The Relative Orientation between Local Magnetic Field and Galactic Plane in Low Latitude Dark Clouds

CB24 is a starless, small spherical cloud at a distance of 293 ± 54 pc (Das et al. 2015). There was no association with an IRAS point source identified. Kane et al. (1994) found that CB24 is a relatively less dense cloud, and the low column density may indicate that Bok globules like CB24 did not undergo significant core contraction and represent an ideal sample of starless small dark clouds.

CB27, also known as L1512, is an isolated Taurus core cloud near the GP. The distance of CB27 is found to be 140 pc (Kenyon et al. 1994). A compact sub-mm source (full width at half maximum, FWHM ∼ 104 au) is found to be present at the center of CB27 (Kirk et al. 2005; Di Francesco et al. 2008). The central density of the cloud is near to the maximum stable density, which is required for a pressure-supported, self gravitating cloud and this makes the cloud indistinguishable whether the core is a starless stable or a prestellar one (Launhardt et al. 2013).

CB188 is an isolated small cloud at a distance 262 ± 49 pc (Das et al. 2015). The bolometric luminosity of this cloud is 2.6 L_⊙ and the envelope mass is about 0.7 M_⊙ obtained from an interferometric study of the N₂H⁺ (1–0) emission (Chen et al. 2007). The mean density of the core is ∼2 × 10⁶ cm⁻³. CB188 is found to be physically associated with L673 (Tsitali et al. 2010), as shown by the dotted rectangle in the lower region of Figure 3. In our study, we only covered the northern region of the cloud CB188.

We conducted the optical polarimetric observations of four fields toward the Bok globules CB27, CB24 and CB188 each on 2017 December 22, 23 and 2019 May 8, respectively. We have selected these three globules because of their close proximity with the GP (−10° < b < 10°) as we aim to study the magnetic field morphology of low latitude clouds. Also, no polarimetric study of these clouds was performed in the past. Moreover, the availability of dust continuum emission maps of CB27 (Herschel SPIRE 500 μm) and CB188 (and SCUBA 850 μm) also motivated us to select these globules. Polarimetric observations were conducted in the R-band (filter: λ = 630 nm, Δλ = 120 nm). The observed region around each globule is divided into four fields of 8' × 8' dimension because the CCD has a field of view of ∼8' in diameter. The observations were carried out with the 104 cm Sampurnanand Telescope (ST) at Aryabhatta Research Institute of Observational Sciences, Nainital, India (for exemplary previous polarimetric observations using this telescope see Chakraborty et al. 2014; Chakraborty & Das 2016; Das et al. 2016). The observation log is presented in Table 1. The 104 cm ST is an f/13 Cassegrain telescope. An Imaging Polarimeter (AIMPOL) is connected to the back-end of the telescope that has a Wollaston prism and a rotating half-wave plate (HWP). The Wollaston prism splits the incoming unpolarized light into two orthogonal components (ordinary and extraordinary), and the HWP rotates the polarization state of light into four angles 0°, 225, 45° and 675 which gives the four polarized components (see Das et al. 2013, for detailed observational procedures). The detailed theory and design of AIMPOL are presented in Rautela et al. (2004) and Medhi et al. (2008).

We conducted the observation using the four rotations of HWP as mentioned in Section 3.1. For a particular rotation of HWP (α), the intensities (extraordinary, I_e and ordinary, I_o ) of the two orthogonal polarized components are determined. If the HWP is rotated by α, the electric vector rotates by 2α. For calculation of the linear polarization it is useful to define the ratio R_α

$$\begin{eqnarray}&&{R}_{\alpha }=\frac{({I}_{e}/{I}_{o})-1}{({I}_{e}/{I}_{o})+1}=p\,\cos (2\theta -4\alpha ),\end{eqnarray} \tag{ 1 }$$

where θ and p are the position angle and degree of linear polarization, respectively (Rautela et al. 2004). This ratio becomes Q/I and U/I when α = 0° and 225 respectively, i.e., the values of normalized Stokes parameters q and u (I: total intensity). The linear polarization (p) and the polarization position angle (θ) are given by

$$\begin{eqnarray}&&p=\sqrt{{q}^{2}+{u}^{2}}\hspace{1cm}\mathrm{and}\hspace{1cm}\theta =\displaystyle \frac{1}{2}{\tan }^{-1}\left(\displaystyle \frac{u}{q}\right).\end{eqnarray} \tag{ 2 }$$

In principle, the linear polarization and the position angle of polarization can be measured from the first two rotations of HWP. However, the two additional rotations 45° and 675 are observed due to non-responsivity of the system.

