B-fields in Star-forming Region Observations (BISTRO): Magnetic Fields in the Filamentary Structures of Serpens Main

Magnetic fields, and specifically the field morphology projected on the plane of sky, can be studied using polarized thermal emission from dust grains. Nonspherical dust grains spin around their minor axes due to radiative anisotropic torques. Within a magnetic field, such spinning grains will have their minor axes aligned parallel to the field (e.g., Lazarian & Hoang 2007). Therefore, the thermal emission of nonspherical dust grains is linearly polarized perpendicular to the field. This allows us to infer magnetic field directions by rotating linear polarization angle measurements by 90°. Note that in contrast, at short wavelengths such as those in the optical regime, polarization is mainly caused by preferential absorption of unpolarized background emission by aligned grains, rather than by the emission from grains. Thus, at those shorter wavelengths, linear polarization and magnetic field directions are parallel to one another.

Figure 2 shows magnetic field directions projected onto the plane of sky across Serpens Main as green segments, inferred by rotating the linear polarization angles by 90°, overlaid on the 850 μm total intensity map. Data points with a signal-to-noise ratio (S/N) in total intensity of greater than 30 and a polarization fraction smaller than 15% have been selected after binning up by a factor of 3 in pixel width. This binning yields a pixel size of 12'', comparable to the angular resolution of ∼ 14''. The lengths of the segments indicate polarization fractions, which are relatively large (up to about 10%) in the low-intensity boundary regions, and get smaller going into the dense core regions. The polarization fractions are discussed quantitatively in the next subsection.

The distribution of polarization directions is shown as a histogram in Figure 3. The most common direction is P.A. = 45, suggesting that an east–west magnetic field dominates. This direction is somewhat consistent with that measured by Planck toward Serpens Main. We estimate an average direction of P.A. = −20° from the Planck polarimetric data at 353 GHz (Planck Collaboration et al. 2016), assuming a 10' beam at the J2000 center position of Serpens Main, (R.A., decl.) = (18:29:55, +01:13:00). Since the histogram shown in Figure 3 is a distribution of polarization angles measured on smaller scales than those of the Planck data, the means of the directions are not expected to agree entirely, and the difference of ∼ 25° can be understood. The polarization direction of Planck data matches the direction of elongation of the Serpens Main molecular cloud, suggesting that the large-scale magnetic field is perpendicular to the overall elongation of the molecular gas structures. This large-scale field has also been reported in near-infrared polarimetric observations sensitive to the periphery of the molecular cloud (Sugitani et al. 2010).

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The magnetic field directions within individual filaments with differing physical properties are one of the key focuses of this study. Figure 2 shows that the southern filaments, FS1, FS2, and FS3, have magnetic fields preferentially oriented in the east–west direction. This trend suggests that the field directions are perpendicular to these filaments overall at the angular resolution of our data, which is too low to separate FS2 and FS3. A similar behavior is observed in the FC2 region. In contrast, the FN1 and FC1 filaments exhibit field directions that are mainly parallel to their major axes. Since FN1 and FC1 are less dense than the other filaments, this difference suggests that magnetic fields are parallel to filaments in less-dense cases but perpendicular in denser ones, a result that has been reported in many previous studies (e.g., Planck Collaboration et al. 2016; Ward-Thompson et al. 2017). However, magnetic field directions change dramatically at the locations of dense cores. For example, the magnetic field seems to surround the core located to the south of FN1, (R.A., decl.) = (18:30:02, +01:15:10). Field direction variations are also clearly seen in the central core regions of both the NW and SE subclusters. A detailed quantitative analysis and discussion of the relative orientations of magnetic fields with respect to the filaments follow in Section 4. Comparisons between the magnetic field on the scales resolved in these data and in interferometric observations of Serpens Main will be undertaken in a separate paper (Y.-W. Tang et al. 2022, in preparation).

The relationship between polarization fraction and column density (or continuum intensity as a proxy for column density) has been used to show where polarized dust emission originates from in star-forming clouds. Polarization fractions depend on both grain properties and local physical conditions (Andersson et al. 2015). In the case of linear polarization of dust thermal emission that is used in this study, there must be a mechanism by which nonspherical dust grains are aligned. Such mechanisms include anisotropic radiation, subsonic or supersonic flows, and/or magnetic fields. In the relevant case of the interstellar medium and molecular clouds whose dust grains are still submicrometer to several micrometers in size, polarization principally arises from spinning dust grains, which are aligned with respect to the magnetic field by torques from an incident anisotropic radiation field; i.e., both a magnetic field and an incident radiation field are needed.

