Two-component Magnetic Field along the Line of Sight to the Perseus Molecular Cloud:
Contribution of the Foreground Taurus Molecular Cloud

We use optical polarimetry of 88 sources in the direction of the Perseus molecular cloud observed by Goodman et al. (1990, Figure 1). The passband of the observations were centered at 762.5 nm with a bandwidth of 245 nm.

We also use near-infrared (NIR) polarimetry data taken toward NGC 1333 in the Perseus molecular cloud, as shown in the inset of Figure 1. We use K-band polarimetry by Tamura et al. (1988, 14 sources) and R- and J-band polarimetry by Alves et al. (2011, 33 sources). The near-infrared $\theta_{\mathrm{star}}$ are mainly distributed from $\theta_{\mathrm{star}}\sim-90^{\circ}$ to $\theta_{\mathrm{star}}\sim-40^{\circ}$.

We estimate the B-field orientation measured in sub-mm dust emission by using Planck data at 353 GHz (Planck Collaboration et al., 2020). We use HFI_SkyMap_353-psb_2048_R3.01_full.fits taken from the Planck Legacy Archive, https://pla.esac.esa.int/. In our analysis, we set the spatial resolution of the Planck polarimetric data as a $10^{\prime}$ FWHM Gaussian to achieve good S/Ns.

We estimate the stellar distances by using Gaia astrometry data (DR2; Gaia Collaboration et al., 2016, 2018), and use SIMBAD (Wenger et al., 2000) positions to cross-match the stars in the Gaia catalog. We set a search radius of $5\arcsec$ and take the star at the closest position for R-band and J-band data by Alves et al. (2011). The identified stars show $M_{G}=14$ – 20 mag, and have good correspondence with the J-band magnitude tabulated by Alves et al. (2011).

For relatively old datasets by Goodman et al. (1990) and Tamura et al. (1988), we set a relatively large search radius of $30\arcsec$ and take the brightest star in the searched region, which gives a good cross-match of the catalogued stars as follows. We find that the stars observed by Goodman et al. (1990) show G-band magnitude in the Gaia catalog between $M_{G}=7$ – 15 mag. We exclude three stars that show $M_{G}=17$ – 21 mag from the following analysis for their possible misidentifications. The stars observed by Tamura et al. (1988) show $M_{G}=10$ – 18 mag. We exclude one star that shows $M_{G}=20$ mag from the following analysis because of its non-reliable distance estimation (negative parallax).

The Gaia parallax in DR2 is known to have a systematic bias of -0.03 mas (see Bailer-Jones, 2015; Bailer-Jones et al., 2018; Lindegren et al., 2018; Arenou et al., 2018; López-Corredoira & Sylos Labini, 2019; Melnik & Dambis, 2020). This bias is negligible in our distance evaluation, as the stellar distances in our analysis are less than 1 kpc and thus their parallaxes $\varpi>1$ mas. Thus, we did not correct the bias, and estimate the distances of each star by $d=1000/\varpi$ (pc).

The Renormalised Unit Weight Error (RUWE) is a parameter that is expected to be around 1.0 for sources where the single-star model provides a good fit to the astrometric observations. A value significantly greater than 1.0 (say, $>1.4$) could indicate that the source is non-single or otherwise problematic for the astrometric solution (see the Gaia data release documentation¹¹1https://gea.esac.esa.int/archive/documentation/GDR2/). We estimate the RUWE for each source following the formulation described in the document “Re-normalising the astrometric chi-square in Gaia DR2”²²2https://www.cosmos.esa.int/web/gaia/public-dpac-documents and use the data only if whose $\mathrm{RUWE}\leq 1.4$.

As a result, we estimate the distances of 111 sources, including 70 sources from Goodman et al. (1990), 20 sources (R-band) and 15 sources (J-band) from Alves et al. (2011), and 6 sources from Tamura et al. (1988). We summarize the identified data in Table A in Appendix A.

Figure 1 compares the spatial distribution of the polarimetry data in our analysis with the inferred B-field morphology from Planck. The polarization position angles, $\theta_{\mathrm{star}}$, show a bi-model distribution, with concentrations at $\theta_{\mathrm{star}}\sim+70^{\circ}$ and $\theta_{\mathrm{star}}\sim-40^{\circ}$ (measured from North to East). Goodman et al. (1990) found that the two populations in $\theta_{\mathrm{star}}$ also have distinct polarization fractions ($P_{\mathrm{star}}$). We display the relationship between the observed $\theta_{\mathrm{star}}$ and $P_{\mathrm{star}}$ in Figure 2.

