The JCMT BISTRO Survey: The Magnetic Field of the Barnard 1 Star-forming Region

The JCMT is a submillimeter observatory equipped with a 15 m dish that is located at an altitude of 4092 m on top of Maunakea in Hawaii, USA. Its continuum instrument is SCUBA-2, a cryogenic 10,000-pixel camera capable of simultaneous observing in the 450 μm and 850 μm atmospheric windows (Holland et al. 2013). The SCUBA-2 beams can be approximated by a two-dimensional Gaussian with an FWHM of 96 at 450 μm and 146 at 850 μm (Dempsey et al. 2013).

The POL-2 polarimeter consists of a rotating half-wave plate and a fixed polarizer placed in the optical path of the SCUBA-2 camera (Bastien et al. 2011; Friberg et al. 2016; P. Bastien et al. 2019, in preparation). POL-2 is the follow-up instrument to the SCUBA polarimeter (SCUPOL), which had a similar basic design (Greaves et al. 2003). While SCUBA-2 always simultaneously observes at both 450 μm and 850 μm, only the 850 μm capabilities of POL-2 were commissioned at the time of writing. In brief, POL-2 observes by scanning the sky at a speed of 8'' s⁻¹ in a daisy-like pattern over a field that is roughly 11'' in diameter. Since the half-wave plate is rotated at a rate of 2 Hz, this scanning rate ensures a full rotation of the half-wave plate for every measurement of a 4'' box position in the map. For this paper, the flux calibration factor (FCF) of POL-2 at 850 μm is assumed to be 725 Jy pW⁻¹ beam⁻¹ for each of the Stokes I, Q, and U parameters (the Stokes parameters are defined in Section 3.1). This value was determined by multiplying the typical SCUBA-2 FCF of 537 Jy pW⁻¹ beam⁻¹ (Dempsey et al. 2013) by a transmission correction factor of 1.35 measured in the laboratory and confirmed empirically by the POL-2 commissioning team using observations of the planet Uranus (Friberg et al. 2016).

Perseus B1 was observed with POL-2 between 2016 September and 2017 March as part of the BISTRO large program at the JCMT (project ID: M16AL004). These observations total 14 hr (or 20 individual sets of ∼40-minute observations) of integration in Grade 2 weather (i.e., for a 225 GHz atmospheric opacity, τ₂₂₅, between 0.05 and 0.08). A 20-minute SCUBA-2 scan of B1 without POL-2 in the beam was also obtained on 2016 September 8 to serve as a reference for pointing corrections during data reduction.

The data were reduced using the starlink (Currie et al. 2014) procedure pol2map (Parsons et al. 2017), which is adapted from the SCUBA-2 data reduction procedure makemap (Chapin et al. 2013). In particular, this routine is used to reduce POL-2 time-series observations into Stokes I, Q, and U maps. We follow the convention set by the International Astronomical Union for the definition of Stokes parameters. The default pixel size of the maps produced by pol2map is 4''. For the analysis presented in this paper, we have instead chosen a pixel size of 12'' at the start of the data reduction process to improve the resulting signal-to-noise ratio (S/N) of the final Stokes I, Q, and U maps.

The data reduction process is divided into three steps to optimize the S/N in the resulting maps: (1) the procedure pol2map is run a first time without applying any masks to obtain an initial Stokes I intensity map directly from the POL-2 time-series observations; (2) this initial Stokes I map is then used as the reference for the automatic masking process of pol2map, which is run a second time on the time-series observations to produce the final Stokes I map; and (3) the masks obtained in step 2 are also applied during a third run of pol2map to reduce the Stokes Q and U maps, which are automatically corrected for the instrumental polarization. The uncertainties in each pixel of the Stokes I, Q, and U maps are taken directly from the variance maps provided by the pol2map procedure. The role of masking in the reduction of SCUBA-2 data, and incidentally POL-2 data, is discussed at length by Mairs et al. (2015).

The correction for instrumental polarization is a crucial step in the analysis of any polarization measurement. If the instrumental polarization is not properly taken into account, then it may lead to erroneous results. For this reason, the latest model (2018 January) for the instrumental polarization of the JCMT at 850 μm was extensively tested by the POL-2 commissioning team with observations of Uranus and Mars (Friberg et al. 2016, 2018; P. Bastien et al. 2019, in preparation). They found that the instrumental polarization can be accurately described using a two-component model combining the optics of the telescope and its protective wind blind. While the level of instrumental polarization is dependent on elevation, it is typically ∼1.5% of the measured total intensity (Friberg et al. 2018).

We also use 850 μm polarization data of Perseus B1 from the SCUPOL Legacy Catalog. Matthews et al. (2009) built this legacy catalog by systematically re-reducing SCUPOL 850 μm observations toward 104 regions, including previously published observations of B1 (Matthews & Wilson 2002), to provide reference Stokes cubes of comparable quality for all the astronomical sources with at least a 2σ detection of polarization. For this paper, the SCUPOL Stokes I, Q, and U cubes for B1 were downloaded from the legacy catalog's online archive hosted by the CADC. To match the POL-2 results, we resampled the SCUPOL polarization vectors onto a 12'' pixel grid.

The JCMT is also equipped with the Heterodyne Array Receiver Program (HARP)/Auto-Correlation Spectral Imaging System (ACSIS) high-resolution heterodyne spectrometer capable of observing molecular lines between 325 and 375 GHz (or 922 μm to 799 μm). The HARP is a 4 × 4 detector array that can be used in combination with the ACSIS to rapidly produce large-scale velocity maps of astronomical sources (Buckle et al. 2009). In this paper, we use the previously published ∼14'' resolution integrated intensity map of the ¹²CO J = 3−2 molecular line toward Perseus B1 (project ID: S12AC01) (Sadavoy et al. 2013). This intensity map was integrated over a bandwidth of 1.0 GHz centered on the rest frequency of the ¹²CO J = 3−2 line at 345.796 GHz. The noise added by integrating over such a large bandwidth has no effect on the results presented in this work since the ¹²CO J = 3−2 data are used only to indicate the presence of outflows in Figure 1.

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It is important to note that SCUBA-2, POL-2, and HARP are not sensitive to exactly the same spatial scales. This difference is due to a combination of the different scanning strategies for each instrument and their associated data reduction procedures (e.g., Chapin et al. 2013). Hence, this difference must be kept in mind when combining results from different instruments, such as correcting for molecular contamination using HARP or comparing source intensities between POL-2 and SCUBA-2. While this difference is not an issue for the results presented in this paper, it may need to be taken into account in future studies using BISTRO data (see the appendix  for more details).

Finally, this project makes use of spectroscopic data from the Green Bank Ammonia Survey (GAS; Friesen et al. 2017). GAS uses the K-Band Focal Plane Array and the VErsatile GBT Astronomical Spectrometer at the Green Bank Telescope to map ammonia lines, among others, in nearby star-forming regions. In this work, we specifically use measurements of the NH₃ (1,1) and (2,2) lines toward Perseus B1 (GAS Consortium 2019, in preparation). These observations of NH₃ molecular lines at ∼23.7 GHz have a spatial resolution of 32'' and a velocity resolution of ∼0.07 km s⁻¹.

