The Role of Magnetic Fields in the Formation of the Filamentary Infrared Dark Cloud G11.11–0.12

The data were taken with SIRPOL on the 1.4 m InfraRed Survey Facility (IRSF) telescope at the South African Astronomical Observatory, Sutherland, South Africa. SIRPOL is a single-beam polarimeter capable of simultaneous imaging polarisation observations in the near-IR $JHK_{s}$ bands with an $\sim 8\arcmin\times 8\arcmin$ field of view (FOV)(Nagayama et al., 2003; Kandori et al., 2006). The observations were made during the night of 2018 July 21. To cover the entire extent of G11, we designed a mosaic of three frame positions around the location of G11, as shown in Figure 1. In each frame position, three sets of observations were taken. In each set, the telescope was dithered at 10 positions, each with 30 s exposure time at 4 wave-plate angles in a sequence of $0\degr$, $45\degr$, $22.5\degr$, and $67.5\degr$. The total integration time was 1200 s per wave-plate angle in each of the three frame positions. The total observation time in each of the three frame positions was 83 min, taking observation overheads into account. The total observation time for all the three frame positions was 4.1 hr. The airmass during the observations was in the range of $1.03-1.60$. Observations with lower airmass generally have narrower full width at half maximum (FWHM). The average FWHM for the point sources in all the three frame positions were determined be ranging from around 3 to 4 pixels, or $\sim 1\aas@@fstack{\prime\prime}5$–$1\aas@@fstack{\prime\prime}8$ ($0\aas@@fstack{\prime\prime}45$ per pixel).

In the same night of the IRSF observations, a field from the Galactic Plane Infrared polarisation Survey (GPIPS) (Clemens et al., 2012) was observed with SIRPOL just before the observations for G11. The original field GP0006 from GPIPS is centered on $R.A.(J2000)=18^{\mathrm{h}}26^{\mathrm{m}}11\aas@@fstack{s}2$, $Decl.(J2000)=-13\degr 19\arcmin 57\arcsec$, and has a $10\arcmin\times 10\arcmin$ FOV. The SIRPOL images for the GP0006 field are centered on the inner $\sim 8\arcmin\times 8\arcmin$ region, slightly smaller than the original FOV of the GP0006 field.

The data reduction and aperture photometry for the $JHK_{s}$ polarimetric observations of G11 and the GP0006 field are referred to in Chen et al. (2022b). Following the procedures used in determining polarisation measurements in Chen et al. (2022b), we obtained the $JHK_{s}$ polarisation measurements of sources towards G11 and the GP0006 field. The polarisation measurements have a positive bias because the polarisation degree derived from $\sqrt{Q^{2}+U^{2}}$ is always positive, even though the normalized Stokes parameters $Q$ and $U$ can be either positive or negative. Following the methods of correcting this positive bias (Wardle & Kronberg, 1974; Hull & Plambeck, 2015), the bias-corrected polarisation is $P=\sqrt{Q^{2}+U^{2}-\delta P^{2}}$. In the following texts, the bias-corrected polarisation is labeled as $P$, unless otherwise noted. The polarisation position angle $\theta_{\mathrm{PA}}$ is measured counterclockwise from the north to the east using equatorial coordinates. The $1\sigma$ uncertainty of $\theta_{PA}$ depends on the polarisation signal-to-noise ratio $\frac{P}{\delta P}$. For simplicity, the $1\sigma$ uncertainty of $\theta_{PA}$ is obtained from the standard formula $\delta\theta_{PA}=28\aas@@fstack{\circ}6\,\delta P/P$. For sources with $\frac{P}{\delta P}\geqslant 6$, the $1\sigma$ uncertainty of $\theta_{PA}$ obtained as outlined above is reliable (Naghizadeh-Khouei & Clarke, 1993). For lower SNR ($2\leqslant\frac{P}{\delta P}<6$), the $1\sigma$ uncertainty of $\theta_{PA}$ is underestimated, with smaller deviation for higher $\frac{P}{\delta P}$. More than 85% of the observed sources towards G11 in all the $JHK_{s}$ bands meet the criterion $\frac{P}{\delta P}\geqslant 3$, which means that majority of the sources considered in this work have $1\sigma$ uncertainty of $\theta_{PA}$ of less than $10\degr$.

The $H$-band starlight polarisation obtained with SIRPOL towards the GP0006 field of GPIPS are compared to the GPIPS catalog (Clemens et al., 2012; Clemens et al., 2020). We found 13 polarised stellar sources common in both our SIRPOL observations and the GPIPS catalog. These stellar sources satisfy $P_{H}/\delta P_{H}\geqslant 6$ and 2MASS $J-H$ $\geqslant 0.9$, and have $P_{H}$ in the range from $1\%$ to $9\%$ and $\theta_{\mathrm{PA}}$ in the range from $10\degr$ to $180\degr$. For the 13 stellar sources, the mean difference in $P_{H}$ relative to the GPIPS value is $-6.2\%$ with a standard deviation of 16.3%; the mean difference in $\theta_{\mathrm{PA}}$ relative to the GPIPS value is $7\aas@@fstack{\circ}8$ with a standard deviation of $7\aas@@fstack{\circ}6$. The SIRPOL observations of the GP0006 field show tight consistency with the GPIPS catalog. For the simultaneous observations in the $JHK_{s}$ bands with the IRSF telescope, the $JK_{s}$-band starlight polarisation are aligned in polarisation orientations with the $H$-band polarisation. For tracing the on-sky magnetic field component, $B_{\mathrm{sky}}$, only the polarisation orientations are utilized. The starlight polarisation in the $JHK_{s}$ bands of the H ii regions M17 and RCW 120 show consistency in polarisation orientations in all three bands (Chen et al., 2012; Chen et al., 2022b), though a dependence of polarisation degree with the wavelength was observed (Chen et al., 2012).

