Velocity Gradient and Stellar Polarization: Magnetic Field Tomography towards the L1688 Cloud

In this work, we employ the ¹²CO (1-0) emission line provided by the COMPLETE survey (Ridge et al., 2006). The data were acquired via an observation using the 14 m Five College Radio Astronomy Observatory (FCRAO) telescope. ¹²CO emission line has an effective velocity resolution of 0.07 km/s. The full width at half maximum (FWHM) The Half Power Beam Width (HPBW) of the ¹²CO observation is $\approx$ 46". However, the final data cube is convolved onto a regular 23" per pixel resolution to satisfy the Nyquist sampling. The RMS noise level per channel is $\approx$ 0.98 K for ¹²CO in unit of antenna temperature $T^{*}_{A}$. The radial velocity of the cloud’s bulk motion ranges from 0 to 7 km/s. This velocity range was chosen for our analysis in this work.

In this work, we obtained our HI data from the 100 m NRAO Green Bank Telescope (GBT) in West Virginia which observed Ophiuchus in 2011 (Li & Goldsmith, 2003)¹¹1Data archive: doi:10.7910/DVN/R7DYQP. The HI data utilized from the GBT has a pixel size or effective resolution of $1^{\prime}$, an angular resolution of $\sim 9^{\prime}$, a spectral resolution of 0.32 km s^-1, and a normal RMS value of 0.15 K per channel (Ridge et al., 2006). In this work, we analyze the velocity range from -60 km/s to 30 km/s.

In this study, we have employed the magnetic field orientations deduced from the Planck polarization data and conducted a comparative analysis with the VGT and stellar polarization. Specifically, we utilized the 353 GHz polarized dust signal data from the Planck 3rd Public Data Release (DR3) in 2018 of the High-Frequency Instrument (Planck Collaboration et al., 2020).

The Planck observations determine the polarization angle, $\phi$, using the Stokes parameter maps for intensity $I$, and the polarization states $Q$ and $U$. The angle $\phi$ is mathematically defined by the following relation:

where the notation $-U$ is employed to align the angle with the IAU convention, differing from the HEALPix standard. To improve the signal-to-noise ratio, the Stokes parameter maps were convolved with a Gaussian kernel to smooth the data from the original angular resolution of 5^′ to 10^′. The magnetic field angle, denoted as $\phi_{B}$, is subsequently derived by $\phi_{B}=\phi+\pi/2$.

In this study, we have harnessed stellar polarization data from a variety of surveys to map the magnetic field (Heiles, 2000; Kwon et al., 2015; Vrba et al., 1976; Wilking et al., 1979; Sato et al., 1988). The polarization measurements for the stars included in our analysis were taken in the K band from each respective survey. The selection criteria for inclusion were a signal-to-noise ratio greater than 3, and a location within the bounds of the L1688 star-forming region.

Distance information for the stars is directly accessible from the survey conducted by Heiles (2000) online. For the remaining surveys (Vrba et al., 1976; Wilking et al., 1979; Sato et al., 1988), which only provide data in paper-table form, we have compiled the relevant information—including stars’ coordinates, polarization angle, and distance—into Tab. 1 for ease of reference.

VGT is a new method of mapping magnetic fields in the ISM theoretically based on MHD turbulence theories (Goldreich & Sridhar, 1995) and fast turbulent reconnection theory (Lazarian & Vishniac, 1999). It has been applied in the study of magnetic fields in a multitude of interstellar clouds, including neutral clouds, ionized clouds, and molecular clouds. (Hu et al., 2020b; Lu et al., 2020; Hu et al., 2021; Tram et al., 2022; Hu et al., 2022a).

