# Studying magnetic fields and dust in M17 using polarized thermal dust
emission observed by SOFIA/HAWC+

Thuong Duc Hoang University of Science and Technology of Hanoi (USTH), Vietnam
Academy of Science and Technology (VAST), 18 Hoang Quoc Viet, Hanoi, Vietnam
Nguyen Bich Ngoc Department of Astrophysics, Vietnam National Space Center,
Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam
Graduate University of Science and Technology, Vietnam Academy of Science and
Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam  Pham Ngoc Diep Department of
Astrophysics, Vietnam National Space Center, Vietnam Academy of Science and
Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam  Le Ngoc Tram Stratospheric
Observatory for Infrared Astronomy, Universities Space Research Association,
NASA Ames Research Center, MS 232-11, Moffett Field, 94035 CA, USA  Max-
Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53-121, Bonn, Germany
Thiem Hoang Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro,
Yuseong-gu, Daejeon 34055, Republic of Korea  University of Science and
Technology, Korea, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113, Republic of
Korea  Wanggi Lim Stratospheric Observatory for Infrared Astronomy,
Universities Space Research Association, NASA Ames Research Center, MS 232-11,
Moffett Field, 94035 CA, USA  Ngan Le Institute of Astronomy, Faculty of
Physics, Astronomy and Informatics, Nicolaus Copernicus University,
Grudziadzka 5, 87-100 Torun, Poland  Kate Pattle National University of
Ireland Galway, University Road, Galway, Ireland H91 TK33  Department of
Physics and Astronomy, University College London, Gower Street, London WC1E
6BT, United Kingdom  Dieu D. Nguyen Department of Physics, International
University, Quarter 6, Linh Trung Ward, Thu Duc City, Ho Chi Minh City,
Vietnam  Vietnam National University, Ho Chi Minh City, Vietnam  Nguyen Thi
Phuong  Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro,
Yuseong-gu, Daejeon 34055, Republic of Korea  Department of Astrophysics,
Vietnam National Space Center, Vietnam Academy of Science and Technology, 18
Hoang Quoc Viet, Hanoi, Vietnam  Nguyen Fuda Department of Physics,
International University, Quarter 6, Linh Trung Ward, Thu Duc City, Ho Chi
Minh City, Vietnam  Vietnam National University, Ho Chi Minh City, Vietnam
Tuan Van Bui  University of Science and Technology of Hanoi (USTH), Vietnam
Academy of Science and Technology (VAST), 18 Hoang Quoc Viet, Hanoi, Vietnam
Gia Bao Truong Le Department of Physics, International University, Quarter 6,
Linh Trung Ward, Thu Duc City, Ho Chi Minh City, Vietnam  Vietnam National
University, Ho Chi Minh City, Vietnam  Hien Phan University of Science and
Technology of Hanoi (USTH), Vietnam Academy of Science and Technology (VAST),
18 Hoang Quoc Viet, Hanoi, Vietnam  Nguyen Chau Giang  Korea Astronomy and
Space Science Institute, 776 Daedeokdae-ro, Yuseong-gu, Daejeon 34055,
Republic of Korea

###### Abstract

We report the highest spatial resolution measurement of magnetic fields
(B-field) in M17 using thermal dust polarization taken by SOFIA/HAWC+ centered
at 154 $\mu$m wavelength. Using the Davis-Chandrasekhar-Fermi method, we found
the presence of strong B-fields of $980\pm 230\;\mu$G and $1665\pm 885\;\mu$G
in lower-density (M17-N) and higher-density (M17-S) regions, respectively. The
B-field morphology in M17-N possibly mimics the fields in gravitational
collapse molecular cores while in M17-S the fields run perpendicular to the
density structure and display a pillar and an asymmetric large-scale hourglass
shape. The mean B-field strengths are used to determine the Alfvénic Mach
numbers, revealing B-fields dominate turbulence. We calculate the mass-to-flux
ratio, $\lambda$, and obtain $\lambda=0.07$ for M17-N and $0.28$ for M17-S.
The sub-critical values of $\lambda$ are in agreement with the lack of massive
stars formed in M17. To study dust physics, we analyze the relationship
between dust polarization fraction, $p$, and emission intensity, $I$, gas
column density, $N({\rm H_{2}})$, polarization angle dispersion function $S$,
and dust temperature, $T_{\rm d}$. $p$ decreases with intensity as
$I^{-\alpha}$ with $\alpha=0.51$. $p$ also decreases with increasing $N(\rm
H_{2})$, which can be explained by the decrease of grain alignment by
radiative torques (RATs) toward denser regions with a weaker radiation field
and/or tangling of magnetic fields. $p$ tends to first increase with $T_{\rm
d}$ and then decreases at higher $T_{\rm d}$. The latter feature seen in M17-N
at high $T_{d}$ when $N(\rm H_{2})$ and $S$ decrease is evidence of the RAT
disruption effect.

magnetic fields and star formation, M17, SOFIA/HAWC+, dust polarization,
turbulence, Alfvénic Mach number, mass-to-flux ratio, radiative torques

††journal: ApJ††facilities: SOFIA HAWC+, Nobeyama 45m Telescope, GLIMPSE
Spitzer Data, Herchel††software: spectral_cube (Ginsburg et al., 2019), aplpy
(Robitaille, 2019), astropy (Astropy Collaboration et al., 2013, 2018),
reproject (Robitaille et al., 2020)

##  1 Introduction

Star formation is a complex process that involves self-gravity, turbulence,
magnetic fields, and stellar feedback. Understanding the exact role of
magnetic fields in the evolution of molecular clouds (MCs) and star formation
process is a challenge of modern astrophysics. In the past decades, there are
emerging evidences suggesting the importance of magnetic fields (B-fields) in
the evolution of MCs and star formation (McKee & Ostriker, 2007; Derek Ward-
Thompson & McKee, 2020). There are two types of B-field models in which
B-fields play contrasting roles. First, the strong B-field models support a
paradigm of magnetic pressure acting against gravitational collapse of MCs.
This magnetic-driven support is suppressed when the ratio of the core mass to
the magnetic flux exceeds a critical value that turns the cloud into a state
of gravitational collapse to form a new star (Nakano & Nakamura, 1978).
Second, in the weak-field models, B-fields are sufficiently weak in MCs and
dominated by turbulence. Star formation takes place in filaments that are
likely produced by intersection of turbulent supersonic flows (e.g., Padoan &
Nordlund, 1999; Elmegreen, 2000; Mac Low & Klessen, 2004; Crutcher, 2012). It
is, thus, necessary to carry out observations of B-fields in specific MCs
(e.g., M17 in this work) to investigate their effects on star formation and
the subsequent evolution of the entire clouds to testify theoretical
predictions (Seifried & Walch, 2015; Federrath, 2016; Li & Klein, 2019; Derek
Ward-Thompson & McKee, 2020).

Dust polarization induced by aligned grains is widely used to map B-fields
(see e.g., Crutcher, 2012). This method is based on the fact that the
polarization direction of thermal dust emission is perpendicular to the
B-fields. The strength of B-fields can then be estimated using the Davis-
Chandrasekhar-Fermi (DCF) method (Davis, 1951; Chandrasekhar & Fermi, 1953).
The DCF method measures indirectly the strength based on the fluctuations of
B-fields that are encoded in the dispersion of dust polarization directions.
Although we have a full-sky map of dust polarization provided by Planck at 353
GHz (see e.g., Planck Collaboration: et al., 2015) and many studies of
B-fields at MC scales (see e.g., filament structures and star-forming cores;
Dotson, 1996; Houde et al., 2002; Pellegrini et al., 2007; Chen et al., 2012;
Pattle et al., 2017, 2018; Wurster & Li, 2018; Chuss et al., 2019; Hennebelle
& Inutsuka, 2019; Sugitani et al., 2019), dust polarization data at higher
resolutions are still lacking. Recently, we start to have high-resolution
polarisation observations made with the Atacama Large Millimeter/sub-
Millimeter Array (Liu et al., 2020; Beuther et al., 2020; San, 2021).

Dust polarization also allows us to get insights into fundamental properties
of dust grains such as grain shapes, size distribution, and alignment. Grain
alignment is a longstanding problem of astrophysics, and the leading theory
for grain alignment is the Radiative Torque Alignment theory (RATA; Draine &
Weingartner 1997; Lazarian & Hoang 2007; Hoang & Lazarian 2016) (see reviews
e.g., Andersson et al., 2015; Lazarian et al., 2015). Note that radiative
torques (RATs) arising from the interaction of an anisotropic radiation field
with an irregular grain were first suggested by Dolginov & Mytrophanov (1976)
and later numerically demonstrated by Draine & Weingartner (1996), and
analytically modelled by Lazarian & Hoang (2007). The grain alignment
efficiency by RATs is found to increase with increasing radiation field or
dust temperature (see Hoang et al. 2021), which results in the increase of the
polarization fraction with the dust temperature (Lee et al., 2020).
Furthermore, Hoang et al. (2019) realized that large grains will be disrupted
and depleted around a very strong radiation source via a mechanism, the so-
called Radiative Torque Disruption (RATD; see Hoang 2020 for a review). The
basic idea of the RATD mechanism is that an intense radiation field can spin
up dust grains to extremely fast rotation, such that the centrifugal stress
can exceed the tensile strength of the grain material and disrupts the dust
grain into smaller fragments. The RATD effect is found to decrease the
polarization fraction predicted by the RATA theory (Lee et al., 2020). The
combination of RATA and RATD was demonstrated to successfully reproduce the
observed dust polarization data in various regions of strong radiation fields
such as Oph-A (Tram et al., 2021a), 30 Doradus (Tram et al., 2021). Therefore,
dust polarization observations toward strong radiation sources like M17 are
crucial to test grain alignment and disruption by RATs.

Figure 1: An RGB image of M17 observed by Spitzer (R = 8 $\rm\mu m$, G = 4.5
$\rm\mu m$, and B = 3.6 $\rm\mu m$). The white rectangle is the observed
region of SOFIA/HAWC+ used in this work.