The population of $\theta_{\mathrm{star}}\sim-40^{\circ}$ show relatively larger $P_{\mathrm{star}}\gtrsim 1$%, while the other population that has $\theta_{\mathrm{star}}\sim+70^{\circ}$ show relatively smaller $P_{\mathrm{star}}\lesssim 1$%.

In Figure 3, we display the $\theta_{\mathrm{star}}$ and $P_{\mathrm{star}}$ dependences as a function of the estimated stellar distances.

As seen in the figure, there is a clear jump in both distributions at a distance of about 300 pc, which is the distance to the Perseus molecular cloud. In addition to that, another jump is noticeable at a distance of about 150 pc. Hereafter, we call the polarimetry data whose stellar distances $d>300$ pc as Group 1, those with $150<d<300$ pc as Group 2, and those with $d<150$ pc as Group 3, respectively. We summarize $\theta_{\mathrm{star}}$ and $P_{\mathrm{star}}$ values of each group in Table 1.

Figure 3 shows that $\theta_{\mathrm{star}}$ and $P_{\mathrm{star}}$ are both consistent within their own distance group. That is, the stars in Group 1 and the stars in Group 2 are tracing the polarization from ISM clouds located at distances of 300 pc and 150 pc, respectively. In particular, the consistency of $P_{\mathrm{star}}$ indicates that the ISM between and behind the two clouds does not significantly contribute to stellar polarization. Group 3 can be thought as foreground stars with little interstellar extinction. We note that these stars have very high uncertainties in their polarization measurements due to having very low polarization fractions.

To quantitatively estimate the distance of the two ISM clouds that cause polarization, we perform a breakpoint analysis on $\theta_{\mathrm{star}}$ and $P_{\mathrm{star}}$ distributions as a function of $d$ shown in Figure 3. We assume that $\theta_{\mathrm{star}}$ and $P_{\mathrm{star}}$ are constant as a function of $d$, which corresponds to the assumption that the observed polarization is caused by 2D sheet(s) of ISM at specific distance(s). In addition, we assume the $\theta_{\mathrm{star}}$ and $P_{\mathrm{star}}$ distributions have a certain number of step-wise changes (i.e., breakpoints), which correspond to the positions of the 2D sheets. We perform least-squares fits to the data and make most likelihood estimations (MLE) of the positions of breakpoints. We then repeat the fit with different number of breakpoints, and compare the goodness-of-fit values based on the Bayesian information criterion, to get the most likely numbers of breakpoints and their positions.

We perform the breakpoint analysis for each distance dependence of $\theta_{\mathrm{star}}$ and $P_{\mathrm{star}}$ shown in Figure 3 by using the R library ‘strucchange’ (Zeileis et al., 2002, 2003). For $\theta_{\mathrm{star}}$ distribution analysis, we combine optical and NIR polarimetry data as shown in Figure 3 and use the shortest waveband data if multiple waveband data are available for a single star. On the other hand, we use only optical polarimetry data to analyze $P_{\mathrm{star}}$ distribution because different wavelengths give different polarization fractions. We used $P_{\mathrm{star}}$’s logarithm values in our analysis to detect breakpoints of widely different $P_{\mathrm{star}}$ values (see Figure 3) with comparable sensitivity to each other.

The estimated distances of the breakpoints are shown in Figure 3 and Table 2. The breakpoint analysis shows that there are two breakpoints in both $\theta_{\mathrm{star}}$ and $P_{\mathrm{star}}$ distributions. The estimated distances are $150^{+4}_{-34}$ pc and $303^{+12}_{-7}$ pc for $\theta_{\mathrm{star}}$, and $150^{+4}_{-136}$ pc and $295^{+22}_{-67}$ pc for $P_{\mathrm{star}}$, respectively. Here we show the MLE positions and their 95% confidence intervals. Although the breakpoints in $P_{\mathrm{star}}$ have larger errors comparing to that in $\theta_{\mathrm{star}}$ due to smaller jumps in $P_{\mathrm{star}}$ values, the estimated distances are fully compatible between $\theta_{\mathrm{star}}$ and $P_{\mathrm{star}}$.