Magnetic fluctuations are anisotropic, which was originally proposed in (Goldreich & Sridhar, 1995)(denoted as GS95 hereafter). GS95 also discussed the relation of the anisotropy with the "critical balance" condition, where the cascading time $(k_{\perp}v_{l})^{-1}$ must be equal to the wave period $(k_{\parallel}v_{A})^{-1}$. $k_{\perp}$ and $k_{\parallel}$ denote wavevectors measured is perpendicular and parallel to the magnetic field. $v_{A}$ is defined as Alfvén velocity and $v_{l}$, where $v_{l}\propto l^{1/3}$ is defined as turbulent velocity, accounting for Kolmogorov-type turbulence. From this, the turbulent eddies of the magnetic field can be shown to have anisotropic properties where they elongate along the magnetic field. The scaling presented in originally in Goldreich & Sridhar (1995) (henceforth GS95) in the form of wavenumbers, must be rewritten to account for the anisotropy of magnetic field that must be evaluated locally, rather than in respect to the mean magnetic field direction. This can be understood from the theory of turbulent reconnection,²²2The time taken for the reconnection of eddies should be mentioned as equal to the eddy turnover time, which was predicted by fast turbulent reconnection (Lazarian & Vishniac, 1999). Consequently, the magnetic field doesn’t stop the evolution of these eddies, and almost all of the energy of the cascade is channeled into them (Zhao et al., 2023). and it corresponds to numerical testing (Lazarian & Vishniac, 1999; Cho & Vishniac, 2000)

In the global system of reference related to the mean magnetic field, largest scale eddies dominate the anisotropy making it scale-independent. From (Lazarian & Vishniac, 1999), it was found that eddies obey the hydrodynamic Kolmogorov scaling because magnetic fields give little resistance to eddy motion that is perpendicular to the local direction of the magnetic field. This scaling law can be defined as $v_{l,\bot}\propto l_{\bot}^{1/3}$ where $v_{l,\bot}$ is the turbulence’s component which is perpendicular to the local magnetic field at scale $l$. Additionally, we know because eddies are affected only by the local magnetic field, both $l_{\parallel}$ and $l_{\perp}$ should be established with regards to the magnetic field’s local direction. The previously mentioned critical balance condition can be modified and it can be shown $l_{\perp}/v_{l}$) should be equal to the wave period ($l_{\parallel}/v_{\rm A}$).

From this, the relationship between the parallel and perpendicular scales of the eddies in the local reference frame can be acquired Lazarian & Vishniac 1999):

where $L_{\rm inj}$ denotes the turbulent injection scale, $M_{\rm A}=v_{\rm inj}/v_{A}$ denotes the Alfvén Mach number, and $v_{\rm inj}$ denotes the turbulence’s injection velocity. Because of the anisotropy $l_{\bot}\ll l_{\parallel}$, the amplitude of velocity fluctuation and its corresponding gradient can be found via:

Here, $v_{\rm inj}$ refers to injection velocity. The $\nabla v_{l}$ here points in the direction of maximum variation in the velocity’s fluctuation amplitude which will be perpendicular to the local magnetic field. Thus, this provides a convenient way to map magnetic fields.

In this work, we utilize the Velocity Channel Gradients (VChGs). VChGs work by using a thin velocity channel, which is mapped as Ch($x$,$y$). Fluctuations in these velocity channels in Postion-Position-Velocity (PPV) space were theorized in Lazarian & Pogosyan (2000). It was shown, that the narrower the velocity channel width, the greater the contribution of the velocity fluctuations is (Hu et al., 2023). When the velocity channel width $\Delta v$ is smaller than the velocity dispersion $\sqrt{\delta(v^{2})}$ of the turbulent eddies being studied, then the intensity fluctuation in the thin channel will be dominated by velocity fluctuations. Conversely, in a thick channel, the intensity fluctuation will be dominated by density fluctuation. The thick and thin channels can be defined by the following criteria:

The minimum value of velocity channel width $\Delta v$ is constrained by the spectral or velocity resolution of the observation, while it is possible to combine several channels to get larger $\Delta v$. $\sqrt{\delta(v^{2})}$ is typically calculated from the dispersion of the velocity centroid map, i.e., the moment-one map.

