarius arm. The foreground component of A_G < 1 mag can be attributed to the cloud(s) in the outskirts of the Aquila Rift at d < 200 pc (Section 1), and it is likely related to the Local Bubble shell (Lallement et al. 2019; Pelgrims et al. 2020).

Doi et al. (2021b) showed that breakpoint analysis, a statistical technique that detects the points at which data values make stepwise changes, can effectively recover the distance dependence of stellar polarimetry data. Based on this breakpoint analysis, Doi et al. (2021b) characterized the distribution of dust clouds as a function of distance along the LOS and the 3D structure of the magnetic field associated with those clouds. The details of the breakpoint analysis are described in Appendix C. We apply the breakpoint analysis to our observed q_Gal and u_Gal, assuming a step change at each breakpoint and constant values between them, as was done by Doi et al. (2021b). We identify four breakpoints, as shown in Table 1 (Polarimetry) and by the dashed lines in Figure 3 (top two panels), together with 68% confidence intervals of the estimation.

Table 1. Breakpoints Estimated in q_Gal, u_Gal, and A_G

			Breakpoints
	(pc)
Polarimetry	${1225}_{-34}^{+29}$	${1470}_{-28}^{+10}$	${1632}_{-42}^{+2}$	${2229}_{-61}^{+3}$
AG	${1276}_{-19}^{+10}$	${1497}_{-27}^{+3}$	${1654}_{-22}^{+8}$	${1840}_{-81}^{+33}$	${2638}_{-83}^{+83}$

Notes. The error values indicate the 68% confidence intervals of the estimation. See text for the breakpoint analysis.

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We also perform breakpoint analysis for the A_G values similar to that for the polarimetry data. The results are shown in Table 1 (A_G) and Figure 3 (the bottom panel). We can find reasonable agreement between the two independent evaluations. In particular, the three breakpoint distances on the near side show good consistency. On the other hand, the A_G analysis finds an extra breakpoint at larger distances and these two farther breakpoints are roughly on either side of the polarimetry breakpoint. We note that there are fewer stars with Gaia-estimated A_G values than stars with polarimetry data in this distance range (d > 1.6 kpc; the number of stars in each distance range is listed in Table 2). Also, we find a significant step change in PA at 2.2 kpc (Figure 3). As the focus of this paper is on the magnetic field structure inferred from the polarization data, we utilize the breakpoints detected in the polarization data analysis, which are expected to directly trace changes in the magnetic field structure, in the subsequent analyses.

In the breakpoint analysis, as in Doi et al. (2021b), we assume that the values of q_Gal and u_Gal are both constant between neighboring breakpoints. To validate this assumption, we perform a linear fitting on each parameter between breakpoints to test if the slope is statistically consistent with a value of 0. We perform the Student t-test for q_Gal and u_Gal, and A_G as well. The statistical p-values, which are for the null hypothesis that the slope of the distribution is equal to 0, are shown in Table 2. The null hypothesis that the slope of the distribution is equal to 0, i.e., a constant value, cannot be rejected as the p-values are all greater than 5% for the tested cases.

Among the test results, in the two distance ranges under d ≥ 1.63 kpc, where the breakpoint estimate of A_G differs from that of the polarimetry data, A_G shows smaller p-values. However, the p-values are larger than 15% and are still consistent with the assumption of constant A_G values at each distance range defined by the polarimetry. Therefore, these analysis results can be considered as supporting evidence for the validity of the breakpoint analysis of the polarimetry data.

The constancy of the q_Gal and u_Gal values within each distance range implies that there is a discrete contribution of polarizing dust sheets at the breakpoints, while there is no significant contribution between breakpoints. In Figure 4, we visually confirm this discrete polarization along the LOS by presenting a cumulative sum plot of the q_Gal–u_Gal vectors with increasing distance. In this plot, the sum vector defines a straight line while the PAs of the polarization remain constant if the PAs of the vectors are aligned. This is because the vector sum averages out the random component of each vector. On the other hand, if they are not aligned, a change in the polarization PA turns the direction of the path of the cumulative sum plot.

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As shown in Figure 4, the cumulative sum can be described by the combination of five sections, including four line segments and a clump between 1.23 and 1.47 kpc. The four line segments indicate that the q_Gal–u_Gal vectors are well aligned in each distance range. The phase angle of the q_Gal–u_Gal vector on the ∑q_Gal–∑u_Gal plane corresponds to twice the PA, and therefore, it should be noted that vectors pointing in opposite directions (e.g., the green and light green vectors in the figure) differ by 90° in PA. The clump between 1.23 and 1.47 kpc shows that the length of the q_Gal–u_Gal vectors is 0 on average, which indicates that the vectors are aligned in one orientation in this distance range (due to the complete depolarization by the foreground cloud in this case). In summary, Figure 4 shows that the polarization vectors as a whole are well aligned in a specific direction for each of the five distance ranges, with discrete contributions of thin polarizing dust sheets at the breakpoints. Different colors depict the distance range between the breakpoints, which correspond well to each line segment and clump.

The scattered distribution of dust clouds and their discrete contribution to the polarization (Figure 4) is comparable to the finding for the Perseus and (foreground) Taurus molecular clouds (Doi et al. 2021b) and the thin-layer model developed for high Galactic latitude clouds (Pelgrims et al. 2023). This suggests that the thin-layer model is also applicable to LOSs at low Galactic latitudes. Therefore, in the following, we will assume that the discrete dust sheets/clouds at the four breakpoints, in addition to a foreground component before the first breakpoint, generate polarization in each distance range—that is, "foreground," "1.23 kpc cloud," "1.47 kpc cloud," "1.63 kpc cloud," and "2.23 kpc cloud."