M17 is a well-known star-forming region (Povich et al., 2009; Lim et al.,
2020) which is located in the Omega Nebula or Swan Nebula (also known as
Horseshoe Nebula) in the constellation of Sagittarius at a distance of 1.98
kpc (Xu et al., 2011). Figure 1 is a RGB image of the region using
Spitzer111https://www.spitzer.caltech.edu Galactic Legacy Infrared Midplane
Survey Extraordinaire (GLIMPSE) mosaic data. Besides the study of B-fields and
dust physics, since M17 is the closest giant H II region to Earth it is thus
an excellent laboratory for the investigation of stellar feedback from a
nearby massive star cluster and a photodissociation region (PDR) as well as
infrared sources in the regions such as UC 1, IRS 5, CEN 92, Anon 1, and Anon
3 (Lim et al., 2020). In the present work, we use data taken by the High-
resolution Airborne Wideband Camera Plus (HAWC+; Harper et al., 2018)
accommodated on Stratospheric Observatory for Infrared Astronomy (SOFIA; Temi
et al., 2018) to serve our purposes.

The structure of the paper is organized as follows. The SOFIA/HAWC+
observations of M17 are presented in Section 2. In Section 3, we describe the
use of the DCF method to estimate the strengths of B-fields. We then present
the obtained results, including the B-field morphologies and strengths, and
discuss the implications of dust polarization fraction for grain alignment and
disruption in Section 4. A summary of our main findings is presented in
Section 5.

Figure 2: From left to right: $I$, $Q$, and $U$ maps.

##  2 Observations

The thermal dust polarization data of M17 were obtained by SOFIA/HAWC+
instrument centered at 154 $\rm\mu m$ with a designed beam size of $13.6$″.
The observed region is shown in Figure 1. The inferred maps have an original
pixel-size of $6.9$″(Harper et al., 2018). Then, Nyquist samplings were
applied during the data reduction processes to have the final Stokes parameter
maps with a re-sampled pixel-size of $\sim 3.4$″. The Stokes $I$, $Q$, and $U$
maps are shown in Figure 2.

The biased polarization fraction, $p_{\rm bias}$, is calculated as follows
(Gordon et al., 2018):

$p_{\rm
bias}=100\sqrt{\left(\frac{Q}{I}\right)^{2}+\left(\frac{U}{I}\right)^{2}}=100\frac{I_{p}}{I}\quad[\%],$
(1)

where $I_{p}=\sqrt{Q^{2}+U^{2}}$ is the polarization intensity. The associated
error on the biased polarization fraction is

$\sigma_{p}=p_{\rm
bias}\left[\left(\frac{\sigma_{I_{p}}}{I_{p}}\right)^{2}+\left(\frac{\sigma_{I}}{I}\right)^{2}\right]^{1/2},$
(2)

where $\sigma_{I_{p}}$ and $\sigma_{I}$ are the uncertainties on $I_{p}$ and
$I$, respectively. The error propagation on the non-linear function
$I_{p}(Q,U)$ is expanded as

$\sigma_{I_{p}}^{2}=\left|\frac{\partial I_{p}}{\partial
Q}\right|^{2}\sigma_{Q}^{2}+\left|\frac{\partial I_{p}}{\partial
U}\right|^{2}\sigma_{U}^{2}+2\frac{\partial I_{p}}{\partial Q}\frac{\partial
I_{p}}{\partial U}\sigma_{QU}.$ (3)

Assuming $Q$ and $U$ are uncorrelated, namely the covariance term
$\sigma_{QU}=0$, we obtain the error for the polarization intensity

$\sigma_{I_{p}}=\left[\frac{Q^{2}\sigma_{Q}^{2}+U^{2}\sigma_{U}^{2}}{Q^{2}+U^{2}}\right]^{1/2},$
(4)

where $\sigma_{Q}$ and $\sigma_{U}$ are the errors on Stokes $Q$ and $U$,
respectively. The debiased polarization fraction, $p$, is calculated following
the approach by Wardle & Kronberg (1974)

$p=\sqrt{p_{\rm bias}^{2}-\sigma_{p}^{2}}.$ (5)

The polarization angle, $\theta$, is defined as

$\theta=\frac{1}{2}\tan^{-1}\left(\frac{U}{Q}\right),$ (6)

and the error on the polarization angle is calculated as follows

$\sigma_{\theta}=\frac{\sqrt{(Q\sigma_{U})^{2}+(U\sigma_{Q})^{2}}}{2(Q^{2}+U^{2})}.$
(7)

A detailed exploration of the raw data is presented in Appendix A. In the
following, we apply two quality cuts to the data, requiring a high signal-to-
noise ratio (SNR) on the measurements of (1) the intensities $\rm SNR_{I}>236$
and (2) the polarization fraction $\rm SNR_{p}>3$. In what follows, we call
this ‘master cut’. This quality cut automatically satisfies the third
criterion recommended by the SOFIA collaboration for high-quality scientific
data, namely $p<50\%$ (Gordon et al., 2018). The choice of the cut on $\rm
SNR_{I}$ is to have the measurement uncertainties on the polarization
fraction, $\sigma_{p}$, better than $0.6\%$. The approximate correlation
between these two quantities is ${\rm
SNR_{I}}=\dfrac{\sqrt{2}}{\sigma_{p}}\approx 236$ (Gordon et al., 2018). After
applying the master cut, the remaining pixels are 5139, corresponding to 23%
of an original map of the size 138$\times$162 (22356 pixels). The master cut
excludes the low level emission and significantly improves the quality of the
data. Its performance is illustrated in Figure A.1 and Table A.1. Subsequent
analyses are carried out using this master cut.

Figure 3: The inferred B-field orientation map of the M17 region observed by
SOFIA/HAWC+. The half-vectors are proportional to the polarization fraction
and rotated by $90^{\circ}$ from the polarization angles to trace the
B-fields. The color scale shows the total intensity in units of $\rm
Jy\;arcsec^{-2}$. The intensity map is smoothed with a kernel beam of
3$\times$3 pixels across. A line segment of $10\%$ polarization is shown as a
reference in the upper right corner. The beam size is shown in the lower left
corner. The blue ellipses are the M17-N and M17-S regions mentioned in the
text.

Figure 3 shows the inferred B-field orientation map of M17. The lengths of the
line segments are proportional to the polarization fraction, and their
orientations are obtained by rotating 90∘ from the polarization angles to
trace the B-fields. These segments are called ‘half-vectors’ since the
directions of the vectors are not known. The map gives the first impression of
the general B-field morphology and the polarization properties of the 154
$\mu$m emission of the region.

Figure 4: Structure functions $\langle\Delta\theta^{2}(\ell)\rangle^{1/2}$ for
M17-N (left), and M17-S (right): black dot lines are the data and black curves
are the fitted models. In the inserts,
$\langle\Delta\theta^{2}(\ell)\rangle^{1/2}$ of M17-N and M17-S are shown in
black curves for the full range of $\ell$, respectively. The horizontal dashed
lines are the random field value of $\pi/\sqrt{12}\sim 52^{\circ}$ (Serkowski,
1962; Planck Collaboration: et al., 2015). The vertical dashed lines are the
SOFIA/HAWC+ beam size of 13.6″at 154 $\rm\mu m$.

##  3 Data analysis

The Davis-Chandrasekhar-Fermi method is one of the most well-known techniques
for the estimation of the B-field strength. There is some controversy around
the method (e.g., Pattle & Fissel, 2019; Liu et al., 2021), however the DCF
method still provides a mean for estimating the magnetic field strength from
dust polarization measurements. The method is based on the fact that turbulent
motions induce a turbulent component on top of the mean, large-scale magnetic
field and the assumption that the turbulent kinetic energy is balanced by the
induced magnetic energy. The strength of B-fields in the plane-of-sky, $B_{\rm
POS}$, is calculated using the following equation (Crutcher, 2004)

$B_{\rm POS}=Q_{c}\sqrt{4\pi\rho}\frac{\sigma_{\nu}}{\sigma_{\theta}}\approx
9.3\sqrt{n(\rm H_{2})}\frac{\Delta V}{\sigma_{\theta}}\ [\rm\mu G],$ (8)

where $Q_{c}$ is a factor of the order of unity to correct for the light-of-
sight and beam-integration effects, $\rho$ is the gas density in $\rm
g\;cm^{-3}$, $\Delta V=\sigma_{\nu}\sqrt{8\ln{2}}=2.355\sigma_{\nu}$ is the
FWHM of the one dimensional non-thermal velocity dispersion, $\sigma_{\nu}$,
in $\rm km\;s^{-1}$, $n(\rm H_{2})$ is the volume density of the molecular
hydrogen in $\rm cm^{-3}$, and $\sigma_{\theta}$ is the polarization angle
dispersion in degrees. In general, the magnetic field strength in MCs is in
the range of micro- to milligauss.

Figure 3 indicates two distinct regions encompassed by the two ellipses in
terms of physical conditions, including density, temperature, and dynamics, as
can be seen in Figures 5, 7, and 11 in the next sections. The ellipse in the
north has a center at RA $\sim$ $\rm 18^{h}20^{m}31^{s}.6$ and DEC $\sim$
$\rm-16^{\circ}08\arcmin 04\arcsec.3$, major $\times$ minor axes of
137″$\times$ 69″, and a position angle of $14^{\circ}$. The ellipse in the
south has a center at RA $\sim$ $\rm 18^{h}20^{m}23^{s}.2$ and DEC $\sim$
$\rm-16^{\circ}12\arcmin 24\arcsec$, major $\times$ minor axes of 200″$\times$
143″, and a position angle of $0^{\circ}$. Therefore, we divide the map into
two corresponding northern and southern regions coined M17-N and M17-S for
further analyses. Subsequently, we are going to identify the input parameters
for the DCF method to calculate the B-field strengths of the two regions.