The observed polarimetric data have been reduced using the Image Reduction and Analysis Facility (IRAF) package (see Rautela et al. 2004 for detailed data reduction procedures). The uncertainties associated with p and θ are calculated using the relations (Ramaprakash et al. 1998)

$$\begin{eqnarray}&&{e}_{p}=\frac{\sqrt{N+{N}_{b}}}{N}\,\,\,\,\,\mathrm{and}\,\,\,\,\,{e}_{\theta }=28\mathop{.}\limits^{^\circ }65\times \frac{{e}_{p}}{p},\end{eqnarray} \tag{ 3 }$$

where N and N_b represent the flux counts corresponding to the source and background, respectively.

3.2.1. Instrumental Calibration

Our observed clouds are situated at low galactic latitude close to GP, which allows us to map the magnetic field morphology of the star-forming clouds near the GP. We collected 14 additional low galactic latitude star-forming clouds for which polarimetric observations at optical wavelength are available in literature allowing us to perform a systematic statistical analysis. These include 12 Bok globules (viz. CB3, CB4, CB17, CB25, CB26, CB34, CB39, CB56, CB60, CB69, CB130 and CB246) and two Lynd's clouds (viz. L1014 and L1415). All these clouds are located in the galactic latitude (b) range from −10° < b < 10°. Optical polarimetric observation of CB26 was performed by our group (P. Halder et al. 2022, in preparation). The details of the 17 clouds are compiled in Table 3.

Various studies were carried out to find a correlation between the orientation of envelope magnetic field in molecular clouds with the orientation of the GP (e.g., Sen et al. 2000; Soam et al. 2015; Chakraborty & Das 2016; Das et al. 2016; Choudhury et al. 2019). The magnetic lines of force in the spiral arm of our Galaxy are parallel to the arm everywhere (Ireland & Hoyle 1961). In our previous work, the polarimetric study of CB17 reveals that the projected envelope magnetic field of the globule is oriented along the GP (Choudhury et al. 2019).

Based on the results of 17 clouds presented here, the mean value of position angle of polarization (${\theta }_{B}^{\mathrm{env}}$) determined for 13 clouds shows that the envelope magnetic field is almost aligned along the position angle of GP (θ_GP) (see Figure 4). The offset between the orientation of magnetic field and the GP (θ_off) for these 13 clouds is within 20° with an average offset of 78. However, for four other clouds, a decoupling in the relative orientation between the magnetic field and the GP is observed with an average offset of 676. The details are listed in Table 7.

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In this section, we discuss the possible correlation in the relative orientation between the magnetic field and the GP with the galactic longitude. In Figure 4, we show the variation in the offset between the orientation of envelope magnetic field and the GP, θ_off ($=| {\theta }_{B}^{\mathrm{env}}-{\theta }_{\mathrm{GP}}|$), with the galactic longitude (l). In this plot, we have considered only the magnitude of the offset and not the sign.

In our sample size, we have adequate data points in longitude range 0° to 250°. A second order polynomial fitting is done with these data points (solid curve) and a strong correlation is observed between l and θ_off, with an equation, θ_off = a₁. l² − b₁. l + c₁, where a₁ = 0.0020 ± 0.0006, b₁ = 0.8380 ± 0.1841 and c₁ = 90.5358 ± 14.09. The fitting is done by including the error (standard error of the mean) in θ_off. To test the goodness of fit, we estimated the coefficient of determination (R²) of the best fitted equation. R² is a key output of regression analysis that may be interpreted as the proportion of the variance in the dependent variable predicted from the independent variable, which lies between 0 and 1. The higher the coefficient, the better is the goodness of fit. In this case, R² is estimated to be ≈0.87. It is evident from this figure that the offset between the orientation of GP and the envelope magnetic field is relatively low in the region 115° < l < 250° (which corresponds to the region toward the galactic anti-center, GAC). This indicates that the local magnetic field of the clouds situated toward the GAC region is oriented along the GP. The significant offset between the ${\theta }_{B}^{\mathrm{env}}$ and θ_GP observed in the region l < 100° (corresponding to the region toward the galactic center, GC) shows that the orientation of the magnetic field in the clouds lying in those regions (CB130, CB188 and L1014) tends to become perpendicular to the GP. Hence, for the clouds located at those longitudes, the envelope magnetic field has a local deflection of its own irrespective of the orientation of GP.

Note that no CB cloud is located in the region 250° < l < 350° in the molecular cloud catalog of Clemens & Barvainis (1988). However, since there is one single cloud (CB69) available between 350° and 360° which has a huge offset between the orientation of envelope magnetic field and GP, we have included this cloud in our sample size to observe the best fitting parameters. A second order polynomial fitting is done which is represented by the dashed curve. The curve appears to follow the same trend as in the region l < 100°, which indicates that, toward the GC region, the clouds possess their own local deflection irrespective of the orientation of GP. In this case, R² is estimated to be ≈0.81 and the fitting equation is θ_off = a₂. l² − b₂. l + c₂, where a₂ = 0.0032 ± 0.0004, b₂ = 1.1177 ± 0.1484 and c₂ = 105.559 ± 13.82. Although we have made an attempt to fit the equation including CB69, the fitting parameters obtained may not be significant enough due to insufficient data points beyond l > 250°.