We used the near-IR photometry from the extension of the Vista Variables in the Vía Láctea (hereafter VVVX) survey (Minniti et al., 2010). G11 was covered in tile e0954 as part of the VVVX Data Release 1. The $JHK_{s}$ images and catalogs of the e0954 tile were retrieved from the European Southern Observatory (ESO) Science Archive Facility ¹¹1The VVVX datasets are available at https://doi.eso.org/10.18727/archive/68. The VVVX $JHK_{s}$ catalogs include magnitudes from a series of aperture diameters, ranging from $1\arcsec$ to $24\arcsec$ (aperture 1 to aperture 13). The aperture 3 corresponds to an aperture diameter of $2\arcsec$. We utilized the aperture 3 measurements of the VVVX photometric catalog. The calibrated $JHK_{s}$ magnitudes can be derived from the aperture 3 fluxes using the following equation:

where the zero point calibration $MAGZPT$ and aperture correction term $APCORN$ of aperture 3 are integrated in the same VVVX catalog ²²2See https://www.eso.org/rm/api/v1/public/releaseDescriptions/187.

The CO and CO isotopologues $J$=1-0 data of G11 were obtained from the Milky Way Imaging Scroll Painting (MWISP) project, an unbiased Galactic plane CO survey in the northern sky using the 13.7 m millimeter-wavelength telescope at Delingha, China (see Su et al., 2019, for more details). A linear baseline was subtracted from the ¹²CO, ¹³CO, and C¹⁸O $J$=1-0 spectra. After the calibration of the main beam efficiency of $\sim 50\%$, the reduced position-position-velocity (PPV) cubes with a $30\arcsec\times 30\arcsec$ grid are on the intensity scale of main beam temperature, $T_{\mathrm{MB}}$. The measured root-mean-square (rms) noise is $\sim 0.5\,\mathrm{K}$ for ¹²CO $J$=1-0 spectra and $\sim 0.3\,\mathrm{K}$ for ¹³CO/C¹⁸O $J$=1-0 spectra at a channel width of $61\,\mathrm{kHz}$ ($\approx 0.16-0.17\,\mathrm{kms^{-1}}$). The angular resolutions of the CO PPV cubes are close to $50\arcsec$.

Similar to the velocity range of $26-35\,\mathrm{km\,s^{-1}}$ of ¹²CO $J$=3-2 emission presented in Schneider et al. (2015), we utilized a wider velocity range of $25-37\,\mathrm{km\,s^{-1}}$, based on the ¹²CO and ¹³CO $J$=1-0 emission. With this velocity range, we derived the moment 0 (integrated intensity), moment 1 (velocity), and moment 2 (line width) maps of the ¹²CO, ¹³CO and C¹⁸O $J$=1-0 emissions from the MWISP CO PPV cubes. The excitation temperature, $T_{\mathrm{ex}}$, of the CO molecular gas was derived from the ¹²CO $J$=1-0 spectra, following Equation 9 in Chen et al. (2022b). The pixel-wise $T_{\mathrm{ex}}$ map across G11 is uniform with a median of $12\,\mathrm{K}$, and a standard deviation of $1\,\mathrm{K}$. Assuming that the molecular gas is in the state of local thermal equilibrium (LTE), the ¹³CO and C¹⁸O gas share the same $T_{\mathrm{ex}}$ of the ¹²CO gas. In the state of LTE, we derived the column densities of ¹³CO and C¹⁸O gas ($N(\mathrm{{}^{13}CO})$ and $N(\mathrm{C^{18}O})$) according to Equation 10 in Chen et al. (2022b). Following Equation 11 in Chen et al. (2022b), the optical depth $\tau_{\mathrm{13}}$ of the ¹³CO emission reaches a maximum value $\approx 1$ towards the densest regions of G11. In the remaining parts of lower density in G11, $\tau_{\mathrm{13}}$ varies mostly between 0.3 and 0.7. The $N(\mathrm{{}^{13}CO})$ was then multiplied by the optical depth correction factor, $\frac{\tau_{13}}{1-\exp{(-\tau_{13})}}$. We simply assumed that the optical depth of C¹⁸O $J$=1-0 emission is $\ll 1$. Thus the optical depth correction factor for the C¹⁸O emission is $\approx 1$. Assuming the H₂ to ¹³CO and C¹⁸O abundance ratios as $5\times 10^{5}$ and $3.5\times 10^{6}$, respectively (Chen et al., 2022b, and references therein), we derived molecular hydrogen column density $N(\mathrm{H_{2}})$ from $N(\mathrm{{}^{13}CO})$ and $N(\mathrm{C^{18}O})$ respectively. The moment 2 map of the ¹³CO emission evaluates the velocity dispersion $\delta v$ across G11, which has a narrow distribution with a median of $1.9\,\mathrm{km\,s^{-1}}$ and a standard deviation of $0.2\,\mathrm{km\,s^{-1}}$. Meanwhile the sound speed $c_{s}$ is around $0.2\,\mathrm{km\,s^{-1}}$ across G11, corresponding to a rather uniform $T_{\mathrm{ex}}\sim 12\,\mathrm{K}$ derived from the ¹²CO line emission. The molecular gas is supersonic, characterized by the Mach number ($\mathcal{M}=\delta v/c_{s}$) $>6$ across G11. The column density, velocity field and velocity dispersion of molecular gas derived from CO PPV cubes and magnetic fields traced by near-IR starlight polarisation are the fundamental parameters characterizing the dynamical state of G11.