###  3.1 Polarization angle dispersion: Structure function

Structure function method was initially proposed by Falceta-Gonçalves et al.
(2008) and Hildebrand et al. (2009) for evaluating the polarization angle
dispersion, $\sigma_{\theta}$, in the plane-of-sky. Fundamentally, this method
calculates a two-point correlation function for pairs of the polarization
angles as follows,

$\langle\Delta\theta\left(\ell\right)\rangle^{1/2}=\Bigg{\\{}\frac{1}{N\left(\ell\right)}\sum_{i=1}^{N\left(\ell\right)}\left[\theta\left(\mathbf{x}\right)-\theta\left(\mathbf{x}+\boldsymbol{\ell}\right)\right]^{2}\Bigg{\\}}^{1/2},$
(9)

where $\langle...\rangle$ denotes an average; $N(\ell)$ is the number of pairs
of pixels having a displacement between the two pixels
$\lvert\boldsymbol{\ell}\rvert=\ell$; $\mathbf{x}$ and
$\mathbf{x}+\boldsymbol{\ell}$ are the location vectors of the two considered
pixels with corresponding polarization angles $\theta(\mathbf{x})$ and
$\theta(\mathbf{x}+\boldsymbol{\ell})$, respectively. We note that only the
pairs with
$\Delta\theta(\ell)=\left|\theta(\mathbf{x})-\theta(\mathbf{x}+\boldsymbol{\ell})\right|<90^{\circ}$
are retained for further analyses because the directions of the half-vectors
are unknown. Assuming that the B-fields have two independent components (a
large-scale structure field, $B_{0}$, and a turbulent one, $\delta B$), the
structure function can be written as below for a small displacement $\ell$
(Hildebrand et al., 2009; Crutcher, 2012)

$\langle\Delta\theta\left(\ell\right)\rangle=b^{2}+m^{2}\ell^{2}+\sigma^{2}_{M}(\ell),$
(10)

where $b$ is the contribution of turbulence, $m\ell$ of the large-scale
structure component; and $\sigma_{M}(\ell)$ is the measurement uncertainties
calculated for each pair of polarization angles. In practice,
$\sigma_{M}(\ell)$ is taken into account while calculating
$\langle\Delta\theta\left(\ell\right)\rangle^{1/2}$ in Equation 9. The ratio
of the turbulent to large-scale structure components is expressed as
(Hildebrand et al., 2009)

$\frac{\delta B}{B_{0}}=\dfrac{b}{\sqrt{2-b^{2}}}.$ (11)

From the definition of polarization angle dispersion, $\sigma_{\theta}$, in
Equation 9 combining with Equation 10 neglecting the contribution of the
large-scale structure component, $m\ell$, we have
$\sigma_{\theta}=b/\sqrt{2}$.

We calculate the structure functions for M17-N and M17-S separately. In
general, they display similar rising tendencies at small displacements $\ell$
and reach the value of random field of $52^{\circ}$ at large displacements
(see Figure 4). The extent of M17-N is smaller than that of M17-S, hence the
maximum displacement of M17-N is smaller. We then fit the calculated structure
functions with the $\ell$-dependence of the polarization angle dispersion
using Equation 10. The fits were carried out for $\ell>13.6\arcsec$ to avoid
the beam effects. The upper $\ell$-limits were chosen to achieve the best fits
(black curves in Figure 4). Table 1 lists the obtained values $b$ = $4.9\pm
0.2$ and $8.8\pm 0.7$ which means $\sigma_{\theta}$ = $3.5^{\circ}\pm
0.2^{\circ}$ and $6.2^{\circ}\pm 0.5^{\circ}$ for M17-N and M17-S,
respectively. The uncertainties on $b$ are from the fits. We note these values
are smaller than that found by Hildebrand et al. (2009), $\sigma_{\theta}\sim
10^{\circ}$, using data from Caltech Submillimeter Observatory at $350\ \rm\mu
m$ with a spatial resolution of $\sim 20\arcsec$ for M17. Table 1 also lists
the ratios of the turbulent to large-scale structure B-field components
calculated using Equation 11 which shows that the turbulent fields are much
smaller than the large-scale one in M17.

###  3.2 Velocity dispersion

We estimate the velocity dispersion of M17 using the publicly available
archival data $\rm{}^{13}CO\ (J=1\rightarrow 0)$ taken by Nobeyama 45-m
telescope (Nakamura et al., 2019). The measurements were made at the central
frequency $\nu_{\rm central}\sim 110.201\ \rm{GHz}$ with a spectral resolution
of 0.1 km s-1. The resulting map has a pixel size of 7.5″and a beam size of
14.9″which was then cropped to match with the M17 observed region by
SOFIA/HAWC+. Figure 5 presents the velocity-integrated intensity (left) and
the mean velocity (right) maps. The integration was done over the entire
velocity range from $-20$ to $60\ \rm km\;s^{-1}$ in the local standard of
rest (LSR) frame. Figure 6 displays the integrated spectra averaged over the
number of pixels for M17-N (left) and M17-S (right). They can be well-
described by two-Gaussian fits giving the mean positions and corresponding
standard deviations, $\sigma_{\nu;{\rm{}^{13}CO}}$, of the peaks. The spectra
of M17-N and M17-S show the main peaks at $22.7\pm 1.6$ and $19.9\pm 2.3$ km
s-1, respectively. These main peaks are much higher than the second ones.
Therefore, we only use the velocity dispersion of these two peaks for the
calculation of B-field strengths in M17-N and M17-S. Table 1 summarizes the
results of the fits which are in agreement with the results obtained by
Nakamura et al. (2019); Nguyen-Luong et al. (2020), although their results are
for wider areas of M17. The statistical uncertainties of the velocity
dispersion from the two-Gaussian fits are smaller than the spectral resolution
of Nobeyama, therefore, we take the uncertainties equal to the spectral
resolution of 0.1 km s-1. We note that M17-N is affected by the outflows and
shocks from G015.128222G015.128 is a massive young stellar object (MYSO) and
likely an A-type super-giant star (Pomohaci et al., 2017). The current
SOFIA/HAWC+ data does not fully cover this source. (Lim et al., 2020) which is
expected to be more turbulent than M17-S.

Figure 5: Left: velocity-integrated intensity map. Right: mean velocity map. The contour levels are at 20, 35, 70, 100, and 200 K km s-1 for both maps. Figure 6: Average integrated spectra (black) with two-Gaussian fits (red) for M17-N (left) and M17-S (right). Figure 7: Map of the column density, $N(\rm H_{2})$, of M17. The contours display the dust temperature, $T_{\rm d}$, values as shown in Figure 11. The blue ellipses are the same as in Figure 3. In M17-S, the relation of $T_{\rm d}$ and $N(\rm H_{2})$ is complex. $T_{\rm d}$ is positively correlated to $N(\rm H_{2})$ up to $\sim 43\,\rm K$, but turns to negatively otherwise. Table 1: Summary of the obtained results. $b$ is the rms contribution of the turbulent component and $m$ of the large-scale B-fields to the polarization angle dispersion, $\sigma_{\theta}$. $\delta B/B_{0}$ is the ratio of the turbulent to the large-scale fields. Detailed description of the parameters can be found in the text. Parameters | M17-N | M17-S
---|---|---
$b$ | $4.9\pm 0.2$ | $8.8\pm 0.7$
$m$ | $0.55\pm 0.002$ | $0.55\pm 0.01$
Polarization angle dispersion, $\sigma_{\theta}$ [deg] | $3.5\pm 0.2$ | $6.2\pm 0.5$
$\delta B/B_{0}$ | 0.06 | 0.11
Peak position, $\nu_{\rm LSR}\ \rm[km\;s^{-1}]$ | 22.7 $\pm$ 0.1 | 19.9 $\pm$ 0.1
Velocity dispersion, $\sigma_{\nu,{\rm{}^{13}CO}}\ \rm[km\;s^{-1}]$ | 1.6 $\pm$ 0.1 | 2.3 $\pm$ 0.1
Column density, $\langle N({\rm H_{2}})\rangle\ \rm[cm^{-2}]$ | $(9.1\pm 4.0)\times 10^{21}$ | $(6.2\pm 6.5)\times 10^{22}$
Volume density, $n({\rm H_{2}})\ \rm[cm^{-3}]$ | $(9.3\pm 4.1)\times 10^{3}$ | $(4.2\pm 4.4)\times 10^{4}$

The obtained standard deviations are then converted to the non-thermal
velocity dispersion using
$\sigma^{2}_{\nu}=\sigma^{2}_{\nu;{\rm{}^{13}CO}}-\dfrac{k_{\rm
B}T}{m_{{}^{13}\rm CO}}$, where $m_{{}^{13}\rm CO}$ is the $\rm{}^{13}CO$
molecule mass equal to 29 amu, $k_{\rm B}$ is the Boltzmann constant, and $T$
is the gas temperature. It turns out that the thermal contribution to the
velocity dispersion is negligible if we adopt the average gas temperature of
20 K according to Nguyen-Luong et al. (2020). Therefore, we take the non-
thermal velocity dispersion for the M17-N and M17-S regions equal to the
standard deviations of their corresponding main peaks listed in Table 1.

###  3.3 Column and volume densities

The column density, $N({\rm H_{2}})$, has been derived based on a graybody
(i.e., modified blackbody) fit with the filter weighted opacity,
$\kappa_{\rm\nu}$, and blackbody radiation, $B_{\nu}(T_{\rm d})$. As following
the techniques described in Lim et al. (2016), Herschel 160, 250, and
$350\;\micron$ data of M17 have been convolved to the beam size of Herschel
$500\;\micron$ images ($\sim 36\arcsec$) before executing the pixel-by-pixel
graybody fitting via all four band data. The template $T_{\rm d}$ of angular
resolution $\sim 36\arcsec$ is re-gridded to match to the pixel geometry of
SCUBA2 $850\;\micron$ map (angular resolution $\sim 14\arcsec$ and pixel
size=$4\arcsec$). We then repeated the graybody fit by utilizing SCUBA2
$850\;\micron$ map (Reid & Wilson, 2006) of M17 with the re-gridded $T_{\rm
d}$ map. Figure 7 shows the resulting $N({\rm H_{2}})$ map superimposed by the
dust temperature (Figure 11). Using this map, we calculate the total column
densities, $N_{\rm total,H_{2}}$, the average column densities, $\langle
N({\rm H_{2}})\rangle$, as well as their associated uncertainties,
$\sigma_{N({\rm H_{2}})}$, for the M17-N and M17-S regions. The results are
presented in Table 1.