The distance of the breakpoint at 300 pc is consistent with the distance to the Perseus molecular cloud (Ortiz-León et al., 2018; Zucker et al., 2018; Pezzuto et al., 2021). Thus, we conclude that the polarization of Group 1 traces the B-field of the Perseus molecular cloud. The distance of another breakpoint at 150 pc is thought to indicate the distance to the foreground ISM cloud that causes the polarization of Group 2. This 150 pc foreground ISM also contribute to depolarize Group 1. We will discuss this depolarization effect in Section 3.4. Group 3 are the foreground stars that show no clear indication of interstellar extinction.

In the previous section (Section 3.1), we estimated the distances of the breakpoints found in polarimetric data. The accuracy of estimation is limited to a few tens of parsecs because of small number of data points. In this section, we perform a breakpoint analysis using all the available stellar photometric data in the Perseus molecular cloud direction to make a more accurate estimation of the distances to the breakpoints, including the foreground ISM. We then compare our result with other existing estimates of cloud distances in this direction to deomonstrate the reliability of our method for measuring dust clouds’ distance.

We use G-band extinction ($A_{G}$) values catalogued in Gaia DR2 with $d\leq 1$ kpc and $\mathrm{RUWE}\leq 1.4$, and fit these values as a function of distances obtained from Gaia parallax. The spatial distribution of the stellar data used in this analysis is shown in Figure 4.

We note the paucity of stars in the direction of the dense molecular cloud. This is due to the larger extinction in the visible G-band compared with the NIR bands.

Zucker et al. (2018) combined Gaia distance and NIR photometry to obtain the distance of each velocity component of the ${}^{12}\mathrm{CO}~{}(1-0)$ line emission for each cloud core in the Perseus molecular cloud. We perform breakpoint analyses for similar spatial directions and compare the result with their estimation. Since the exact spatial region analyzed in Zucker et al. (2018) is not indicated in the literature, we estimate the breakpoints at $1^{\circ}$ diameter regions centered at each molecular cloud core. The regions are shown as red circles in Figure 4.

We show the comparison of the results of our breakpoint analysis with the estimation by Zucker et al. (2018) in Figure 5 and Table 3. Our results are in reasonable agreement with those of Zucker et al. (2018) for each component of the Perseus cloud at a distance of about 300 pc. The estimated distances show a gradual increase from west to east as pointed out by Zucker et al. (2018, also see , ). Therefore, we can conclude that our G-band breakpoint analysis can measure the distance to the cloud with sufficient accuracy.

On the other hand, the position of the foreground component is less consistent. Therefore, we further check our estimated foreground distance’s consistency, comparing that with 3D dust maps based on Gaia distance data by Lallement et al. (2019) and Leike & Enßlin (2019). We show a comparison between these results and ours for the Perseus molecular cloud direction in Figure 6.

Figure 6a shows the results of our breakpoint analysis. The spatial resolution of this analysis is set as $55\arcmin$. The estimated cloud distances are concentrated at 150 pc and 300 pc (Figure 6a).

Figures 6b and c show the results by Lallement et al. (2019) and Leike & Enßlin (2019), respectively. Both two dust maps successfully detect the foreground cloud in addition to the Perseus cloud. Lallement et al. (2019, Figure 6b) combine 2MASS photometric data with Gaia DR2. Their map has limited LOS resolution of 50 pc and thus the estimated extinction for the Perseus cloud is smoothed out, resulting in relatively lower extinction per unit length value ($\sim 0.01$ mag/pc) compared to Leike & Enßlin (2019, Figure 6c). Lallement et al. (2019) indicated plans to improve the LOS resolution as the Gaia data is updated.

Leike & Enßlin (2019, Figure 6c, also see , ) used the Gaia DR2 $A_{G}$ values as photometric data to create a 3D dust map of the solar vicinity ($d\lesssim 300$ pc). They achieved higher resolution in LOS comparing to the map by Lallement et al. (2019) that used NIR photometric data (Figure 6b). NIR photometry is effective for tracing dense molecular clouds’ interior but has limited sensitivity for tracing faint clouds with a high spatial resolution. On the other hand, the disadvantage of tracing the shape of the dust cloud using only photometry of visible wavebands is also apparent; in their analysis shown in Figure 6c, the 300 pc Perseus molecular cloud cannot be correctly detected due to saturation of the $A_{G}$ value and is split into two distances before and after the cloud.

Our results shown in Figure 6a are based on the simple 2D sheet assumption and have limited spatial resolution of about $1^{\circ}$, but the use of the $A_{G}$ value achieves a good sensitivity to faint clouds as well as the correct determination of the distances of dense molecular clouds.

