11institutetext: Department of Astronomy, School of Physics, Peking
University, Beijing, 100871, People’s Republic of China 22institutetext: Kavli
Institute for Astronomy and Astrophysics, Peking University, Beijing 100871,
People’s Republic of China
22email<EMAIL_ADDRESS>33institutetext: Max Planck Institute for
Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany 44institutetext:
National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun
Road, Chaoyang District, Beijing 100101, China

# Relative alignment between gas structures and magnetic field in Orion A at
different scales using different molecular gas tracers

Wenyu Jiao 112233 Ke Wang 22 Fengwei Xu 1122 Chao Wang 44 Henrik Beuther 33

(Received 9 January 2024; accepted 27 February, 2024)

###### Abstract

Context. Magnetic fields can play crucial roles in high-mass star formation.
Nonetheless, the significance of magnetic fields at various scales and their
relationship with gas structures is largely overlooked.

Aims. Our goal is to examine the relationship between the magnetic field and
molecular gas structures within the Orion A giant molecular cloud at different
scales and density regimes.

Methods. We assess the gas intensity structures and column densities in Orion
A by utilizing 12CO, 13CO, and C18O from Nobeyama observations. Through
comparing Nobeyama observations with Planck polarization observations on large
scales ($\sim 0.6$ pc) and JCMT polarization observations on small scales
($\sim 0.04$ pc), we investigate how the role of magnetic fields change with
scale and density.

Results. We find a similar trend from parallel to perpendicular alignment with
increasing column densities in Orion A at both large and small spatial scales.
Besides, when changing from low-density to high-density tracers, the relative
orientation preference changes from random to perpendicular. The self-similar
results at different scales indicate that magnetic fields are dynamically
important in both cloud formation and filament formation. However, magnetic
fields properties at small scales are relative complicated, and the interplay
between magnetic field and star-forming activities needs to be discussed case-
by-case.

###### Key Words.:

Molecular cloud, high-mass star formation, magnetic fields, Polarimetry

##  1 Introduction

The formation and evolution of molecular clouds are intricate processes driven
by the interplay of several crucial factors, which include gravity,
turbulence, magnetic fields, thermal instability, feedback, and cosmic-ray
ionization. They play different roles in different activities such as
filamentary structure formation and star formation process (e.g., Bergin &
Tafalla 2007; McKee & Ostriker 2007; Wang 2015). The magnetic field is
considered as one of the key components in slowing down the star formation
rate, but it is difficult to measure the magnetic field because of
observational limitations (Hennebelle & Falgarone 2012).

Till now, there are two primary methods for measuring magnetic field in
molecular clouds. The observation of Zeeman effect is the only direct way to
measure the strength of magnetic field in interstellar clouds. Some
observations have succeeded in calculating the value of magnetic fields along
the line of sight (e.g., Crutcher et al. 1996; Falgarone et al. 2008).
However, because the Doppler broadening effect is always larger than Zeeman
splitting effect, this method is limited by observational techniques and can
not be widely used (Crutcher 2012).

An alternative way to infer the morphology of magnetic fields is through the
observations of linear polarization induced by interstellar dust. One possible
and longstanding method is to measure the polarization from the background
starlight, which assumes aspherical dust grains align with their long axis
perpendicular to the local magnetic field and produce polarization parallel to
the field projected on the plane of sky (see Andersson et al. 2015 for a
review). However, it is difficult to reproduce the projected magnetic field
morphology fully inside molecular clouds. In recent years, a more general
method to measure the orientation of magnetic fields within the molecular
clouds is through the measurement of linearly polarized radiation emitted from
dust, which can map the morphology of magnetic fields on the plane of sky on
small scales around the star-forming regions (e.g., Matthews et al. 2009;
Zhang et al. 2014) or on large scales across the whole sky (Planck
Collaboration et al. 2020a), helping us to know more about magnetic fields.

The strength of the magnetic field projected on the plane of sky can be
estimated using the Davis-Chandrasekhar-Fermi method (Davis 1951;
Chandrasekhar & Fermi 1953), but the uncertainty and applicability of the DCF
method are always discussed (e.g., Liu et al. 2021; Skalidis & Tassis 2021;
Skalidis et al. 2021; Chen et al. 2022). Some studies seek to derive
additional information about the role of magnetic field by comparing the
relative orientation of magnetic field inferred from dust polarization with
the orientation of molecular cloud structures, which find strong correlation
between magnetic field morphology and the geometry of the molecular cloud
structures (e.g., Tassis et al. 2009; Koch et al. 2013; Planck Collaboration
et al. 2016a, b). Multiscale magnetic fields are also explored in different
researches, suggesting that the role of the magnetic field may vary at
different scales (e.g., Li et al. 2015; Chen et al. 2020; Liu et al. 2023).

In recent years, the histograms of relative orientation (HRO) technique is
introduced to compare the relative orientation between column density
structures and magnetic field (Soler et al. 2013). Applying this method to
observations, the transition from parallel to perpendicular with increasing
column density has been found, indicating that the magnetic field plays an
important role for the gas dynamics at cloud scales (Planck Collaboration et
al. 2016b; Soler et al. 2017).