The polarization vectors are defined by the polarization fraction P and the polarization angle Φ measured eastward from celestial north. These properties are determined directly from the Stokes I, Q, and U parameters, which is the commonly accepted parameterization for partially polarized light. The Stokes I parameter is the total intensity of the incoming light, and the Stokes Q and U parameters are respectively defined as $Q=I\,P\,\cos \left(2{\rm{\Phi }}\right)$ and $U=I\,P\,\sin \left(2{\rm{\Phi }}\right)$.

When Q and U are near zero, these values will be dominated by the noise in our measurements. This noise contribution always leads to a positive bias in the calculation of the polarization fraction P due to the quadratic nature of the polarized intensity I_P = [Q² + U²]^1/2 (e.g., Wardle & Kronberg 1974; Montier et al. 2015; Vidal et al. 2016). The amplitude of this positive bias can be approximated from the uncertainty ${\sigma }_{{I}_{P}}$ given in Equation (2), which is used in Equation (1) to debias the polarization fraction P (e.g., Naghizadeh-Khouei & Clarke 1993).

The debiased polarization fraction P (in per cent) can therefore be written as

$$\begin{eqnarray}&&P=\displaystyle \frac{100}{I}\,\sqrt{{Q}^{2}+{U}^{2}-{\sigma }_{{I}_{P}}^{2}}=\displaystyle \frac{100}{I}\,{I}_{P},\end{eqnarray} \tag{ 1 }$$

where we redefine I_P as the debiased polarized intensity with uncertainty ${\sigma }_{{I}_{P}}$. This uncertainty ${\sigma }_{{I}_{P}}$ is given by

$$\begin{eqnarray}&&{\sigma }_{{I}_{P}}={\left[\displaystyle \frac{{\left(Q{\sigma }_{Q}\right)}^{2}+{\left(U{\sigma }_{U}\right)}^{2}}{{Q}^{2}+{U}^{2}}\right]}^{1/2},\end{eqnarray} \tag{ 2 }$$

where σ_Q and σ_U are the uncertainties on the Stokes Q and U parameters, respectively. The uncertainty σ_P of the polarization fraction P is given by

$$\begin{eqnarray}&&{\sigma }_{P}=P{\left[{\left(\displaystyle \frac{{\sigma }_{{I}_{P}}}{{I}_{P}}\right)}^{2}+{\left(\displaystyle \frac{{\sigma }_{I}}{I}\right)}^{2}\right]}^{1/2},\end{eqnarray} \tag{ 3 }$$

where σ_I is the uncertainty on the Stokes I total intensity.

Finally, the expression for the polarization angle Φ is

$$\begin{eqnarray}&&{\rm{\Phi }}=\displaystyle \frac{1}{2}\,\arctan \left(\displaystyle \frac{U}{Q}\right),\end{eqnarray} \tag{ 4 }$$

where Φ is defined between 0 and π (0° and 180°) for convenience, and its related uncertainty σ_Φ is given by

$$\begin{eqnarray}&&{\sigma }_{{\rm{\Phi }}}=\displaystyle \frac{1}{2}\,\displaystyle \frac{\sqrt{{\left(U{\sigma }_{Q}\right)}^{2}+{\left(Q{\sigma }_{U}\right)}^{2}}}{{Q}^{2}+{U}^{2}}.\end{eqnarray} \tag{ 5 }$$

Figure 1 (left) shows the BISTRO 850 μm linear polarization map of Perseus B1 for a pixel size of 12''. The catalog of polarization vectors is calculated for every pixel of the POL-2 Stokes I, Q, and U maps, but only vectors passing a set of predetermined selection criteria are shown. These selection criteria are as follows: an S/N of I/σ_I > 3 for Stokes I and its uncertainty σ_I, an S/N of P/σ_P > 3 for the polarization fraction P and its uncertainty σ_P, and an uncertainty σ_P < 5% for the polarization fraction. The criterion of σ_P < 5% was chosen arbitrarily as a precaution against potentially spurious vectors with anomalously high polarization fractions. These criteria provide a catalog of 224 polarization vectors for Perseus B1.

The mean values of the Stokes uncertainties σ_I, σ_Q, and σ_U for the polarization vectors shown in Figure 1 are 1.6, 1.3, and 1.3 mJy beam⁻¹, respectively. At best, we achieve a sensitivity of 0.1% in polarization fraction and an uncertainty of 21 in polarization angle, with mean values for σ_P of 1.9% and for σ_Φ of 57 for the entire catalog of vectors.

Assuming that interstellar dust grains are aligned with their long axis perpendicular to the magnetic field, the plane-of-sky field morphology in Perseus B1 is obtained by rotating the vectors in the polarization map by 90°. Figure 1 (right) shows the inferred plane-of-sky magnetic field map for B1. To help highlight the magnetic field structure, the rotated vectors are normalized to the same length. A contour plot of the HARP ¹²CO J = 3−2 integrated intensity map from the JCMT Gould Belt Survey (Sadavoy et al. 2013) is also included in the right panel of Figure 1.

Selected submillimeter sources are identified in both panels of Figure 1 to serve as references for the discussion in Section 5 (Bally et al. 2008). These sources are embedded young stellar objects that have been associated with molecular outflows (Evans et al. 2009; Hatchell & Dunham 2009; Hirano & Liu 2014; Carney et al. 2016). Specifically, the lobes of the precessing molecular outflow originating from the protostellar core B1-c (Matthews et al. 2006) are particularly well defined by the ¹²CO J = 3−2 contour plot shown in the right panel of Figure 1.

The top panel of Figure 2 compares the fraction of polarization P with the Stokes I total intensity for each of the POL-2 vectors shown in the left panel of Figure 1. There is a clear trend of decreasing fraction P as a function of increasing Stokes I. If the total intensity is correlated with the column density (Hildebrand 1983), this behavior can be understood as the result of a depolarization effect toward higher-density regions of the cloud. The origin of this depolarization effect is discussed in Section 5. This trend does not mean, however, that the polarized intensity I_P itself is decreasing. Indeed, the bottom panel of Figure 2 shows that I_P may be in fact increasing slowly with Stokes I.

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We fitted a power law (P ∝ I^α) to the data in Figure 2 (top) using an error-weighted least-squares minimization technique. We find a power-law index α = −0.85 ± 0.01, with a reduced chi-squared ${\chi }_{r}^{2}$ = 3.4. This power law is shown in both panels of Figure 2 as a solid line. The spread of data points relative to their uncertainties is responsible for the large ${\chi }_{r}^{2}$ value obtained, which indicates that fitting a single power law may not be sufficient to account for the entire data set. The detailed effects of measurement uncertainties on the power-law fit between P and I are currently under investigation (K. Pattle et al. 2019, in preparation).

The power-law index α ∼ −0.85 we find for B1 is nearly identical to the value measured in ρ Ophiuchus B by Soam et al. (2018) and relatively close to the index α ∼ −0.8 measured by Kwon et al. (2018) in ρ Ophiuchus A, both obtained from BISTRO data. Similarly, Matthews & Wilson (2002) previously found a power-law index α ∼ −0.8 in B1 using SCUPOL 850 μm measurements. The differences between POL-2 and SCUPOL polarization maps of B1 are quantified in Section 3.3.