Within the distance range where we identify dust clouds along the LOS (d = 1.2–2.2 kpc from the Sun), the vertical offset from the Galactic plane is ∣Z∣ = 32–57 pc. This vertical distance is comparable to or less than the scale height of the Galactic thin disk component (50–70 pc; Nakanishi & Sofue 2006; Kalberla et al. 2007; Yao et al. 2017) and well below that of the disk component of Galactic magnetic field models (100–400 pc; Sun et al. 2008; Jansson & Farrar 2012; Jaffe et al. 2013; Han et al. 2018). Therefore, we are likely observing the Galactic disk component of the magnetic field.

We estimate the 3D distribution of dust clouds in the Galactic disk by applying the breakpoint analysis to the A_G values, as described in Appendix D. The color scale in Figure 5 represents the surface density of these dust clouds within a range of ±100 pc from the Galactic plane. The red dashed line in the figure indicates the LOS of the observation. The positions of the four identified dust clouds along the LOS are indicated by their respective distances.

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The high dust surface density structure observed around the 1.23, 1.47, and 1.63 kpc clouds in Figure 5 corresponds to the Sagittarius arm. The surface density around the 2.23 kpc cloud appears to be relatively low. However, this does not necessarily imply the absence of dust clouds or the absence of the Sagittarius and Scutum arm structures, as dust clouds located on the far side within the Sagittarius arm may go undetected, being hidden behind dust clouds on the near side of the Sagittarius arm or other foreground clouds.

In summary, our observations identify multiple clouds in the Sagittarius arm and detect polarization at each distance range.

We show the distribution of polarization pseudovectors and their PAs for each distance range in Figures 6 and 7. As in Figures 1 and 2, we plot only the data points with good PA determination (δPA ≤ 10°), which correspond to 105 objects. The overall distribution of PAs in Figures 1 and 2 appears spatially uncorrelated with a large scatter. However, if we plot the data independently for each distance bin, as shown in Figures 6 and 7, the polarization pseudovectors instead show a well-ordered pattern.

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We present the mean orientation and angular dispersion of the PAs for each distance range in Table 3, calculated using the circular mean and circular standard deviation. The circular mean and the circular standard deviation (hereafter ${\breve{\sigma }}_{\mathrm{PA}}$) account for the 180° ambiguity of the polarization pseudovectors. This approach allows for an unbiased estimation of the standard deviation of the PAs, even if the deviation exceeds 50°, and is capable of capturing a wider, though not infinite, range of deviations in the PA measurements compared to the usual arithmetic standard deviation, which saturates at $\pi /\sqrt{12}\,(\mathrm{rad})=51\buildrel{\circ}\over{.} 96$ (Doi et al. 2020). ¹⁹

Table 3. Angular Mean and Standard Deviation of PAs within Each Distance Range

Distance Range	Cloud	Circular Mean				Circular SD $({\breve{\sigma }}_{\mathrm{PA}})$
		Equatorial Coordinates		Galactic Coordinates
		Observed	Intrinsic	Observed	Intrinsic	Observed	Differential
(kpc)		(deg)	(deg)	(deg)	(deg)	(deg)	(deg)
<1.23	foreground	${72.6}_{-2.9}^{+2.9}$	a ${72.6}_{-2.9}^{+2.9}$	${134.5}_{-2.8}^{+2.8}$	a ${134.5}_{-2.8}^{+2.8}$	${29.3}_{-2.3}^{+2.4}$	a ${29.3}_{-2.3}^{+2.4}$
1.23–1.47	1.23 kpc	${22.9}_{-49.0}^{+47.1}$	${164.2}_{-5.1}^{+5.1}$	${85.2}_{-48.3}^{+48.4}$	${46.1}_{-4.8}^{+4.7}$	${63.2}_{-6.9}^{+8.9}$	${33.6}_{-3.7}^{+4.1}$
1.47–1.63	1.47 kpc	${176.7}_{-2.4}^{+2.5}$	${176.2}_{-3.1}^{+3.0}$	${58.6}_{-2.2}^{+2.2}$	${58.1}_{-2.8}^{+2.8}$	${34.5}_{-2.6}^{+2.8}$	${34.8}_{-3.0}^{+3.1}$
1.63–2.23	1.63 kpc	${90.1}_{-1.6}^{+1.6}$	${88.3}_{-1.5}^{+1.5}$	${152.0}_{-1.4}^{+1.5}$	${150.2}_{-1.4}^{+1.4}$	${37.2}_{-1.3}^{+1.4}$	${22.3}_{-1.4}^{+1.4}$
>2.23	2.23 kpc	${146.0}_{-1.3}^{+1.4}$	${158.4}_{-1.3}^{+1.3}$	${28.0}_{-1.3}^{+1.3}$	${40.3}_{-1.2}^{+1.2}$	${29.2}_{-1.6}^{+1.6}$	${22.4}_{-1.5}^{+1.7}$
All		${109.1}_{-3.6}^{+3.6}$	a ${109.1}_{-3.6}^{+3.6}$	${171.0}_{-3.3}^{+3.2}$	a ${171.0}_{-3.3}^{+3.2}$	${54.9}_{-1.9}^{+2.1}$	a ${54.9}_{-1.9}^{+2.1}$

Notes. The raw observed values for each distance range are shown in the "Observed" columns. The mean intrinsic PA for each cloud, calculated by subtracting the foreground contributions of all components in front of the respective cloud, is presented in the "Intrinsic" columns. The circular standard deviation values minus the foreground contribution are shown in the "Differential" column. Note that they are biased by the variation caused by the observational error and do not show the intrinsic magnetic field angular variation of each cloud. See text and Doi et al. (2020) for the definitions of the circular mean and the circular standard deviation.