Figure 4 presents the relationship between polarization fraction and continuum intensity in Serpens Main at 850 μm on a log–log scale. Only data points that have an S/N in total intensity larger than 30 and a polarization fraction smaller than 15% have been selected. The black dashed line denotes a slope of −1 for reference, i.e., p ∝ I⁻¹. The blue and red solid lines show the trend of the highest polarization data points at each intensity (p ∝ I^−0.8; i.e., the behavior of the upper envelope of the data points) and the result of fitting a power law to the full sample (p = 75.6 I^−0.634), respectively. Note that the slopes of both the upper-envelope trend and the fit are shallower than −1. If polarization were limited to the outer shell area of the molecular cloud, the relationship between polarization fraction and intensity would follow the reference slope of −1 (e.g., Jones et al. 2015). The data points in Serpens Main, however, show a shallow slope, so polarization is not limited to the periphery of the molecular cloud; some must originate from the dense inner regions of the cloud as well. This overall trend of polarization fraction decreasing with intensity is well known, and often found in molecular clouds (e.g., Kwon et al. 2018; Soam et al. 2018; Coudé et al. 2019; Liu et al. 2019; Pattle et al. 2019; Wang et al. 2019). This trend is understood to result from weakening of dust grain alignment mechanisms in high-density regions, due to decreasing radiation fields and increasing gas density and grain sizes (e.g., Hoang et al. 2021). In addition, varying magnetic field directions along the line of sight depolarize the signal, due to integration over multiple magnetic field components. Studies of alternative polarization mechanisms and dust properties, which are also interesting and important, have been carried out (e.g., Planck Collaboration et al. 2020; Le Gouellec et al. 2020; Pattle et al. 2021b), but are unlikely to apply on the size scales that we consider.

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We note that Pattle et al. (2019) showed that as polarization intensities are calculated using the squared sum of Stokes Q and U values, resulting in the noise characteristics of polarization fractions following a Ricean distribution, an observed power-law index of −1 of polarization fraction as a function of intensity may result from the non-detection of polarized emission, raising the need for caution when interpreting such a relationship. Our results, however, clearly show a shallower power-law index, indicating real detections. Note that in the regime with a power-law index slope shallower than −1, i.e., well-detected polarization, both the Rice distribution and the power-law approaches are consistent with one another.

The relationship between magnetic fields and filamentary structures can be studied quantitatively using the histogram of relative orientations (HRO) technique (Soler et al. 2013; Planck Collaboration et al. 2016), which compares polarization directions ($\hat{E}$) against optical depth gradients (∇τ). For the HRO technique, a histogram of relative orientations (ϕ) between $\hat{E}$ and ∇τ needs to be produced at each density to be investigated. In this work, instead of using optical depth gradients, we use intensity gradients for simplicity, which is reasonable in a small region where temperatures and dust properties are not varying significantly. Note that in the optically thin regime, which is our case, intensity is proportional to the product of the Planck function for blackbody radiation and the optical depth. Thus, a 15% variation of temperature, e.g., between 12 and 14 K, produces a 25% variation in intensity at 850 μm. In addition, note that the directions of the gradients are analyzed here, not the amplitudes of the gradients. Next, within the HRO, the central area (A_c , where ∣ϕ∣ < 225) and the extreme area (A_e , where ∣ϕ∣ > 675) are calculated. Finally, the histogram shape parameter (ξ) is calculated:

$$\begin{eqnarray}&&\xi =\displaystyle \frac{{A}_{c}-{A}_{e}}{{A}_{c}+{A}_{e}}.\end{eqnarray} \tag{ 1 }$$