However, in the context of grain alignment theory, it is more meaningful to take the optical depth into account when studying depolarization effects in molecular clouds. While an accurate modeling of the alignment efficiency of dust grains in Perseus B1 will require a detailed analysis beyond the scope of this work, we can nonetheless begin to characterize the relationship between the polarized dust thermal emission and the visual extinction A_V in the cloud by fitting a power law of the form $P\propto {A}_{V}^{\beta }$ (e.g., Alves et al. 2014). Specifically, we know that the polarization fraction P of dust thermal emission obtained from submillimeter observations is proportional to the polarization efficiency P_ext/A_V derived from measurements of the polarization fraction P_ext due to extinction at visible wavelengths (Andersson et al. 2015).

Figure 3 shows the relation between the polarization fraction P and the derived visual extinction A_V for the polarization vectors shown in the left panel of Figure 1 that also have an associated opacity measurement in the 300 μm τ₃₀₀ opacity map from Chen et al. (2016). We estimate the visual extinction A_V using Equation (A5) from Jones et al. (2015) and a version of the τ₃₀₀ opacity map from Chen et al. (2016) that has been regridded from a pixel scale of 14'' to 12'' to match our observations. We also assume a reddening R_V of 3.1, which may be more representative of the diffuse interstellar medium (Weingartner & Draine 2001) but should nonetheless serve as a reasonable lower limit for our estimation of the visual extinction A_V across the cloud.

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We fitted a power-law $P\propto {A}_{V}^{\beta }$ to the data shown in Figure 3 using an error-weighted least-squares minimization technique. We find a power-law index β = −0.51 ± 0.03, with a reduced chi-squared ${\chi }_{r}^{2}=26.3$. This power law is shown in Figure 3 as a solid line. The large reduced chi-squared ${\chi }_{r}^{2}$ value we find clearly indicates a poor fit to the data considering the spread of values and their uncertainties for the polarization fraction P in Figure 3. This could be explained in part by our use of a single reddening value to derive the visual extinction A_V. Indeed, the reddening R_V depends on the size distribution and composition of the dust grains, and so we do not expect this value to be constant across the cloud.

Nevertheless, the power-law index β ∼ −0.5 we find in B1 is shallower than the power indices obtained from submillimeter observations in the Pipe-109 starless core (β ∼ −0.9; Alves et al. 2014, 2015) and in the LDN 183 starless core (β ∼ −1.0; Andersson et al. 2015). In fact, a power-law index β ∼ −0.5 is closer to the power-law index β ∼ −0.6 measured toward lower-extinction regions (A_V < 20) of LDN 183 using visible and near-infrared observations (Andersson et al. 2015). Although Figure 2 clearly shows a depolarization effect with increasing total intensity I, the power-law index β ∼ −0.5 we find using the data in Figure 3 suggests that dust grains in Perseus B1 are aligned more efficiently than in starless cores with comparable measures of visual extinction A_V. Since B1 is a site of ongoing star formation, this may provide evidence that radiation from embedded young stellar objects can compensate for the expected loss of grain alignment with increasing visual extinction.

As mentioned in Section 2.1, Perseus B1 was previously observed at 850 μm with the SCUPOL polarimeter (Matthews & Wilson 2002). Here we specifically compare the BISTRO results presented in Section 3.2 to the polarization data of B1 found in the SCUPOL Legacy Catalog (Matthews et al. 2009).

Figure 4 compares the BISTRO observations to their equivalent data set in the SCUPOL Legacy Catalog, with the POL-2 polarization vectors (same as Figure 1) in red and the SCUPOL vectors in blue. To have a significant number of SCUPOL vectors for this comparison, we relaxed their selection criteria compared to POL-2. For the SCUPOL data, we use I/σ_I > 2, P/σ_P > 2, and σ_P < 10%. These relaxed criteria provide a total catalog of 69 vectors, compared to only 17 when applying the same selection criteria as for the POL-2 data.

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At best, the relaxed catalog of SCUPOL vectors achieves a sensitivity of 0.5% in polarization fraction and an uncertainty of 55 in polarization angle, with mean values for σ_P of 2.7% and for σ_Φ of 103.

Figure 5 shows the distribution of angles for both the POL-2 and SCUPOL polarization maps. The top panel shows the histogram including all the POL-2 and SCUPOL polarization vectors shown in Figure 4, normalized by the maximum value in each distribution. Both distributions peak between 65° and 85°. The bottom panel shows the normalized distributions only for those vector positions that are common (i.e., spatially overlapping within the same pixel) to both SCUPOL and POL-2. There are 52 such positions in the maps.

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We used a Kolmogorov–Smirnov test to compare the distributions shown at the bottom of Figure 5. Specifically, a two-sample Kolmogorov–Smirnov test provides the probability that two independent data samples are drawn from the same intrinsic distribution by measuring the maximum distance between the cumulative probability distribution of each sample. For example, if both the SCUPOL and POL-2 values for the selected cospatial vectors were exact measurements of the 850 μm polarization toward Perseus B1, then we would expect the two catalogs of polarization angles, and therefore their respective cumulative probability distributions, to be identical and the Kolmogorov–Smirnov test to return a 100% probability that they are drawn from the same intrinsic distribution of polarization angles. In reality, the POL-2 and SCUPOL distributions shown in the bottom panel of Figure 5 are not identical even though they probe the same positions in B1, and so the Kolmogorov–Smirnov test becomes a way of quantifying the difference between them since it makes no assumption about the nature of the aforementioned intrinsic distribution.

In this case, we find a low likelihood (0.6%) that both POL-2 and SCUPOL distributions in the bottom panel of Figure 5 are drawn from the same intrinsic distribution of polarization angles (with a maximum deviation D = 0.39 between the cumulative probability distributions). In other words, based only on the 52 available cospatial vectors in each sample, a two-sample Kolmogorov–Smirnov test shows that the distributions of POL-2 and SCUPOL polarization angles are significantly different from each other. If we set the selection criteria for POL-2 vectors to be identical to those applied for SCUPOL vectors, we find instead 64 positions with vectors common to both catalogs. This relaxed data set does not, however, improve the results of the Kolmogorov–Smirnov test.

Figure 6 expands the comparison shown in Figure 5 (bottom) between the POL-2 and SCUPOL polarization angles for pairs of spatially overlapping vectors. The top panel of Figure 6 shows that most outliers from the 1:1 correspondence line are found toward lower-intensity regions (I < 200 mJy beam⁻¹), as measured from POL-2 Stokes I. Furthermore, in Figure 6 (bottom), the vector pairs displaying the largest angular difference ($\left|{{\rm{\Phi }}}_{\mathrm{SCUPOL}}-{{\rm{\Phi }}}_{\mathrm{POL}-2}\right|$) are found near or below an S/N of 3 for the polarization fraction (${P}_{\mathrm{SCUPOL}}/{\sigma }_{{P}_{\mathrm{SCUPOL}}}$ ≲ 3) measured with SCUPOL. Although the pairs of vectors at high S/N (${P}_{\mathrm{SCUPOL}}/{\sigma }_{{P}_{\mathrm{SCUPOL}}}$ > 4) also exhibit a non-negligible angular difference, this effect is not nearly as pronounced as for the low-S/N vectors (${P}_{\mathrm{SCUPOL}}/{\sigma }_{{P}_{\mathrm{SCUPOL}}}\lesssim 3$). This disparity between POL-2 and SCUPOL could therefore be explained by the relatively high noise levels in the SCUPOL Legacy data.