The $N(\mathrm{H_{2}})_{\mathrm{dust}}$ map of G11 was constructed from the modified blackbody fit to the Herschel maps at 160, 250, 350 and 500 $\mu$m (see Schneider et al., 2015, for further details). The $N(\mathrm{H_{2}})_{\mathrm{dust}}$ map of G11 was smoothed to an angular resolution of $\sim 36\arcsec$, slightly better than the angular resolution of the MWISP CO data.

The IRSF observations are not deep enough to detect sufficient background stars, especially in the $J$ band. Similar to the approach used in Chen et al. (2017, 2022b), we combine the $JHK_{s}$ starlight polarisation ($P_{\lambda}\geqslant 2\delta P_{\lambda}$) with the VVVX photometry. Applying a search radius of $1\arcsec$, there are 637 sources in the $J$ band, 749 sources in the $H$ band, and 624 sources in the $K_{s}$ band, that have starlight polarisation $P_{\lambda}\geqslant 2\delta P_{\lambda}$ and $JHK_{s}$ photometry from VVVX. The VVVX sources with starlight polarisation have a mean FWHM of $0\aas@@fstack{\prime\prime}9$, a value not too much smaller than the mean FWHM ($\sim 1\aas@@fstack{\prime\prime}5-1\aas@@fstack{\prime\prime}8$) of the IRSF observations. Although the IRSF observations have larger FWHM than the mean FWHM of VVVX sources, the cross-match between the starlight polarisation and VVVX catalog is mostly one-to-one. In the $K_{s}$ band sample of 624 sources with polarisation and VVVX photometry, as an example, only one source has another fainter source from the VVVX catalog within a $2\arcsec$ radius. The fractions of blended sources are believed to be very low for the starlight polarisation samples in the $JHK_{s}$ bands.

We applied the classical $J-H$ versus $H-K_{s}$ colour-colour diagram to classify background stars that mostly show large $J-H$ and $H-K_{s}$ colours. Figure 2 shows the colour-colour diagram for the starlight polarisation with the VVVX photometry in the $JHK_{s}$ bands. Two populations exist in the colour-colour diagrams, stars with low $H-K_{s}$ and $J-H$ colours, and those reddened to higher colours along the extinction line. Similar to the criteria in Chen et al. (2022b), the criteria for background stars with respect to G11 are

and are displayed as the red dashed lines in Figure 2. Stars enclosed by the blue dashed lines are classified as the foreground stars with respect to G11. The $J-H\geqslant 1.1$ threshold for classifying background stars is a compromise value neither too high nor too low, and is chosen mainly based on our experience and the source distributions in the colour-colour diagram in Figure 2. Because the distance to G11 is 3.6 kpc, 2 times that to the RCW 120 region (Chen et al., 2022b), the interstellar extinction caused by diffuse ISM towards G11 is likely higher than that for RCW 120. The higher interstellar extinction by diffuse ISM is represented by the $J-H\geqslant 1.1$ criterion for G11, higher than $J-H\geqslant 0.9$ applied for RCW 120 (Chen et al., 2022b). The numbers of foreground stars are 341, 232, 86 in the $J$, $H$ and $K_{s}$ bands, respectively. In contrast, the numbers of background stars are 97, 334, and 364 in the $J$, $H$ and $K_{s}$ bands, respectively. In the short wavelengths, starlight polarisation is dominated by foreground stars. The numbers of background stars with polarisation in more than two bands are largely reduced. For example, 60 background stars meet the criterion, $P_{\lambda}\geqslant 2\delta P_{\lambda}$, in all the three bands. The number of background stars with polarisation of $P_{\lambda}\geqslant 2\delta P_{\lambda}$ in both $H$ and $K_{s}$ bands is 237, a value much lower than that in the $H$ or $K_{s}$ band.

The Gaia mission provides accurate stellar parallaxes up to several kilo parsecs (Gaia Collaboration et al., 2016, 2021), depending on the level of interstellar extinction along the LOS. From the star distance catalog (Bailer-Jones et al., 2021), we retrieved distance estimates of $JHK_{s}$ starlight polarisation by using a match radius of $1\arcsec$. We considered the sources with narrow range of distance estimates, defined by $(B_{rgeo}-b_{rgeo})/rgeo<0.2$, where $B_{rgeo}$, $b_{rgeo}$, $rgeo$ are the 84th percentile, the 16th percentile, and the median value of the geometric distance posterior (Bailer-Jones et al., 2021).

The fraction of foreground stars with available distance estimates from Gaia is between $1/3$ to $1/2$ of all foreground stars detected in the $JHK_{s}$ bands. We found no background stars observed in the $JHK_{s}$ bands with accurate distance estimates, indicating that the observed background stars are most likely too far and/or highly reddened (e.g. Chen et al., 2013). A robust classification of background stars relative to G11 relies on distance. Nevertheless, the smaller extinction in the $K_{s}$ band would, therefore, lead to more stars detected behind G11. In contrast, fewer stars behind G11 are detectable in the $J$ and $H$ bands, which have relatively larger extinction. Background stars observed in the $K_{s}$ band are generally more distant and more likely to be lying behind G11. In fact, background stars observed in the $K_{s}$ band (364 stars) are comparatively more than those detected in the $J$ and $H$ bands (97 and 334 stars, respectively). The starlight polarisation of background stars in the $K_{s}$ band is, therefore, more likely to be resulting from the dust grains within G11. In this paper, we analyse the $K_{s}$-band starlight polarisation of 364 background stars.