We can calculate the gradient map mathematically from these equations: $\psi_{\rm g}$:

In these equations, $\nabla_{x}$Ch${}_{i}(x,y)$ and $\nabla_{y}$Ch${}_{i}(x,y)$ denote the $x$ and $y$ components of the gradients in the thin channel map, respectively. The subscript of these equations, $i=1,2,...,n_{v}$, indicates the $i^{\rm th}$ channel map where $n_{v}$ is the total number of velocity channels. These equations are utilized and applied to each pixel that has a spectral line emission with a signal-to-noise ratio $>$ 3.

Even with these equations, the gradient orientation applied to each pixel, can statistically derive turbulence’s property. The gradient map $\psi_{\rm g}$ is further processed via the sub-block avveraging method introduced in Yuen & Lazarian (2017). The gradient map is divided into sub-blocks and a Gaussian fitting is applied within each sub-block to find the most probable orientation of gradient. The peak value of the each Gaussian distribution is utilized as the mean gradient for each corresponding specific sub-block. The processed gradient map is thus denoted as $\psi^{i}_{gs}(x,y)$.

Similar to Planck polarization measurements, we formulate the Pseudo-Stokes-parameters $Q_{g}$ and $U_{g}$ from $\psi^{i}_{gs}(x,y)$ using the following equations:

where the pseudo polarization angle is represented by $\psi_{g}$ and is perpendicular to the POS magnetic field. The magnetic field orientation can then be inferred from orthogonality to be $\psi_{B}=\psi_{g}+\pi/2$

In order to acquire a spatial distribution of HI gas, an accurate Galactic rotational curve of the Milky Way Galaxy is required. In this work, we utilize the high-order polynomial curve obtained by (Clemens, 1985). The equation of the rotational curve is represented in the following lines:

where $R$ denotes the distance from a point of interest in the Galaxy to the Galactic center, $R_{0}=8.5$ kpc and denotes the distance from the Galactic center to the Sun in our solar system, and $V$ denotes the circular velocity of the point of interest. $A,B,C,$ and $D$ are fitted coefficients of the curve using the assumption that our Sun’s circular velocity is $V_{0}=220$ km/s. The specific values of these coefficients can be found in Table 3 in (Clemens, 1985). Rather than listing the numerous coefficients, we reproduce the rotational curve as shown in Fig. 1. After 1.6$R_{0}$, we assume that the curve will flatten.

Using the curve, the spatial distribution of the HI gas at the point of interest can easily be found using the relative velocity $V_{r}$ and the angular velocity $\omega=V/R$ of the Galactic rotation. Relative velocity can be expressed as (McClure-Griffiths & Dickey, 2007):

where $l$ denotes Galactic longitude and $b$ denotes Galactic latitude in the Galactic coordinate system. For $l$ and $b$, we use the central coordinates of the L1688 cloud. We determine $V_{r}$ via the central velocity of each HI channel.

Fig. 2 delineates the orientation of the magnetic field as a function of stellar distances. Predominantly, the magnetic field angles span between [30^∘, 110^∘], with the median orientation settling at $60^{\circ}$ east of the north. Notably, a significant shift is observed at the location corresponding to L1688, approximately 139 pc away and up to 200 pc, where the angle exhibits a bifurcation—either escalating to nearly $180^{\circ}$ or plummeting to $0^{\circ}$. This bifurcation is indicative of a potential difference in the magnetic field orientation between the foreground/background and that within the L1688 cloud itself.

To ascertain the relative contributions of the foreground/background and the molecular cloud, we analyze their hydrogen column densities, assuming dust grains are uniformly mixed with hydrogen. Instead of using the typical SED fitting method, the column density of hydrogen for the foreground/background, denoted as $N_{\rm H}^{\rm f}$, is calculated using the X-factor method from (Panopoulou et al., 2019):

where $N_{\rm HI}$ represents the column density of atomic hydrogen (H I), $N_{\rm H_{2}}$ denotes the column density of molecular hydrogen (${\rm H_{2}}$), and $T_{\rm MB}^{\rm HI}$ signifies the brightness temperature of H I emission.

