# B-fields And dust in interstelLar fiLAments using Dust POLarization (BALLAD-
POL): II. Testing the Radiative Torque Paradigm in Musca and OMC-1

Nguyen Bich Ngoc Department of Astrophysics, Vietnam National Space Center,
Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam,
<EMAIL_ADDRESS>Graduate University of Science and Technology, Vietnam
Academy of Science and Technology, 18 Hoang Quoc Viet, Hanoi, Vietnam  Thiem
Hoang Korea Astronomy and Space Science Institute, 776 Daedeokdae-ro, Yuseong-
gu, Daejeon 34055, Republic of Korea<EMAIL_ADDRESS>University of
Science and Technology, Korea, 217 Gajeong-ro, Yuseong-gu, Daejeon 34113,
Republic of Korea  Pham Ngoc Diep 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  Le Ngoc Tram
Max-Planck-Institut für Radioastronomie, Auf dem Hügel 69, 53-121, Bonn,
Germany

###### Abstract

Polarization of starlight and thermal dust emission caused by aligned dust
grains is a valuable tool to characterize magnetic fields (B-fields) and
constrain dust properties. However, the physics of grain alignment is not
fully understood. To test the popular paradigm of radiative torque (RAT)
theory, including RAT alignment (RAT-A) and disruption (RAT-D), we use thermal
dust polarization data observed by Planck and SOFIA/HAWC+ toward two filaments
with contrasting physical conditions: Musca, a quiet filament without an
embedded source, and OMC-1, a highly dynamic filament due to feedback. We
analyze various relations of the observed polarization fraction, $P$, with gas
column density, $N(\rm H_{2})$, dust temperature, $T\rm{{}_{d}}$, and
polarization angle dispersion function, $\mathcal{S}$. We found that $P$
decreases with increasing $\mathcal{S}$ and increasing $N(\rm H_{2})$, as
expected from the RAT-A theory. On the other hand, the $P-T\rm{{}_{d}}$
relation is more complicated; it is a linear correlation at low $T\rm{{}_{d}}$
but turns into an anti-correlation when $T\rm{{}_{d}}$ reaches a certain high
value. Next, we compute the polarization fraction on a pixel-by-pixel basis
for the ideal model with B-fields in the plane of the sky using the DustPOL
code based on the RAT paradigm, incorporate the depolarization effect by
B-field tangling using the polarization angle dispersion, and compare the
realistic polarization model with observations of Musca and OMC-1. For Musca
with well-ordered B-fields, our numerical model successfully reproduces the
decline of $P$ toward the filament spine (aka. polarization hole), having
higher $N(\rm H_{2})$ and lower $T\rm{{}_{d}}$, indicating the loss of grain
alignment efficiency due to RAT-A. For OMC-1, with stronger B-field variations
and higher temperatures, our model can reproduce the observed $P-T\rm{{}_{d}}$
and $P-N(\rm H_{2})$ relations only if the depolarization effect resulting
from B-field tangling and RAT-D effect are taken into account. Our results
provide more robust observational evidence for the RAT paradigm, particularly
the recently discovered RAT-D.

dust, formation – magnetic fields – dust – ISM: individual objects (Musca,
OMC-1, Orion A, BN-KL)

††software: Astropy (Robitaille et al., 2013), APLpy (Robitaille & Bressert,
2012; Robitaille, 2019)

##  1 Introduction

Dust is a crucial element of the interstellar medium (ISM) and has a
significant impact on a variety of astrophysical processes, such as gas
heating and cooling, the formation of stars and planets, and surface
astrochemistry (see Draine 2003 for an overview). Moreover, the polarization
of starlight (Hall, 1949; Hiltner, 1949) or polarized thermal dust emission
(Hildebrand, 1988) induced by aligned dust grains are widely used to observe
magnetic fields (B-fields), which are predicted to regulate the formation and
evolution of molecular clouds and filaments, star formation, and stellar
feedback via outflows/jets (see Crutcher 2012; Pattle & Fissel 2019; Pattle et
al. 2022 for reviews). Thermal dust polarization has also been used to map
B-fields in nearby galaxies (e.g., Lopez-Rodriguez et al. 2022) and even in
high-z galaxies (Geach et al., 2023). Moreover, dust polarization is a
valuable tool to probe dust physics and dust properties (size, shape, and
composition, see e.g., Draine & Hensley 2021).