The interstellar polarisation arising from foreground dust in front of G11.11-0.12 is not negligible and therefore should not be ignored. For the 39 foreground stars with distances $<3\,\mathrm{kpc}$, no trend is found for the $K_{s}$-band polarisation degree with respect to distance. The weighted $K_{s}$-band mean polarisation degree of these 39 foreground stars is $1.3\pm 0.1\%$. Those contributions should be removed from the starlight polarisation of background stars prior to the analyses of magnetic fields. This correction of foreground polarisation has been widely applied when the foreground polarisation is significant and ordered (e.g. Clemens et al., 2013; Eswaraiah et al., 2019; Devaraj et al., 2021). The foreground polarisation can be neglected in cases where it is small and random (e.g. Chen et al., 2017).

Foreground polarisation towards G11 in the $K_{s}$ band was computed from the starlight polarisation of 86 foreground stars. The spatial distributions of foreground polarisation are split into grids with a spacing of $5\arcmin$. The weighted Stokes mean values of the foreground contribution in a grid is computed from the foreground polarisation within the same grid. Every background star is assigned to a nearest grid with computed foreground polarisation. The corrected Stokes values of background stars are $Q_{\mathrm{cor}}=Q_{\mathrm{bg}}-Q_{\mathrm{fg}}$ and $U_{\mathrm{cor}}=U_{\mathrm{bg}}-U_{\mathrm{fg}}$. The foreground-corrected polarisation degree ($P_{K}$), and position angle ($\theta_{\mathrm{PA}}$) are calculated using Equations (6) and (7) in Chen et al. (2022b). The foreground-corrected starlight polarisation of the 364 background stars in the $K_{s}$ band are used to trace the local $B_{\mathrm{sky}}$ of G11. Table 1 lists the foreground-corrected polarisation in the $K_{s}$ band and VVVX photometry in $JHK_{s}$-bands of the 364 background stars. The values of $\theta_{\mathrm{PA}}$ in Table 1 are read counterclockwise from the equatorial north, following the IAU definition.

Figure 3 shows the orientation of the foreground-corrected $K_{s}$-band polarisation (green segments) of the background stars superimposed on the $N(\mathrm{H_{2}})$ map derived from ¹³CO emission. The $K_{s}$-band polarisation segments were rotated by a specific angle to match the Galactic coordinate frame of Figures 3, 4, 7 and 8. Besides the polarization orientation of individual background stars shown in Figure 3, we also split the starlight polarisation into each cell of $2\arcmin\times 2\arcmin$ size. Table 2 lists the properties of each cell, including positions of the cell center, weighted mean polarisation (degrees and orientations), average gas velocity, the absolute difference between polarisation orientation and the filament orientation, and the number of polarisation vectors. We keep only the cells with at least three polarisation vectors. The $2\arcmin\times 2\arcmin$ cell is moderate, so that each effective cell contains eight starlight polarisation in average. In Figure 4 the mean orientation of starlight polarisation in each cell was displayed as green segments, overlaid on the map of polarisation angle dispersion in each cell.

The spine of G11, better traced by the $N(H_{2})_{dust}$ contours, clearly shows a positive gradient of column density, from left to right. The spine of G11 is, in fact, twisted along the Galactic plane, and can be divided into four regions with different orientations relative to the Galactic plane. The spine orientations of the four regions, read counterclockwise from the equatorial north, were determined by the visual inspection of the highest $N(H_{2})_{dust}$ isocontours in each region.

Region A: $11\aas@@fstack{\circ}24<l<11\aas@@fstack{\circ}34$; the spine here has the lowest $N(H_{2})_{dust}$ of $\sim 2\times 10^{22}\,\mathrm{cm^{-2}}$, with a peak value of $3.6\times 10^{22}\,\mathrm{cm^{-2}}$. The envelope of region A has even lower column density, $\sim 0.5\times 10^{22}\,\mathrm{cm^{-2}}$, derived from the ¹³CO $J$=1-0 emission.

Region B: $11\aas@@fstack{\circ}15<l<11\aas@@fstack{\circ}24$; the spine of region B has a peak $N(H_{2})_{dust}$ of $4.7\times 10^{22}\,\mathrm{cm^{-2}}$, higher than that of region A. The envelope here has column density $\sim 0.8\times\gtrsim 10^{22}\,\mathrm{cm^{-2}}$.

Region C: $11\aas@@fstack{\circ}04<l<11\aas@@fstack{\circ}15$; the spine here contains the densest and most massive dust condensation, P1 (Johnstone et al., 2003), in which massive star formation is evident (e.g. Pillai et al., 2006; Shipman et al., 2014; Rosero et al., 2014). The peak $N(H_{2})_{dust}$ reaches $7.7\times 10^{22}\,\mathrm{cm^{-2}}$ towards the dust condensation P1. The magnetic fields in the spine of this region are perpendicular to the filament itself (Pillai et al., 2015), as indicated by the white segment in Figure 3.

Region D: $10\aas@@fstack{\circ}93<l<11\aas@@fstack{\circ}04$; the spine of region D has a peak $N(H_{2})_{dust}$ of $8.3\times 10^{22}\,\mathrm{cm^{-2}}$ towards its densest part, and is almost as dense as that of region C. It contains the massive dust condensation P6 (Johnstone et al., 2003; Wang et al., 2014) and the $70\,\mu$m dark clump G10.99–0.08 (Pillai et al., 2019; Sanhueza et al., 2019), in which star formation is still on-going.