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The magnetic field strength in molecular clouds can be estimated through the DCF method (Davis 1951; Chandrasekhar & Fermi 1953). This technique relies on the assumption that turbulent motions in the gas will locally inject randomness in the observed morphology of a large-scale magnetic field. Since polarization vectors are expected to trace the plane-of-sky component of the magnetic field, we can infer the strength of this component by measuring the dispersion of polarization angles relative to the large-scale field orientation. This technique, however, also requires the velocity dispersion and the density of the gas in the cloud to be known beforehand.

According to Crutcher et al. (2004), the DCF equation for the plane-of-sky magnetic field strength B_pos can be written as

$$\begin{eqnarray}&&{B}_{\mathrm{pos}}=A\,\sqrt{4\pi \rho }\,\displaystyle \frac{\delta V}{\delta {\rm{\Phi }}},\end{eqnarray} \tag{ 6 }$$

where ρ is the density, δV is the velocity dispersion of the gas in the cloud, δΦ is the dispersion of polarization angles (in radians), and A is a correction factor usually assumed to be ∼0.5. The correction factor A is included to account for the three-dimensional nature of the interplay between turbulence and magnetism (e.g., Ostriker et al. 2001). There is, however, a caveat to Equation (6), namely, that it cannot intrinsically account for changes in the large-scale field morphology. As a consequence, the technique from Crutcher et al. (2004) was modified by Pattle et al. (2017) to take large-scale variations in field morphology into account when calculating the magnetic field strength in Orion A.

Specifically, Pattle et al. (2017) calculate the dispersion δΦ of polarization angles in Equation (6) with an unsharp-masking technique. First, the large-scale component of the field is found by smoothing the map of polarization angles using 3 × 3 pixel boxes. This smoothed map is then subtracted from the original to obtain a map of the residual polarization angles. Finally, the dispersion δΦ is obtained from the mean value of the residual angles fitting a specific set of conditions. This approach therefore cancels the contribution of a changing field morphology to the dispersion of polarization angles at scales larger than the smoothed mean field map.

In our work, we instead apply the improved DCF method developed by Hildebrand et al. (2009) and Houde et al. (2009), which was also adapted for polarimetric data obtained by interferometers such as the SMA and CARMA (Houde et al. 2011, 2016). This technique avoids the problem of spatial changes in field morphology by using an angular dispersion function (sometimes called structure function) rather than the dispersion of polarization angles around a mean value. Furthermore, the angular dispersion technique from Houde et al. (2009) was independently tested using both R-band (e.g., Franco et al. 2010) and submillimeter (e.g., Ching et al. 2017) polarimetric observations to characterize the magnetic and turbulent properties of star-forming regions.

This angular dispersion function is calculated by taking the angular difference between every pair of polarization vectors in a given map as a function of the distance between them. This technique effectively traces the ratio between turbulent and magnetic energies, which can then be fitted without any prior assumptions on the turbulence in the cloud or the morphology of the large-scale field (Hildebrand et al. 2009). As before, this analysis can be used to estimate the strength of the plane-of-sky magnetic field component if the density and velocity dispersion of the cloud are known. Additionally, it can be used to measure the effect of integrating turbulent cells along the line of sight within a telescope beam, effectively constraining the theoretical factor A included in Equation (6) (Houde et al. 2009).

We first need to define the relevant quantities for the dispersion analysis presented in this paper. The difference in polarization angle between two vectors as a function of distance ℓ is defined as ${\rm{\Delta }}{\rm{\Phi }}({\ell })\equiv {\rm{\Phi }}(x)-{\rm{\Phi }}(x+{\boldsymbol{\ell }})$, where Φ(${\boldsymbol{x}}$) is the angle Φ of the polarization vector found at a position ${\boldsymbol{x}}$ in the map and ${\boldsymbol{\ell }}$ is the angular displacement between two vectors. With this quantity, we can define the angular dispersion function as formulated by Houde et al. (2009):

$$\begin{eqnarray}&&1-\left\langle \cos [{\rm{\Delta }}{\rm{\Phi }}({\ell })]\right\rangle ,\end{eqnarray} \tag{ 7 }$$

where $\left\langle ...\right\rangle$ is the average over every pair of vectors separated by a distance ℓ. Since Equation (7) is essentially a measure of the mean difference in polarization angles as a function of distance, it is accurate to describe it as an angular dispersion function.

The magnetic field ${\boldsymbol{B}}$(${\boldsymbol{x}}$) in the cloud at a position ${\boldsymbol{x}}$ can be written as a combination of a large-scale (or ordered) component ${\boldsymbol{B}}$_o(${\boldsymbol{x}}$) and a turbulent component ${\boldsymbol{B}}$_t(${\boldsymbol{x}}$), i.e., ${\boldsymbol{B}}$(${\boldsymbol{x}}$) = ${\boldsymbol{B}}$_o(${\boldsymbol{x}}$) + ${\boldsymbol{B}}$_t(${\boldsymbol{x}}$). Furthermore, we define the ratio of the average energy of the turbulent component to that of the large-scale component as $\left\langle {B}_{t}^{2}\right\rangle /\left\langle {B}_{o}^{2}\right\rangle$ and the ratio of the average energy of the turbulent component to that of the total magnetic field as $\left\langle {B}_{t}^{2}\right\rangle /\left\langle {B}^{2}\right\rangle$. Both quantities can be obtained from fitting the angular dispersion function.

To relate the magnetic fields and turbulence, we also need to define the turbulent properties of the cloud. Specifically, we require the number N of independent magnetic turbulent cells observed for a column of dust along the line of sight and within a telescope beam from

$$\begin{eqnarray}&&N={\rm{\Delta }}^{\prime} \,\displaystyle \frac{\left({\delta }^{2}+2{W}^{2}\right)}{\sqrt{2\pi }\,{\delta }^{3}},\end{eqnarray} \tag{ 8 }$$

where δ is the turbulent correlation length scale of the magnetic field, W is the radius of the circular telescope beam (specifically, $\mathrm{FWHM}=2\,\sqrt{2\,\mathrm{ln}2}\,W$), and Δ' is the effective thickness of the cloud (see Equation (52) in Houde et al. 2009). The turbulent correlation length scale δ can be understood as the typical size of a magnetized turbulent cell in the cloud. In this specific case, the turbulence is supposedly isotropic and the turbulent correlation length scale δ is assumed to be smaller than the thickness Δ' of the cloud.

If the physical depth of the cloud is not known beforehand, the effective thickness Δ' can be estimated from the autocorrelation function of the integrated polarized intensity across the cloud (see Equation (51) in Houde et al. 2009). This autocorrelation function is defined as

$$\begin{eqnarray}&&\left\langle {I}_{P}^{2}({\ell })\right\rangle \equiv \left\langle {I}_{P}(x)\,{I}_{P}(x+{\boldsymbol{\ell }})\right\rangle ,\end{eqnarray} \tag{ 9 }$$

from which we use the width at half-maximum to evaluate Δ'. This approach, however, assumes that the spatial distribution of polarized dust emission on the plane of the sky is an adequate probe of the cloud's properties along the line of sight, which we believe to be reasonable in the case of dense molecular clouds.