We then follow the approach described in Lee et al. (2012); Li et al. (2014);
Ngoc et al. (2020) to estimate the volume density, $n(\rm H_{2})$. The mass of
a region is calculated by $M=\beta m_{\rm H_{2}}N_{\rm
total,H_{2}}(D\Delta)^{2}$, where $\beta=1.39$ accounts for $\rm He$ of $10\%$
abundance in addition to $\rm H_{2}$ in the total mass, $m_{\rm H_{2}}$ is the
mass of hydrogen molecules, $D=1.98\ \rm kpc$ is the distance to M17 (Xu et
al., 2011), and $\Delta=4\arcsec$ is the pixel size of the $N({\rm H_{2}})$
map. Then the volume density $n(\rm H_{2})$ is expressed as

$n({\rm H_{2}})=\frac{3M}{4\pi R^{3}m_{\rm H_{2}}}=\frac{3\beta N_{\rm
total,H_{2}}(D\Delta)^{2}}{4\pi R^{3}}\quad\rm[cm^{-3}],$ (12)

where $R$ is the radius of a considered clump equal to $R=\sqrt{(ab)/2}$ ($a$
and $b$ are the major and minor axes of an ellipse covering the clump (Li et
al., 2014)). Equation 12 is deduced assuming spherical geometry of the
molecular clump. As shown in Figure 3 and 7, we define two ellipses that
encompass the observed M17-N and M17-S regions with corresponding angular
radii of $69\arcsec$ and $120\arcsec$, respectively. The physical radii of the
two clumps are obtained by multiplying $R$ with the distance $D$ to M17.
Employing Equation 12, the volume densities, $n(\rm H_{2})$, are calculated,
and the results are shown in Table 1 for both M17-N and M17-S regions. The
uncertainty on $n(\rm H_{2})$ is calculated assuming that its relative
uncertainty is the same as that of $N_{\rm total,H_{2}}$, namely
$\dfrac{\sigma_{n(\rm H_{2})}}{n(\rm H_{2})}=\dfrac{\sigma_{N(\rm
H_{2})}}{\langle N(\rm H_{2})\rangle}$.

##  4 Results and Discussion

In this section, we first describe the magnetic field morphology of the
observed region and report the magnetic strengths in the plane of the sky,
$B_{\rm POS}$, using the DCF method for the M17-N and M17-S regions. We then
explore the relative contribution of B-fields, gravity, and turbulence by
calculating the mass-to-flux ratios and Alfvénic Mach numbers. Finally, we
investigate dust grain alignment and disruption based on the polarization
fraction in the observed regions.

###  4.1 Magnetic field morphology

Figure 8: The B-field morphology of M17 superimposed on the intensity map.
Half-vectors are rotated by $90^{\circ}$ from the polarization angles to
represent the magnetic field orientations. Here, the half-vectors are plotted
with uniform length. The star symbols represent the most massive ($>$O7) stars
in the open cluster NGC 6618 (Hanson et al., 1997). There are several infrared
sources present in M17-S such as UC 1, IRS 5, CEN 92, Anon 1, and Anon 3. The
KW (Kleinmann-Wright) object is a binary-star system Kleinmann & Wright
(1973). The locations of the sources are adopted from Lim et al. (2020).
Figure 9: A RGB image of M17 using Spitzer data as described in Figure 1. The
white box is the observed area of SOFIA/HAWC+ toward M17. The half-vectors are
plotted with uniform lengths using SOFIA/HAWC+ data and rotated by 90∘ to
trace the magnetic fields. In this figure, we can see the larger scale
structure mentioned in the text connected to the M17-N region. Figure 10: Same
as Figure 8 with the ‘drapery’ patterns produced by the line integral
convolution (LIC) tool (Cabral & Leedom, 1993).

The magnetic field morphology is imprinted on star formation processes. We
thus can use the observational data to test theoretical predictions of star
formation.

Overall, over the whole region, B-fields in the outer, low-density regions are
perpendicular to the density structure. Figure 3 and its close-up version
(Figure A.3) illustrate well this point with many half-vectors in the low-
density regions orthogonal to the intensity contours. Another prominent
feature is that when the field lines pass through the H II region, east of the
observed region (see Figure 1 of Povich et al. (2009) for a more precise
location of the H II region), they tend to run parallel threading M17-N and
M17-S. However, in M17-S, the fields run perpendicular to these fields behind
the high-density areas at the center (see Figure 8).

In the M17-N region, the main B-fields line up along the north-south direction
through the highest density area and curve in the eastern and western sides of
the region (see Figures 8 and A.3). The morphology of the B-fields in M17-N
mimics the fields of a gravitationally collapsed molecular core with the
presence of strong magnetic fields where we see the classical hourglass
geometry with straight fields in the middle and curved ones at the two wings.
This is a common structure which we often find in observations and simulations
(see e.g., Kandori et al., 2018; Wurster & Li, 2018; Pattle et al., 2019). On
the other hand, M17-N is only a small part of a much larger M17-N region (see
Figure 9) which is not fully covered by the current SOFIA/HAWC+ observations.
Therefore, the observed structure may be just a coincidence with the fields
following the emission by Polycyclic Aromatic Hydrocarbons (PAHs) at 8 $\mu$m
wavelength observed by Spitzer (the bright pink color structure in the
northern region in Figure 9). Distinguishing these two scenarios requires
observations of higher angular resolution and larger areas of M17-N.

In the M17-S region, generally, the fields run perpendicular to the elongation
of the high-density structure. In the region, the fields also form an
asymmetric large-scale hourglass shape (see Figures 8, 9, and 10). In the
southern side of the hourglass, just below Anon 1 and Anon 3, the fields run
from east toward west while gradually bending the to south. The fields are
highly curved in the most southern part of the region and perturbed in the
high density regions at the center. On the northern side of the hourglass,
just above UC 1 and CEN 92, we see the curved fields (see Figures 8 and 10).
This hourglass structure seems to be clearly caused by the gravitational
contraction of the massive cores at the center of M17-S. Moreover, the
asymmetry is due to the complex distribution of matter in the center and
north-western part of the region. Another prominent feature of M17-S, which is
commonly found in PDR regions (see e.g., Pattle et al. (2018)), is the pillar
structure. Here, we find a ‘triangular’ pillar with the top-end coincident
with the positions of UC 1 and IRS 5 and a ‘base’ in the western direction
(see Figures 8 and 10). It is interesting to note that the magnetic field
morphology found here is similar to the ones reported by Pattle et al. (2018)
where the fields run parallel to the pillar in the H II region side and
perpendicular to the pillar on and behind the base.

Dotson (1996) reported that magnetic field lines are elongated into the cloud
core and bulged away from the H II region, heated by OB stars. This could be
an evidence that the H II region is expanding into its surrounding medium
(Zeng et al., 2013). As a conclusion, our magnetic field morphology is
consistent with the scenario that the magnetic field is distorted by H II
region (see more in Section 4.5), several IR sources, and stellar clusters.

###  4.2 Magnetic field strengths

Given the estimated values of the polarization angle dispersion,
$\sigma_{\theta}$, the one dimensional non-thermal velocity dispersion,
$\sigma_{\nu}$, and the volume densities, $n(\rm H_{2})$, in Table 1, we
calculate the strengths of B-fields in the plane of the sky using Equation 8,
yielding $B_{\rm POS}=980\pm 230\ \rm\mu G$ and $1665\pm 885\ \rm\mu G$ for
M17-N and M17-S, respectively (see more details in Table 2). The magnetic
field strength of M17-S is greater than that of M17-N because of its higher
density.

We note that the DCF method is applicable here since all the polarization
angle dispersion is smaller than 25∘ (Crutcher, 2004). The uncertainties are
propagated from the uncertainties on $\sigma_{\theta}$, $\sigma_{\nu}$, and
$n(\rm H_{2})$ using the following equation

$\frac{\delta B_{\rm POS}}{B_{\rm POS}}=\sqrt{\left(\frac{1}{2}\frac{\delta
n(\rm H_{2})}{n(\rm H_{2})}\right)^{2}+\left(\frac{\delta\Delta V}{\Delta
V}\right)^{2}+\left(\frac{\delta\sigma_{\theta}}{\sigma_{\theta}}\right)^{2}},$
(13)

where $\delta n(\rm H_{2}),\ \delta\Delta V$, and $\delta\sigma_{\theta}$ are
the uncertainties on $\sigma_{\nu}$, $\Delta V=2.355\sigma_{\nu}$, and
$\sigma_{\theta}$, respectively.

Table 2: Results of the magnetic field strengths, Alfvénic Mach numbers, mass-to-flux ratios for M17-N and M17-S. $\gamma$ is the inclination angle of the B-fields with respect to the line of sight. Region | $B_{\rm POS}$ | Mach number | Mass-to-flux ratio
---|---|---|---
| $[\rm\mu G]$ | $\mathcal{M_{A}}$ | $\lambda$
M17-N | $980\pm 230$ | $0.12\sin\gamma$ | 0.07 $\pm$ 0.04
M17-S | $1665\pm 885$ | $0.22\sin\gamma$ | 0.28 $\pm$ 0.33

Previous study by Pellegrini et al. (2007) found the magnetic field strength
around the southwestern part of the M17 photodissociation region with the peak
value of $\sim$ 600 $\rm\mu G$. Brogan et al. (1999) measured directly the
magnitude of the B-fields along the line-of-sight (LOS) using the Zeeman
observations of H I absorption lines toward the H II region and obtained
$B_{\rm LOS}\sim-$450-550 $\rm\mu G$. Chen et al. (2012) carried out a rough
estimation of the magnetic field strength based on the polarization of point
sources in NIR and FIR using a sampling rectangle in the M17-S region, and
they found the total magnetic field strength of $B=\sqrt{B_{\rm
LOS}^{2}+B_{\rm POS}^{2}}\sim 230\ \mu G$ and the inclination angle of the
magnetic field vector with respect to the plane-of-sky of $\sim 40^{\circ}$.
Therefore, our estimated values of $B_{\rm POS}=980\pm 230$ $\rm\mu G$ and
$1665\pm 885$ $\rm\mu G$ with rather large uncertainties are still comparable
with previous results.

Several factors affect the estimation of the magnetic field strengths. One is
that we assumed spherical geometries of the molecular clumps to estimate the
radii of the observed regions when estimating the volume densities. Depending
on the real 3D geometry of the clouds, this assumption could lead to a large
uncertainty in $B_{\rm POS}$ values. In addition, polarization observations
integrate over all of the structure along the line of sight, which leads to
the reduction of $\sigma_{\theta}$, therefore, an overestimate of B-field
strengths. Also it is important to note that, statistically, margin of
overestimation by the DCF method on the magnetic field strengths can be up to
a factor of 2 for an individual cloud (Crutcher, 2004). There are efforts to
improve the DCF method such as of Cho & Yoo (2016) who modified the method to
reduce the overestimate of $B_{\rm POS}$ strength by a factor equal to the
ratio of the average line-of-sight velocity dispersion and the standard
deviation of centroid velocities. We calculated the standard deviation of
centroid velocities for the M17-N and M17-S which are equal to 0.59 $\rm
km\;s^{-1}$ and 1.46 $\rm km\;s^{-1}$, respectively. This means that the
conventional DCF method overestimates the strength of the mean magnetic field
in the plane-of-the-sky by a factor of (1.6/1.46)=1.1 and (2.3/0.59)=3.9 in
M17-N and M17-S, respectively (the line of sight velocity dispersion is taken
from Table 1). Therefore, the strengths of B-fields now are $891\pm 209\
\rm\mu G$ and $427\pm 227\ \rm\mu G$ for M17-N and M17-S, respectively.