The orientations of the foreground-corrected $K_{s}$-band polarisation in the four regions of G11 are displayed in the top rows of Figure 5. The orientations of mean starlight polarisation in cells in the four regions are shown in the bottom rows. In regions A, B and C, the overall orientations of starlight polarisation are in the range $130-150\degr$. The starlight polarisation in region A shows a narrow distribution, with a peak at $132\degr$. The orientations of starlight polarisation in region B are similar to those in region A, but show a larger scatter. In areas of region B adjacent to region A, the orientations of starlight polarisation are parallel to those in region A. The orientations of starlight polarisation in region C have a peak at $147\degr$, roughly parallel to those in regions A and B. The orientations of starlight polarisation in region D peak at $32\degr$, roughly perpendicular to those in regions A, B and C.

The cutout 353 GHz Stokes $I,Q,U$ maps with pixel scale of $1\aas@@fstack{\prime}7$ for G11 were retrieved from the Planck Legacy Archive ³³3http://http://pla.esac.esa.int/pla. The Planck $I,Q,U$ maps were smoothed to $5\aas@@fstack{\prime}1$ resolution using a Gaussian kernel with FWHM = 3 pixels. The 353 GHz dust polarisation degree and polarisation orientation in the IAU definition were computed from the Stokes $I,Q,U$ maps, following Equations 1 and 2 in Planck Collaboration et al. (2015). The magnetic field orientations on the plane of the sky are displayed as the blue segments in Figures 3 and 4. The magnetic field orientations, rotated $90\degr$ from the Planck dust polarisation angle, are compared with the field orientations traced by the foreground-corrected starlight polarisation angles. The average offsets in magnetic field orientations traced by the two methods are $68\degr$, $57\degr$, $50\degr$, and $4\degr$ in regions A, B, C, and D respectively, an indication that magnetic fields traced by starlight polarisation are very different from the field orientations traced by dust polarisation in most parts of the G11 filament. G11 is lying within the Galactic plane, along which direction the Planck dust polarisation data record all the polarised dust emission along the same LOS. The LOS contamination might be much less severe for the filaments at high Galactic latitude. Soler et al. (2016) compared the orientation of magnetic field projected on the plane of the sky inferred from the Planck 353 GHz dust polarisation and optical/infrared starlight polarisation for four nearby molecular clouds at high Galactic latitude. The consistency between field orientation sampled in the dust polarised emission and dust extinction suggests considerable agreement between the magnetic field orientation on the plane of the sky estimated with both techniques at comparable scales (Soler et al., 2016). Because of the severe LOS contamination, the Planck dust polarised emission towards G11 are inadequate in tracing the local magnetic field of this filament.

Figure 5 compares the polarisation orientations with the spine orientations in the four regions. In both regions A and C, the orientations of starlight polarisation are perpendicular to their spines. The mean starlight polarisation in the cells within region B show a bimodal distributions. In areas of region B adjacent to region A, the starlight polarisation is roughly parallel to the spine of region B. In the rest parts of region B, the grid-averaged starlight polarisation tends to be perpendicular to the spine of region B, but with higher dispersion, as shown by Figure 4. The projection effects of the magnetic field and filaments can produce small relative orientation angles between the filament and the magnetic field projected onto the plane of the sky, even though the magnetic field is orthogonal to the filament in a 3D space (Doi et al., 2020). The observed parallel orientation between the starlight polarisation and the spine in region B is likely to be due to the projection effects. The orientations of starlight polarisation in region D are almost perpendicular to those in regions A, B and C. However, the grid-averaged starlight polarisation in region D is close to being perpendicular to the spine of region D. The orientations of the foreground-corrected $K_{s}$-band polarisation in the four regions are generally perpendicular to the spines in each region, despite the fact that the spines in each region have different orientations.

The $N(\mathrm{H_{2}})_{\mathrm{dust}}$ in the spine of region C is $\gtrsim 4\times 10^{22}\,\mathrm{cm^{-2}}$. Few starlight polarisation was detected towards the spine of region C. Starlight polarisation traces better the $B_{\mathrm{sky}}$ in the surrounding envelope of region C. The $B_{\mathrm{sky}}$ traced by starlight polarisation in the envelope is consistent with the magnetic fields on the spine of region C obtained from dust polarisation (Pillai et al., 2015). The consistency of $B_{\mathrm{sky}}$ in the envelope of parsec scales, and the dense core of sub-parsec scales, indicates that magnetic fields are perpendicular to the spine of region C. Combining the near-IR starlight polarisation in this work and the dust polarisation in Pillai et al. (2015), the magnetic fields are found to be always perpendicular to G11 across the parsec-scale envelope surrounding the filament to the dense spine in which star formation is on-going.

The variable extinction across G11 leads to the non-uniform spatial distribution of starlight polarisation. The number of starlight polarisation towards the low-density envelope of G11 is much higher than that towards the high-density spine. Therefore, the magnetic field strength is more representative of the envelope rather than the dense spine of G11. Pillai et al. (2015) estimated a total magnetic strength of $>267\pm 26\,\mu$G for the dense condensation in region C (the rectangle in Figures 3 and 4). In the envelope of G11, the gas density is much lower than that in the spine. Assuming energy equipartition between turbulence and perturbed magnetic fields, given by

the polarisation angle dispersion, $\delta\phi\approx\frac{\delta B}{B}$, yields the Davis-Chandrasekhar-Fermi (DCF) method (Davis & Greenstein, 1951; Chandrasekhar & Fermi, 1953) for estimating the on-sky strength of magnetic field

where $Q$ is the correction factor, $\rho$ is the gas density, and $\delta v$ is the gas velocity dispersion.