We compare the polarization of the clouds with the polarization of the stars by Heiles (2000) to interpret the relative orientation of the local magnetic field and the GP. The stellar polarization catalogs compiled by Heiles (2000) contain polarization data of 9268 stars in the whole sky (−90° < b < 90°) with p and θ ranging from 0% to 12.47% and 0°–180°. As the region of interest in our study is −10° < b < 10°, we have extracted 2386 polarization data of stars from the Heiles catalog within this range. In Figures 5–7, the mean degree of polarization (〈p〉) and position angle of polarization vectors (〈θ〉) of the field stars located in the clouds are overlaid along with the stellar polarization vectors obtained from the Heiles catalog on the DSS image. The red lines represent stellar polarization vectors obtained from the Heiles catalog and yellow lines signify the mean polarization vectors of the stars background to the clouds (indicated by the cloud IDs). In Table 8, we have presented the mean position angle of stellar polarization vectors (〈θ_Heiles〉) and the standard deviation (${\sigma }_{{\theta }_{\mathrm{Heiles}}}$) in the selected longitude range where the studied clouds are located. The polarization vectors of 1523 stars obtained from the Heiles catalog are plotted in Figure 5 along with 〈p〉 and 〈θ〉 of twelve clouds, viz., CB3, CB4, CB17, CB24, CB25, CB26, CB27, CB34, CB39, CB246, L1014 and L1415 in the longitude range 88° < l < 195°. In Figure 6, 156 stellar polarization vectors are plotted along with 〈p〉 and 〈θ〉 of CB56 and CB60, situated in the longitude range 235° < l < 250° that corresponds to the region of GAC. We have also presented polarization vectors of 724 stars along with 〈p〉 and 〈θ〉 of CB69, CB130, and CB188 situated in the longitude range l > 350° and l < 60° corresponding to the region of GC in Figure 7. Misalignment among stellar polarization vectors at particular longitude ranges is noted to be strong when $\tfrac{\langle {\theta }_{\mathrm{Heiles}}\rangle }{{\sigma }_{{\theta }_{\mathrm{Heiles}}}}$ < 2 (see Table 8). As revealed from Figure 5 and Table 8, the observed nine star-forming clouds in the longitude range 110° < l < 180° $(\tfrac{\langle {\theta }_{\mathrm{Heiles}}\rangle }{{\sigma }_{{\theta }_{\mathrm{Heiles}}}}\gt \,2)$ are found to be almost aligned with the stellar polarization vectors at their local regions which correspond to the region of GAC. A slight randomness is observed in the region 185° < l < 195° $(\tfrac{\langle {\theta }_{\mathrm{Heiles}}\rangle }{{\sigma }_{{\theta }_{\mathrm{Heiles}}}}=\,1.67)$ where CB34 and CB39 are located. However, in the region around l ∼ 90° toward GC $(\tfrac{\langle {\theta }_{\mathrm{Heiles}}\rangle }{{\sigma }_{{\theta }_{\mathrm{Heiles}}}}=\,1.08)$, a significant randomness in the alignment of stellar polarization vectors is seen where the cloud L1014 is located. In Figure 6, almost unidirectional orientation of the stellar polarization vectors is observed in the region 235° < l < 250° $(\tfrac{\langle {\theta }_{\mathrm{Heiles}}\rangle }{{\sigma }_{{\theta }_{\mathrm{Heiles}}}}\approx 2)$. However, in Figure 7, misalignment in the orientation of stellar polarization vectors as well as the star-forming clouds is observed in the regions 345° < l < 355° and 20° < l < 50° $(\tfrac{\langle {\theta }_{\mathrm{Heiles}}\rangle }{{\sigma }_{{\theta }_{\mathrm{Heiles}}}}\,\lt \,2)$. Thus, a local irregularity is observed toward GC as revealed from the stellar polarization data of Heiles, which is found to be consistent with our results.