The detailed derivations given by Hildebrand et al. (2009) and Houde et al. (2009) show that the relationship between the angular dispersion function and the magnetic and turbulent properties of a molecular cloud can be expressed by the following equation:

$$\begin{eqnarray}&&1-\left\langle \cos [{\rm{\Delta }}{\rm{\Phi }}({\ell })]\right\rangle \simeq \displaystyle \frac{1}{N}\,\displaystyle \frac{\left\langle {B}_{t}^{2}\right\rangle }{\left\langle {B}_{o}^{2}\right\rangle }-{b}^{2}({\ell })+a\,{{\ell }}^{2},\end{eqnarray} \tag{ 10 }$$

where a is the first Taylor coefficient of the ordered autocorrelation function and b²(ℓ) is the autocorrelated turbulent component of the dispersion function (see Equations (53) and (55) in Houde et al. 2009). Specifically, the Taylor coefficient a is related to the large-scale structure of the magnetic field. Additionally, we can write this autocorrelated turbulent component as

$$\begin{eqnarray}&&{b}^{2}({\ell })=\displaystyle \frac{1}{N}\,\displaystyle \frac{\left\langle {B}_{t}^{2}\right\rangle }{\left\langle {B}_{o}^{2}\right\rangle }\,{e}^{-{{\ell }}^{2}/2({\delta }^{2}+2{W}^{2})}.\end{eqnarray} \tag{ 11 }$$

Since the beam radius W and the effective cloud thickness Δ' can be considered as known quantities, we only need to fit three parameters to the angular dispersion function: the ratio of turbulent energy to large-scale magnetic energy $\left\langle {B}_{t}^{2}\right\rangle /\left\langle {B}_{o}^{2}\right\rangle$, the turbulent correlation length scale δ of the magnetic field, and the first Taylor coefficient a of the ordered autocorrelation function.

Finally, Houde et al. (2009) rewrote the DCF equation (see Equation (6)) for the plane-of-sky strength of the magnetic field to calculate it directly from the ratio of turbulent energy to total magnetic energy $\left\langle {B}_{t}^{2}\right\rangle /\left\langle {B}^{2}\right\rangle$ in the cloud. This new formulation of the DCF equation can be written as

$$\begin{eqnarray}&&{B}_{\mathrm{pos}}\simeq \sqrt{4\pi \rho }\ \delta V{\left[\displaystyle \frac{\left\langle {B}_{t}^{2}\right\rangle }{\left\langle {B}^{2}\right\rangle }\right]}^{-1/2},\end{eqnarray} \tag{ 12 }$$

where, as previously, ρ is the density and δV is the one-dimensional velocity dispersion for the gas (see Equation (57) in Houde et al. 2009 and Equation (26) in Houde et al. 2016). The gas density ρ takes the form ρ = μ m_H n(H₂), where μ = 2.8 is the mean molecular weight of the gas (Kauffmann et al. 2008), m_H is the mass of a hydrogen atom, and n(H₂) is the number density of hydrogen molecules in the cloud.

Once the strength of the plane-of-sky component of the magnetic field has been calculated with Equation (12), it becomes possible to evaluate the magnetic critical ratio λ_c of the studied molecular cloud (Crutcher et al. 2004). The critical ratio λ_c can be estimated from the plane-of-sky amplitude of the magnetic field with the following equation:

$$\begin{eqnarray}&&{\lambda }_{c}\simeq 7.6\times {10}^{-21}\,\displaystyle \frac{N({{\rm{H}}}_{2})}{{B}_{\mathrm{pos}}},\end{eqnarray} \tag{ 13 }$$

where N(H₂) is the typical column density of molecular hydrogen in the cloud. If λ_c < 1, then the molecular cloud is magnetically subcritical and the magnetic field is sufficiently strong to stop its gravitational collapse. If λ_c > 1, the cloud is instead magnetically supercritical and the magnetic field alone cannot support the cloud against its self-gravity.

Following Section 4.1, we determine the angular dispersion function from the POL-2 data of Perseus B1. We include in this analysis all the POL-2 polarization vectors found in a 240''-wide square centered on the position (03^h33^m2045, +31°07'5016), as illustrated in the right panel of Figure 1. This region covers most of the embedded young stellar objects in the densest parts of Perseus B1. The resulting angular dispersion function is shown in the top panel of Figure 7 as a function of the distance ℓ in bins of 12''. The observed steady increase of this function with ℓ at small spatial scales (0.01 to 1.0 pc) is also a behavior seen in other studies using this technique (e.g., Houde et al. 2009, 2016; Franco et al. 2010; Ching et al. 2017; Chuss et al. 2019).

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The angular dispersion function was fitted with Equation (10) to obtain δ and $\left\langle {B}_{t}^{2}\right\rangle /\left\langle {B}_{o}^{2}\right\rangle$ using an effective cloud depth Δ' of 84'' and a beam radius W of 62 (or a FWHM of 146) at 850 μm. The reduced chi-squared value for this fit is χ_r² = 1.5. The results of the fit to the angular dispersion, including $\left\langle {B}_{t}^{2}\right\rangle /\left\langle {B}^{2}\right\rangle$, are given in Table 1. Additionally, the resulting turbulent autocorrelation function b²(ℓ) is shown in the bottom panel of Figure 7.

Table 1.  Derived Magnetic and Turbulent Properties, and Other Related Parameters in Perseus B1

Parameter	Value	Description
δ	50 ± 25	Turbulent correlation length scale
N	27.3 ± 0.3	Number of beam-integrated turbulent cells along the line of sight
$\left\langle {B}_{t}^{2}\right\rangle /\left\langle {B}_{o}^{2}\right\rangle$	0.9 ± 1.1	Turbulent-to-ordered magnetic energy ratio
$\left\langle {B}_{t}^{2}\right\rangle /\left\langle {B}^{2}\right\rangle$	0.5 ± 0.3	Turbulent-to-total magnetic energy ratio
a	(2.4 ± 0.2) × 10−6 −2	First Taylor coefficient of the ordered autocorrelation function
δV	(2.9 ± 1.1) × 104 cm s−1	Velocity dispersion of the gas along the line of sighta
n(H2)	(1.5 ± 0.3) × 105 cm−3	Mean number density of the gasa
N(H2)	(4.7 ± 0.9) × 1022 cm−2	Estimated column density for a cloud depth of ∼0.1 pc
ρ	(7.0 ± 1.4) × 10−19 g cm−3	Estimated density of the gas for a molecular weight μ = 2.8
Bpos	120 ± 60 μG	Plane-of-sky amplitude of the magnetic field
λc	3.0 ± 1.5	Criticality ratiob

Notes.

^aFriesen et al. (2017); GAS Consortium (2019, in preparation). ^bCrutcher et al. (2004).