For the molecular cloud L1688, the hydrogen column density is derived similarly using the CO-${\rm H_{2}}$ conversion factor $X_{\rm CO}=2\times 10^{20}$, commonly applied in giant molecular clouds (Narayanan et al., 2011):

where $T_{\rm MB}^{\rm CO}$ is the brightness temperature of CO emission. Differing from Eq. 10, the contribution of H I is considered negligible here as ${\rm H_{2}}$ emission predominates.

Our findings (see Appendix D) indicate that the average values of $N_{\rm H}^{\rm f}$ and $N_{\rm H}^{\rm mc}$ are comparably significant, with $N_{\rm H}^{\rm f}\approx 2\times 10^{21}{\rm cm}^{-2}$ and $N_{\rm H}^{\rm mc}\approx 7\times 10^{21}{\rm cm}^{-2}$. This suggests that both the foreground/background and the molecular cloud contribute notably to stellar polarization. At a distance of approximately 139 pc, the molecular cloud’s contribution predominates, significantly affecting the direction of stellar polarization to reflect the magnetic fields within the cloud. Conversely, at larger distances, the background’s contribution gradually increases, leading to stellar polarization being influenced by both the foreground/background and the molecular cloud.

The VGT has proven to be effectively applicable to atomic HI emissions. When compared to molecular emissions, the VGT applied to HI (VGT-HI) has the distinctive advantage of mapping the magnetic fields that exist in the cloud’s foreground and background (González-Casanova & Lazarian, 2019; Lu et al., 2020; Hu & Lazarian, 2023b). In Fig. 3, we compare the integrated VGT-HI map with stellar polarization data ³³3We plotted all available stellar polarization data from multiple surveys (Heiles, 2000; Kwon et al., 2015; Vrba et al., 1976; Wilking et al., 1979; Sato et al., 1988), including data for stars without available parallax measurements. Figs. 4, 5, and 8 exclusively use data from stars with known distance information.. This comparison extends beyond the stars for which distance data are available (as depicted in Fig. 2) to include those without known distances. It is observed that the global magnetic field patterns deduced from VGT-HI broadly align with those inferred from stellar polarization. Nevertheless, notable differences are present in the southern reaches of the map, especially the central dense clump of L1688 (see Fig. 6). Given the magnetic field variations in L1688 identified in Fig. 2, it is plausible that the stellar polarization is significantly influenced by the molecular cloud, accounting for the observed misalignments.

Furthermore, to quantify our comparison between the magnetic field inferred by stellar polarization and VGT-HI, we use the Alignment Measure (AM; González-Casanova & Lazarian 2017), expressed as:

here, $\theta_{\rm r}=|\phi_{\rm B}-\psi_{\rm B}|$. An AM value of 1 implies parallel alignment of $\phi_{\rm B}$ and $\psi_{\rm B}$, while -1 indicates perpendicular alignment. The AM map presented in Fig. 3 demonstrates a concentration of positive AM values in the northeastern, lower-intensity tail of the cloud, whereas regions of negative AM predominantly coincide with the central dense clump.

Fig.4 illustrates the variation in AM as a function of distance along the LOS to the stars. It is crucial to note that the VGT-HI maps from which these measurements are derived represent an integration along the LOS. We observe that the AM is predominantly positive at smaller distances (approximately $<200$ pc), which is indicative of the foreground magnetic field alignment. Beyond 200 pc, where the influence of the L1688 cloud becomes more pronounced, there is a tendency for the AM to increase again at larger distances ($>200$ pc). An evident decrease in AM to negative values occurs around $\sim 200$ pc. The data is binned uniformly in logarithmic space, and the bins are of equal size in non-logarithmic space, resulting in a physical bin interval of $\sim 80$ pc between the second and third bins from the left, as shown in Fig. 2. The bin at $\sim 200$ pc, thus, should contain also the contribution from the cloud.