A more detailed study attempts to connect the intensity structure of molecular
lines from different gas tracers with the magnetic field in Vela C Giant
Molecular Cloud (Fissel et al. 2019). When comparing the results with
different tracers, the authors find a transition from parallel to
perpendicular as the critical density of the tracers increases, indicating a
correlation between gas intensity structures and magnetic fields, which is
consistent with the result of recent Magnetohydrodynamics (MHD) simulations
(Mazzei et al. 2023). However, it is just a case study, and the findings in
this work need to be examined in a larger sample.

The Orion Giant Molecular Cloud is the nearest high-mass star forming region
(d=414 pc, Menten et al. 2007), which is well known for its hierarchical
filamentary structure and strong star-forming activities (Bally 2008).
Magnetic field properties in Orion A are deeply investigated both on large
scales (e.g., Houde et al. 2004; Clark & Johnson 1974) and small scales (e.g.,
Poidevin et al. 2010; Pattle et al. 2017; Soler 2019). In this work, we aim at
examining the relationship between molecular gas structure and magnetic field
at different scales. We make use of the spectral cube of three different
molecular lines and then calculate different gas intensity structure maps to
make a comparison with the magnetic field111We note that we compare the local
relative orientation between intensity structures and magnetic field pixel by
pixel. This is different from the Velocity Gradient Technique (VGT), which
requires an additional subblock-averaging step (Yuen & Lazarian 2017). Due to
this difference in methodology, our results cannot be directly compared to
those obtained using the VGT technique.. Our goal is to find whether there is
similar trend in relative orientation with the increasing column density
within Orion A Giant Molecular Cloud, whether the correlation between gas
structures and magnetic field change when comparing them at different scales,
and whether there are clear differences in relative orientation when using
different gas tracers.

We first introduce Nobeyama-, SCUPOL- and Planck\- derived maps used in our
analysis in Sect. 2. The methods of calculation and the results are described
in Sect. 3. We show the main results in Sect. 4. In Sect. 5, we compare our
results with the previous study and discuss the implications of our work. A
brief summary of our results is given in Sect. 6.

##  2 Observations

###  2.1 Planck dust polarization data

For the analyses in this work, we calculate the magnetic field orientations
from linearly polarized dust emission. Planck observations provide the linear
polarization maps (Stokes Q and U) from 30 to 353 GHz in multiple frequency
bands across the entire sky (Planck Collaboration et al. 2020a). These maps
serve as valuable tools for studying polarized emission from interstellar
dust. On large scales we adopt the publicly available PR3 data from the High-
Frequency Instrument (HFI; Lamarre et al. 2010) at 353 GHz (Planck
Collaboration et al. 2020b). The maps of Stokes Q and U are initially at 5′
spatial resolution ($\sim 0.6$ pc) and the pixel size of 1.71′. We calculate
the polarization position angle projected on the plane of sky from the Stokes
parameters as

$\theta_{\mathrm{p}}^{\mathrm{g}}=\frac{1}{2}\arctan(U,Q)$ (1)

The magnetic field position angle can be obtained by adding 90∘ to the
polarization angle:
$\theta_{\mathrm{B}}^{\mathrm{g}}=\theta_{\mathrm{p}}^{\mathrm{g}}+\frac{\pi}{2}$.
The downloaded Planck Q and U maps222We downloaded the data from
http://pla.esac.esa.int/. If the data is downloaded from
https://irsa.ipac.caltech.edu/applications/planck/, the derived polarization
position angle should be $\theta_{\mathrm{p}}=\frac{1}{2}\arctan(-U,Q)$.  are
in galactic coordinates and the polarization position angle in equatorial
coordinates can be derived using
$\theta_{\mathrm{p}}^{\mathrm{e}}=\theta^{\mathrm{g}}_{\mathrm{p}}-\Delta\theta_{\mathrm{p}}^{\mathrm{g}-\mathrm{e}}$,
where

$\Delta\theta^{\mathrm{g-e}}_{\mathrm{p}}=\arctan[\frac{\cos\left(l-32.9^{\circ}\right)}{\cos
b\cot 62.9^{\circ}-\sin b\sin\left(l-32.9^{\circ}\right)}]$ (2)

is the angle difference between galactic and equatorial coordinates (Corradi
et al. 1998). The debiased polarization intensity $PI$ and the corresponding
uncertainties $\sigma_{PI}$ can be derived using the modified asymptotic
estimator (pMAS; Plaszczynski et al. 2014). The rms noise level of stokes Q
and U map is similar, with a value of 0.26 $\mathrm{mK_{CMB}}$ (hereafter
$\sigma_{QU})$. In our analysis we only use the polarization data with enough
signal-to-noise ratio ($PI/\sigma_{PI}>3$). The uncertainties of polarization
position angle can be calculated using
$\sigma_{\theta}\sim\frac{\sigma_{QU}}{2}\sqrt{{1}/{(Q^{2}+U^{2})}}$.

###  2.2 JCMT dust polarization data

On small scales we use the data to measure relative orientation of magnetic
field from the JCMT/SCUPOL catalog (Matthews et al. 2009). This catalog is a
combination of calibrated and reduced data observed between 1997 to 2005 at
850 $\mu$m by the polarimeter for Submillimeter Common-User Bolometer Array
(SCUBA) on the James Clerk Maxwell Telescope. There are two regions with high-
resolution polarization data in Orion A region: OMC-1 and OMC-2/3. The
individual observations were combined to create composite Stokes I, Q and U
maps. The data are sampled on a 10′′ per pixel grid in J2000 coordinates and
the effective beam size of the map is 20′′ ($\sim 0.04$ pc). We calculate the
polarization position angle projected on the plane of sky from the Stokes
parameters as

$\theta_{\mathrm{p}}^{\mathrm{e}}=\frac{1}{2}\arctan(U,Q)$ (3)

The derived position angle is in equatorial coordinates, and there is no need
for transformation. We apply the same method mentioned in Sect. 2.1 to
calculate the uncertainties and the rms noise of stokes Q and U maps is
approximately 0.1 mJy/beam within OMC-1 and 0.03 mJy/beam within OMC-2/3.