^a The "intrinsic" and "differential" values of the foreground cloud and the "all" values are the same as the observed values because there is no foreground component to be subtracted.

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We utilize all 184 objects selected according to the criteria described in Section 2.4, including those with large δPA, for estimating the circular mean and ${\breve{\sigma }}_{\mathrm{PA}}$. This is in contrast to Figures 1, 2, 6, and 7, which display data from only 105 objects. To estimate the uncertainty of each parameter, we perform 10,000 Monte Carlo simulations. In each simulation, we add Gaussian random errors independently to the relative Stokes parameters q and u based on their respective uncertainties. From the generated samples, we calculate P and PA and obtain the required quantities for the analysis. We show the median value of the 10,000 estimates as the maximum likelihood value and the 15.9% and 84.1% quantiles as the negative and positive errors in Table 3 and in succeeding estimations in this paper.

The angular dispersions (${\breve{\sigma }}_{\mathrm{PA}}$) of the observed polarization pseudovectors are found in the "Observed" column in Table 3. Except for the 1.23–1.47 kpc distance range, where the polarization pseudovectors are almost of zero length due to the geometrical depolarization, the angular dispersion for each distance range is significantly smaller than that of the total data, confirming that the polarization pseudovectors of each distance bin are better aligned.

To accurately evaluate the magnetic field structure associated with each cloud, it is important to consider that the observed polarization is a result of integrating all contributions along the optical path to the stars. The relative Stokes parameters q_Gal and u_Gal can be approximated as an addition of the contributions from each element along the LOS, particularly in the case of low polarization levels (say, ≪10%; e.g., Patat et al. 2010; Panopoulou et al. 2019; Pelgrims et al. 2023). By subtracting the foreground contribution from the observed polarization in each distance range, we can obtain a more reliable approximation of the intrinsic magnetic field structure associated with each cloud. This allows us to isolate the specific magnetic field characteristics within each cloud, independently of the foreground effects.

The observed q_Gal and u_Gal data for the nth distance range on the LOS are the sum of the contributions from all distance ranges from the first to the nth. Similarly, the observed q_Gal and u_Gal data for the (n − 1)th distance range are the sum of the contributions from the first to the (n − 1)th distance ranges. Therefore, to obtain the q_Gal and u_Gal values of the nth distance range, we can subtract the (n − 1)th data from the nth data, i.e., we can differentiate the observed q_Gal and u_Gal values of each distance range.

Figure 8 shows the q_Gal–u_Gal data distribution for all distance groups. We also plot the 1σ contours of the q_Gal–u_Gal data scatter for each distance range. We can see that the data are discriminated by distance. We estimate the average intrinsic polarization of each interstellar cloud by subtracting the average observed data of the immediately preceding cloud from the average observed data of a particular cloud. The average intrinsic polarization vector is represented by each black line segment in Figure 8. For each data point, similarly, we can obtain a better approximation of the q_Gal and u_Gal values of individual clouds by subtracting the average values of q_Gal and u_Gal of the immediately preceding cloud, which represents the integration of the contributions of foreground clouds. The subtraction of foreground contributions is thus equivalent to shifting the coordinate origin of the q_Gal–u_Gal plane to the average of the q_Gal and u_Gal values of the immediately preceding cloud. We will discuss the connection between this shift of origin and an anticorrelation between P and ${\breve{\sigma }}_{\mathrm{PA}}$ in Section 4.1.

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Figures 9 and 10 depict the distribution of polarization pseudovectors specific to each distance range, obtained by subtracting the mean foreground polarization. Comparing them to the raw observed values plotted in Figures 6 and 7, we observe that the polarization pseudovectors in each distance range exhibit better alignment. This alignment enhancement can be attributed to the subtraction of the mean foreground polarization, which effectively shifts the origin of the q_Gal–u_Gal plane to the average foreground value, as discussed earlier. Consequently, this adjustment often elongates the q_Gal–u_Gal vectors (resulting in increased P) and aligns them more coherently. The improved alignment of these polarization pseudovectors indicates a well-ordered magnetic field associated with each dust cloud. The spatial scale of the observed region is approximately 5–10 pc, as indicated by the scales shown in Figure 9. This suggests that the spatial structure of the magnetic field associated with each cloud appears smooth at scales smaller than 5–10 pc, with a scale length of the magnetic field structure larger than 10 pc.

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However, it is important to note that in Figure 9, we only subtract the mean foreground polarization, which means that the depicted vectors are corrected for the mean foreground contributions and not their variances. The contribution of the foreground component to the variance of PA and P can only be estimated statistically, and polarization pseudovectors cannot be corrected individually for this contribution.