As we have previously discussed, this decline in AM corresponds to distances along the LOS that are near or slightly beyond the cloud. In Fig. 2, we note that the polarization angles for stars in the vicinity of L1688 range widely from 0 to 180 degrees. In contrast, the polarization angles for the remainder of the stars are generally confined between approximately 30 to 110 degrees with a median value of 60 degrees. Such a variation within the magnetic field orientations associated with a molecular cloud, however, would not be captured by VGT-HI due to the lack of HI in the cloud, where H₂ and other molecules are dominant.

To facilitate a more direct comparison between VGT-HI mapping and stellar polarization, we employed the Galactic rotation curve method detailed in § 3 to ascertain the spatial positioning of each HI channel within the Galaxy. Fig. 11 displays the original HI spectrum alongside its corresponding distance from the Galactic center.

In Fig. 5, stars are superimposed onto 11 distinct velocity channels, each aligned with their respective distances from the Sun. Across these channels, we observe subtle variations in the magnetic field, reflecting the nuanced 3D magnetic field mapping that polarization studies alone had previously found challenging. A closer examination reveals that, compared to the integrated HI map, there is a more pronounced agreement between the magnetic field gradient orientations and the stars within each channel. Most notably, there is a consistent positive correlation between the VGT-HI-derived magnetic field orientations and those of the stars at comparable distances. Exceptions are present in a few channels—for instance, sub-figures a (65 - 155 pc), b (155 - 335 pc), d (335 - 585 pc), and g (585 - 665 pc)— have some stars which exhibit discrepancies, typically within high-intensity regions. Nevertheless, these instances of disagreement may be attributed to the limited number of stars in those samples. When examining larger samples, the statistical agreement improves, reinforcing the effectiveness of VGT-HI in tracing magnetic field orientations across different Galactic scales.

As established in our investigations, while VGT-HI is proficient at reconstructing the magnetic field within 3D space—offering a form of magnetic field tomography—it does not encompass the nuanced contributions and variations of the magnetic fields within molecular clouds. This caveat can be mitigated by integrating VGT with molecular emission lines, such as ¹²CO.

Fig. 6 presents the magnetic fields inferred by stellar polarization within the L1688 region, compared with magnetic field estimations derived from two distinct methodologies: VGT-¹²CO and Planck dust polarization. Collectively, these measurements suggest that the magnetic fields tend to align with the northeastern low-intensity tails of the region. Nonetheless, within the central, high-density core, the magnetic field configurations become markedly complex. Divergences among the three methods are discernible. The stellar polarization vectors are relatively sparse and exhibit a more disordered pattern of magnetic fields.

Fig. 7 presents the AM between the magnetic field orientations deduced from stellar polarization and those inferred from VGT-¹²CO, alongside a comparison with Planck polarization. For VGT-¹²CO, regions of low intensity predominantly show positive AM values, suggesting alignment. In contrast, high-intensity clumps, especially at the cloud’s center, exhibit a mix of both positive and negative AM values. However, it is within these denser regions that the most significant perpendicular orientations—or misalignments—of the magnetic fields are observed.

Several factors may contribute to this misalignment. Primarily, stellar polarization may include influences from both the foreground and background, which are not represented in VGT-¹²CO measurements. This is further corroborated by the alignment discrepancies observed between Planck and stellar polarization (see Fig. 7), where Planck data reflect the cumulative magnetic field integrated along the LOS. Additionally, VGT-¹²CO tends to trace magnetic fields within specific volume densities, as optically thick ¹²CO generally maps to a critical volume density of 10² cm^-3 (Draine, 2011), whereas stellar polarization, driven by dust grain alignment with magnetic fields, might encompass contributions from denser regions (Seifried et al., 2019; Hoang et al., 2021). Despite this, the overall agreement between VGT-¹²CO and Planck data (see Hu & Lazarian 2023c and Figs. 9 and 10) attests that VGT-¹²CO is capable of reliably mapping the magnetic fields associated with diffuse molecular gas.