###  2.3 Nobeyama molecular line data

To study the gas structures in Orion A, we select data from the NRO Star
Formation Legacy Project, based on observations at the Nobeyama Radio
Observatory (NRO). The observations covered ${}^{12}\mathrm{CO}$ ($J=1-0$),
${}^{13}\mathrm{CO}$ ($J=1-0$), $\mathrm{C}^{18}\mathrm{O}$ ($J=1-0$),
$\mathrm{N}_{2}\mathrm{H}^{+}$ ($J=1-0$), CCS ($J_{N}=8_{7}-6_{7}$) lines
toward some nearby star-forming regions (M17, Orion A and Aquila Rift).
Detailed information of observations is summarized in Nakamura et al. (2019).
In this paper, we use three different molecular lines with significant
extended emission: ${}^{12}\mathrm{CO}$ (J=1-0), ${}^{13}\mathrm{CO}$ (J=1-0),
$\mathrm{C}^{18}\mathrm{O}$ (J=1-0). The ${}^{12}\mathrm{CO}$,
${}^{13}\mathrm{CO}$, $\mathrm{C}^{18}\mathrm{O}$ data are convolved to 21.7′′
beam size and reprojected to a common 7.5′′$\times$7.5′′ grid. The velocity
resolution is $0.1\mathrm{\leavevmode\nobreak\ km}\mathrm{\leavevmode\nobreak\
s}^{-1}$ and the noise levels of ${}^{12}\mathrm{CO}$, ${}^{13}\mathrm{CO}$,
$\mathrm{C}^{18}\mathrm{O}$ lines are at a range of $0.5-1.5$ K, $0.2-0.3$ K
and $0.26-0.3$ K, respectively. For the sake of comparison with Planck
observations, we use the maps smoothed to a angular resolution of 5′, and we
directly compare the map between SCUPOL and Nobeyama observations on small
scales without smoothing because of similar angular resolution.

##  3 Methods

###  3.1 Detailed procedure of comparison

Figure 1: Nobeyama Moment-0 maps for 12CO, 13CO, C18O with the resolution of
21.7′′. The white vectors represent the orientation of Planck magnetic fields.
Only segments with enough signal-to-noise ratio are plotted
(PI/$\sigma_{\mathrm{PI}}>$3). The blue circle in the bottom left marks the
resolution of Planck magnetic fields. The black rectangles represent the
specific regions with high-resolution polarization observations. Figure 2: The
magnetic field vectors overlaid on the 12CO moment-0 map of OMC-1. Only the
segments with stokes I signal-to-noise ratio (S/N$>25$) and
PI/$\sigma_{\mathrm{PI}}>$3 are plotted. The black circle in the bottom left
marks the resolution of SCUPOL magnetic fields.  Figure 3: The magnetic field
vectors overlaid on the 12CO moment-0 map of OMC-2/3. Only the segments with
stokes I signal-to-noise ratio (S/N$>25$) and PI/$\sigma_{\mathrm{PI}}>$3 are
plotted. The black circle in the bottom left marks the resolution of SCUPOL
magnetic fields.

Here is the detailed procedure in our work. On large scales, we initially
smoothed the spectral line data cube to match the angular resolution of Planck
observations and then compared the relative orientation of gas intensity
gradients with the polarization position angle (perpendicular to the magnetic
field) at low angular resolution. In special regions with high-resolution
SCUPOL polarization observations, we directly calculated the gradient vector
field of each map and compared it with the orientation of polarization.

###  3.2 Column density maps

The 0th Moment map was created by integrating the emission between 0 and
$18\mathrm{\leavevmode\nobreak\ km}\mathrm{\leavevmode\nobreak\ s}^{-1}$. To
exclude the impacts of unreliable pixels, we selected pixels whose spectra
exhibit at least five velocity channels with a brightness temperature greater
than 8$\sigma$ for ${}^{12}\mathrm{CO}$, 5$\sigma$ for ${}^{13}\mathrm{CO}$,
and 3$\sigma$ for $\mathrm{C}^{18}\mathrm{O}$. The derived moment-0 maps for
Orion A region are shown in Fig. 1, and zoom-in ${}^{12}\mathrm{CO}$ moment-0
maps for OMC-1 and OMC-2/3 with overlaid JCMT-derived magnetic field vectors
are shown in Figs. 2 and 3.