The ¹³CO $J$=1-0 emission is mostly optically thin within the envelope of G11. The moment 2 map of ¹³CO $J$=1-0 emission provides the velocity dispersion $\delta v$ of molecular gas across G11. The median $\delta v$ values of the areas overlapping with starlight polarisation are extracted from the moment 2 map of ¹³CO emission. The density of molecular gas, or molecular hydrogen number density $n(\mathrm{H_{2}})$ multiplied by $\mu m_{\mathrm{H}}$ ($\mu=2.83$ is the average molecular gas weight (Kauffmann et al., 2008), $m_{\mathrm{H}}$ is the mass of a single hydrogen atom), is always a challenging task in characterizing the fundamental properties of molecular clouds. The mean gas density of G11 was estimated to be as $n(\mathrm{H_{2}})\approx 0.5\times 10^{3}\,\mathrm{cm^{-3}}$ at a scale of $\sim 5$ pc (see Fig.4 in Kainulainen et al., 2013). Assuming the LOS extent $\sim 5$ pc for the envelope of G11, $n(\mathrm{H_{2}})$ is approximately $N(\mathrm{H_{2}}$)/5 pc, where $N(\mathrm{H_{2}})$ is the hydrogen column density derived from ¹³CO $J$=1-0 emission. The median $n(\mathrm{H_{2}})$ of region A is $0.4\times 10^{3}\,\mathrm{cm^{-3}}$, roughly consistent with the large scale $n(\mathrm{H_{2}})\approx 0.5\times 10^{3}\,\mathrm{cm^{-3}}$ in Kainulainen et al. (2013).

The starlight polarisation orientations show ordered mean orientation in the four regions of the G11 filament, an indication of ordered magnetic fields. Therefore, the polarisation angle dispersion $\delta\phi$ due to turbulence can be determined from the angular dispersion function (ADF; Falceta-Gonçalves et al., 2008; Hildebrand et al., 2009; Houde et al., 2009). The ADF method obtains a measure of the difference in polarisation angle of $N(l)$ pairs of starlight polarisation separated by displacement $l$ (Hildebrand et al., 2009):

The polarisation angle dispersion attributed to the turbulent magnetic field, $B_{t}(x)$, and the large scale structure magnetic field, $B_{0}(x)$, are folded into the $\langle\Delta\Phi^{2}(l)\rangle^{1/2}$. If the large scale and turbulent magnetic fields are statistically independent, and the turbulent dispersion of polarisation angle is a constant $b$ for the displacement $l$ larger than the turbulent correlation length $\delta$, characterizing $B_{t}(x)$, the following equation holds

when $\delta<l\ll d$, where $d$ is the typical scale length of variations in $B_{0}(x)$, and $\sigma_{M}(l)$ is the measurement uncertainty (Hildebrand et al., 2009). Houde et al. (2009) obtained a turbulent correlation length $\delta\approx 0.016$ pc for the massive star-forming cloud OMC–1 at a distance of 450 pc. Following the same method as Houde et al. (2009), Franco et al. (2010) estimated the turbulent correlation length $\delta$ to be of a few 0.001 pc for the Pipe nebula at a distance of 145 pc. For the G11 filament, at a distance of 3.6 kpc, the closest separation between any pair of starlight polarisation is $\approx 0.1$ pc, at least one order of magnitude higher than the turbulent correlation length $\delta$ found for OMC–1 cloud and Pipe nebula. The length of G11 is about 30 pc and the sizes of regions A, B, C and D are $\gtrsim 6$ pc. The starlight polarisation towards G11 satisfies the condition $\delta\ll l\ll d$, which implies that Equation 8 is suitable for deriving the turbulent dispersion of polarisation angle.

Figure 6 shows the ADF of the four regions along with the best fit from Equation 8 using the first five data points ($l\leqslant 2\aas@@fstack{\prime}5$ or $\leqslant 2.6$ pc). The constant $b$ is determined by the zero intercept of the best fit to the ADF, at $l=0$. According to the work by Houde et al. (2009), the ratio of turbulent to large scale magnetic fields is a function of the constant $b$:

We use $\delta B$ to replace $\langle B_{t}^{2}\rangle^{1/2}$, and define the polarisation angle dispersion as $\delta\phi=\frac{\delta B}{B_{0}}$. Table 3 shows the results of ADF analyses of the four regions in G11.

We utilize the $\delta\phi$ obtained from the ADF analyses to estimate the strength of $B_{\mathrm{sky}}$ in the four regions. MHD simulations found that the original DCF method would overestimate the field strength. Therefore, a correction factor, $Q$ in Equation 6, was proposed to bring the field strength derived from the DCF method closer to the model field strength (Ostriker et al., 2001; Heitsch et al., 2001). For the ordered magnetic fields with $\delta\phi<25\degr$, $Q\approx 0.5$ yields a reasonable estimate on the field strength (Ostriker et al., 2001; Heitsch et al., 2001). For regions C and D with $\delta\phi\gtrsim 25\degr$, we still assume $Q\approx 0.5$, but the field strength within them is quite uncertain. From Equation 6, we estimate the $B_{\mathrm{sky}}$ strength of the four regions in G11. We present the $B_{\mathrm{sky}}$ strength estimated from the DCF method, and the corresponding Alfvén speed $v_{\mathrm{A}}$ and Alfvén Mach number $\mathcal{M}_{\mathrm{A}}$ in Table 3 for the four regions.

The DCF method is based on the assumption that turbulence is sub-Alfvénic. MHD simulations demonstrated that the field strength obtained by the DCF method is reliable when molecular gas is sub-Alfvénic or equivalently $\delta\phi<25\degr$ (Ostriker et al., 2001; Falceta-Gonçalves et al., 2008; Liu et al., 2021). The field lines close to the LOS would cause large polarisation angle dispersion even if the turbulence is sub-Alfvénic (Falceta-Gonçalves et al., 2008). The polarisation angle dispersion, $\delta\phi$, is smallest in region A. The highly ordered starlight polarisation in region A implies that magnetic fields in region A are close to the plane of the sky. The projection effect due to the field orientation is small in region A. The field strength in region A, estimated from the DCF method, is $108\,\mu$G, while much lower field strength is found in regions B, C, and D.