Additionally, the observed variance of PA, or ${({\breve{\sigma }}_{\mathrm{PA}})}^{2}$, does not arise from a linear sum of contributions from each element along the LOS, as will be discussed in Section 4. Moreover, the observed values of ${\breve{\sigma }}_{\mathrm{PA}}$ are positively biased due to observation errors. Therefore, we compute the variance of q_Gal–u_Gal vectors (${({\sigma }_{q,u})}^{2}$) specific to individual clouds by removing the foreground cloud's contribution as follows:

$$\begin{eqnarray}\begin{array}{ccc}{\left({\sigma }_{q,u,\mathrm{intrinsic}}^{n}\right)}^{2} & = & \left[{\left({\sigma }_{q,u,\mathrm{observed}}^{n}\right)}^{2}-{\left({\sigma }_{q,u,\mathrm{uncertainty}}^{n}\right)}^{2}\right]\\ & & -\left[{\left({\sigma }_{q,u,\mathrm{observed}}^{n-1}\right)}^{2}-{\left({\sigma }_{q,u,\mathrm{uncertainty}}^{n-1}\right)}^{2}\right],\end{array}\end{eqnarray} \tag{ 1 }$$

where the variance of q_Gal–u_Gal for the nth cloud is denoted as ${\left({\sigma }_{q,u}^{n}\right)}^{2}$, and the variance of q_Gal–u_Gal for the immediately preceding cloud is denoted as ${\left({\sigma }_{q,u}^{n-1}\right)}^{2}$.

Subsequently, this derived variance of q_Gal–u_Gal vectors is employed to determine the variance in PA specific to individual clouds. For a more precise evaluation of the variance of PA, we will provide further discussion in Section 4.2.

Table 3 presents the circular mean and circular standard deviation of the polarization PAs for both the raw observed values (listed in the "Observed" columns) and the differential values. The differential values of the circular means are considered intrinsic to the magnetic field associated with each cloud, and we label these estimations as "intrinsic" values in the table.

On the other hand, the differential values of ${\breve{\sigma }}_{\mathrm{PA}}$ in Table 3 do not represent the angular dispersions specific to individual clouds, as explained previously. Therefore, in Table 3, we label the differential values of ${\breve{\sigma }}_{\mathrm{PA}}$ as "differential" instead of "intrinsic."

In the following discussions, our primary focus will be on the intrinsic properties of the magnetic field associated with each cloud, unless stated otherwise.

Table 4 shows the polarization fraction (P) for each cloud. To obtain these intrinsic P values, we subtract the average observed q_Gal and u_Gal values of the immediately preceding cloud from the average observed q_Gal and u_Gal values of the specific cloud, and subsequently convert them into the polarization fraction (P). This estimation can be visualized as the length of the black line segments in Figure 8. The average P values of the raw observed data used for evaluating the intrinsic P values are listed in Appendix E.

Table 4. Polarization Fraction and Polarization Efficiency within Each Distance Range

Cloud	Polarization Fraction (P)	AG	NH a	Polarization Efficiency
	(%)	(mag)	(1021 cm−2)	(% mag−1)
Foreground	${0.22}_{-0.02}^{+0.02}$	${0.53}_{-0.00}^{+0.00}$	${1.48}_{-0.01}^{+0.01}$	${0.42}_{-0.04}^{+0.04}$
1.23 kpc	${0.22}_{-0.04}^{+0.04}$	${0.17}_{-0.01}^{+0.01}$	${0.47}_{-0.02}^{+0.02}$	${1.35}_{-0.23}^{+0.24}$
1.47 kpc	${0.52}_{-0.05}^{+0.05}$	${0.41}_{-0.01}^{+0.01}$	${1.14}_{-0.02}^{+0.02}$	${1.28}_{-0.12}^{+0.13}$
1.63 kpc	${0.99}_{-0.05}^{+0.05}$	${0.98}_{-0.01}^{+0.01}$	${2.76}_{-0.03}^{+0.03}$	${1.00}_{-0.05}^{+0.05}$
2.23 kpc	${1.02}_{-0.04}^{+0.04}$	${0.75}_{-0.02}^{+0.02}$	${2.09}_{-0.06}^{+0.05}$	${1.37}_{-0.06}^{+0.07}$
All b	${0.11}_{-0.01}^{+0.01}$	${1.15}_{-0.00}^{+0.00}$	${3.21}_{-0.01}^{+0.01}$	${0.10}_{-0.01}^{+0.01}$

Notes.

^a N_H = A_G · 2.21 × 10²¹/0.789 is assumed (Güver & Özel 2009; Wang & Chen 2019). ^b Average of all the observed data.

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To estimate the column density of each cloud, we utilize the Gaia DR3 cataloged interstellar extinction (A_G; Andrae et al. 2023). We calculate the average A_G within the ranges corresponding to each cloud and subtract the average A_G value of the immediately preceding cloud from the average A_G value of the specific cloud.

We estimate the column density (N_H) of each cloud based on these A_G values, assuming A_V = A_G/0.789 (mag; Wang & Chen 2019) and N_H/A_V = 2.21 × 10²¹ (H atoms cm⁻² mag⁻¹; Güver & Özel 2009). The estimated A_G and N_H values are presented in Table 4. The average A_G and N_H values of the raw observed data used for evaluating these intrinsic A_G and N_H values can be found in Appendix E.