When the relative orientation of most data points is small, i.e., when polarization directions are parallel to optical depth (here intensity) gradients, ξ is positive since A_c > A_e . In contrast, when polarizations are perpendicular to intensity gradients, ξ is negative. Since the magnetic field direction is perpendicular to the polarization and the filament direction is perpendicular to the intensity gradient, positive values of the histogram shape parameter ξ also imply that magnetic fields are parallel to filaments. In the same way, negative ξ values indicate that magnetic fields are perpendicular to filaments. The uncertainties on ξ are dominated by the number of data points in each bin of the histogram and are calculated as (Planck Collaboration et al. 2016)

$$\begin{eqnarray}&&{\sigma }_{\xi }^{2}=\displaystyle \frac{4({A}_{e}^{2}{\sigma }_{{A}_{c}}^{2}+{A}_{c}^{2}{\sigma }_{{A}_{e}}^{2})}{{\left({A}_{c}+{A}_{e}\right)}^{4}}.\end{eqnarray} \tag{ 2 }$$

Here, ${\sigma }_{{A}_{c}}$ and ${\sigma }_{{A}_{e}}$ are obtained following ${\sigma }_{k}^{2}={h}_{k}(1-{h}_{k}/{h}_{\mathrm{tot}})$, where h_k corresponds to the number of data points in the central or extreme bin, and h_tot is the total number of data points.

Figure 5 presents the histogram shape parameters for individual filaments as well as for all filaments combined. The data used for the plot are gridded to 4'' pixels, with an S/N in total intensity larger than 3, an S/N in polarized intensity larger than 2, a polarization angle uncertainty smaller than 15°, and a polarization fraction smaller than 15%. Data points for individual filament areas have been selected by eye based on the identifications of Lee et al. (2014) and are presented in the left panel of Figure 5. Note that the main trends of the HRO analysis do not change depending on whether a few segments are included or excluded. The column densities (${N}_{{{\rm{H}}}_{2}}$) of molecular hydrogen used in this analysis are calculated from the 850 μm dust thermal intensities (I_ν ):

$$\begin{eqnarray}&&{N}_{{{\rm{H}}}_{2}}=\displaystyle \frac{{I}_{\nu }}{{\kappa }_{\nu }{B}_{\nu }({T}_{d})\,{\mu }_{{{\rm{H}}}_{2}}{m}_{{\rm{H}}}{\rm{\Omega }}},\end{eqnarray} \tag{ 3 }$$

where κ_ν is the mass absorption coefficient at 850 μm assuming a gas-to-dust mass ratio of 100 (0.02 cm² g⁻¹), T_d is the dust temperature (12 K; Lee et al. 2014), ${\mu }_{{{\rm{H}}}_{2}}$ is the molecular weight per hydrogen molecule (2.8; Kauffmann et al.2008), m_H is the atomic hydrogen mass (1.67 × 10⁻²⁴ g), and Ω is the beam solid angle.

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As shown in Figure 5 by the black solid line, the ξ values using all filament data decrease with column density. The transition from positive to negative ξ is not clear, since it happens in the first bin where the mean intensity corresponds to ${N}_{{{\rm{H}}}_{2}}\approx 0.93\times {10}^{22}{{\rm{cm}}}^{-2}$. Therefore, although individual filaments reveal a range of properties, which are addressed below, the overall transition of the magnetic field from parallel to perpendicular within the Serpens Main filaments tends to occur around this column density.

Individual filaments show a range of relative orientations for their associated magnetic fields, as shown in various colors in Figure 5. First, all of the histogram shape parameters of the filament FN1 are positive (blue), which means that the magnetic fields in FN1 are parallel to the filamentary structure. Also, note that FN1 is the least-dense filament, ${N}_{{{\rm{H}}}_{2}}\,\lt 3.1\times {10}^{22}$ cm⁻². Next, FC1 is the second least-dense filament and shows a transition from positive to negative ξ values (green), which indicates that magnetic fields are parallel to the filament in its less-dense regions but perpendicular in the relatively denser regions. Note that the mass-per-length values of these two filaments are subcritical (FN1) or marginally supercritical (FC1) even without considering magnetic fields, and they have no embedded YSOs (Table 1). All of the other filaments (FC2, FS1, and FS2/3) have negative ξ values, suggesting that their magnetic fields are perpendicular to their major axes. Filaments FS2 and FS3 are not easy to separate from each other, so we consider them together. Indeed, these two filaments are very similar in density as the densest filaments among the sample (Table 1). We also note that there is no significant magnetic field variation over the region in response to the different observed filament directions.