Moreover, our analysis extends to a comparison between VGT-¹²CO and LOS-decomposed stellar polarization. As depicted in Fig. 8, we observe that the negative AM observed with VGT-HI near L1688 dissipates, and the AM remains consistently positive across all distances. This finding indicates a superior alignment between stellar polarization and VGT-¹²CO within L1688, leading to the conclusion that the negative AM associated with VGT-HI can be attributed to the magnetic field changes within the molecular cloud.

The quest to map the POS magnetic field in three dimensions within our Galaxy poses a significant scientific challenge. Traditional techniques that trace magnetic fields via dust polarization—such as those employed by the Planck mission—provide invaluable insights into the integrated magnetic fields along the LOS (Planck Collaboration et al., 2020; Andersson et al., 2015; Lazarian & Hoang, 2007; Hoang & Lazarian, 2016; Hu & Lazarian, 2023a; Hoang & Truong, 2023). Despite their utility, these methods alone do not offer the detailed information needed to fully reconstruct the Galactic magnetic field in 3D.

Recent advancements have been made by coupling stellar polarization with precise astrometric data (Clemens et al., 2012; Panopoulou et al., 2019), and by integrating the VGT-HI with Galactic rotation curves (González-Casanova & Lazarian, 2019; Hu & Lazarian, 2023b). These innovative approaches represent a leap forward in our ability to visualize magnetic fields within the three-dimensional expanse of the Galaxy. This study delves into the magnetic fields of the star-forming region L1688, applying the VGT-HI methodology. We use VGT-HI in conjunction with the Galactic rotation curve to create a tomographic map of the magnetic field, which encompasses the regions foreground and background to L1688, and we benchmark the VGT-HI with stellar polarization observations. VGT-HI advantageously provides more extensive field coverage than stellar polarization and we find a significant correlation with stellar polarization data.

A prior investigation into the relationship between VGT-HI and stellar polarization noted a prevalence of negative AM values within regions situated on the Galactic plane and within the spiral arms (González-Casanova & Lazarian, 2019). This pattern was postulated to originate from variations in the magnetic fields associated with molecular clouds. Echoing these findings, our current study also records a decline in the agreement between VGT-HI mapping and stellar polarization near the L1688 region. To unravel the origins of this negative AM, we examined the magnetic field angle variation along the LOS, as shown in Fig. 2. Our analysis revealed a change in the stellar polarization angles from a range of [30^∘, 110^∘], with a median at $60^{\circ}$ east of north, to [0^∘, 180^∘]. This latter range is distinct at L1688’s specific LOS position (139 pc). The VGT-HI approach does not account for this rotation, thus making it the probable cause for the observed negative AM.

To rectify this negative AM, we advocate the implementation of VGT-¹²CO, where we noted a statistically significant positive AM that corresponds with both stellar and Planck polarization data (see Fig. 10 and Fig. 8). The alignment found in VGT-¹²CO lends credence to the hypothesis that the negative AM detected in VGT-HI stems from the molecular cloud’s contribution on stellar polarization. Incorporating VGT-¹²CO, which adeptly maps the local magnetic field within molecular clouds, alongside VGT-HI, offers a promising pathway to synthesize a more comprehensive depiction of the Galactic magnetic field in three dimensions.

Mapping the Galaxy’s pervasive 3D magnetic field with greater precision is crucial for effectively modeling the Galactic foreground polarization, which is a well-known confounding factor in discerning B-mode within the CMB polarization (Ade et al., 2023; Macellari et al., 2011; Abril-Cabezas et al., 2023; Hensley & Bull, 2018; Planck Collaboration et al., 2020). Several studies (Clark & Hensley, 2019) also proposed the use of HI emission to model the Galactic foreground polarization, while the change of magnetic fields within molecular clouds, as identified in this work, was not accounted for. Here we showed that the combined application of the VGT-HI and VGT-¹²CO presents a promising avenue for accurately modeling the Galactic foreground dust polarization. As shown in Figs. 5, 8, and 9, this syngengy adeptly delineates the POS magnetic field across the foreground, background, and the L1688 region. Theoretically, this method is scalable and could be applied to broader swathes of the Galaxy, potentially enhancing the fidelity of the Galactic foreground polarization modeling.