According to the discussions above, we judge that our method is a simple and effective one for measuring dust clouds. Using G-band photometry, we can obtain the distance to both faint and dense clouds stably and accurately. Our results described in Section 3.1 (Figure 3) are thus consistent that the stellar polarization in the Perseus molecular cloud’s direction originates in two isolated clouds at 150 pc and 300 pc, which are both detected by Lallement et al. (2019) and Leike & Enßlin (2019).

Figure 7 shows the spatial distribution of the dust cloud at 150 pc and 300 pc by Leike & Enßlin (2019) as contours.

To avoid the effect of LOS splitting of the dense molecular cloud, we show the spatial distribution of the integrated $A_{G}$ magnitude in the 100–200 pc range for the 150 pc component and in the 250–350 pc range for the 300 pc component, respectively. In addition to the Perseus cloud (the top panel of Figure 7), we also show the Taurus-Perseus molecular cloud complex in the bottom panel of Figure 7. As is evident in Figure 7, the 150 pc foreground cloud lying in front of the Perseus molecular cloud corresponds to the outer edge of the Taurus molecular cloud at a distance of $\sim 140$ pc (Yan et al., 2019; Zucker et al., 2019; Roccatagliata et al., 2020), as was proposed by Ungerechts & Thaddeus (1987) and Cernis (1990). The estimated extinction of the foreground cloud based on Leike & Enßlin (2019) is $A_{G}^{\mathrm{150pc}}=0.1$ – 0.9 mag.

We overlay the B-field orientation measured by Planck ($\theta_{\mathrm{Planck}}$) in Figure 7 as black line segments. $\theta_{\mathrm{Planck}}$ toward the Perseus cloud is generally aligned to the northwest-southeast direction. The circular mean and the circular standard deviation are $\theta_{\mathrm{Planck}}=-56^{\circ}\pm 25^{\circ}$ if we estimate them in the region where the 300 pc dust cloud component $A_{G}^{\mathrm{300pc}}>0.6$ mag. This $\theta_{\mathrm{Planck}}$ is consistent with the optical $\theta_{\mathrm{star}}$ of Group 1 ($-37.6^{\circ}\pm 35.2^{\circ}$). Outside the Perseus molecular cloud but on the foreground component, we note a significant change in the B-field orientation. If we estimate $\theta_{\mathrm{Planck}}$ on the foreground cloud whose $A_{G}^{\mathrm{150pc}}>0.2$ mag, $A_{G}^{\mathrm{300pc}}<0.6$ mag, and R.A. $<4^{\mathrm{h}}00^{\mathrm{m}}$, the position angle is $\theta_{\mathrm{Planck}}=-89^{\circ}\pm 19^{\circ}$, which is considerably different from $\theta_{\mathrm{Planck}}$ on the Perseus molecular cloud and is consistent with the $\theta_{\mathrm{star}}$ of Group 2 ($+66.8^{\circ}\pm 19.1^{\circ}$) within the range of errors.

As a result, we conclude that there are two B-field components observed in the Perseus molecular cloud’s direction: one is the Perseus molecular cloud’s B-field, and the other is that of a tenuous cloud at the outer edge of the Taurus molecular cloud with $A_{G}<1$ mag.

The two-component B-field of Perseus and Taurus clouds, described in the previous section, is traced by Group 1 ($d>300$ pc) and Group 2 ($150<d<300$ pc) stellar polarization data. Group 2 traces only the Taurus cloud, while Group 1 sees through both Perseus and Taurus clouds. Here we estimate the Perseus molecular cloud’s B-field by removing the Taurus contribution from Group 1.

The observed polarization fraction is $\ll 10$ % (Figure 3 and Table 1). In such a low polarization condition, the relative Stokes parameters $q$ (= Q/I) and $u$ (= U/I) in the polarized flux can be approximated as additive (e.g., Panopoulou et al., 2019b and the references therein). In other words, we can separate the contribution to the polarization from individual clouds as follows.

where $q_{\mathrm{\,Group1}}$ and $q_{\mathrm{\,Group2}}$ are the relative Stokes $q$ parameter observed for Group 1 and Group 2 stars, and $q_{\mathrm{\,Taurus}}$ and $q_{\mathrm{\,Perseus}}$ are the relative Stokes $q$ parameter originated in Taurus and Perseus clouds, respectively. These formulas also hold for the relative Stokes $u$ parameter. We estimate $q_{\mathrm{\,star}}$ and $u_{\mathrm{\,star}}$ values of Group 1 and Group 2 from the observed $\theta_{\mathrm{star}}$ and $P_{\mathrm{star}}$ as follows.