Previous studies show that the relative orientation between gas structures and
magnetic field change from parallel to perpendicular with increasing column
densities (e.g., Planck Collaboration et al. 2016b; Soler et al. 2017). In our
study, we estimated the column density structures the using
${}^{12}\mathrm{CO}$ and ${}^{13}\mathrm{CO}$ lines (see detailed derivation
from Mangum & Shirley 2015; Li et al. 2018, and references therein). Firstly,
assuming LTE and ${}^{12}\mathrm{CO}$ line always optically thick, we
calculated the excitation temperature of ${}^{12}\mathrm{CO}$ according to the
following formula:

$T_{\text{ex}}=5.532\left[\ln\left(1+\frac{5.532}{T_{\text{peak}}+0.819}\right)\right]^{-1}$
(4)

where $T_{\text{peak}}$ is the maximum brightness temperature of the
${}^{12}\mathrm{CO}$ line. Then the optical depth of ${}^{13}\mathrm{CO}$ can
be calculated as

$\tau^{13}(v)=-\ln\left[1-\frac{T_{\mathrm{mb}}(^{13}\mathrm{CO})}{5.29}\left(\left[e^{5.29/T_{\mathrm{ex}}}-1\right]^{-1}-0.164\right)^{-1}\right]$
(5)

where $T_{\mathrm{mb}}(^{13}\mathrm{CO})$ is the brightness temperature of the
${}^{13}\mathrm{CO}$ line. Assuming that the emission of ${}^{13}\mathrm{CO}$
is optically thin, we can use the approximate relation (Pineda et al. 2010):

$T_{\mathrm{ex}}\int\tau dv\approx\frac{\tau_{0}}{1-e^{-\tau_{0}}}\int
T_{\mathrm{mb}}dv$ (6)

where $\tau_{0}$ is the peak optical depth of the ${}^{13}\mathrm{CO}$
emission line. Assuming $\mathrm{H_{2}}/{}^{13}\mathrm{CO}=7.1\times 10^{5}$
(Frerking et al. 1982), we can obtain the total column density of molecular
hydrogen:

$N(\mathrm{H}_{2})^{{}^{13}\mathrm{CO}}=1.72\times
10^{20}\frac{\tau_{0}}{1-e^{-\tau_{0}}}\\\
\times\frac{1+0.88/T_{\mathrm{ex}}}{1-e^{-5.29/T_{\mathrm{ex}}}}\int
T_{\mathrm{mb}}{(^{13}\mathrm{CO})}dv$ (7)

One thing we must note is that the ${}^{13}\mathrm{CO}$ line might be
optically thick in some dense regions within Orion A, but the derived column
density maps were only used to examine the relation between the orientation
and column density, which would not affect our results much.

###  3.3 Calculation of relative orientations

We determined the relative angle between the intensity structures and the
magnetic field using similar methods described in Soler et al. (2013). The
angle $\phi$, between the tangent to the local intensity structure contours
and magnetic field, can be calculated using the following formula (Fissel et
al. 2019):

$\phi=\arctan\ (|\nabla I\times\theta_{\mathrm{p}}|,\nabla
I\cdot\theta_{\mathrm{p}})$ (8)

Here, $\theta_{\mathrm{p}}$ marks the polarization position angle,
perpendicular to the orientation of magnetic fields. $\nabla I$ represents the
orientation of the intensity gradient that is perpendicular to the contours of
intensity. The relative orientation angle $\phi$ is within the range
$[0^{\circ},90^{\circ}]$ and angle outside the range can be transformed to
uniquely determined angle within the limit.

The gradient technique was employed to characterize the orientation of
structures in a scalar field. Within nearby $3\times 3$ pixels, the gradient
of gas intensity structures was derived using the similar method described in
Sokolov et al. (2019). We utilized a first-degree bivariate polynomial
function to describe the gradient field:
$f(\alpha,\delta)=I_{0}+a\Delta\alpha+b\Delta\delta$, where $I_{0}$ represents
the intensity of the central point, $\Delta\alpha$ and $\Delta\delta$
represent the pixel offsets. To ensure precision, only pixels with sufficient
signal-to-noise ratio were included in the gradient analysis. In addition, we
selected pixels with more than 7 good points in the nearby $3\times 3$ kernel
size. A least-squares method was employed to derive the gradient field toward
each pixel:

$p({r})={\mathop{argmin}\limits_{a,b}{\sum_{\Delta\alpha,\Delta\delta\leq
1}{(I({r}^{\prime})-f({r}^{\prime},a,b))^{2}}}}$ (9)

The orientation of gradient and its uncertainty can be given by $\nabla
I=\arctan\ (b,a)$ and $\sigma_{\nabla
I}=\frac{1}{a^{2}+b^{2}}\sqrt{a^{2}\sigma_{b}^{2}+b^{2}\sigma_{a}^{2}}$
(Planck Collaboration et al. 2016b).

We assessed the relative orientation of the magnetic field in relation to the
gas structures using the HRO technique (Soler et al. 2013). The map was
segmented into seven intensity bins, each containing an equal number of
selected pixels based on their intensity values. We used 9 angle bins, each
with a width of $10^{\circ}$. Besides, we replaced the intensity structures
with column density to examine whether there is a transition from parallel to
perpendicular alignment with increasing column density. Uncertainties in the
HRO plot were estimated through a Monte Carlo method by adding the
uncertainties of each parameter pixel by pixel. We repeated this method for
10000 times, and took the mean value and standard deviation as the results and
uncertainties.