The calculated B-field strengths have quite large uncertainties which are
mostly propagated from the uncertainties on column densities averaged over
large regions (see the results in Table 1). We, therefore, examine the B-field
strength limiting to the highest density regions with $N(\rm
H_{2})>10^{22}\;cm^{-2}$. With this new cut, only the densest regions of M17-S
are retained (see Figure A.4) and the new results are $N(\rm H_{2})=(7.6\pm
6.5)\times 10^{22}\rm\;cm^{-2}$, $n(\rm H_{2})=(6.9\pm 5.9)\times
10^{4}\rm\;cm^{-3}$, while $\Delta V=(2.3\pm 0.1)\rm\;km\;s^{-1}$ and
$\rm\sigma_{\theta}=6.5^{\circ}\pm 0.5^{\circ}$ are the same as before the cut
is applied. Therefore, the magnetic field strength obtained is $B_{\rm
POS}=2134\pm 738\rm\;\mu G$ and $\lambda=0.27\pm 0.25$ with smaller
uncertainties as expected. This calculation serves as a reference providing a
more precise measurement of the average magnetic strength in the dense region
of M17 and a sense of possible changes of its value depending on the choices
of the regions of interest. In reality, the magnetic field strength is
expected to change with the local gas density.

###  4.3 Alfvénic Mach number $\mathcal{M_{A}}$

The Alfvénic Mach number, $\mathcal{M_{A}}$, represents the relative
contribution of turbulence to magnetic fields. $\mathcal{M_{A}}$ is an
important parameter for describing the evolution of molecular clouds (Kritsuk
et al., 2017). A sub-Alfvénic Mach ($\mathcal{M_{A}}<1$) means the cloud has a
strong magnetic field while a super-Alfvénic ($\mathcal{M_{A}}>1$) implies a
weak magnetic field compared to turbulence. In the super-Alfvénic case, the
magnetic field morphology is significantly affected by turbulence due to it
subdominance. In this scenario, the morphology of the magnetic field is
therefore expected to be random.

$\mathcal{M_{A}}$ can be calculated from the velocity dispersion following
Padoan et al. (2001); Nakamura & Li (2008); Wang et al. (2019) as

$\mathcal{M_{A}}=\frac{\sigma_{\nu}}{v_{A}}=\frac{\sigma_{\nu}\sqrt{4\pi\rho}}{B},$
(14)

where $v_{A}$ is the Alfvénic velocity. Combining with Equation 8, we obtain

$\mathcal{M_{A}}=\frac{\sigma_{\theta}\sin\gamma}{Q_{c}},$ (15)

where $\gamma$ is the inclination angle of B-fields with respect to the line-
of-sight in the range of $[0^{\circ},90^{\circ}]$ with $B_{\rm
POS}=B\sin\gamma$, and $Q_{c}=0.5$ (Ostriker et al., 2001).

Using the inferred polarization angular dispersion in Table 1, we obtained
$\mathcal{M_{A}}=0.12\sin\gamma$ and $0.22\sin\gamma$ for M17-N and M17-S,
respectively. Thus, M17 is sub-Alfvénic, implying that the magnetic fields in
the region dominate turbulence. The sub-Alfvénic Mach numbers are also in
agreement with the generally well-ordered magnetic field morphology in the
region (see Figure 8).

###  4.4 Mass-to-flux ratio

The mass-to-flux ratio, $M/\Phi$, refers the ratio of the mass to the magnetic
flux. This is a crucial parameter describing the role of B-fields relative to
gravity in star formation (Crutcher, 2012). Usually, it is determined by the
critical value, defined as follows (Crutcher, 2004)

$\lambda=\frac{(M/\Phi)_{\rm observed}}{(M/\Phi)_{\rm critical}}=7.6\times
10^{-21}\frac{\langle N(\rm H_{2})\rangle}{B_{\rm POS}},$ (16)

where the critical mass-to-flux ratio $(M/\Phi)_{\rm
critical}=\dfrac{1}{2\pi\sqrt{G}}$ (Nakano & Nakamura, 1978), $\langle N(\rm
H_{2})\rangle$ is the average column density of the considered region in units
of $\rm cm^{-2}$, and $B_{\rm POS}$ is given in units of $\rm\mu G$.333We note
that an analysis by Crutcher (2004) indicated that statistically the true
mass-to-flux ratio can be over-estimated by a factor of three. The errors on
the mass-to-flux ratio are calculated using the following equation

$\frac{\sigma_{\lambda}}{\lambda}=\sqrt{\left(\dfrac{\sigma_{N(\rm
H_{2})}}{\langle N(\rm H_{2})\rangle}\right)^{2}+\left(\frac{\delta B_{\rm
POS}}{B_{\rm POS}}\right)^{2}}.$ (17)

A super-critical value of $\lambda>1$ means that the gravity dominates the
magnetic field pressure, and the cloud can undergo gravitational collapse to
form a protostar. Conversely, a sub-critical value of $\lambda<1$ indicates
that the magnetic field is strong enough to counteract the gravitational
collapse (Crutcher, 2004; Pattle et al., 2017).

We obtained $\lambda=0.07$ and 0.28 for the M17-N and M17-S regions,
respectively (see Table 2). Overall, the M17 cloud is sub-critical. It means
that M17 is magnetically supported and belongs to the strong magnetic field
model of star formation theory (Nakano & Nakamura, 1978). However, the
$\lambda$ values here are calculated by averaging over the whole considered
regions. Therefore, the highest density cores can still become super-critical
and then gravitationally collapse to form new stars. The inferred values of
the mass-to-flux ratio are compatible with theoretical predictions of the am-
bipolar diffusion model where the B-field morphology is dragged inward from
the outer layers of the cloud to the massive cores in both M17-N and M17-S. In
addition, these results are compatible with the deficiency of forming massive
stars and the lack of gravitationally bound clumps in the regions (Nguyen-
Luong et al., 2020).

Figure 11: Dust temperature map of M17 with the contours indicating the
corresponding temperatures. In M17-S, the dust temperature tends to decrease
from the east to the west, in the direction away from the star cluster.

###  4.5 Stellar feedback from the massive star cluster

As shown in Figures 8 and 10, the magnetic fields in the vicinity of the star
cluster (yellow symbols in Figure 10) appear to be bent and aligned with the
density structure (see more in Section 4.1). This suggests the effects of
stellar feedback of the massive star cluster.

To understand the importance of feedback, we estimate the ratio of the ram
pressure, $P_{\rm ram}$, by stellar winds (or expansion of HII region) to the
magnetic pressure, $P_{\rm mag}$, which is given by

$\displaystyle\frac{P_{\rm ram}}{P_{\rm mag}}$ $\displaystyle=$
$\displaystyle\frac{\rho v_{\rm wd}^{2}}{B^{2}/8\pi}$ (18)
$\displaystyle\simeq$ $\displaystyle 4.7\left(\frac{v_{\rm wd}}{20~{}\rm
km\;\rm s^{-1}}\right)^{2}\left(\frac{1000~{}\mu{\rm
G}}{B}\right)^{2}\left(\frac{n({\rm H}_{2})}{10^{4}~{}\rm
cm^{-3}}\right),~{}~{}~{}~{}~{}~{}~{}$

where the wind speed $v_{\rm wd}\sim 20\rm\;km~{}s^{-1}$ is adopted from
Pellegrini et al. (2007). The mass density $\rho=2.8m_{\rm H}n({\rm H}_{2})$.
The volume densities, $n({\rm H}_{2})$, and magnetic field strengths, $B$,
close to those from Table 1 and 2 have been used. The dominance of the ram
pressure over the magnetic pressure as shown in Equation 18 implies that the
winds can bend the magnetic fields frozen in the gas. Stellar feedback due to
the star cluster may also trigger star formation in the M17-S region, as
revealed by the presence of numerous IR sources.

###  4.6 Dust temperature map

As described in Section 3.3, the dust temperature, $T_{\rm d}$, map has been
derived via the graybody fit with Herschel 160-$500\;\micron$ data. Then, the
template $T{\rm{}_{d}}$-map has been re-gridded to match to SCUBA2
$850\;\micron$ map where the final map has the pixel size of 4$\arcsec$.

Figure 11 shows the temperature map of the observed region by Herschel. The
average temperatures are $74\pm 11$ and $43\pm 10$ K for the M17-N, and M17-S,
respectively. Globally, the M17 dust temperature gradually decreases from east
to west and from north to south. This general trend is consistent with the
scenario that dust is heated by the massive star cluster (see e.g., Figure
10). The higher temperatures in M17-N and the eastern and northeastern parts
of M17-S arise from the fact that these regions are located next to the star
cluster and thus strongly heated by this intense radiation source as well as
the H II region in the east. M17-N is additionally affected by shocks from
G015.128 resulting in an obvious temperature excess. In the M17-S, the dust
temperatures decrease along the direction away from the star cluster, which
stems from dust absorption within dense structures of an order of magnitude
higher than that of M17-N.

Figure 12: The variation of the polarization fraction with the total
intensity. The orange, yellow and red lines are the best fit to a power law
model for M17-N, M17-S and the whole map, respectively. The black solid line
is the same power law model with $\alpha=1$.

###  4.7 Polarization fraction vs. emission intensity, column density, and
dust temperature

We now study the dependence of the polarization fraction, $p$, on the total
intensity, $I$, the column density, $N({\rm H_{2}})$, and the dust
temperature, $T_{\rm d}$, which reveal basic properties of grain alignment and
disruption in M17.

Figure 13: Upper-left: the relationship of the polarization fraction and
column density. Upper-right: the relationship of the polarization fraction and
temperature. $\alpha$ is the power index of a power-law model. The SOFIA/HAWC+
polarization map is smoothed to 14″angular resolution of SCUBA2 $850\;\micron$
which were used together with Herschel data to get the $N({\rm H_{2}})$ and
$T_{\rm d}$ maps. Bottom: the variation of the gas column density, $N({\rm
H_{2}})$, with the dust temperature, $T_{\rm d}$.