The estimated $q_{\mathrm{\,star}}$ and $u_{\mathrm{\,star}}$ are shown in Figure 8a.

The $q$ and $u$ values of Group 1 show a significant scatter, reflecting the local variation in $\theta_{\mathrm{star}}$ and $P_{\mathrm{star}}$ of Group 1. On the other hand, the data of Group 2, which represents the $q$ and $u$ values of the Taurus cloud, show a small variation. This is because $\theta_{\mathrm{star}}$ and $P_{\mathrm{star}}$ of Group 2 have a small spatial variation, and the value of $P_{\mathrm{star}}$ is also small (see Table 1).

Since $q$ and $u$ are additive, the observed values of $q$ and $u$ in Group 1 can be expressed as a vector sum of the contributions from Perseus and Taurus clouds on the $q$-$u$ plane. This relationship is shown in Figure 8b. Here we show the relationship between the averaged $q$ and $u$ values of Taurus and Perseus.

The estimated $q$-$u$ vectors of Perseus and Taurus are nearly opposite to each other. That is, the estimated orientations of the POS B-field of these two clouds are nearly perpendicular to each other ($\theta_{\mathrm{Perseus}}\perp\theta_{\mathrm{Taurus}}$). Furthermore, the Perseus contribution to the Group 1 $q$-$u$ vector is dominant compared to that of Taurus. This may reflect the fact that the column density of the Perseus main cloud is dominant compared to that of the foreground Taurus cloud’s outskirt (see Figure 6). As a result, the Taurus contribution to the Group 1 polarization does not significantly change the Perseus’ q-u vector direction (Figure 8b). Thus, we conclude the followings.

On the other hand, we estimate that $P_{\mathrm{Perseus}}$ is slightly larger than $P_{\mathrm{Group1}}$ due to the depolarization by the foreground Taurus cloud. We estimate $\theta_{\mathrm{Perseus}}$ and $P_{\mathrm{Taurus}}$ by subtracting the averaged Group 2 $q$-$u$ vector from the observed Group 1 $q$-$u$ vectors, as shown in Figure 8b. The results are shown in Table 4.

Figure 9 shows $P_{\mathrm{Perseus}}$ and $P_{\mathrm{Taurus}}$ as a function of $A_{G}$ to investigate individual clouds’ polarization efficiency ($P_{\mathrm{star}}/A_{G}$). We assume that the Taurus cloud’s $A_{G}$ values are equal to the Group 2 data. For the Perseus cloud, we estimate $A_{G}$ by subtracting that of 150 pc cloud component estimated by Leike & Enßlin (2019, Figure 7) at each position of the stars from the observed Group 1 $A_{G}$ values. Note that the maximum observed $A_{G}$ value is 3 mag (also see Figure 13 and the discussion in Sections 4.2 and 4.3), indicating that the stellar extinction data are biased to regions in the cloud where the extinction is smaller ($A_{G}<3$ mag).

In Figure 9, we also show the observed maximum polarization efficiency taken from the literature ($P/E(B-V)=13\%$; Panopoulou et al., 2019a; Planck Collaboration et al., 2020). The polarization efficiency of Taurus and Perseus corresponds to less than $\sim 50\%$ of the maximum efficiency. We do not find a significant difference between the polarization efficiencies of Taurus and Perseus clouds.

In Section 3.3, we identified that the 150 pc foreground cloud lying in front of the Perseus molecular cloud is the outer edge of the Taurus molecular cloud. Both Taurus and Perseus molecular clouds are thought to be on a common large HI shell’s outer edge (Sancisi, 1974; Sun et al., 2006; Shimajiri et al., 2019). In Figure 6, we find a low-extinction region between these two clouds. In the following, we estimate the spatial extent of this low-extinction region.

Figure 10 shows the 3D distribution of the dust cloud in the region containing Taurus and Perseus.

We estimate the 3D shape of the dust shell by using the cloud positions determined by our breakpoint analysis. We assume the shell’s profile as a simple ellipsoid and perform a least-square fit of the shell to the cloud positions weighted by their extinction (mag/cloud). The estimated size and position of the shell are shown in Figure 11.