The estimated intrinsic A_G of each cloud ranges from 0.17 to 0.98 mag, corresponding to relatively low column densities of N_H ≲ 2.76 × 10²¹ (H atoms cm⁻²). This is because we have selected an observational FOV with relatively low interstellar extinction and with high-accuracy measurements from Gaia's optical trigonometry. In other words, the observed magnetic field is not associated with star-forming regions within dense molecular clouds, but rather with the diffuse gas that likely surrounds the molecular gas in isolated clouds. In fact, no corresponding CO molecular cloud is found in our FOV in catalogs (Rice et al. 2016; Miville-Deschênes et al. 2017), indicating that the gas is primarily atomic. H i surveys (e.g., Kalberla et al. 2005; Kalberla & Haud 2015) do not resolve the clouds due to low spatial and spectral resolutions, so the velocity dispersion of each dust cloud is unknown. Chen et al. (2020) identified dust clouds by referring to Gaia DR2 interstellar extinction data. Their cloud No. 505 at l = 14821, b = −1107 and cloud No. 506 at l = 13370, b = −0212 may correspond to our observed cloud(s) because of their spatial proximity to our FOV (l = 1415, b = −147). The angular distances between the outer edges of their clouds and our FOV are ∼05 (for the spatial extent of the clouds, see their Figures 505 and 506 available online). ²⁰ The distance estimate of cloud No. 505 is 1815.2 ± 42.8 pc and that of cloud No. 506 is 1793.0 ± 42.3 pc. The average distance of our four detected clouds (1.23, 1.47, 1.63, and 2.23 kpc), weighted by their column densities, is estimated to be ${1767.0}_{-1.3}^{+1.3}$ pc. This average distance is almost identical to the distances of cloud No. 505 and No. 506. We find more overlapping clouds in the LOS than in the literature, suggesting that we have detected tenuous dust clouds thanks to the distinct change in the magnetic fields' PAs as a function of distance.

We estimate the polarization efficiency (e.g., Whittet 2022) by dividing P by A_G, as tabulated in Table 4. The estimated polarization efficiency specific to individual clouds is 0.4% mag⁻¹ for the foreground cloud and 1.0%–1.4% mag⁻¹ for the clouds in the Sagittarius arm. In a similar analysis, Doi et al. (2021b) estimated a polarization efficiency of 1.5% mag⁻¹ for the Taurus and Perseus molecular clouds. Taking into account the difference between the two observations (0.7625 μm for Taurus and Perseus (Goodman et al. 1990) and 0.65 μm for this work) and assuming a wavelength dependence of the fractional polarization of P ∝ λ^−1.8 (Mathis 1990), it corresponds to approximately 2.0% mag⁻¹. Therefore, the observed efficiencies in our study are relatively lower than those estimated for the Taurus and Perseus molecular clouds using the same method.

The pitch angle (the angle relative to the direction of the Galactic rotation) of the Sagittarius arm around the observed region is estimated to be ψ ≃ 17° (Reid et al. 2019). Assuming the magnetic field follows the spiral arm structure, the magnetic field in the observation field is inclined to the POS by i = 35° (see Figure 5). In this case, the expected polarization fraction is approximately 0.7 times the maximum value, or 1.4% mag⁻¹, if the maximum value is ∼2.0% mag⁻¹ as observed in the Taurus and Perseus clouds, based on the relation $P\propto {\cos }^{2}i$. That is, the lower polarization efficiency found in the Sagittarius arm compared to that in the Taurus and Perseus molecular clouds may be partly due to the tilted magnetic field orientation to the POS in the Sagittarius arm and the nearly parallel orientation to the POS in Taurus and Perseus (e.g., Jansson & Farrar 2012). The difference in polarization efficiency of clouds in the Sagittarius arm may indicate that the magnetic field structure in the arm has a substantial variation in the in-plane direction of the Galaxy in addition to the direction perpendicular to the Galactic plane. This variation in polarization efficiency may arise from a combination of factors, including differences in the alignment of dust particles and the intricate geometry of the magnetic field.

Planck Collaboration et al. (2020b) reported an anticorrelation between the dispersion of polarization angles and the polarization fraction (also see Fissel et al. 2016). They attributed this anticorrelation to variations in the magnetic field structure along the LOS. Figure 11 illustrates the mean observed polarization fraction (P) and PA dispersion (${\breve{\sigma }}_{\mathrm{PA}}$) estimated for each distance range in the optical polarimetry data. These values, presented as the "observed" values of ${\breve{\sigma }}_{\mathrm{PA}}$ and P in Tables 3 and 4, represent measurements of multiple magnetic field components superimposed along the LOS at their respective distances. In other words, Figure 11 showcases the relationship between P and ${\breve{\sigma }}_{\mathrm{PA}}$ associated with different numbers of magnetic field layers along the LOS.

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The data presented in Figure 11 show an anticorrelation, albeit one with a slightly shallower slope compared to the correlation reported by Planck Collaboration et al. (2020b) as ${\breve{\sigma }}_{\mathrm{PA}}\times P=\mathrm{const}.$ In the following analysis, we will investigate whether this shallower anticorrelation can be attributed to the same correlation reported in Planck Collaboration et al. (2020b).

4.1.1. Theoretical Curve

A geometrical depolarization caused by multiple magnetic field layers along the LOS is equivalent to shifting the coordinate origin of the q–u plane, as discussed in Section 3.3. When the origin of the coordinate system deviates further from the distribution of q and u data, the polarization fraction P increases proportionally. At the same time, the polarization PA dispersion ${\breve{\sigma }}_{\mathrm{PA}}$ decreases approximately inversely, particularly when P is sufficiently large. This dependence of ${\breve{\sigma }}_{\mathrm{PA}}$ on P is the same as that of the estimation error of PA derived from the observed q_Gal and u_Gal when the standard deviations of q_Gal and u_Gal (σ_q and σ_u ) are interpreted as uncertainties in q_Gal and u_Gal, respectively, rather than as standard deviations. For isotropic uncertainty distributions where σ_q ≈ σ_u ≡ σ_q,u , the marginal probability distribution G of PAs can be expressed as follows (Naghizadeh-Khouei & Clarke 1993; Quinn 2012):