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At a distance of 295 pc (Ortiz-León et al. 2018), the effective cloud depth Δ' of 84'' in B1 represents a physical depth of ∼0.1 pc. While this effective cloud depth Δ' ∼ 0.1 pc was derived independently from the autocorrelation function of the polarized intensity I_P (see Section 4.1), it is nonetheless comparable to the typical width of dense filaments in star-forming regions (e.g., Arzoumanian et al. 2011; André et al. 2014, 2016; Koch & Rosolowsky 2015). For reference, the square region shown in the right panel of Figure 1 has a width of ∼0.4 pc (∼270'').

The exact distance to the Perseus molecular cloud, and to B1 in particular, is still subject to some ambiguity. Indeed, different methods provide a wide range of values from 235 pc (22 GHz water maser parallaxes; Hirota et al. 2008, 2011) to 315 pc (photometric reddening; Schlafly et al. 2014). Furthermore, Schlafly et al. (2014) found a gradient of distances from the western (260 pc) to the eastern (315 pc) parts of the Perseus molecular cloud complex. However, recent parallax measurements with the Gaia space telescope instead suggest a smaller range of distances between NGC 1333 (295 pc) and IC 348 (320 pc) (Ortiz-León et al. 2018). According to these Gaia results, the distance to B1 is similar to that of NGC 1333 at 295 pc. This distance to B1 assumes that the young stellar objects used for these parallax measurements provide a good estimate of the clump's true position along the line of sight.

Perseus B1 was mapped in emission from several NH₃ inversion transitions at ∼24 GHz by GAS (the first data release of the survey was presented by Friesen et al. 2017). NH₃ is a commonly used selective tracer of moderately dense gas (n ≳ a few × 10³ cm⁻³; Shirley 2015). The NH₃ (1,1) emission closely follows the intensity detected with POL-2 across the cloud (GAS Consortium 2019, in preparation). The velocity dispersion of the gas along each line of sight was obtained through simultaneous modeling of hyperfine structure of the detected NH₃ (1,1) and (2,2) inversion line emission. Assuming that the (1,1) and (2,2) lines share the same line-of-sight velocity, velocity dispersion, and excitation temperature, the analysis produces maps of the aforementioned parameters along with the gas kinetic temperature and the total column density of NH₃. Further details of the modeling are given in Friesen et al. (2017).

For the region delimited by the square in the right panel of Figure 1, we find an average velocity dispersion δV = 0.29 km s⁻¹, with a standard deviation σ_δV = 0.11 km s⁻¹. The uncertainties for individual line width measurements are typically <0.05 km s⁻¹. We therefore use the velocity dispersion δV = (2.9 ± 1.1) × 10⁴ cm s⁻¹ to calculate the plane-of-sky amplitude of the magnetic field with Equation (12).

The number density n(H₂) of the gas in Perseus B1 is also calculated from the same GAS NH₃ data (Friesen et al. 2017; GAS Consortium 2019, in preparation). Specifically, we follow the relation described by Ho & Townes (1983) between density, excitation temperature, and gas kinetic temperature to estimate the number density n(H₂) in B1, assuming that the NH₃ emission in B1 can be approximated by a two-level system. First, for the denser regions associated with polarized emission, we find a mean gas temperature of 11.6 K with a standard deviation of 1.2 K and a mean excitation temperature of 6.5 K with a standard deviation of 0.4 K. Using these temperatures, we calculate a mean density n(H₂) = (1.5 ± 0.3) × 10⁵ cm⁻³. If the typical depth of the dense material in B1 is indeed ∼0.1 pc, we then find a column density N(H₂) = (4.7 ± 0.9) × 10²² cm⁻², in agreement with the values obtained from fitting far-infrared and submillimeter measurements of dust thermal emission (Sadavoy et al. 2013; Chen et al. 2016). Finally, assuming a molecular weight μ = 2.8 (Kauffmann et al. 2008), we derive an average gas density ρ = (7.0 ± 1.4) × 10⁻¹⁹ g cm⁻³.

The ratio $\left\langle {B}_{t}^{2}\right\rangle /\left\langle {B}^{2}\right\rangle$ of turbulent to total magnetic energy given in Table 1 can be used to calculate the plane-of-sky strength of the magnetic field in Perseus B1 using Equation (12). Combined with the values given previously for the density ρ and velocity dispersion δV, we calculate the plane-of-sky strength of the magnetic field in Perseus B1 to be 120 ± 60 μG.

We compare the plane-of-sky strength of the magnetic field derived from the angular dispersion analysis (Houde et al. 2009) with the one obtained from the classical DCF method (Crutcher et al. 2004). First, we fit a Gaussian curve to the histogram of POL-2 polarization angles shown in the top panel of Figure 5 and find a dispersion δΦ_obs = 0.213 rad (122). We then evaluate the dispersion δΦ_err due to instrumental errors using the mean uncertainty in polarization angle of 0.099 rad (57) given in Section 3.2. This allows us to calculate the intrinsic angular dispersion $\delta {\rm{\Phi }}=\sqrt{\delta {{\rm{\Phi }}}_{\mathrm{obs}}^{2}-\delta {{\rm{\Phi }}}_{\mathrm{err}}^{2}}=0.188$ rad (108). We then use Equation (6), assuming a correction factor A = 0.5 (e.g., Pattle et al. 2017; Kwon et al. 2018; Soam et al. 2018), to derive a plane-of-sky magnetic field amplitude B_pos ∼ 230 μG. This larger value for B_pos suggests that a more appropriate correction factor for B1 would be A ∼ 0.25. However, this derived field strength of 230 μG could even be a lower limit (in the context of the classical DCF method) since the polarization vectors around B1-c are also included in the Gaussian fit, and so the appropriate correction factor to use would in fact be A ≲ 0.25.

With the magnetic field amplitude B_pos = 120 ± 60 μG we have obtained from the angular dispersion analysis, it becomes possible to estimate the criticality criterion λ_c of Perseus B1 with Equation (13). Using the hydrogen column density N(H₂) = (4.7 ± 0.9) × 10²² cm⁻² derived previously, we find λ_c = 3.0 ± 1.5. Since λ_c > 1, Perseus B1 is a magnetically supercritical molecular cloud, i.e., magnetic pressure alone cannot support the cloud against gravity.

Perseus B1 is among a few molecular clouds with a detection of OH Zeeman splitting, and thus a measurement of its magnetic field's line-of-sight component. With observations of the OH lines at 1665 and 1667 MHz using the Arecibo telescope and a beamwidth of 29, Goodman et al. (1989) found a line-of-sight amplitude of 27 ± 4 μG for the magnetic field toward IRAS 03301+3057 (B1-a). While this value might have been overestimated relative to the line-of-sight amplitude of the magnetic field at large scales (Crutcher et al. 1993; Matthews & Wilson 2002), it nonetheless supports the idea that the orientation of the magnetic field in B1 might be mostly parallel to the plane of the sky (i.e., an inclination θ < 15° relative to the plane of the sky).