As shown in the figure, we can identify a low-density dust cavity surrounded by the dust shell. The estimated diameters in the heliocentric galactic cartesian coordinates (X, Y, and Z directions; see Figure 10 for the definition of X, Y, and Z) are $D_{X}\simeq 156$ pc, $D_{Y}\simeq 110$ pc, and $D_{Z}\simeq 105$ pc, centering at $X\simeq-220$ pc, $Y\simeq 38$ pc, and $Z\simeq-80$ pc. The ellipsoid’s estimated shape is also shown in Figure 10, and the outline of the ellipsoid projected on the POS is shown in Figure 7 (bottom panel). The Taurus molecular cloud is located in front of this cavity, and the Perseus molecular cloud is located behind it. Thus, we conclude that the two-component B-fields observed in the Perseus cloud’s direction at distances of 150 pc and 300 pc show the B-field structure in front of and behind the dust cavity, respectively.

The HI shell and dust cavity are thought to be formed by the Per OB2 association. The velocity field that is consistent with the assumed expanding motion of the shell was found for the Perseus molecular cloud in the HI (Sancisi, 1974) and CO (Ridge et al., 2006b; Sun et al., 2006) line emissions (see, e.g., Figure 10 of Sun et al., 2006). Tahani et al. (2018) found that the LOS B-field directions are different from each other in the north and south of the east-west extending Perseus molecular cloud. This LOS B-field is consistent with a scenario where the environment has impacted and influenced the field lines to form a bow-shaped magnetic field morphology as explored in Heiles (1997) and Tahani et al. (2019, also see , ). $\theta_{\mathrm{star}}$ of the Perseus cloud B-field component is $-30.0^{\circ}\pm 25.2^{\circ}$ (Section 3.4 and Table 4). In the galactic coordinates, $\theta_{\mathrm{star}}^{\mathrm{GAL}}$ (measured from Galactic North to Galactic East) of the Perseus cloud B-field is $-68.5^{\circ}\pm 24.9^{\circ}$. This $\theta_{\mathrm{star}}^{\mathrm{GAL}}$ is nearly parallel to the galactic plane (see Figure 3, top panel), i.e., it is consistent with the global B-field component of the galaxy (see, e.g., Jansson & Farrar, 2012, Figure 9). This $\theta_{\mathrm{star}}^{\mathrm{GAL}}$ is, therefore, in accordance with the formation scenario of the Perseus molecular cloud described above. The impact of feedback on the B-fields is explored on a forthcoming paper (Tahani et al., in prep).

On the other hand, the orientation of the Taurus B-field is close to perpendicular to the galactic plane ($\theta_{\mathrm{star}}^{\mathrm{GAL}}=28.7^{\circ}\pm 19.8^{\circ}$; Figure 3, top panel). It is difficult to form a B-field structure close to perpendicular to the galactic plane (i.e., perpendicular to the global B-field of the galaxy) if the ISM is compressed simply along the LOS. Thus, the observed $\theta_{\mathrm{star}}$ in front of and behind the dust cavity is not compatible with the simple expansion of HI bubble in a uniform B-field.

As shown in the discussion in this section, thanks to the advent of Gaia data, we can now isolate the B-fields associated with each of the multiple clouds superimposed along the LOS and map them to the 3D structure of the ISM (see also Panopoulou et al., 2019b, and the references therein). This isolation of the B-fields is important in studying the B-field properties of individual clouds, e.g., applying the DCF method (Davis, 1951; Chandrasekhar & Fermi, 1953) to estimate the B-field strength.

In Figure 1, we showed the spatial distribution of $\theta_{\mathrm{star}}$ and $\theta_{\mathrm{Planck}}$. We note that the Group 1 $\theta_{\mathrm{star}}$ ($d>300$ pc; red points in Figure 1) has a good one-to-one correspondence with $\theta_{\mathrm{Planck}}$ in the same position, despite their spatial variations. Figure 12 shows the offset angle between $\theta_{\mathrm{star}}$ and $\theta_{\mathrm{Planck}}$ as a function of stellar distance.