$$\begin{eqnarray}&&\begin{array}{l}G(\mathrm{PA}\,| \,{P}_{0},{\mathrm{PA}}_{0},{\sigma }_{q,u})\\ =\displaystyle \frac{1}{\sqrt{\pi }}\left(\displaystyle \frac{1}{\sqrt{\pi }}+{\eta }_{0}{e}^{{\eta }_{0}^{2}}[1+\mathrm{erf}({\eta }_{0})]\right){e}^{-\tfrac{{P}_{0}^{2}}{2{\sigma }_{q,u}^{2}}},\\ \mathrm{where}\,{\eta }_{0}=\displaystyle \frac{{P}_{0}}{\sqrt{2}{\sigma }_{q,u}}\cos \left[2(\mathrm{PA}-{\mathrm{PA}}_{0})\right].\end{array}\end{eqnarray} \tag{ 2 }$$

Here, P₀ and PA₀ represent the average values of P and PA, respectively, and "erf" denotes the Gaussian error function.

We can estimate the angular dispersion ${\breve{\sigma }}_{\mathrm{PA}}$ from the probability distribution of PA based on the function G (hereafter ${\breve{\sigma }}_{G(\mathrm{PA})}$) for each value of σ_q,u , or more precisely, for each value of P₀/σ_q,u (see Equation (2)). Since we cannot solve the function G analytically, we numerically estimate the dependence of ${\breve{\sigma }}_{G(\mathrm{PA})}$ on P, shown in Figure 11. The dashed lines in Figure 11 show the ${\breve{\sigma }}_{G(\mathrm{PA})}$ dependence on P for several example σ_q,u values. We observe a general agreement between the angular dispersion ${\breve{\sigma }}_{\mathrm{PA}}$ obtained from observations and the theoretical ${\breve{\sigma }}_{G(\mathrm{PA})}$ values within the range of σ_q,u values of 0.2% to 0.8%.

In the q–u plane, the angular dispersion ${\breve{\sigma }}_{G(\mathrm{PA})}$ corresponds to the spread of q–u data, measured in radians from the origin of the q–u plane. This angle can be approximated by the tangent of σ_q,u with respect to P. This is why ${\breve{\sigma }}_{G(\mathrm{PA})}$ in Equation (2) is a function of P₀/σ_q,u . We illustrate the comparison between P₀/σ_q,u and ${\breve{\sigma }}_{G(\mathrm{PA})}$ in Figure 12. When normalizing the mean observed polarization fraction (P) estimated for each distance range in the optical polarimetry data shown in Figure 11 by the values of σ_q,u for the same distance range, this normalization removes the dependence of all the observed ${\breve{\sigma }}_{\mathrm{PA}}$ values and the data points should fall on the same theoretical curve of ${\breve{\sigma }}_{G(\mathrm{PA})}$ represented by the solid line in Figure 12.

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The theoretical curve of ${\breve{\sigma }}_{G(\mathrm{PA})}$ follows the relation ${\breve{\sigma }}_{\mathrm{PA}}\times P/{\sigma }_{q,u}=0.5$ rad when P is sufficiently large and ${\breve{\sigma }}_{\mathrm{PA}}\ll 10^\circ$. This is because the phase angle standard deviation of the q–u vectors in radians is approximately equal to the ratio between σ_q,u and P if P is sufficiently large compared to σ_q,u . Thus, ${\breve{\sigma }}_{\mathrm{PA}}$ is approximately 0.5 × σ_q,u /P radians. On the other hand, when P is small and ${\breve{\sigma }}_{\mathrm{PA}}\gg 10^\circ$, the slope of the theoretical curve becomes larger than –1 and closer to 0.

We present a comparison of the optical polarimetry data with ${\breve{\sigma }}_{G(\mathrm{PA})}$ after normalizing P by σ_q,u in the inset of Figure 12. The observations show a general agreement with the theoretical ${\breve{\sigma }}_{G(\mathrm{PA})}$ curve.

4.1.2. Influence of Nonisotropic σ_q and σ_u Distributions

4.1.3. Anticorrelation Induced by Superposition of Multiple Magnetic Field Layers

4.2.1. Intrinsic σ_PA Values of Each Cloud

As described in Section 4.1, σ_q,u is a function of ${\breve{\sigma }}_{\mathrm{PA}}$ and P derived from Equation (2) and can be approximated as ${\sigma }_{q,u}\simeq ({\breve{\sigma }}_{\mathrm{PA}}/0.5\,\mathrm{rad})\times P$. Thus, we see that σ_q,u is a function of three physical quantities—the magnetic turbulence amplitude ($={\breve{\sigma }}_{\mathrm{PA}}$), the dust alignment efficiency (∝P/A_G), and the extinction or gas column density ($\propto {A}_{{\rm{G}}}\propto \mathrm{log}{N}_{{\rm{H}}}$)—as follows:

$$\begin{eqnarray}&&{\sigma }_{q,u}\simeq \displaystyle \frac{{\breve{\sigma }}_{\mathrm{PA}}}{0.5\,\mathrm{rad}}\times \displaystyle \frac{P}{{A}_{{\rm{G}}}}\times {A}_{{\rm{G}}}\,.\end{eqnarray} \tag{ 3 }$$

If we estimate σ_q,u , P, and A_G from observations, we can evaluate these physical quantities, including fluctuations in the PA of the turbulent magnetic field in the POS.