Skalidis & Tassis (2021) proposed a new method for estimating magnetic field strength in the compressible ISM. Unlike the DCF method assuming the cross term $\delta B\cdot B_{0}=0$, Skalidis & Tassis (2021) included this cross term, and assumed the turbulent kinetic energy equal to the magnetic energy fluctuations. They obtained

and the new method (hereafter ST) for estimating magnetic field strength

Although the polarisation angle dispersion $\delta\phi$ is caused by the perpendicular fluctuations of magnetic energy rather than the parallel fluctuations, Skalidis et al. (2021) demonstrated that perpendicular fluctuations are comparable to the parallel fluctuations, thus $\delta\phi$ can also be utilized in the ST method. The magnetic field strength in the four regions, estimated by the ST method, are given in the parentheses after the DCF estimates in Table 3. The field strength estimated by the ST method is lower than that by the DCF method. The reduction factor is nearly two for region A, in which $\delta\phi$ is small. In regions B, C, and D with much higher $\delta\phi$ and higher density, the ST method returns field strength comparable to the DCF estimates. In contrast to the DCF method, the square root of $\delta\phi$ used in the ST method will weaken the effect of $\delta\phi$ variation on calculating the magnetic field strength. The ST method derives rather constant field strength, about $50\,\mu$G, for the four regions.

With the field strength estimated from the DCF or ST method, $v_{\mathrm{A}}$ in the four regions are computed using $v_{\mathrm{A}}=|B_{\mathrm{sky}}|/\sqrt{4\pi\rho}$. Table 3 lists the $v_{\mathrm{A}}$ from the DCF and ST methods. The Alfvénic Mach number $\mathcal{M}$ is a function of the polarisation angle dispersion $\delta\phi$, depending on the DCF or ST method. Following the DCF method, $\mathcal{M}_{\mathrm{A}}=\frac{\delta\phi}{Q}$, depends only on $\delta\phi$ when the correction factor $Q\approx 0.5$ is fixed. Note that the ST method derives $\mathcal{M}_{\mathrm{A}}=\sqrt{2\delta\phi}$, higher than that derived from the DCF method. For instance, $\mathcal{M}_{\mathrm{A}}$ of region A estimated from the ST method is 0.6, twice that estimated from the DCF method. The difference in $\mathcal{M}_{\mathrm{A}}$ between the two methods is small for regions B, C, and D. The turbulence within the four regions of G11 is generally sub-Alfvénic to trans-Alfvénic. If the turbulence is super-Alfvénic, the isotropic turbulence would twist the field lines. The average field orientations traced by starlight polarisation would be close to random distribution, and the average starlight polarisation degree is likely to be small. The average starlight polarisation per cell (see Table 2) is aligned perpendicular to the G11 filament, and the average polarisation degree per cell is typically 2-3%. The turbulence within G11 is sub-Alfvénic to trans-Alfvénic, indicating that magnetic fields are generally more important than, or as important as, turbulence.

The starlight polarisation within the $2\arcmin\times 2\arcmin$ grid spacing across G11, in principle, provides the information about the $\mathcal{M}_{\mathrm{A}}$ distribution across G11. Figure 4 shows the map of polarisation angle dispersion $\delta\phi$ with a grid spacing of $2\arcmin\times 2\arcmin$. The $\mathcal{M}_{\mathrm{A}}$ map, derived from $\mathcal{M}_{\mathrm{A}}=\frac{\delta\phi}{0.5}$ following the DCF method, is shown in the top panel of Figure 7. In region A, the $\mathcal{M}_{\mathrm{A}}$ values obtained from the ST method are roughly twice the values from the DCF method; in regions B, C, and D, the $\mathcal{M}_{\mathrm{A}}$ values obtained from both methods are overall consistent (see also Table 3). The $\mathcal{M}_{\mathrm{A}}$ distributions across G11 is sub-Alfvénic to trans-Alfvénic, an indication that magnetic field channels the gas flow from the envelope to the spine of G11. To better explore the gas flow from the envelope to the spine, we present the moment 1 map from the ¹³CO $J$=1-0 emission along with magnetic fields in Figure 8. The moment 0 and moment 2 maps of ¹³CO $J$=1-0 emission are shown in Figures 9 and  10 in Appendix A. In region A the gas velocity is fairly uniform, indicating little velocity gradient along the LOS. The magnetic fields in region A are likely oriented on the plane of the sky, implied by the highly aligned starlight polarisation in region A. Such oriented magnetic fields channel gas flow from the envelope to the spine without producing observable velocity gradient along LOS, which is consistent with the uniform velocity field in region A. Different from region A, the LOS components of magnetic fields in region B might be more pronounced. The gas flow along the LOS components of magnetic fields from the envelope to the spine in region B would lead to observable velocity gradient perpendicular to the spine. The gas flow from the envelope in both side onto the spine in middle would cause red-shifted and blue-shifted components relative to the gas velocity in the spine. In Figure 8 we note the $32\,\mathrm{km\,s^{-1}}$ area roughly parallel to spine, the area with velocity below $31\,\mathrm{km\,s^{-1}}$ on the opposite site, and the spine with velocity in between $31-32\,\mathrm{km\,s^{-1}}$. The blue-shifted and red-shifted areas on the two sides of the spine in region B suggest that gas flow from the envelope onto the spine is threaded by the magnetic fields.