For stars closer than 300 pc, the offset angle is $\theta_{\mathrm{star}}-\theta_{\mathrm{Planck}}=-71^{\circ}\pm 45^{\circ}$ and shows large variation, while for stars farther than 300 pc, the offset angle is $\theta_{\mathrm{star}}-\theta_{\mathrm{Planck}}=2^{\circ}\pm 28^{\circ}$. $\theta_{\mathrm{Planck}}$ traces the B-field approximately in proportion to the column density along the LOS. Thus, $\theta_{\mathrm{Planck}}$ mainly traces the Perseus molecular cloud, with an additional contribution from the foreground cloud (see Figure 6). As discussed in Section 3.4, this also applies to $\theta_{\mathrm{star}}$ of Group 1. As a result, $\theta_{\mathrm{Planck}}$ agrees well with the Group 1 $\theta_{\mathrm{star}}$ as they trace the similar ISM on the LOS. On the other hand, $\theta_{\mathrm{star}}$ of Group 2 and Group 3 has no contribution from the Perseus cloud, and the foreground contribution traced by them is much smaller than that of the Perseus cloud in the background. Therefore, $\theta_{\mathrm{star}}$ of Group 2 and Group 3 does not correlate with $\theta_{\mathrm{Planck}}$. Thus, we conclude that $\theta_{\mathrm{star}}$ and $\theta_{\mathrm{Planck}}$ observations of the Perseus molecular cloud’s direction with stellar distances $d>300$ pc are in good agreement with each other, despite the fact that their spatial resolutions are largely different from each other. The spatial resolution of $\theta_{\mathrm{Planck}}$ is $10^{\prime}$, which corresponds to $\sim 1$ pc at 300 pc from the sun. On the other hand, the spatial resolution of $\theta_{\mathrm{star}}$ is the size of stellar photospheres, which is much smaller than the Planck’s beam size.

The good one-to-one correspondence between $\theta_{\mathrm{star}}$ and $\theta_{\mathrm{Planck}}$ was first pointed out by Planck Collaboration et al. (2015b, see also , ). Soler et al. (2016) further analyzed the $\theta_{\mathrm{star}}$–$\theta_{\mathrm{Planck}}$ correlation and investigated possible small-scale structures of the B-field inside the Planck beam. They concluded that the B-field inside the Planck beam is uniform or randomly fluctuated around the mean $\theta$, because $\theta_{\mathrm{Planck}}$ agrees with $\theta_{\mathrm{star}}$. Our observed good correlation between $\theta_{\mathrm{Planck}}$ and $\theta_{\mathrm{star}}$ supports their conclusion.

We show the offset angle between $\theta_{\mathrm{star}}$ and $\theta_{\mathrm{Planck}}$ as a function of the stellar extinction in Figure 13. The data show good agreement (angle differences consistent with zero) with no clear dependence on column density. The data shown here suggest that the small scale B-field is smooth in the column density range up to $A_{G}\sim 3$ mag or hydrogen column density $N_{\mathrm{H}}\sim 10^{22}~{}(\mathrm{cm}^{-2})$.

In addition, Figure 12 suggests that the correlation between $\theta_{\mathrm{Planck}}$ and $\theta_{\mathrm{star}}$ doesn’t change with the stellar distance beyond 300 pc. As shown in Section 3.1 and Figure 3, $\theta_{\mathrm{star}}$ and $P_{\mathrm{star}}$ show no clear variation as a function of the stellar distance. The good correlation between $\theta_{\mathrm{star}}$ and $\theta_{\mathrm{Planck}}$ thus indicate that the ISM between and behind the clouds does not significantly contribute to polarization in either emission ($\theta_{\mathrm{Planck}}$) or extinction ($\theta_{\mathrm{star}}$). In other words, the polarization is produced only in dense clouds along the LOS.

Doi et al. (2020) performed sub-mm polarimetry observations of NGC 1333, a dense star-forming region in the Perseus molecular cloud. They observed the region with POL-2 on JCMT, and revealed the B-field orientation ($\theta_{\mathrm{JCMT}}$) with a high spatial resolution of $\simeq 0.02$ pc. The observed region has ISM column density $N_{\mathrm{H}}\gg 10^{23}~{}\mathrm{(cm^{-2}}$; also see Figure 1). In contrast to the good agreement between $\theta_{\mathrm{star}}$ and $\theta_{\mathrm{Planck}}$ described in the previous section, $\theta_{\mathrm{JCMT}}$ is not well correlated with $\theta_{\mathrm{Planck}}$. Instead, $\theta_{\mathrm{JCMT}}$ shows significantly more complex structure on $<0.5$ pc scales. Doi et al. (2020) found that the small-scale B-field appears to be twisted perpendicular to the gravitationally super-critical massive filaments, most probably due to the dense filaments’ formation.