The intrinsic values of P and A_G for individual clouds can be found in Table 4. We can estimate the intrinsic σ_q,u values specific to each cloud by subtracting the contributions from foreground polarization and observational uncertainties from the observed values (Equation (1)). This estimation assumes that the observed σ_q,u is the squared sum of the intrinsic σ_q,u and the contributions from foreground and observational uncertainties. We measure σ_q,u⊥, which represents the σ_q,u values in the direction perpendicular to each cloud's mean q–u vector. The estimated intrinsic σ_q,u⊥ values are presented in Table 5. The σ_q,u⊥ values of the raw observed data and their observational uncertainties used for evaluating the intrinsic σ_q,u⊥ values are listed in Appendix E.

Table 5. The Turbulent Magnetic Field's Angular Amplitude

Cloud	σq,u⊥	${\breve{\sigma }}_{\mathrm{PA}}$
	(%)	(rad)	(deg)
Foreground	${0.19}_{-0.03}^{+0.03}$	${0.45}_{-0.15}^{+0.22}$	${25.9}_{\ -8.4}^{+12.4}$
1.23 kpc	${0.14}_{-0.14}^{+0.07}$	${0.37}_{-0.37}^{+0.29}$	${21.0}_{-21.0}^{+16.9}$
1.47 kpc	${0.43}_{-0.06}^{+0.05}$	${0.43}_{-0.08}^{+0.10}$	$24.4{\ }_{-4.8}^{+5.6}$
1.63 kpc	${0.26}_{-0.14}^{+0.08}$	${0.14}_{-0.08}^{+0.06}$	$8.1{\ }_{-4.4}^{+3.2}$
2.23 kpc	${0.29}_{-0.13}^{+0.09}$	${0.13}_{-0.06}^{+0.06}$	$7.3{\ }_{-3.4}^{+3.4}$
All	${0.64}_{-0.02}^{+0.02}$	${0.83}_{-0.12}^{+0.16}$	$47.8{\ }_{-6.9}^{+9.0}$

Note. The standard deviation of q and u is measured in the direction perpendicular to the mean q–u vector (σ_q,u⊥) of each cloud, and the amplitude of the turbulent magnetic field (${\breve{\sigma }}_{\mathrm{PA}}$) is estimated from σ_q,u⊥ and the polarization fraction (P) (the intrinsic P value of each cloud is taken from Table 4), by referencing the theoretical function ${\breve{\sigma }}_{G(\mathrm{PA})}$ (Equation (2)), and considering the influence of the nonisotropic distribution of σ_q,u .

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Similar to σ_q,u , the observed ${\breve{\sigma }}_{\mathrm{PA}}$ is determined by the summation of contributions from multiple clouds along the LOS. However, it should be noted that the addition of these contributions is not a simple linear sum of squares, as evident from the deviation of ${\breve{\sigma }}_{G(\mathrm{PA})}$ from the relation ${\breve{\sigma }}_{\mathrm{PA}}\times P=\mathrm{const}.$ in Figures 11 and 12. Therefore, we estimate the intrinsic ${\breve{\sigma }}_{\mathrm{PA}}$ values specific to individual clouds by referencing the σ_q,u and P values and the theoretical function ${\breve{\sigma }}_{G(\mathrm{PA})}$.

In the evaluation of ${\breve{\sigma }}_{G(\mathrm{PA})}$, we also take into account the nonisotropic distribution of σ_q,u , as described in Section 4.1.2. For each cloud, we determine the aspect ratio of the σ_q,u distribution's major and minor axes, the rotation angle between the major axis and the average direction of the q–u vectors, and the skewness of the σ_q,u distribution in both the radial and tangential directions. The obtained results are listed in Table 9 in Appendix F. We numerically calculate the deviation from the theoretical curve given by Equation (2), taking into account the nonisotropic distribution of measured σ_q and σ_u , and use the obtained theoretical curve to calculate ${\breve{\sigma }}_{\mathrm{PA}}$. The values of ${\breve{\sigma }}_{\mathrm{PA}}$ obtained considering the anisotropy of σ_q and σ_u vary within the range of estimation errors compared to the case where this consideration is omitted. The comparison of ${\breve{\sigma }}_{\mathrm{PA}}$ estimation values with and without considering anisotropy is shown in Table 9. Finally, we present the obtained values of ${\breve{\sigma }}_{\mathrm{PA}}$, taking into account the anisotropy in the distribution of σ_q and σ_u , in Table 5.

4.2.2. Turbulent-to-uniform Magnetic Field Intensity Ratio

4.2.3. Potential Correlation between ${\breve{\sigma }}_{\mathrm{PA}}$ and N_H

Figure 13 presents the dependence of ${\breve{\sigma }}_{\mathrm{PA}}$ (which corresponds to B_turb/B_unif) on N_H. The two more distant clouds, where the magnetic fields remain undisturbed, exhibit relatively high column densities of N_H = 2.1–2.8 × 10²¹ cm⁻². In contrast, the three closer clouds, where the magnetic fields show a high degree of perturbation, have relatively low column densities of N_H = 0.5–1.5 × 10²¹ cm⁻² (Table 4). As a result, N_H and ${\breve{\sigma }}_{\mathrm{PA}}$ demonstrate a rough anticorrelation.