Figure 14: Left: Variation of the polarization angle dispersion function, $S$,
with the dust temperature, $T_{\rm d}$. The black curve shows the mean and the
standard deviation per bin. Right: Variation of $S$ with the polarization
fraction, $p$. Red dots are pixels with $T_{\rm d}>~{}90$ K. The orange and
red lines are the fit to a power law model for M17-N and M17-S, respectively.

Firstly, we investigate the relationship between the polarization fraction,
$p$, and the total intensity, $I$. It is clearly seen from Figure 3 that
polarization fraction decreases when intensity increases. Generally, the
dependence of $p\propto I^{-\alpha}$ describes the variation of the grain
alignment efficiency and the magnetic field geometry across the cloud. For a
uniform magnetic field, the slope $\alpha=1$ implies that the grain alignment
is present only in the outer layer of the cloud and becomes completely lost in
the inner region. Figure 12 shows a fitted power-law model $p\propto
I^{-\alpha}$ with the power index $\alpha=0.51\pm 0.01$ for the whole M17
region. The values of $\alpha$ do not change much when we fit only with pixels
from M17-N ($\alpha=0.54\pm 0.02$) or M17-S ($\alpha=0.55\pm 0.01$)
separately. A best-fitted value of $\alpha=0.51$ reveals that grain alignment
is significant toward the high emission intensity. As a conclusion, we observe
that the polarization fraction, $p$, decreases with increasing the emission
intensity, $I$.

To better understand the variation of $p$, we now study the relationship
between the polarization fraction, $p$, and the column density, $N({\rm
H_{2}})$, as well as the dust temperature, $T_{\rm{d}}$. Figure 13 (top-left)
shows the $p-N(\rm{H_{2}})$ relation. Here we derive the slopes by fitting
piece-wise linear functions to the data. In the M17-S region, the polarization
fraction gradually decreases with increasing the gas column density, before it
experiences a steep drop with a slope of $-0.71$ for $N_{\rm H_{2}}>3\times
10^{22}\rm\;cm^{-2}$ or the visual extinction444 $A_{\rm V}(\rm
mag)=R_{V}\times 2N({\rm H_{2}})/(5\times 10^{21}\;{\rm cm^{-2}})$ $A_{\rm
V}>37$ mag for the typical total-to-selective extinction $R_{V}\approx 3.1$.
In the M17-N region, a steep decrease with slope of $-0.67$ occurs at $N({\rm
H_{2}})>5\times 10^{21}\;\rm cm^{-2}$ or $A_{\rm V}>6$ mag. The strong
depolarization occurs in the region where the gas density is relatively low
and the dust temperature is relatively high compared to that of M17-S (see
Figure 11), which is unexpected from the RATA theory.

The top-right panel of Figure 13 shows the $p-T_{\rm d}$ relation. It appears
that the polarization fraction first decreases, then increases before
decreasing again as the dust temperature increases. The decrease-increase
feature originates from the M17-S region, while the depolarization is from
M17-N.

The bottom panel of Figure 13 shows the gas column density as a function of
the dust temperature. The column density decreases rapidly with increasing
$T_{d}$ in M17-S, but it varies slowly in M17-N. The first decrease of $p$ for
$T_{\rm d}<40\,\rm K$ corresponds to the highest gas density ($A_{\rm V}\sim
100$ mag). This depolarization is caused by decrease of grain alignment due to
the attenuation of the radiation field and the enhancement of collisional
damping. Towards higher dust temperatures, the gas density drops, and the
grain alignment efficiency enhances toward the central luminous source, which
results in the following increment of $p$. These features are expected from
the context of the RATA theory. In M17-N, the dust temperature could be up to
150 K and the gas density becomes more diffuse at $A_{\rm V}<10$ mag, the
polarization fraction, however, appears to monotonically decrease to higher
$T_{\rm d}$, which is completely contradictory to what expected from RATA
theory.

###  4.8 Implications for grain alignment and rotational disruption by RATs

We now discuss the implications of the observed polarization fraction toward
M17 for physics of grain alignment and disruption based on radiative torques.

According to the RATA theory (see Lazarian & Hoang 2007), the polarization
fraction of thermal emission increases with decreasing alignment size, $a_{\rm
align}$, which is the minimum size of aligned grains. The alignment size is
determined by the balance between spin-up by RATs and spin-down by gas
collisional damping, which is a function of the local dust temperature (or
radiation intensity) and gas density as $a_{\rm align}\sim n^{2/7}_{\rm
H}T^{-12/7}_{\rm d}$ (Tram et al. 2021). As a result, a denser gas and lower
dust temperature increase $a_{\rm align}$ (see details in Hoang et al. 2021),
which results in the decrease of the polarization fraction. Such a prediction
by the RATA theory can explain the decrease of the polarization fraction with
increasing the gas column density observed in the M17-S region (see Figure 13;
left panel). However, it can not explain the decrease of $p$ with increasing
$T_{\rm d}$ in the M17-N region where the gas density changes slowly (see
Figure 13; right panel), which reveals evidence of the RATD.

The polarization fraction at far-IR/submm is very sensitive to the maximum
size of the grain size distribution because large grains dominate long-
wavelength emission. Following the RATD mechanism (Hoang et al. 2019), the
maximum grain size above which large grains are disrupted by RATs is
determined by the local dust temperature (radiation field), gas density, and
the grain tensile strength, $S_{\rm max}$, which follows by $a_{\rm disr}\sim
n^{1/2}_{\rm H}T^{-3}_{\rm d}S^{1/4}_{\rm max}$ (Tram et al. 2021). For the
average estimated volume density $n_{\rm H_{2}}\simeq 3\times 10^{4}\,\rm
cm^{-3}$ (see Table 1), and the temperature at the breaking point in Figure 13
of $T_{\rm d}\simeq 53\,\rm K$, one obtains $a_{\rm disr}=0.16$, $0.28$, and
$0.49\,\mu$m for $S_{\rm max}=10^{7}$, $10^{8}$, and $10^{9}\,\rm
erg\,cm^{-3}$, assuming the mean wavelength of radiation field of $1\,\mu$m.
The decrease of the disruption size by RATD leads to the decrease of the dust
polarization as the dust temperature increases (Lee et al. 2020; Tram et al.
2021b), which successfully reproduces the observed trend toward the M17-N (see
Figure 13).

To quantify the magnetic field tangling at small scales on the depolarization,
we calculate the polarization angle dispersion function, $S$, (see Section 3.3
in Planck Collaboration: et al. 2015). For each pixel at location $x$, $S$ is
calculated as the standard deviation of the polarization angle difference,
$S_{xi}$, of pixel $x$ and pixel $i$ which lies on a circle having $x$ as the
center and a radius of $\delta$

$S^{2}(x,\delta)=\frac{1}{N}\sum_{i=1}^{N}S_{xi}^{2},$ (19)

where $S_{xi}=\theta(x)-\theta(x+\delta)$ is the polarization angle difference
and $N$ is the number of pixels lying on the circle.

For the current SOFIA/HAWC+ data set, we calculate $S$ for
$\delta=27.2\arcsec$ ($\sim$ 2 beam sizes). The left panel of Figure 14 shows
the $S$ \- $T_{\rm d}$ relation where it can be seen that the angular
dispersion function $S$ strongly correlates with the dust temperatures but not
the highest ones, $T_{\rm d}>~{}90$ K, in M17-N. Thus, the decline of $p$ at
$T_{\rm d}>~{}90$ K (see the upper-right panel of Figure 13) cannot be
explained by the magnetic field tangling and presents an evidence for the RATD
effect. The right panel of Figure 14 shows a general anti-correlation of the
angular dispersion function $S$ and polarization fraction $p$, except, again,
for the highest temperatures pixels (red dots) whose temperatures are greater
than 90 K. For $T_{\rm d}<~{}35$ K located in M17-S (blue dots in Figure 13
and 14), we can see that the rapid decrease of $p$ is due to B-field tangling
in the high density region (see bottom panel of Figure 13). For the
temperature range from 35 K to 65 K, in both M17-N and M17-S, $N({\rm H_{2}})$
and $S$ decrease, therefore, $p$ increases. This clearly shows the important
contribution of field tangling to the depolarization at low dust temperatures.

A detailed modeling of dust polarization using the grain alignment and
disruption by RATs for M17 is beyond the scope of this paper, and will be
addressed in a follow-up study.

##  5 Summary

In this study, we use the dust polarization data taken by SOFIA/HAWC+ at $154\
\rm\mu m$ to study magnetic fields and dust physics in the M17 nebula. Our
main results are summarized as follows:

  1. 1.

Using the DCF method, we estimated the magnetic field strengths to be $980\pm
230\ \rm\mu G$ in the lower-density region (M17-N) and $1665\pm 885\ \rm\mu G$
in the higher-density region (M17-S). The strong magnetic field could be a
result of the pressure exerted by the H II region in the eastern part of the
observed region by SOFIA/HAWC+ toward M17. In the M17-N, the B-field
morphology can be that of a gravitational collapse molecular core. The
morphology of B-fields in M17-S displays a well-organized elongated structure
along with the gravitational collapse directions from the outer regions to the
denser central regions. The fields are dragged inward to the gravitational
center. The whole B-field morphology represents an asymmetric large-scale
hourglass structure. We also found a pillar structure which is one of the
common features of PDR regions. A rough estimation of the contribution of ram
over the magnetic pressure suggests that the wind from the PDR region is
strong enough to impact on the B-field morphology of M17.

  2. 2.

The Mach numbers are determined to be sub-Alfvénic ($\mathcal{M_{A}}<1$) which
indicate that the magnetic field dominates turbulence. In addition, the
inferred sub-critical values of mass-to-flux ratios, $\lambda<1$, imply that
the magnetic fields in the regions are strong enough to resist gravitational
collapse. These results are consistent with the deficiency of the formation of
massive stars in the region from previous studies.

  3. 3.

To study dust physics, we analyzed the relation of the polarization fraction,
$p$, with the emission intensity, $I$, gas column density, $N(\rm H2)$, and
dust temperature, $T_{d}$. The power index of the $p$ vs $I$ relation,
$\alpha=0.51$, implies that dust grains could still be aligned by radiation in
the region. The decrease of the dust polarization with the column density in
the M17-S can be explained by the RATA theory as well as the tangling of the
magnetic field.

  4. 4.