###  3.4 Statistical study of relative orientation

####  3.4.1 Histogram Shape Parameter $\xi$

We analyzed the changes in the HRO shape with column density, employing the
histogram shape parameter (Planck Collaboration et al. 2016b):

$\xi=\frac{A_{\mathrm{c}}-A_{\mathrm{e}}}{A_{\mathrm{c}}+A_{\mathrm{e}}}$ (10)

Given the angle $\phi$ ranges from 0 to 90 degrees in our analysis,
$A_{\mathrm{c}}$ represents the area concentrated around 0 degrees in the
histogram ($0^{\circ}<\phi<20^{\circ}$) and $A_{\mathrm{e}}$ is the area
concentrated around 90 degrees in the histogram
($70^{\circ}<\phi<90^{\circ}$). A positive $\xi$ ($>$ 0) indicates a parallel
alignment between intensity structure contours and the magnetic field, while a
negative $\xi$ ($<$ 0) suggests that the intensity structure is more likely to
be perpendicular to $\theta_{\mathrm{B}}$. We use a linear function to
investigate the correlation between $\xi$ and the total gas column density for
each tracer:

$\xi=C_{\mathrm{HRO}}[\
\log_{10}(N_{\mathrm{H}}/\mathrm{{cm}^{-2}})-X_{\mathrm{HRO}}]$ (11)

where $C_{\mathrm{HRO}}$ is the slope of the linear relation, and
$N_{\mathrm{H}}$ is the total gas column density. The molecular gas column
density $N_{\mathrm{H_{2}}}$ can be transferred to the $N_{\mathrm{H}}$
through the relation $N_{\mathrm{H}}=N_{\mathrm{HI}}+2\times
N_{\mathrm{H_{2}}}$. However, because we only focus on regions with high
column density, the contribution of $N_{\mathrm{HI}}$ can be neglected
(Sternberg et al. 2014). Thus, we derived the total gas column density using
$N_{\mathrm{H}}=2\times N_{\mathrm{H_{2}}}$. The value of $X_{\mathrm{HRO}}$
for the $\log_{10}(N_{\mathrm{H}}/\mathrm{{cm}^{-2}})$ represents the
transition point where the relative orientation changes from parallel to
perpendicular. The Pearson Correlation Coefficient and p-value are computed to
measure the significance of the linear correlation. The Histogram Shape
Parameter is used to measure the preference of the angle groups for pure
parallel and perpendicular alignments.

####  3.4.2 Alignment Measure Parameter

Jow et al. (2017) introduced the Projected Rayleigh Statistics (PRS) as a test
for non-uniform relative orientation between two pseudo-vector fields, and it
has been widely used (e.g., Soler 2019; Beuther et al. 2020). However, when
comparing the results of different gas tracers, direct comparison of PRS
values is not feasible since it is not a normalized parameter and can be
influenced by the number of data points. Here we use the normalized Alignment
Measure (AM) parameter, as described by González-Casanova & Lazarian (2017)
and Lazarian & Yuen (2018):

$\mathrm{AM}=2\left\langle\cos^{2}(\phi)\right\rangle-1$ (12)

If the results of AM $\rightarrow 1$, it indicates a significant parallel
alignment. If AM $\rightarrow-1$, it indicates a significant perpendicular
alignment. In cases where there is no preference for angle distribution, the
value of AM will be close to 0. We use the same method as in Eq. 11 to test
the linear correlation between AM and column density. The physical meaning of
the Alignment Measure parameter is similar to the physical meaning of the
Histogram Shape Parameter. The difference lies in $\xi$ being employed to
measure the preference of angle groups for purely parallel and perpendicular
alignments, while AM is used to characterize global orientation distributions.
If we find similar trends in both $\xi$ and AM, the alignment preference will
be more reliable.

(a)

(b)

(c)

Figure 4:  (a): Histogram of relative orientations between intensity
structures for different tracers and magnetic field in the whole Orion A
region. The red, green, blue solid lines represent the angle distribution in
the lowest, medium, highest intensity bins. (b): Histogram Shape Parameter
calculated in different column density bins in the whole Orion A region. The
red solid line is the optimal linear fitting results between $\xi$ and column
density. The gray area represents the 1$\sigma$ confidence interval of the
fitted line. (c): The same as (b), but for the relation between AM and column
density.

##  4 Results

###  4.1 Results on large scales

We examine the relative orientation between gas intensity structures and the
magnetic field across the entire Orion A region at a low resolution ($\sim$
0.6 pc). The HROs in different intensity bins are presented in Fig. 4(a). The
figure illustrates a clear difference in angle distribution among various
intensity bins for 12CO, ranging from a preference for parallel alignment to a
weak preference for perpendicular alignment. This difference becomes less
clear with decreasing optical depth of gas tracers. Both $\xi$ and AM exhibits
a clear anti-correlation with column densities for 12CO and 13CO. Regarding
C18O, the orientation between intensity structures and magnetic field prefers
to be perpendicular across all column density bins, and there is no
correlation between $\xi$/AM and column density. The transition from parallel
to perpendicular is consistent with some previous works (Planck Collaboration
et al. 2016b; Soler 2019) and MHD simulations (Soler et al. 2013; Chen et al.
2016), indicating a trans-to-sub-Alfvénic state (see Liu et al. 2022 for a
review). The variation in results from different gas tracers can be explained
by the fact that different gas molecules trace different density regions. In
high-density regions, the orientation between intensity structures and the
magnetic field is more likely to be perpendicular. The result is consistent
with the deduction described in Fissel et al. (2019).

###  4.2 Results on small scales

(a)

(b)

(c)

Figure 5: Same as Fig. 4, but for the relative orientation in OMC-1.

We compare the relative orientation between gas intensity structures and
magnetic field in two specific areas (OMC-1, OMC-2/3) within the Orion A
region, where high-resolution polarization observations are available ($\sim$
0.04 pc). There are significantly higher column densities traced in these
areas. In this section we present the respective results for each region.