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Table 8. Heiles Polarization Data Averaged over Particular Galactic Longitude Ranges: Longitude Range over which the mean Polarization is Estimated (l-range), mean Position Angle of Heiles Polarization Vectors (〈θ_Heiles〉), Standard Deviation in the Position Angle of Heiles Polarization (${\sigma }_{{\theta }_{\mathrm{Heiles}}}$), Number of Polarization Vectors found (n), Position Angle of Envelope Magnetic Field Averaged over Star-forming Clouds Located in the Longitude Range given in Column 1 Along with the Standard Deviation ($\langle {\theta }_{B}^{\mathrm{env}}\rangle \pm {\sigma }_{{\theta }_{B}^{\mathrm{env}}}$) and Cloud IDs

l	〈θHeiles〉	${\sigma }_{{\theta }_{\mathrm{Heiles}}}$	〈θHeiles〉/${\sigma }_{{\theta }_{\mathrm{Heiles}}}$	na	$\langle {\theta }_{B}^{\mathrm{env}}\rangle \pm {\sigma }_{{\theta }_{B}^{\mathrm{env}}}$	Cloud IDs
(°)	(°)	(°)			(°)
20–30	89.47	52.39	1.71	43	80 ± 3.20	CB130
40–50	72.77	48.01	1.52	65	98.5 ± 2.30	CB188
88–100	53.90	49.81	1.08	89	15.0 ± 2.2	L1014
110–122	77.01	27.69	2.78	129	67.8 ± 2.58	CB3, CB4, CB246
147–157	130.49	29.70	4.39	81	147.10 ± 7.14	CB17, CB24, CB25, CB26, L1415
167–177	122.13	54.91	2.22	92	145.5 ± 3.7	CB27
185–195	113.72	68.26	1.67	58	146.79 ± 4.93	CB34, CB39
235–250	103.83	53.13	1.95	156	153.05 ± 2.98	CB56, CB60
345–355	79.06	67.66	1.17	177	155.8 ± 3.3	CB69

Note.

^a Number of Heiles polarization vectors present in the longitude range given in column 1.

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Based on the theory given by Davis & Greenstein (1951), Ireland & Hoyle (1961) found that the polarization effect tends to attain maximum intensity in galactic longitudes close to 102°. At such longitude, the direction of polarization is close to being parallel to the plane of the Galaxy. Ireland & Hoyle (1961) also found that, for 70° < l < 130°, polarization is high and the magnetic field lines are oriented along the GP in the Orion Arm. However, for 170° < l < 220°, the polarization is much weaker, and there is a marked tendency for a few polarizations to be normal to the GP. Berkhuijsen et al. (1964), while studying the linear polarization of the galactic background, observed that there is a homogeneous magnetic field parallel to the GP around l = 140°, b = 6°. Generally, the magnetic field lines of the Milky Way galaxy are ascertained to follow the orientation of the spiral arms (Han et al. 2006). Fosalba et al. (2002) observed a net alignment of the magnetic field with galactic structures on large scales. Beck & Wielebinski (2013) found evidence of turbulence in polarized intensity toward the inner Galaxy (270° < l < 90°, ∣b∣ < 30°). The region toward the GC holds the most uniform fields of up to milligauss strength that are oriented normal to the plane (Beck 2004). Thus, the results obtained from this study are in good agreement with the previous studies of the homogeneous magnetic field parallel to the GP for a certain longitude range as discussed above.

The GC is considered to have higher turbulence, indicating high activities in star-forming regions (Boldyrev & Yusef-Zadeh 2006). So, the observed misalignment in the orientation between the envelope magnetic field and the GP toward the GC led us to study the effect of turbulence. The ¹²CO line width or velocity dispersion values are considered to be a good measure of turbulence in molecular clouds. We listed the ¹²CO line width (ΔV km s⁻¹ in column 7 of Table 7) taken from Wang et al. (1995), Clemens et al. (1991), Lippok et al. (2013), Crapsi et al. (2005) and Soam et al. (2017). The uncertainties associated with ΔV taken from Clemens et al. (1991) are the dispersion of the distribution and not the standard error of the mean. They provided the dispersion based on three cloud categories viz. Group A (uncertainty = 0.5), Group B (uncertainty = 0.4) and Group C (uncertainty = 1.1). In our sample, the clouds CB24, CB25, CB39, CB56, CB69 and CB246 fall into Group A, CB60 falls into Group B and CB188 into Group C (see Table 3 of Clemens et al. 1991 for details).

It can be seen from Table 7 (column 7) that the clouds which show noticeable misalignment between the envelope magnetic field and the orientation of GP (θ_off > 30°) are seen to have comparatively higher ΔV (>2 km s⁻¹). Thus, it can be commented that the clouds having higher ΔV, which is an indication of more dynamical activities within the cloud, are seen to have weaker alignment among the polarization vectors. However, most of the clouds with ΔV < 2 km s⁻¹ appear to have low θ_off (<20°), which shows that the polarization vectors of the molecular clouds with less dynamical activities display comparatively better alignment among themselves as well as with the orientation of the GP. The clouds having more dynamical activities exhibit randomness in the alignment of polarization vectors. This is because the regions with high turbulent activities are supposed to be warmer and have better grain alignment. However, precise measurements of the turbulence velocity are not available for these clouds, preventing us from reaching firm conclusions.