To study the effect of magnetic field tangling on the dust polarization, we
also analyzed the variation of the polarization angle dispersion function,
$S$, with dust temperature and gas column density. In the M17-N, the decrease
of $p$ with $T_{d}$ at high temperatures when both $N(H_{2})$ and $S$ decrease
is most consistent with the theoretical predictions of dust polarization by
both RATA and RATD effects.

  5. 5.

There are large statistical biases on the estimation of the polarization angle
dispersion, $\sigma_{\theta}$, as well as the volume density, $n(\rm H_{2})$,
due to the fact that the magnetic field strengths are estimated over a large
areas. These biases lead to large uncertainties on the measurement of the
magnetic field strengths. It is also known that for a random sample of
magnetic field orientations, using the DCF method the mass-to-flux ratio will
on average be overestimated by a factor $\sim 3$. In fact, using a method
proposed by Cho & Yoo (2016) we found the overestimate factors equal to 1.1
and 3.9 for the M17-N and M17-S, respectively. Therefore, new methods of
estimating magnetic field strength which are able to identify the fields and
accompanied useful parameters such as Alfvénic Mach numbers and mass-to-flux
ratios in the basis of pixel by pixel would provide detailed and more precise
conditions of star formation in the M17 molecular cloud.

This research is based on observations made with the NASA/DLR Stratospheric
Observatory for Infrared Astronomy (SOFIA). SOFIA is jointly operated by the
Universities Space Research Association, Inc. (USRA), under NASA contract
NNA17BF53C, and the Deutsches SOFIA Institut (DSI) under DLR contract 50 OK
0901 to the University of Stuttgart. Nobeyama Radio Observatory is a branch of
the National Astronomical Observatory of Japan, National Institutes of Natural
Sciences, part of the data were retrieved from the JVO portal
(http://jvo.nao.ac.jp/portal/) operated by ADC/NAOJ. We thank Joseph M.
Michail for his line integral convolution (LIC) Python tool. T.H. acknowledges
the support by the National Research Foundation of Korea (NRF) grants funded
by the Korea government (MSIT) through the Mid-career Research Program
(2019R1A2C1087045). P.N.D., N.B.N., and N.T.P. are grateful to the funding
from the Vietnam National Foundation for Science and Technology Development
(NAFOSTED) under grant number 103.99-2019.368. N.L. acknowledges the support
from the First TEAM grant of the Foundation for Polish Science No.
POIR.04.04.00-00-5D21/18-00. K.P. is a Royal Society University Research
Fellow.

## References

  * San (2021) 2021, 915, L10, doi: 10.3847/2041-8213/ac081c
  * Andersson et al. (2015) Andersson, B.-G., Lazarian, A., & Vaillancourt, J. E. 2015, Annual Review of Astronomy and Astrophysics, 53, 501, doi: 10.1146/annurev-astro-082214-122414
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., SipHocz, B. M., et al. 2018, aj, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Beuther et al. (2020) Beuther, H., Soler, J. D., Linz, H., et al. 2020, The Astrophysical Journal, 904, 168, doi: 10.3847/1538-4357/abc019
  * Brogan et al. (1999) Brogan, C. L., Troland, T. H., Roberts, D. A., & Crutcher, R. M. 1999, The Astrophysical Journal, 515, 304, doi: 10.1086/306996
  * Cabral & Leedom (1993) Cabral, B., & Leedom, L. C. 1993, in Proceedings of the 20th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’93 (New York, NY, USA: Association for Computing Machinery), 263–270, doi: 10.1145/166117.166151
  * Chandrasekhar & Fermi (1953) Chandrasekhar, S., & Fermi, E. 1953, ApJ, 118, 113, doi: 10.1086/145731
  * Chen et al. (2012) Chen, Z., Jiang, Z., Wang, Y., et al. 2012, Publications of the Astronomical Society of Japan, 64, doi: 10.1093/pasj/64.5.110
  * Cho & Yoo (2016) Cho, J., & Yoo, H. 2016, The Astrophysical Journal, 821, 21, doi: 10.3847/0004-637x/821/1/21
  * Chuss et al. (2019) Chuss, D. T., Andersson, B.-G., Bally, J., et al. 2019, The Astrophysical Journal, 872, 187, doi: 10.3847/1538-4357/aafd37
  * Crutcher (2004) Crutcher, R. M. 2004, in The Magnetized Interstellar Medium, ed. B. Uyaniker, W. Reich, & R. Wielebinski, 123–132
  * Crutcher (2012) Crutcher, R. M. 2012, Annual Review of Astronomy and Astrophysics, 50, 29, doi: 10.1146/annurev-astro-081811-125514
  * Davis (1951) Davis, L. 1951, Phys. Rev., 81, 890, doi: 10.1103/PhysRev.81.890.2
  * Derek Ward-Thompson & McKee (2020) Derek Ward-Thompson, Ray S. Furuya, Y. T., & McKee, C. F. 2020, The Role of Magnetic Fields in the Formation of Stars (Frontiers in Astronomy and Space Sciences)
  * Dolginov & Mytrophanov (1976) Dolginov, A. Z., & Mytrophanov, I. G. 1976, Astrophysics and Space Science, 43, 291, doi: 10.1007/BF00640010
  * Dotson (1996) Dotson, J. L. 1996, ApJ, 470, 566, doi: 10.1086/177888
  * Draine & Weingartner (1996) Draine, B. T., & Weingartner, J. C. 1996, ApJ, 470, 551, doi: 10.1086/177887
  * Draine & Weingartner (1997) —. 1997, ApJ, 480, 633, doi: 10.1086/304008
  * Elmegreen (2000) Elmegreen, B. G. 2000, 530, 277, doi: 10.1086/308361
  * Falceta-Gonçalves et al. (2008) Falceta-Gonçalves, D., Lazarian, A., & Kowal, G. 2008, The Astrophysical Journal, 679, 537, doi: 10.1086/587479
  * Federrath (2016) Federrath, C. 2016, Journal of Physics: Conference Series, 719, 012002, doi: 10.1088/1742-6596/719/1/012002
  * Ginsburg et al. (2019) Ginsburg, A., Koch, E., Robitaille, T., et al. 2019, radio-astro-tools/spectral-cube: v0.4.4, v0.4.4, Zenodo, doi: 10.5281/zenodo.2573901
  * Gordon et al. (2018) Gordon, M. S., Lopez-Rodriguez, E., Andersson, B. G., et al. 2018, SOFIA Community Science I: HAWC+ Polarimetry of 30 Doradus.  https://arxiv.org/abs/1811.03100
  * Hanson et al. (1997) Hanson, M., Howarth, I., & Conti, P. 1997, ApJ, 489, 698
  * Harper et al. (2018) Harper, D. A., Runyan, M. C., Dowell, C. D., et al. 2018, Journal of Astronomical Instrumentation, 07, 1840008, doi: 10.1142/S2251171718400081
  * Hennebelle & Inutsuka (2019) Hennebelle, P., & Inutsuka, S.-i. 2019, Frontiers in Astronomy and Space Sciences, 6, 5, doi: 10.3389/fspas.2019.00005
  * Hildebrand et al. (2009) Hildebrand, R. H., Kirby, L., Dotson, J. L., Houde, M., & Vaillancourt, J. E. 2009, The Astrophysical Journal, 696, 567, doi: 10.1088/0004-637x/696/1/567
  * Hoang (2020) Hoang, T. 2020, Galaxies, 8, doi: 10.3390/galaxies8030052
  * Hoang & Lazarian (2016) Hoang, T., & Lazarian, A. 2016, ApJ, 831, 159
  * Hoang et al. (2019) Hoang, T., Tram, L. N., Lee, H., & Ahn, S.-H. 2019, Nature Astronomy, 3, 766–775, doi: 10.1038/s41550-019-0763-6
  * Hoang et al. (2021) Hoang, T., Tram, L. N., Lee, H., Diep, P. N., & Ngoc, N. B. 2021, The Astrophysical Journal, 908, 218, doi: 10.3847/1538-4357/abd54f
  * Houde et al. (2002) Houde, M., Bastien, P., Dotson, J. L., et al. 2002, ApJ, 569, 803, doi: 10.1086/339356
  * Kandori et al. (2018) Kandori, R., Tomisaka, K., Tamura, M., et al. 2018, The Astrophysical Journal, 865, 121, doi: 10.3847/1538-4357/aadb3f
  * Kleinmann & Wright (1973) Kleinmann, D. E., & Wright, E. L. 1973, ApJ, 185, L131, doi: 10.1086/181338
  * Kritsuk et al. (2017) Kritsuk, A. G., Ustyugov, S. D., & Norman, M. L. 2017, New Journal of Physics, 19, 065003, doi: 10.1088/1367-2630/aa7156
  * Lazarian et al. (2015) Lazarian, A., Andersson, B.-G., & Hoang, T. 2015, in Polarimetry of stars and planetary systems, ed. L. Kolokolova, J. Hough, & A.-C. Levasseur-Regourd ((New York: Cambridge Univ. Press)), 81
  * Lazarian & Hoang (2007) Lazarian, A., & Hoang, T. 2007, Monthly Notices of the Royal Astronomical Society, 378, 910, doi: 10.1111/j.1365-2966.2007.11817.x
  * Lee et al. (2020) Lee, H., Hoang, T., Le, N., & Cho, J. 2020, The Astrophysical Journal, 896, 44, doi: 10.3847/1538-4357/ab8e33
  * Lee et al. (2012) Lee, K., Looney, L., Johnstone, D., & Tobin, J. 2012, The Astrophysical Journal, 761, 171, doi: 10.1088/0004-637x/761/2/171
  * Li et al. (2014) Li, D. L., Esimbek, J., Zhou, J.-J., et al. 2014, Astronomy & Astrophysics, 567, A10
  * Li & Klein (2019) Li, P. S., & Klein, R. I. 2019, Monthly Notices of the Royal Astronomical Society, 485, 4509, doi: 10.1093/mnras/stz653
  * Lim et al. (2020) Lim, W., Buizer, J. M. D., & Radomski, J. T. 2020, The Astrophysical Journal, 888, 98, doi: 10.3847/1538-4357/ab5fd0
  * Lim et al. (2016) Lim, W., Tan, J. C., Kainulainen, J., Ma, B., & Butler, M. J. 2016, The Astrophysical Journal, 829, L19, doi: 10.3847/2041-8205/829/1/l19
  * Liu et al. (2021) Liu, J., Zhang, Q., Commercon, B., et al. 2021, Calibrating the Davis-Chandrasekhar-Fermi method with numerical simulations: uncertainties in estimating the magnetic field strength from statistics of field orientations.  https://arxiv.org/abs/2106.09934
  * Liu et al. (2020) Liu, J., Zhang, Q., Qiu, K., et al. 2020, The Astrophysical Journal, 895, 142, doi: 10.3847/1538-4357/ab9087
  * Mac Low & Klessen (2004) Mac Low, M.-M., & Klessen, R. S. 2004, Rev. Mod. Phys., 76, 125, doi: 10.1103/RevModPhys.76.125
  * McKee & Ostriker (2007) McKee, C. F., & Ostriker, E. C. 2007, Annual Review of Astronomy and Astrophysics, 45, 565, doi: 10.1146/annurev.astro.45.051806.110602
  * Nakamura & Li (2008) Nakamura, F., & Li, Z.-Y. 2008, The Astrophysical Journal, 687, 354, doi: 10.1086/591641
  * Nakamura et al. (2019) Nakamura, F., Ishii, S., Dobashi, K., et al. 2019, Publications of the Astronomical Society of Japan, 71, doi: 10.1093/pasj/psz057
  * Nakano & Nakamura (1978) Nakano, T., & Nakamura, T. 1978, PASJ, 30, 671
  * Ngoc et al. (2020) Ngoc, N. B., Diep, P. N., Parsons, H., et al. 2020, Observations of magnetic fields surrounding LkH$\alpha$ 101 taken by the BISTRO survey with JCMT-POL-2.  https://arxiv.org/abs/2012.04297
  * Nguyen-Luong et al. (2020) Nguyen-Luong, Q., Nakamura, F., Sugitani, K., et al. 2020, The Astrophysical Journal, 891, 66, doi: 10.3847/1538-4357/ab700a
  * Ostriker et al. (2001) Ostriker, E. C., Stone, J. M., & Gammie, C. F. 2001, The Astrophysical Journal, 546, 980, doi: 10.1086/318290
  * Padoan et al. (2001) Padoan, P., Goodman, A., Draine, B. T., et al. 2001, The Astrophysical Journal, 559, 1005, doi: 10.1086/322504
  * Padoan & Nordlund (1999) Padoan, P., & Nordlund, A. 1999, The Astrophysical Journal, 526, 279, doi: 10.1086/307956
  * Pattle & Fissel (2019) Pattle, K., & Fissel, L. 2019, Frontiers in Astronomy and Space Sciences, 6, 15, doi: 10.3389/fspas.2019.00015
  * Pattle et al. (2017) Pattle, K., Ward-Thompson, D., Berry, D., et al. 2017, The Astrophysical Journal, 846, 122, doi: 10.3847/1538-4357/aa80e5
  * Pattle et al. (2018) Pattle, K., Ward-Thompson, D., Hasegawa, T., et al. 2018, The Astrophysical Journal, 860, L6, doi: 10.3847/2041-8213/aac771
  * Pattle et al. (2019) Pattle, K., Lai, S.-P., Hasegawa, T., et al. 2019, The Astrophysical Journal, 880, 27
  * Pellegrini et al. (2007) Pellegrini, E. W., Baldwin, J. A., Brogan, C. L., et al. 2007, The Astrophysical Journal, 658, 1119, doi: 10.1086/511258
  * Planck Collaboration: et al. (2015) Planck Collaboration:, Ade, P. A. R., Aghanim, N., et al. 2015, A&A, 576, A104, doi: 10.1051/0004-6361/201424082
  * Pomohaci et al. (2017) Pomohaci, R., Oudmaijer, R. D., Lumsden, S. L., Hoare, M. G., & Mendigutía, I. 2017, Monthly Notices of the Royal Astronomical Society, 472, 3624, doi: 10.1093/mnras/stx2196
  * Povich et al. (2009) Povich, M. S., Churchwell, E., Bieging, J. H., et al. 2009, The Astrophysical Journal, 696, 1278, doi: 10.1088/0004-637x/696/2/1278
  * Reid & Wilson (2006) Reid, M. A., & Wilson, C. D. 2006, ApJ, 644, 990, doi: 10.1086/503824
  * Robitaille (2019) Robitaille, T. 2019, APLpy v2.0: The Astronomical Plotting Library in Python, doi: 10.5281/zenodo.2567476
  * Robitaille et al. (2020) Robitaille, T., Deil, C., & Ginsburg, A. 2020, reproject: Python-based astronomical image reprojection.  http://ascl.net/2011.023
  * Seifried & Walch (2015) Seifried, D., & Walch, S. 2015, Monthly Notices of the Royal Astronomical Society, 452, 2410, doi: 10.1093/mnras/stv1458
  * Serkowski (1962) Serkowski, K. 1962, Advances in Astronomy and Astrophysics, 1, 289, doi: 10.1016/B978-1-4831-9919-1.50009-1
  * Sugitani et al. (2019) Sugitani, K., Nakamura, F., Shimoikura, T., et al. 2019, Publications of the Astronomical Society of Japan, 71, doi: 10.1093/pasj/psz072
  * Temi et al. (2018) Temi, P., Hoffman, D., Ennico, K., & Le, J. 2018, Journal of Astronomical Instrumentation, 07, 1840011, doi: 10.1142/S2251171718400111
  * Tram et al. (2021a) Tram, L. N., Hoang, T., Lee, H., et al. 2021a, ApJ, 906, 115, doi: 10.3847/1538-4357/abc6fe
  * Tram et al. (2021b) Tram, L. N., Lee, H., Hoang, T., et al. 2021b, ApJ, 908, 159, doi: 10.3847/1538-4357/abccbe
  * Tram et al. (2021) Tram, L. N., Hoang, T., Lopez-Rodriguez, E., et al. 2021, SOFIA observations of 30 Doradus: I – Far-Infrared dust polarization and implications for grain alignment and disruption by radiative torques.  https://arxiv.org/abs/2105.09530
  * Wang et al. (2019) Wang, J.-W., Lai, S.-P., Eswaraiah, C., et al. 2019, The Astrophysical Journal, 876, 42, doi: 10.3847/1538-4357/ab13a2
  * Wardle & Kronberg (1974) Wardle, J. F. C., & Kronberg, P. P. 1974, ApJ, 194, 249, doi: 10.1086/153240
  * Wurster & Li (2018) Wurster, J., & Li, Z.-Y. 2018, Frontiers in Astronomy and Space Sciences, 5, 39, doi: 10.3389/fspas.2018.00039
  * Xu et al. (2011) Xu, Y., Moscadelli, L., Reid, M. J., et al. 2011, The Astrophysical Journal, 733, 25, doi: 10.1088/0004-637x/733/1/25
  * Zeng et al. (2013) Zeng, L., Bennett, C. L., Chapman, N. L., et al. 2013, The Astrophysical Journal, 773, 29, doi: 10.1088/0004-637x/773/1/29