####  4.2.1 Results in OMC-1

Figure 5(a) illustrates the HRO plots within various intensity bins, while
Figs. 5(b) and 5(c) display the correlation between the relative orientation
and column densities. In contrast to the findings on large scales, the results
in the OMC-1 region differ. We find that the relative orientation between gas
intensity structures and magnetic field shows no correlation with column
density in the ${}^{12}\mathrm{CO}$ line, but exhibits a clear transition from
parallel to perpendicular alignment with increasing column density in the
${}^{13}\mathrm{CO}$ and $\mathrm{C}^{18}\mathrm{O}$ lines. Despite slightly
increased uncertainties in calculations due to the limited number of selected
pixels, the linear relationship maintains statistical significance after
performing Monte Carlo simulations as outlined in Sect. 3.3 ($|r|>0.7,p<0.05$,
the $1\sigma$ confidence interval of the best-fit line does not spread much).
This phenomenon can be interpreted by the fact that the ${}^{12}\mathrm{CO}$
line is a low-density gas tracer. In the OMC-1 region, ${}^{12}\mathrm{CO}$ is
optically thick, and its intensity structure cannot effectively trace the
density structure but only the surface of the cloud. On the other hand, the
denser gas tracers, ${}^{13}\mathrm{CO}$ and $\mathrm{C}^{18}\mathrm{O}$,
prove to be more suitable to trace the density structures.

####  4.2.2 Results in OMC-2/3

Figure 6(a) presents the HRO plots within various intensity bins, while Figs.
6(b) and 6(c) show the correlation between the relative orientation and column
densities now for the OMC-2/3 regions. From the figure, it is evident that
there is no statistically significant trend in orientation distribution with
increasing column density (all p-values are greater than 0.05). Both $\xi$ and
AM are close to 0 across all column density bins, indicating that the relative
orientation between intensity structures and the magnetic field is more likely
to be randomly distributed in all tracers. We conclude that there is no
systematic trend in angle distribution with increasing column densities in the
OMC-2/3 region.

A possible explanation for this result is that there are numerous star-forming
regions along the long filaments (Chini et al. 1997) and they are at different
evolutionary stages (Castets & Langer 1995). The relative alignment between
gas structures and the magnetic field is complex in star-forming regions at
small scales. The impact of feedback from star-forming activities can lead to
the realignment of structures, such as reversing the orientation to parallel
via gas flows (Pillai et al. 2020), or inducing a orientation through gravity-
dragged rotation (Beuther et al. 2020; Sanhueza et al. 2021). The relative
alignment between gas structures and magnetic field in high-mass star-forming
regions should be investigated case by case, depending on scales, tracers and
evolutionary stages. The results in OMC-2/3 will be discussed further in Sect.
5.1.

(a)

(b)

(c)

Figure 6: Same as Fig. 4, but for the relative orientation in OMC-2/3.

##  5 Discussion

Figure 7: The cumulative distribution functions of relative orientation
between intensity structures and magnetic field vectors and the projected
$\theta_{3D}$ for Orion A, OMC-1 and OMC-2/3.

###  5.1 Projection effects

We have measured the relative orientation between intensity structures and
magnetic field on the plane of sky. Due to the projection effects, the angle
distribution in two dimensions (2D) is not the same with the angle
distribution in three-dimensional (3D) space. To examine the projection
effects on the measured angle distribution, we carried out a Monte Carol
simulation method used in previous study (see Stephens et al. 2017). Here we
briefly introduce the method. Firstly we created $10^{6}$ pairs of two random
unit vectors within a unit sphere and calculated the angle between the two
vectors in 3D space. The angle is within the limit $[0^{\circ},90^{\circ}]$.
Then we projected the angle from 3D to 2D space and plot the cumulative
distribution functions (CDF) in 2D in three angle bins: (1) purely parallel:
angle range from $0^{\circ}$ to $20^{\circ}$ in 3D space, (2) random
distribution: angle range from $0^{\circ}$ to $90^{\circ}$ in 3D space and (3)
purely perpendicular: angle range from $70^{\circ}$ to $90^{\circ}$ in 3D
space. Finally we plot the CDF for our data and compare it with the projected
3D angle.

We present the CDF of $\phi$ and the projected $\theta_{\mathrm{3D}}$ in Fig.
7. In the OMC-2/3 region, a random angle distribution is observed for all
tracers. However, in Orion A at a low resolution and the OMC-1 region at a
high resolution, a clear transition from random to perpendicular orientation
is evident when changing from low-density tracers to high-density tracers,
indicating that magnetic fields are dynamically important at both scales.
Although the results are influenced by the integration effect along the line
of sight, the close-to-perpendicular alignment of the two projected vectors in
2D must also indicate perpendicular alignment in 3D, following the conclusion
of previous studies (Planck Collaboration et al. 2016a, b).