Hence, the small scale ($<1$ pc) B-field appears to be highly complicated for the regions with $N_{\mathrm{H}}\gg 10^{23}~{}(\mathrm{cm}^{-2})$ but smooth for the regions with $N_{\mathrm{H}}<10^{22}~{}(\mathrm{cm}^{-2})$. It is unknown about the small-scale B-field structure for the regions with $N_{\mathrm{H}}=10^{22}$ – $10^{23}~{}(\mathrm{cm}^{-2})$. This is because optical and NIR stellar polarimetry cannot trace $N_{\mathrm{H}}\gg 10^{22}~{}(\mathrm{cm}^{-2})$ ISM, while ground-based and airborne sub-mm telescopes currently lack sensitivity to trace $N_{\mathrm{H}}\ll 10^{24}~{}(\mathrm{cm}^{-2})$ ISM. But as described below, observations of the LOS B-field strength and Planck polarimetry with the lower spatial resolution hint at the increasing complexity of small-scale B-field structure at $N_{\mathrm{H}}>10^{22}~{}(\mathrm{cm}^{-2})$. Crutcher (2012) showed that the LOS B-field strength measured by Zeeman splitting measurements show a general increase with an increasing $N_{\mathrm{H}}$ when $N_{\mathrm{H}}>10^{22}~{}(\mathrm{cm}^{-2})$. In such high column density region, Planck Collaboration et al. (2015a) observed a sharp drop in $P_{\mathrm{Planck}}$ where $N_{\mathrm{H}}\geq 1.5\times 10^{22}$ (cm^-2). They also reported an increase of the polarization angle dispersion at the same column density range. Planck Collaboration et al. (2016) and Soler (2019) found that the relative orientation of the Planck-observed B-fields to the long axes of dense filaments changes systematically from being parallel to perpendicular at $N_{\mathrm{H}}\approx 5\times 10^{21}$ (cm^-2), which is consistent with the result reported by Doi et al. (2020).

Photometric observations also suggest that ISM clouds make a significant change in this column density range. Onishi et al. (1998) found recently formed protostars only in regions with $N_{\mathrm{H_{2}}}>8\times 10^{21}$ (cm^-2) in the Taurus molecular cloud. Johnstone et al. (2004) and Kirk et al. (2006) found no obvious substructures below an $N_{\mathrm{H}}\sim 10^{22}~{}(\mathrm{cm}^{-2})$ in their sub-millimeter continuum observations of the Ophiuchus cloud and the Perseus cloud, respectively. Many authors have pointed out that there is a fiducial threshold of $N_{\mathrm{H}}\simeq 0.6$ – $2\times 10^{22}$ (cm^-2) for the star formation (e.g., Lada et al., 2010, 2012; Heiderman et al., 2010; Evans et al., 2014; Könyves et al., 2015; Marsh et al., 2016; Zhang et al., 2019), though some counterexample are reported by, e.g., Di Francesco et al. (2020, lower threshold value) and Pokhrel et al. (2020, no threshold value). Interestingly, the critical line mass of ISM filaments ($M_{\mathrm{line,crit}}=2c_{s}^{2}/G\simeq 16~{}M_{\odot}~{}\mathrm{pc^{-1}}$ for $T_{\mathrm{gas}}=10$ K; Stodólkiewicz, 1963; Ostriker, 1964; Inutsuka & Miyama, 1997) is consistent with this threshold $N_{\mathrm{H}}$ value if we adopt the typical 0.1 pc width of the filaments (André et al., 2014).

These observations described above suggest that ISM clouds make a critical change above $N_{\mathrm{H}}\sim 10^{22}~{}(\mathrm{cm}^{-2})$. Suppose the small scale B-field becomes complex and the B-field strength increases at $N_{\mathrm{H}}>10^{22}~{}(\mathrm{cm}^{-2})$. In that case, it could potentially show that the molecular clouds become gravitationally supercritical when $N_{\mathrm{H}}\gtrsim 10^{22}~{}(\mathrm{cm}^{-2})$, and thus the small scale structure in the molecular clouds are formed in this column density range. The B-field might be bent and distorted in the formation of gravitationally supercritical small scale structures or dense filaments. Therefore, the B-field structure may record these small-scale structures’ formation history. High spatial resolution (e.g., $<15\arcsec$) interstellar B-field observations in this column density range, if achieved, could provide important information on the formation of cloud structure in the early stages of star formation.