##  Appendix A Appendix

This appendix shows the characteristics of the observational data and the
behavior of the master cut applied to the data.

Figure A.1 shows the distributions of the SNRs of $I$, $p$, and $I_{p}$ before
and after applying the master cut defined in the text. Their corresponding
mean and RMS values are listed in Table A.1. The mean values of the SNRs
increase significantly after the cut by 45%, 18%, and 17% for $I$, $p$, and
$I_{p}$, respectively.

Table A.1: Means and RMS of the distribution of $\rm SNR_{I},\ SNR_{p},\ SNR_{I_{p}}$.  | $\rm SNR_{I}$ | $\rm SNR_{p}$ | $\rm SNR_{I_{p}}$
---|---|---|---
Before cut | Mean | 1477.4 | 22.2 | 22.3
RMS | 2186.3 | 19.6 | 19.6
After cut | Mean | 2138.5 | 26.1 | 26.1
RMS | 2366.7 | 20.8 | 20.9
Figure A.1: Left to right: Distributions of signal-to-noise ratios of the
total intensity ($\rm SNR_{I}$), polarization fraction ($\rm SNR_{p}$), and
polarization intensity ($\rm SNR_{I_{p}}$). The histograms are before cut and
blue parts after cut.

Figure A.2 presents one dimensional histograms of the raw data of $I,\
\sigma_{I}$, $\ p,\ \sigma_{p}$, $I_{p},\ \sigma_{I_{p}}$, $\theta,\
\sigma_{\theta}$. The associated mean and RMS values of these quantities
presented in Table A.2.

Figure A.2: From left to right, top to bottom: Distributions of I, $\sigma_{I}$, p, $\sigma_{p}$, $I_{p}$, $\sigma_{I_{p}}$, $\theta$, and $\sigma_{\theta}$ of the raw SOFIA/HAWC+ data. Table A.2: Mean and RMS values of $I,\ \sigma_{I},\ I_{p},\ \sigma_{I_{p}},\ p,\sigma_{p},\ \theta$, and $\ \sigma_{\theta}$.  | $I$ | $\sigma_{I}$ | $I_{p}$ | $\sigma_{I_{p}}$ | $p$ | $\sigma_{p}$ | $\theta$ | $\sigma_{\theta}$
---|---|---|---|---|---|---|---|---
| $\rm[Jy/pixel]$ | $\rm[Jy/pixel]$ | $\rm[Jy/pixel]$ | $\rm[Jy/pixel]$ | $[\%]$ | $[\%]$ | $[^{\circ}]$ | $[^{\circ}]$
Mean | 9.4 | 0.0078 | 0.247 | 0.0119 | 5.0 | 0.367 | -11.1 | 2.5
RMS | 12.7 | 0.0046 | 0.215 | 0.0054 | 5.9 | 0.507 | 48.9 | 2.8

Figure A.3 defines the upper and lower regions used for the data analysis. We
present in the figure different total intensity scales for better vision of
the emission structure of the two regions.

Figure A.3: A close-up view of B-fields of the SOFIA/HAWC+ M17 regions: M17-N
(top) and M17-S (bottom). Description of the maps is the same for Figure 3.

Figure A.4 presents the highest density region of M17-S when applying a cut on
$N(\rm{H_{2}})>10^{22}\;cm^{-2}$.

Figure A.4: The highest density region of the M17-S with
$N(\rm{H_{2}})>10^{22}\;cm^{-2}$. The ellipse has a center at RA $\sim$ $\rm
18^{h}20^{m}22^{s}.17$ and DEC $\sim$ $\rm-16^{\circ}12\arcmin 48\arcsec.3$,
major $\times$ minor axes of 171″$\times$ 118″, and a position angle of
$350^{\circ}$. The contours are the dust temperature.