The perpendicular alignment can be explained by the contraction scenario of a
magnetized, self-gravitating, static collapsing cloud (Mouschovias 1976a, b).
On small scales, given that the spatial resolution of polarization
observations is comparable to the typical width of filaments (Arzoumanian et
al. 2019; André et al. 2022), the filament formation process may affect the
magnetic field structures. Earlier studies discussed the relationship between
magnetic field orientations and filaments, but no consistent alignment
preference was found (Arzoumanian et al. 2021; Baug et al. 2021). In our work,
perpendicular alignment is only observed in the OMC-1 region. One possible
interpretation is that the magnetic field orientation is almost perpendicular
to the major axis of the filament but no similar alignment in the OMC-2/3
region. The gravitational interaction inside the filaments and the BN/KL
outflow helps shaping the magnetic field geometry in OMC-1, suggesting
feedback from star-forming activities may play an important role on the
evolution of magnetic field (Pattle et al. 2017). Another possible explanation
is that OMC-1 is at a later evolutionary stage (Castets & Langer 1995). After
the long-term evolution, gravitational collapse would be restricted to occur
along field lines, and the perpendicular alignment implies that the magnetic
field can not efficiently resist gravity (Koch et al. 2018). Compared with
OMC-2/3, gravity plays a more important role in OMC-1. The results suggest
that perpendicular alignment may only exist in well-evolved, gravity-dominated
regions, but observational evidence is still lacking. Further research with
larger samples is needed in the future.

###  5.2 Comparison of HROs with previous studies

We find a similar slope at different scales using HRO analysis. The transition
column density, representing the shift from parallel to perpendicular
alignment, varies from $10^{21}-4\times 10^{21}\ \mathrm{cm^{-2}}$ on large
scales and $8\times 10^{22}-2\times 10^{23}\ \mathrm{cm^{-2}}$ on small
scales, depending on the tracers employed. The role pf magnetic field is
complex around star-forming regions, with a turning point in $\xi$ and AM
(Planck Collaboration et al. 2016b; Ching et al. 2022). After removing the
data point of highest column density, the derived transition column density is
approximately $4\times 10^{21}\ \mathrm{cm^{-2}}$ at large scales and $1\times
10^{23}\ \mathrm{cm^{-2}}$ at small scales.

Table 1: Summary of HRO analysis Source | Distance | Resolution | Scale | $X_{\mathrm{HRO}}$ | Reference
---|---|---|---|---|---
| (pc) | (′′) | (pc) |  |
Taurus | 140 | 600 | 0.42 | 21.84 | Planck Collaboration et al. 2016b
Ophiuchus | 140 | 600 | 0.42 | 22.70 | Planck Collaboration et al. 2016b
Lupus | 140 | 600 | 0.42 | 21.72 | Planck Collaboration et al. 2016b
Chamaeleon-Musca | 160 | 600 | 0.48 | 21.67 | Planck Collaboration et al. 2016b
Corona Australia (CrA) | 170 | 600 | 0.51 | 24.14 | Planck Collaboration et al. 2016b
Aquila Rift | 260 | 600 | 0.78 | 22.23 | Planck Collaboration et al. 2016b
Perseus | 300 | 600 | 0.90 | 21.76 | Planck Collaboration et al. 2016b
IC5146 | 400 | 600 | 1.20 | 21.79 | Planck Collaboration et al. 2016b
Cepheus | 440 | 600 | 1.32 | 21.90 | Planck Collaboration et al. 2016b
Orion | 450 | 600 | 1.35 | 21.88 | Planck Collaboration et al. 2016b
Vela C-South-Nest | 933 | 180 | 0.84 | 22.23 | Soler et al. 2017
Vela C-South-Ridge | 933 | 180 | 0.84 | 22.40 | Soler et al. 2017
Vela C-Centre-Nest | 933 | 180 | 0.84 | 22.62 | Soler et al. 2017
Vela C-Centre-Ridge | 933 | 180 | 0.84 | 20.69 | Soler et al. 2017
DR21 | 1400 | 14 | 0.10 | 21.20 | Ching et al. 2022
Serpens Main | 415 | 14 | 0.03 | —– | Kwon et al. 2022
OrionA | 414 | 300 | 0.62 | 20.97-21.62 | this work
OMC-1 | 414 | 20 | 0.04 | 22.92-23.24 | this work
OMC-2/3 | 414 | 20 | 0.04 | —– | this work

Planck Collaboration et al. (2016b) investigated the HROs in ten nearby clouds
(d $<$ 450 pc) with a resolution of $10^{\prime}$. They find a common
transition from parallel to perpendicular with increasing column densities,
and the typical transition column density is approximately $5\times 10^{21}\
\mathrm{cm^{-2}}$. Soler et al. (2017) adopted the HRO analysis in Vela C
region (d $\sim$ 933 pc) with a resolution of $3^{\prime}$ using BLASTPol
balloon-borne telescope. The similar trend is found and the transition column
density ranges from $5\times 10^{21}$ to $2\times 10^{23}\ \mathrm{cm^{-2}}$.

Figure 8: The histogram of relative orientation between Planck and JCMT
derived magnetic field in OMC-1 and OMC-2/3. The red line represents the
Gaussian kernel density estimate of the angle difference.

On small scales, Ching et al. (2022) applied the HRO analysis toward the DR21
filament (d $\sim$ 1.4 kpc) using JCMT with a resolution of
$14^{\prime\prime}$ and the resulting transition column density is $1.6\times
10^{21}\ \mathrm{cm^{-2}}$. Kwon et al. (2022) investigated the HRO of the
Serpens Main region (d $\sim$ 415 pc) using JCMT with a resolution of
$14^{\prime\prime}$. They find a complex relationship between $\xi$ and column
density, featuring several turning points and no clear transition column
density. The detailed information of all the HRO analysis is summerized in
Table 1.