This trend of relative magnetic field orientation changing with density has been reported in multiple previous studies. For example, Planck Collaboration et al. (2016) examined the relative orientations between magnetic field directions and column density structures by applying the HRO technique to Planck polarimetric data at 353 GHz. Those authors found that a transition from parallel to perpendicular orientations appears between N_H ∼ 0.5 × 10²² cm⁻² and ∼ 5 × 10²² cm⁻², consistent with our result above.

We further investigated the magnetic field orientation by applying the HRO analysis to all of the dense region observations. Figure 6 shows the histogram shape parameters (ξ) calculated from the whole of Serpens Main including the central subcluster areas as well as the filamentary structures. In the figure, the regime with a column density less than ${N}_{{{\rm{H}}}_{2}}\sim 4.6\times {10}^{22}$ cm⁻² is mainly that of the filamentary structures, in which the ξ values decrease with density. Between ${N}_{{{\rm{H}}}_{2}}\sim 4.6\times {10}^{22}$ and ∼ 10 × 10²² cm⁻², however, the ξ values increase up to zero. This increase suggests that the magnetic fields are trending parallel to elongated structures around a column density of ${N}_{{{\rm{H}}}_{2}}\sim 10\times {10}^{22}$ cm⁻². In other words, magnetic fields become perpendicular to density gradients (where elongated spines are defined as being orthogonal to the direction of a density gradient). The transition can be understood as a result of the local density gradients becoming more significant than the initial density gradients across filaments, presumably arising from mass accumulation of neutral species crossing magnetic field lines (ambipolar diffusion; e.g., Mouschovias 1979; Hennebelle & Inutsuka 2019). Also, this regime corresponds to the area where filaments transition into the central hub while maintaining their initial perpendicular fields. With increasing density, the ξ values again decrease and get back to the overall decreasing trend around ${N}_{{{\rm{H}}}_{2}}\sim 16\times {10}^{22}$ cm⁻², which indicates that magnetic fields are again parallel to density gradients (i.e., perpendicular to "filaments"). This behavior can be understood as magnetic field lines now being dragged along by gravitational collapse at these high column densities.

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The HRO analysis toward the Serpens Main molecular cloud nicely shows the three transitional boundaries in column density for relative orientations of magnetic fields with respect to the physical structures of filaments and cores. Figure 7 summarizes the four column density regimes within the Serpens Main molecular cloud as a schematic diagram. First, magnetic fields are parallel to the least-dense filaments, particularly where ${N}_{{{\rm{H}}}_{2}}\lesssim 0.93\times {10}^{22}$ cm⁻² (region 1). This column density threshold corresponds to n(H₂) ≈ 1.0 × 10⁵ cm⁻³, assuming a 0.03 pc filament width. In contrast, denser filaments ranging from $0.93\times {10}^{22}\lt {N}_{{{\rm{H}}}_{2}}\lesssim 4.6\times {10}^{22}$ cm⁻² have magnetic fields perpendicular to their filamentary structure (region 2). Where the column density becomes larger than ${N}_{{{\rm{H}}}_{2}}\sim 4.6\times {10}^{22}$ cm⁻², cores form and a significant density gradient along a filament appears (region 3). This column density corresponds to n(H₂) ≈ 5.0 × 10⁵ cm⁻³, assuming cores of about 0.03 pc in size. Note that assuming a filament width of 0.03 pc, the critical mass per length M_L,crit = 20 M_⊙ pc⁻¹ corresponds to ${N}_{{{\rm{H}}}_{2}}=3.0\times {10}^{22}$ cm⁻². Considering the presence of magnetic fields, it is interesting that the column density associated with cores in Serpens Main is larger than this value. Finally, at column densities above ${N}_{{{\rm{H}}}_{2}}\sim 16\times {10}^{22}$ cm⁻², magnetic fields are also dragged in by frequent collisions between charged and neutral species during the dynamical contraction, resulting in magnetic fields that are again parallel to the density gradient (region 4). When assuming the same size scale for cores as previously (0.03 pc), this column density threshold corresponds to n(H₂) ≈ 18 × 10⁵ cm⁻³ or higher.