The magnetic field in Perseus B1, as shown in the right panel of Figure 1, is seen to run roughly north–south (or ∼165° east of north) across the whole region, including SMM3. The orientation of the vectors seen in Figure 1 (right) toward the bulk of the cloud (between B1-b N/S and SMM3) can be explained if B1 is part of a dense, slightly flattened cylindrical filament threaded perpendicularly by a large-scale magnetic field and viewed at an inclined angle to the line of sight (Tomisaka 2015). While it may not be clear from Figure 1 alone, Perseus B1 is indeed part of a large filamentary structure extending toward the southwestern part of the map (Chen et al. 2016). Furthermore, magnetic field lines perpendicular to large-scale filaments have been hypothesized to funnel low-density material into the striations (or subfilaments) observed with Herschel in and around molecular clouds (André et al. 2014). Alternatively, if the cloud is collapsing gravitationally, then the apparent curving of the field lines west of SMM3 could be the sign of an emergent hourglass morphology (e.g., Girart et al. 2006).

The largest discrepancy in the morphology of the large-scale magnetic field is seen toward the protostellar core B1-c, which is the source of a powerful molecular outflow viewed almost edge-on (Matthews et al. 2006). Indeed, the field turns more toward an east–west direction (or ∼120° east of north) in the vicinity of B1-c, where it seems instead better aligned with the orientation of the protostellar outflow traced by the ¹²CO J = 3−2 integrated intensity contour. In fact, the plane-of-sky component of the magnetic field toward B1-c is nearly parallel to the orientation of the outflow at 125°. In contrast, the local magnetic field direction is relatively well aligned with the mean field orientation in Perseus B1 (∼165°) at the locations of the candidate first hydrostatic cores and potentially less evolved B1-bN (∼155°) and B1-bS (∼165°) objects (Pezzuto et al. 2012; Gerin et al. 2017), as well as at the previously identified young stellar objects associated with the submillimeter sources B1-a (∼159°) and SMM3 (∼158°), and to a lesser extent B1-d (∼10°) and HH 789 (∼180°) (Bally et al. 2008). This directional variation suggests that the magnetic field morphology is well ordered at large scales but is potentially locally modified by the motion of the gas at smaller scales.

Perhaps the magnetic field orientation at B1-c originally followed the large-scale field of the molecular cloud but was misaligned with the angular momentum of the initial prestellar core. As the core evolved, the magnetic field lines may have been "dragged" into a modified hourglass configuration (e.g., Kataoka et al. 2012). However, although hourglass structures have been seen toward some protostellar cores (e.g., Girart et al. 2006; Hull et al. 2017a), an alignment between magnetic field and outflow orientations does not appear to be a common occurrence (Hull et al. 2014).

Alternatively, the orientation of the magnetic field at B1-c could be explained by more complex field models that have been shown to produce comparable polarization patterns (Franzmann & Fiege 2017). Indeed, recent ALMA observations of the protostellar core Ser-emb 8 in Serpens Main suggest that the magnetic field of that object, which is similarly misaligned with the large-scale field of the rest of the molecular cloud in which it is embedded, may not possess an hourglass morphology at all (Hull et al. 2017b). However, the protostellar core Serpens SMM1 (also in Serpens Main) nevertheless shows evidence of having an hourglass field morphology while still being misaligned with the magnetic field at larger scales (Hull et al. 2017a). It would therefore be premature to assume that an observed misalignment in magnetic field orientations between core and cloud scales necessarily implies the absence of an hourglass field morphology.

Another peculiar property of B1-c is the orientation of the few polarization vectors found east of the protostellar core and along its outflow, as traced by the ¹²CO J = 3−2 contour in Figure 1. The inferred magnetic field orientation from the vectors found directly in the outflow's path (∼160°) is in better agreement with the large-scale field in B1 (∼165°) than with the field orientation toward B1-c itself (∼120°). Magnetic field orientations that are nearly perpendicular to outflows at large scales are not expected from ideal hourglass field morphologies.

An alternative explanation would be that elongated dust grains found in the vicinity of the outflow are aligned mechanically by the flow of gas instead of radiatively. In this case, the polarization vectors would be parallel (and the inferred magnetic field orientation perpendicular) to the outflow orientation, regardless of the field morphology (Gold 1952; Lazarian 1997, 2007), as is seen. This last scenario, however, has been shown to be unlikely even in the case of explosive outflows such as in Orion BN/KL (Tang et al. 2010).

Indeed, the original mechanical alignment proposed by Gold (1952) requires supersonic flows to be efficient, and it is particularly inefficient for suprathermally rotating grains (see Lazarian 1997; Das & Weingartner 2016). Thus, although its polarization pattern seems to be consistent with the observed polarization map, it is rather difficult to explain the high polarization degree (∼15%) shown in Figure 1. On the other hand, the mechanical torque (MAT) alignment mechanism proposed by Lazarian & Hoang (2007a) and numerically demonstrated by Hoang et al. (2018) predicts that the gas flow can efficiently align grains with the magnetic field. Specifically, the MAT mechanism predicts that the long axis of the grains will be perpendicular to either the magnetic field or the gas flow. Therefore, the polarization vectors found along the outflow's lobes may reveal that the magnetic field in the flow is not much different from the large-scale magnetic field in the rest of the molecular cloud.

Finally, there is the possibility that we are mainly measuring the polarization from dust grains found in the cavity walls of the B1-c outflow. Indeed, it has been suggested that strong irradiation of outflow cavity walls can enhance the polarized emission of the associated dust grains through radiative torques (e.g., Maury et al. 2018). This scenario is supported by ALMA observations of B1-c (or Per-emb-29; Cox et al. 2018) that provide evidence for significantly improved grain alignment (with P > 5%) along outflow cavities near the protostar. Although previous ALMA studies have shown that the magnetic field along comparable outflow cavities tends to be parallel to the outflow orientation (Hull et al. 2017a; Cox et al. 2018; Maury et al. 2018) instead of perpendicular as observed eastward from B1-c in Figure 1, their spatial resolutions were much smaller (140, 60, and 100 au, respectively) than our resolution of ∼3500 au. It could be that the dust grains with potentially enhanced polarized emission farther along the outflow cavity are instead tracing the large-scale field in the cloud, which would fit with the twisted field picture from Kataoka et al. (2012), where the polarization signature becomes less affected by the outflow the farther away you look from the central source.

In Section 4.2, we derived the turbulent and magnetic properties of Perseus B1 from the angular dispersion analysis described by Houde et al. (2009) (see Figure 7). Specifically, we obtain a ratio of turbulent to total magnetic energy $\left\langle {B}_{t}^{2}\right\rangle /\left\langle {B}^{2}\right\rangle =0.5\pm 0.3$, which indicates that a large part of the magnetic energy in the cloud is found in the form of magnetized turbulence. This is larger than the ratio $\left\langle {B}_{t}^{2}\right\rangle /\left\langle {B}^{2}\right\rangle \sim 0.4$ found by Levrier et al. (2018) for the galactic magnetic field using Planck data. As a comparison, a previous study utilizing the angular dispersion analysis presented in Section 4.1 found ratios of turbulent to total magnetic energy $\left\langle {B}_{t}^{2}\right\rangle /\left\langle {B}^{2}\right\rangle$ of 0.6, 0.7, and 0.7 for the high mass star-forming regions W3(OH), W3 Main, and DR21(OH), respectively (Houde et al. 2016).