Considering all the findings, a consistent trend from parallel to
perpendicular alignment with increasing column densities is observed in most
regions. However, the transition column density is scale-, tracer-, and
environment-dependent. We do not find a uniform transition column density,
suggesting that the HRO analysis should be applied case-by-case. We note that
it is a comparison between single region studies (this work, Soler et al.
2017, Ching et al. 2022, Kwon et al. 2022) and statistical studies (Planck
Collaboration et al. 2016b). The conclusion from case studies can not be
directly utilized in other star-forming regions.

The variation in relative orientation among different gas tracers is also
interesting. Fissel et al. (2019) report no changes in relative orientation
with increasing column densities across all tracers. However, they observe a
transition from parallel to perpendicular alignment when changing from low-
density tracers to high-density tracers. Using a three-dimensional, turbulent
collapsing-cloud MHD simulation, Mazzei et al. (2023) compared relative
alignment between magnetic fields and molecular gas structure across different
tracers. They find good agreement between their simulations and the results
obtained in Fissel et al. (2019). However, They suggest 12CO would remain in
parallel alignment across the whole observer space, which is in contrast with
our results. In our analysis, we have not only observed the variation of
relative orientation with different gas tracers but also identified its
changes with column density at different scales. This implies that the
physical processes in the Orion A molecular cloud are hierarchical and highly
complicated, making them challenging to be explained by the simulations.

###  5.3 The relative orientation between Planck- and JCMT- derived magnetic
field

Another interesting question to explore is the relative orientation between
magnetic fields at different scales. Zhang et al. (2014) first investigated
this question, comparing SMA and parsec-scale magnetic fields and discovering
a bimodal orientation distribution. They suggested that the magnetic field at
the core scale could align either parallel or perpendicular to that of the
clump scale. Li et al. (2015) revealed that the magnetic field directions do
not change much from cloud to clump and core scales in NGC 6334. In our study,
we interpolate the Planck data and calculate the magnetic field orientation at
large scales using the transformation in Sect. 2.1, and then compute the
angular difference in specific regions.

Figure 8 illustrates the distribution of angle differences between large-scale
and small-scale magnetic fields. We use the Gaussian kernel density estimate
(KDE) to represent the probability density function of the angle difference.
In the OMC-1 region, small-scale magnetic fields align well with Planck data,
with 79$\%$ of them having an angle difference within 40 degrees of the large-
scale magnetic fields. Regarding the OMC-2/3 region, the angle difference
shows a broader distribution. This region exhibits a bimodal orientation
distribution, similar to findings reported by Zhang et al. (2014), with 59$\%$
of them displaying an angle difference within 40 degrees. The discrepancy in
alignment between large-scale and small-scale magnetic fields could be
attributed to the impact of foreground dust emission from Planck observations,
as suggested by Gu & Li (2019). A more important factor is the geometry of
small-scale magnetic fields. As discussed in Zhang et al. (2014), a random
orientation of fields would lead to a uniform distribution of angle
differences. As the magnetic field properties vary rapidly in different sub-
regions of OMC-2/3 (Poidevin et al. 2010), the orientation distribution is
much flatter than that in OMC-1. Whether star-forming activities affect the
alignment of magnetic fields at different scales needs further discussion in a
larger sample with well-ordered magnetic field geometry in the future.

##  6 Summary

We compare the gas intensity structures of three gas tracers (12CO, 13CO and
C18O) with magnetic fields at different scales. We apply the HRO technique in
our analysis to examine the relation between the relative orientation of gas
structures with respect to magnetic fields and column densities. Our main
findings are as follows:

1\. We find a similar trend in relative orientation between magnetic field and
molecular gas intensity structures with respect to column densities at
different scales in Orion A. Through a comparison with previous studies, we
find the trend is common in molecular clouds at pc scales but not uniform at
$<$ 0.1 pc scales. The relation between gas structures and magnetic field
should be discussed case by case in individual star-forming regions at small
scales.

2\. When the orientation is not randomly distributed, we see a significant
change in the relative orientation distribution between gas intensity
structures and magnetic fields for three different molecular gas tracers. The
perpendicular alignment is more clear when changing from low-density tracers
to high-density tracers. The correlation between gas intensity structures and
magnetic field is better traced by low-density gas tracers on large scales and
high-density tracers on small scales, indicating the different roles of
magnetic field in cloud formation and filament formation. The results found in
this work only applies to Orion A region and can not be directly extrapolated
for other star-forming regions.

###### Acknowledgements.

We are grateful to an anonymous referee for the constructive comments that
helped us improve this paper. We would like to express our sincere gratitude
to Qizhou Zhang, Tao-Chung Ching, Hua-bai Li, Juan Diego Soler, and Mengke
Zhao for their valuable discussions and insightful suggestions that greatly
improved this work. This work has been supported by the National Science
Foundation of China (12041305), the China Manned Space Project (CMS-
CSST-2021-A09, CMS-CSST-2021-B06), and the China-Chile Joint Research Fund
(CCJRF No. 2211). CCJRF is provided by Chinese Academy of Sciences South
America Center for Astronomy (CASSACA) and established by National
Astronomical Observatories, Chinese Academy of Sciences (NAOC) and Chilean
Astronomy Society (SOCHIAS) to support China-Chile collaborations in
astronomy. This publication makes use of data from the NRO Star Formation
Legacy Project with the Nobeyama 45-m telescope.

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