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In addition to the transition of magnetic field relative orientations between less-dense and more-dense filamentary structures, this further transition occurring at a higher density, which we recover, has recently been reported by multiple other studies. Pillai et al. (2020) showed that magnetic fields become parallel to filaments at a visual extinction A_v ≈ 20–30 magnitude in the Serpens South cloud. That extinction is equivalent to N_H ≈ 4.5 × 10²²–6.6 × 10²² cm⁻²(${N}_{{{\rm{H}}}_{2}}\approx 1.6\times {10}^{22}\mbox{--}2.4\times {10}^{22}$ cm⁻²), based on the relation between visual extinction and column density (Güver & Özel 2009). Although their column density threshold differs from ours by a factor of 2, the transition likely corresponds with our boundary between regions 2 and 3 (Figure 7). Also, Wang et al. (2020) found that the filament and hub system G33.92+0.11 has a magnetic field parallel to the gravity vector and the gas motion within the central hub region. Since a gravity direction is comparable to a density gradient, it appears that the hub region corresponds to region 4 of our scenario.

Magnetic field strength, specifically the component projected on the plane of the sky, can be estimated from linearly polarized dust thermal emission using the Davis–Chandrasekhar–Fermi (DCF) method (Davis 1951; Chandrasekhar & Fermi 1953). The DCF method estimates a field strength at a given gas density and turbulence from the scatter of the magnetic field orientation around a mean value. This scatter decreases in a stronger field such that:

$$\begin{eqnarray}&&{B}_{\mathrm{POS}}=Q\sqrt{4\pi \rho }\displaystyle \frac{\delta V}{\delta \phi }\approx 9.3\sqrt{n({{\rm{H}}}_{2})}\displaystyle \frac{{\rm{\Delta }}V}{\delta \phi },\end{eqnarray} \tag{ 4 }$$

where Q is a factor of the order of unity, here taken to be 0.5, ρ is the density, δ V is the turbulent velocity dispersion, and δϕ is the magnetic field position-angle dispersion. In the rightmost expression, n(H₂) is a number density of hydrogen molecules in cm⁻³, ΔV is an FWHM linewidth in kilometers per second, and δϕ is the magnetic field position-angle dispersion in degrees, which results in a field strength measured in microgauss (e.g., Ostriker et al. 2001; Crutcher et al. 2004).

Table 2 lists our estimates for the magnetic field strength within each of the filaments in Serpens Main, using the DCF technique. The number densities are calculated from the mean column density of the individual filaments (Figure 5) divided by the filament widths listed in Table 1 (Lee et al. 2014), assuming no additional line-of-sight modification. Nonthermal velocity dispersions are adopted from spectral-line observations of N₂H⁺ made by Lee et al. (2014), also as shown in Table 1. The model of ordered background magnetic field morphology that is required in order to accurately calculate a field dispersion is obtained from the Stokes Q and U maps smoothed to a resolution of 28'', which is equivalent to the two-times-larger beam. This 28'' resolution corresponds to 0.056 pc at the target distance, which is reasonable for the background ordered fields of filamentary structures with a width ≈0.03–0.05 pc. The field dispersions we measure are distributed from 143 to 294 and are plotted in Figure 8. These values provide magnetic field strengths on the plane of the sky ranging from 64 to 300 μG. When estimating the uncertainty on the magnetic field strength, we assume a 50% error for the densities and use the fitting error for the field position-angle dispersion. Combined, this results in 25%–42% uncertainty in magnetic field strength.

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We also estimate the mass-to-flux ratios (λ) with respect to their critical values, which measure whether the structures are supported by magnetic fields or if they are expected to gravitationally contract:

$$\begin{eqnarray}&&\lambda =\displaystyle \frac{{(M/{\rm{\Phi }})}_{\mathrm{obs}}}{{(M/{\rm{\Phi }})}_{\mathrm{crit}}}=7.6\times {10}^{-21}\displaystyle \frac{{N}({{\rm{H}}}_{2})}{B},\end{eqnarray} \tag{ 5 }$$