Since the ionized and neutral components of the gas in molecular clouds are typically well coupled, this magnetized turbulence is expected to be indistinguishable from the turbulence in the neutral gas as long as ambipolar diffusion remains negligible (e.g., Krumholz 2014). Furthermore, the relatively large turbulent component of the magnetic field in B1 could be explained by the presence of at least five young stellar objects with confirmed molecular outflows (B1-a, B1-bS, B1-c, B1-d, and HH 789) in the main body of the cloud (Hatchell & Dunham 2009). Indeed, such outflows are among the most probable drivers of turbulence in molecular clouds (Bally et al. 2008). However, the signature of this protostellar feedback on the velocity dispersion of NH₃ does not appear to be as pronounced in B1 (GAS Consortium 2019, in preparation) as it is in the more compact B59 in the Pipe Nebula (see Figure 9 in Redaelli et al. 2017), but a more detailed coherence analysis will be required to adequately investigate this effect.

The turbulent cells in B1 have a correlation length δ of 50 ± 25, which for a distance of 295 pc represents a physical length of 1475 au. From Equation (8), we estimate that there are typically ∼30 turbulent cells probed by the telescope's beam along the depth of the cloud (0.1 pc). The number of turbulent cells along the line of sight could potentially be greater in higher-density regions, such as toward prestellar cores. This larger number would explain the observed depolarization effect seen in Figure 2 (top) as the Stokes I intensity increases, which can be roughly understood as an increase in the dust column density. Indeed, an increased number of turbulent cells is expected to randomize dust orientations along the line of sight and thus decrease the measured fraction of polarization P. Additionally, and perhaps counterintuitively, numerical simulations by Cho & Yoo (2016) have also shown that the averaging of a high number of turbulent cells along the line of sight could preserve the appearance of a well-ordered field morphology at large scales, which is an effect initially proposed by Jones et al. (1992).

In Section 4.2, we also find a plane-of-sky amplitude B_pos = 120 ± 60 μG for the magnetic field and a criticality criterion λ_c = 3.0 ± 1.5. Although this magnetic field amplitude is relatively weak when compared to the fields found in high-mass star-forming regions such as Orion A (where B_pos ≳ 1.0 mG; e.g., Houde et al. 2009; Pattle et al. 2017) or in hub-filament structures such as IC 5146 (with B_pos ∼ 0.5 mG; e.g., Wang et al. 2019), it is either comparable to or larger than the field strengths (B_pos ≲ 100 μG) typically found in low-mass prestellar cores (e.g., Crutcher et al. 2004; Kirk et al. 2006; Liu et al. 2019). Above all, these results indicate that Perseus B1 is a supercritical molecular cloud (i.e., magnetic pressure alone cannot support the cloud against gravity). The criticality criterion λ_c defined by Equation (13), however, may be overestimated owing to geometric effects. Indeed, Crutcher et al. (2004) find that, on average, the effective criticality criterion is $\overline{{\lambda }_{c}}\approx {\lambda }_{c}/3$. In the case of B1, this adjustment would lead to $\overline{{\lambda }_{c}}\approx 1.0$, which is the theoretical limit at which the cloud would be subcritical.

Since the inclination of the magnetic field in B1 can be calculated using published Zeeman line splitting measurements (see Section 4.2), we can better estimate the effect of geometry on the criticality criterion $\overline{{\lambda }_{c}}$. Assuming that the line-of-sight component obtained by Goodman et al. (1989) (27 ± 4 μG) is not an overestimation at large scales, we find an inclination θ = 12° relative to the plane of the sky and an amplitude B_tot ≈ 125 μG for the total magnetic field when combined with the plane-of-sky amplitude B_pos = 120 ± 60 μG found in Section 4.2. If the cloud can also be approximated as a mostly prolate filament with a cylindrical symmetry, which is a reasonable assumption for a relatively weak magnetic field in a dense filament, then we get $\overline{{\lambda }_{c}}\approx {\lambda }_{c}$. We therefore find it likely that Perseus B1 is indeed supercritical by a factor of ∼3, although we cannot rule out if a combination of magnetic pressure and turbulence would be sufficient to significantly slow down the fall of additional material onto the central clump.

Fundamentally, the fraction P of polarization can be understood as the alignment efficiency of a mixture of dust grains in the interstellar medium. Even though this fraction P can be affected by purely environmental factors such as the number of integrated turbulent cells along the line of sight and complex magnetic field geometries, or even instrumental factors such as molecular contamination (see the appendix), it is intrinsically linked to the models of grain alignment.

Specifically, the contribution to the continuum emission of different grain sizes and compositions in the dust mixture could explain the apparent dependence of P on the wavelength at far-infrared and submillimeter wavelengths (Vaillancourt & Matthews 2012). For example, grain growth in cold, high-density regions may lead to very large dust grains, with sizes a ≳ 1.0 μm (e.g., Pagani et al. 2010), which align less efficiently through radiative torques than the typical grains (a ∼ 0.1 μm) found in molecular clouds (Hoang & Lazarian 2009). This scenario could potentially explain the apparent drop in polarization fraction P seen in Figure 3 above a visual extinction A_V > 200 mag, as well as toward B1-c, since there is significant evidence for grain growth across Perseus B1 (Sadavoy et al. 2013; Chen et al. 2016).

Furthermore, since the RAT theory of grain alignment depends on the stellar radiation field incident on the grains, the alignment efficiency is expected to be smaller toward regions with high dust opacities (e.g., dense prestellar cores; Andersson et al. 2015). This effect would potentially explain the apparent minimum P of ∼1% seen both in Figure 2 (top) and by Matthews & Wilson (2002) for the highest-opacity regions of the cloud, which in the case of Perseus B1 are associated with embedded young stellar objects such as the first hydrostatic core candidates B1-b N/S (see Figure 1). This alignment efficiency, however, is expected to improve again if there is a significant source of radiation, such as a protostar, within the core itself. Such a scenario would explain the shallower-than-expected power-law index β ∼ −0.5 given in Section 3.2 for the relation between the polarization fraction P and the visual extinction A_V in Perseus B1.

Nevertheless, B1-c, which is known to be a bright and warm protostellar core (Sadavoy et al. 2013), also has among the lowest polarization fractions measured by POL-2 for B1. This behavior suggests that we may not be resolving the improved grain alignment efficiency seen by ALMA near the protostar (Cox et al. 2018). Indeed, Jones et al. (2016) previously observed such an effect when comparing single-dish and interferometric polarization data of the protostellar core G034.43+00.24 MM1. Alternatively, it could be that factors other than alignment efficiency need to be taken into account to explain the polarization toward this object.

As an example, previous studies have found an inverse correlation between the polarization fraction P and the local dispersion of magnetic field orientations at several scales in molecular clouds (Planck Collaboration et al. 2015a, 2015b; Fissel et al. 2016; Koch et al. 2018). Such a measure toward B1-c would support the hypothesis of a complex but unresolved polarization structure, and higher-resolution observations using interferometric facilities would provide further evidence to confirm or disprove this scenario. However, while there exist ALMA data of the linear polarization toward B1-c, only the most highly polarized emission is likely to have been recovered owing to the short integration time (8 minutes) of these observations (Cox et al. 2018). A deeper ALMA polarization map of B1-c might therefore reveal a more complex magnetic field structure comparable to those observed in similar protostellar cores (e.g., Hull et al. 2017a, 2017b; Maury et al. 2018).