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Assuming equipartition between the gas kinetic energy and the magnetic field energy, we can convert ${B}_{\mathrm{turb}}/{B}_{\mathrm{unif}}\,(={\breve{\sigma }}_{\mathrm{PA}})$ to the magnetic field intensity. Based on the discussions in Skalidis & Tassis (2021) and Skalidis et al. (2021), we estimate the magnetic field strength in the POS using the following equation:

$$\begin{eqnarray}&&{B}_{\mathrm{POS}}\approx \sqrt{2\pi \rho }\,\displaystyle \frac{{\sigma }_{v,\mathrm{NT}}}{\sqrt{{\sigma }_{\theta }}},\,\rho =\mu \,{m}_{{\rm{H}}}\,{n}_{{\rm{H}}},\end{eqnarray} \tag{ 4 }$$

where σ_v,NT is the gas nonthermal velocity dispersion, σ_θ is the PA dispersion of the polarization due to the turbulent magnetic field ($={\breve{\sigma }}_{\mathrm{PA}}$), ρ is the gas density, μ is the average particle mass (including hydrogen and helium), m_H is the hydrogen atomic mass, and n_H is the gas number density.

Since the gas is primarily atomic (Section 3), we adopt μ = 1.4 (Pattle et al. 2023). As a result, we obtain the following values for the magnetic field strength in each cloud:

$$\begin{eqnarray*}&&\begin{array}{l}\left[{12.1}_{-2.2}^{+2.6},{13.5}_{-3.4}^{+\infty },{12.5}_{-1.2}^{+1.5},{21.7}_{\ -3.3}^{+10.6},{22.8}_{-3.9}^{+8.4}\right]\\ \times \,{\left(\displaystyle \frac{{n}_{{\rm{H}}}}{{10}^{2}\,{\mathrm{cm}}^{-3}}\right)}^{1/2}\left(\displaystyle \frac{{\rm{\Delta }}{V}_{\mathrm{FWHM}}}{5\,\mathrm{km}\,{{\rm{s}}}^{-1}}\right)\mu {\rm{G}},\end{array}\end{eqnarray*}$$

for the foreground, 1.23 kpc, 1.47 kpc, 1.63 kpc, and 2.23 kpc clouds, respectively. Here we normalize the results by the reference value of the gas number density n_H = 10² cm⁻³ and the full width at half-maximum value of the gas velocity ΔV_FWHM = 5 km s⁻¹, since no estimated values are available (Section 3.4). The assumption of n_H = 10² cm⁻³ is equivalent to assuming an LOS thickness of about 1.5–8.9 pc for the five clouds in the LOS, whose column densities are 0.47–2.76 × 10²¹ cm⁻² (Table 4). We apply ΔV_FWHM = 5 km s⁻¹ as a proxy for the H i gas velocity dispersion around CO cloud cores since Nishimura et al. (2015) observed the Orion A and Orion B molecular clouds in the ¹²CO(J = 2 − 1) emission line and estimated that the line widths (≃ΔV_FWHM) are generally 2–5 km s⁻¹. If we adopt a velocity dispersion of ΔV_FWHM = 2 km s⁻¹, it should be noted that the estimated magnetic field strength will proportionally decrease. On the other hand, if the uniform magnetic field is inclined i = 35° from the POS, the estimated magnetic field strength will be approximately 10% larger (by a factor of $\sqrt{1.22}$; see Section 4.2.2).

Heiles & Troland (1980) measured the global magnetic field strength associated with the H i gas in the Sagittarius arm by the Zeeman splitting of the 21 cm H i emission line in the tangential direction (l = 51°, b = 0°) of the Sagittarius arm. They obtained a value of ∼16 μG. Our estimated magnetic field strengths for the clouds in the Sagittarius arm are roughly consistent with their estimation, although the magnetic field strength estimates given here will change depending on the gas velocity dispersion and density.

Using OH Zeeman effect measurements, Crutcher (2012) found that the magnetic field strength in low column density interstellar clouds is typically ∼10 μG, which makes these clouds magnetically subcritical. In contrast, clouds with higher column densities are often magnetically supercritical; the gas contracts gravitationally and drags the magnetic field lines inward. The column density threshold at which the cloud transitions from subcritical to supercritical is N_H ∼ 2 × 10²¹ cm⁻² (also see Pattle et al. 2023 for a review).

The column density of each cloud we observe corresponds to this threshold value, as indicated in Figure 13. Clouds with higher column densities tend to exhibit more ordered magnetic fields (Figure 13), which can be indicative of higher magnetic field strengths (∼20 μG). The higher degree of order may be the result of gas contraction, indicating that these clouds may be magnetically supercritical and the gravitational forces within the cloud are comparable to or greater than the magnetic pressure and the thermal pressure. The presence of a stronger and ordered magnetic field can have significant implications for the dynamics and evolution of the cloud, influencing processes such as star formation and gas kinematics. Thus, our method of separately estimating the magnetic field turbulence and polarization efficiency of each cloud along the LOS can be a promising means of observing interstellar clouds that are magnetically near-critical.

As shown in Figures 6 and 9, we have successfully detected multiple flips in PA along the LOS in the Sagittarius arm. These flips may be due to the local deformation of the magnetic field around individual clouds. However, each cloud's magnetic field is smooth within the observed region (Figures 6 and 9). The physical size of the observed region is 3.8 pc × 6.1 pc for the 1.23 kpc cloud and 6.8 pc × 11 pc for the 2.23 kpc cloud. Thus, the results indicate the possibility of a global magnetic field flipping on a scale sufficiently larger than the observed area and possibly even larger than the size of individual clouds. This possibility should be investigated by observing several neighboring regions.