The critical mass-to-flux ratio for the magnetic stability, $(M/\Phi)_{\mathrm{crit}}=\frac{1}{2\pi\sqrt{G}}$, is given by Nakano & Nakamura (1978), where $G$ is the gravitational constant. The mass-to-flux ratio normalized by $(M/\Phi)_{\mathrm{crit}}$ is evaluated as follows

where $\mu=2.83$ is the average molecular gas weight (Kauffmann et al., 2008). The $N(\mathrm{H_{2}})$ map used in deriving the $\lambda$ map is from the ¹³CO $J$=1-0 emission across G11. The field strength $|B_{\mathrm{sky}}|$ estimated from the DCF method and $N(\mathrm{H_{2}})$ maps yield the distribution of $\lambda$ across G11, as shown in the bottom panel of Figure 7. We also obtained the $\lambda$ distribution across G11 based on the field strength from the ST method. The two methods show consistency for $\lambda$ in regions B, C, and D, in which field strengths estimated from both methods are comparable to each other (see also Table 3). The $\lambda$ distributions in regions B, C, and D are mostly $\lambda\gtrsim 1$, indicative of magnetically critical envelope in these regions of G11. When the field strength of $\sim 300\,\mu$G and an average $N(\mathrm{H_{2}})_{\mathrm{dust}}\sim 6.5\times 10^{22}\,\mathrm{cm^{-2}}$ are adopted for the spine of region C (Pillai et al., 2015), the $\lambda$ in the spine of region C is reduced to a level of marginally supercritical, which is consistent with the $\lambda$ obtained for region C in this work. The $\lambda$ in region A is around 0.5 from the DCF method or close to 1 from the ST method. Moreover, the $N(\mathrm{H_{2}})$ in region A has a mean value $\sim 6\times 10^{21}\,\mathrm{cm^{-2}}$. Both the $N(\mathrm{H_{2}})$ and $\lambda$ in region A show some consistency with the MHD simulations by Seifried et al. (2020), in which the transition from parallel to perpendicular occurs at a transition column density $\approx 10^{21-21.5}\,\mathrm{cm^{-2}}$ and at a transition $\lambda$ close to 1.

The magnetic field orientations in the envelope of G11 are almost identical to that in the spine of region C (Pillai et al., 2015). This consistency between parsec scale and sub-parsec scale magnetic field orientations suggests the important role of magnetic fields in assembling the mass of G11. The field strength in the envelope of G11 is likely in the range $\sim 50-100\,\mu$G. The median $n(\mathrm{H_{2}})$ of region A is $0.4\times 10^{3}\,\mathrm{cm^{-3}}$. Pillai et al. (2015) estimated magnetic field strength of $>267\pm 26\,\mu$G in the spine of region C. They adopted $n(\mathrm{H_{2}})\approx 3\times 10^{4}\,\mathrm{cm^{-3}}$ in the estimate of magnetic field strength. The density contrast between the spine and the envelope of G11 is $\sim 75$. If adopting the relation $|B|\propto\rho^{0.65}$ used in Crutcher (2012), the magnetic field strength contrast between the spine and the envelope is as high as $\sim 17$, much higher than the observed contrast of $\sim 3-5$. The observed magnetic field contrast between the spine and the envelope of G11 does not follow the nominal scaling relationship between magnetic field strength and gas density. Magnetic fields are not significantly enhanced in the spine of G11 as material is assembled onto the spine. This can be explained by gas flow guided by the field lines. Gas flow along the field lines cannot efficiently compress magnetic fields, leading to a minor enhancement of field strength when gas flows are merging onto the filament and a shallower relation $|B|\propto\rho^{0.4}$ (Li et al., 2015). For G11, we estimate both the magnetic field strength and density for the envelope, called $B_{\mathrm{env}}$ and $\rho_{\mathrm{env}}$. Meanwhile, Pillai et al. (2015) estimate both the magnetic field strength and density for the densest part of the G11 filament, called $B_{\mathrm{fil}}$ and $\rho_{\mathrm{fil}}$. The power index of the scaling relation can be derived by $\frac{\log B_{\mathrm{fil}}-\log B_{\mathrm{env}}}{\log\rho_{\mathrm{fil}}-\log\rho_{\mathrm{env}}}$. Adopting a field strength $B_{\mathrm{env}}=75\,\mu$G between the DCF and ST estimates and $\rho_{\mathrm{env}}=0.4\times 10^{3}\,\mathrm{cm^{-3}}$ estimated in this work, as well as $B_{\mathrm{fil}}=267\,\mu$G and $\rho_{\mathrm{env}}=3\times 10^{4}\,\mathrm{cm^{-3}}$ (Pillai et al., 2015), the field strength is proportional to $\rho^{0.3}$, similar to that measured for NGC 6334, in which the elongated gas structures at all scales are nearly perpendicular to the magnetic fields (Li et al., 2015). The flat scaling relation $|B|\propto n^{0.3}$ found for G11 suggests the important role of the magnetic fields in the formation of G11. Firstly, the dense spine can easily be formed without much magnetic field resistance. Secondly, the spine becomes magnetically supercritical, because the field strength is increasing slower than gas density. The spine could reach a high volume density, resulting in a small thermal Jeans mass that is conducive to forming low-mass stars (Li et al., 2010). The outflows, traced by high-velocity CO $J$=2-1 emission towards the dust condensation G10.99–0.08 in region D (Sanhueza et al., 2019), are mostly driven by low-mass protostars with masses below their mass sensitivity of $\sim 1-2\,M_{\odot}$ (Pillai et al., 2019). This overabundance of low-mass protostars in the densest region D of G11, and the perpendicular magnetic fields as shown in this work, are in agreement with the predictions drawn from the MHD simulations (Li et al., 2010).