where N(H₂) is the column density of hydrogen molecules in units of cm⁻² , and B is the magnetic field strength in microgauss (e.g., Crutcher et al. 2004). Regions with λ > 1 can gravitationally contract. The uncertainties on λ are calculated as ${\sigma }_{\lambda }=\sqrt{{({\sigma }_{N}/N{({{\rm{H}}}_{2}))}^{2}+({\sigma }_{B}/B)}^{2}}$, so they are in the range of 56%–65% when adopting the uncertainties used in the previous paragraph. Note that this measure of uncertainty provides only a rough estimate, as we assume Gaussian distributions for the uncertainties on column density and the magnetic field strength, which may in fact not be independent of each other. Table 2 shows the λ values for each filament. Note that the values are based on only the projected magnetic field component. Therefore, depending on the three-dimensional (3D) orientation of the magnetic fields, λ may be smaller. We also compare the magnetic field energy density (P_B ) and the turbulence energy density (P_turb) of individual filaments, which are, respectively, calculated as

$$\begin{eqnarray}&&{P}_{B}=\displaystyle \frac{{B}^{2}}{8\pi },\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&{P}_{\mathrm{turb}}=\displaystyle \frac{3}{2}\rho {\sigma }_{\mathrm{nt}}^{2}.\end{eqnarray} \tag{ 7 }$$

Here σ_nt is nonthermal velocity dispersion. Finally, the last column of Table 2 presents the ratio of mass-per-length to the critical value for a cylindrical structure ignoring magnetic fields, as reported by Lee et al. (2014; our Table 1). Values greater than 1 indicate that such a structure can gravitationally contract.

Table 2. Magnetic Field Strength Measures for Filaments

Filament	${N}_{{{\rm{H}}}_{2}}$	n(H2)	δ ϕ	BPOS	λ	P ${}_{{B}_{\mathrm{POS}}}$	Pturb	ML /ML,crit
	(×1022 cm−2)	(×105 cm−3)	(deg)	(μG)		(×10−10 erg cm−3)	(×10−10 erg cm−3)
FN1	2.6	2.8	29.4	83 ± 35	2.4 ± 1.6	2.73	6.21	0.76
FC1	2.2	1.8	22.9	97 ± 32	1.7 ± 1.0	3.74	5.14	1.4
FC2	4.4	2.9	28.0	84 ± 26	4.0 ± 2.4	2.78	5.72	2.4
FS1	3.0	2.4	20.1	64 ± 17	3.5 ± 2.0	1.65	1.75	2.6
FS2/3	6.5	7.0	14.3	300 ± 75	1.7 ± 0.95	34.7	18.6	3.9

Note. Only the component of the magnetic field on the plane of the sky has been considered for the λ and ${P}_{{B}_{\mathrm{POS}}}$ measures.

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From these measurements, we classify the filaments into three groups: {FN1, FC1}, {FC2, FS1}, and {FS2/3}. In FN1 and FC1, the magnetic and turbulent energy densities are comparable to each other. Note that λ and the magnetic field energy density are calculated from partial components of the fields, specifically the component projected on the plane of the sky. When the 3D orientation of the structure is considered, the total magnetic field may be significantly larger. For example, if the 3D magnetic field strength is larger than the plane-of-the-sky component by a factor of 2, λ and the magnetic field energy density become halved and quadrupled, respectively. The FN1 and FC1 filaments both have λ and line mass ratios either smaller than or close to 1, which suggests that these filaments may not be able to contract gravitationally. It is noteworthy that these filaments have no star formation activity (no YSOs) and that they have large velocity gradients, of 3–5 km s⁻¹ pc⁻¹ (Table 1).

FC2 and FS1 also have magnetic field and turbulence energy densities comparable to one another. These filaments, however, have both λ and mass ratios larger than two, which suggests that they are likely to gravitationally contract. Indeed, there are YSOs found in FS1 (Lee et al. 2014). In the FC2 filament, there are no YSOs found, but it seems to be gravitationally unstable. If FC2 is in fact not gravitationally unstable, our results suggest that significant magnetic support would be required in order to stabilize the structure. For such support to exist, based on the plane-of-sky values of λ that we measure, the filament would require a significant fraction of its magnetic field to be directed along the line of sight.

Lastly, FS2/3, which is the combined region of FS2 and FS3, has the strongest magnetic field among the filaments. The magnetic field energy density is also twice as large as the turbulence energy density. Therefore, magnetic fields dominate over turbulence in this combined structure. Even with the relatively strong magnetic field compared to turbulence, the filaments have λ > 1 and evidence for YSOs, indicating that the regions remain gravitationally unstable.