Dust polarimetry is based on the assumption that dust grains are
systematically aligned with the ambient B-fields with their longest axes
perpendicular to the fields. Hence, the polarization vectors of thermal dust
emission are perpendicular to the B-fields projected on the plane of the sky
(POS). Therefore, by rotating the observed polarization angle by 90∘, we can
infer the morphology of the B-fields (e.g., Hildebrand 1988).

To establish dust polarimetry as a reliable diagnostic tool to probe B-fields
and dust properties, it is necessary to fully understand the detailed physics
of grain alignment. The leading mechanism of grain alignment is based on the
Radiative Torques (RATs), arising from the differential scattering and
extinction of an anisotropic radiation field by non-spherical grains (Dolginov
& Mitrofanov, 1976; Draine & Weingartner, 1997; Lazarian & Hoang, 2007). The
key prediction of the Radiative Torque Alignment (RAT-A) theory is that the
degree of grain alignment depends on the local conditions, including the gas
density and radiation, such that the alignment degree decreases with
increasing the local density but increases with the radiation field intensity
or dust temperature, $T\rm{{}_{d}}$ (Hoang & Lazarian, 2014; Lazarian & Hoang,
2019). As a result, the RAT-A theory predicts two key features, including (1)
the anti-correlation of the polarization degree, $P$, with the gas density,
and (2) a correlation between $P$ and the intensity of the radiation field (or
dust temperature) (see Hoang et al. 2021a).

Previous tests of RAT-A using starlight polarization support the key
predictions of the RAT theory (see Andersson et al. 2015). For instance,
observations toward different regions usually report the general trend of the
decrease in the polarization fraction with the gas column density, which is
known as the polarization hole, by both optical-NIR (e.g., Whittet et al.
2008) and far-IR/submillimeter polarization observations (Ward-Thompson et
al., 2000). Numerical modeling by Whittet et al. (2008) and Hoang et al.
(2021a) showed that the polarization hole could be explained by the loss of
grain alignment toward high-density and low radiation field regions based on
the RAT-A theory. For a comprehensive review of the successful tests for the
RAT-A theory using starlight polarization, please refer to a review by
Andersson et al. (2015).

Nevertheless, to date, there is still a lack of comprehensive testing of the
RAT-A mechanism using thermal dust polarization. Specifically, there is an
ongoing debate regarding the exact cause of the polarization hole observed in
far-IR/sub-mm toward molecular clouds and star-forming regions. For the
diffuse and translucent clouds, it is found that the B-field fluctuations are
likely the main cause of the polarization hole, instead of grain alignment
loss predicted by the RAT-A, as found from both observations (e.g., Planck
Collaboration XIX 2015) and numerical simulations (e.g., Seifried et al.
2019). Furthermore, the inclination angle of the B-fields and the LOS
significantly affects the polarization fraction due to the projection effect
(Hoang & Truong, 2023). In particular, observations revealed that $P$ does not
always increase with $T\rm{{}_{d}}$ (Planck Collaboration XII, 2020), which is
opposite to the key prediction of the RAT-A theory.

Hoang et al. (2019) proposed a new mechanism called RAdiative Torque
Disruption (RAT-D), which is expected to resolve the later puzzle. The basics
of the RAT-D mechanism is that large grains illuminated by an intense
radiation field are spun up to suprathermal rotation by RATs, the same driver
of grain alignment. When the centrifugal stress exceeds the grain’s tensile
strength, large grains will be disrupted into smaller ones. Numerical modeling
of thermal dust polarization by Lee et al. (2020) using DustPOL code that is
based on the RAT-A and RAT-D mechanisms (so-called RAT paradigm) showed that
the depletion of large grains by RAT-D causes the decrease in the polarization
fraction at long wavelengths when the dust temperature is sufficiently large,
which successfully reproduce the anti-correlation seen by Planck. To test the
RAT paradigm, previous studies (Tram et al., 2021a, b; Hoang et al., 2021b;
Ngoc et al., 2021) analyzed far-IR and submillimeter polarization data
observed toward star-forming regions by the High-resolution Airborne Wideband
Camera Plus (HAWC+) instrument (Harper et al., 2018) on board the Stratosphere
Observatory for Infrared Astronomy (SOFIA) and James Clark Maxwell Telescope
(JCMT). They found that $P$ tends to increase with $T\rm{{}_{d}}$ but then
decreases when $T_{\rm d}$ exceeds some high temperatures, supporting the RAT
paradigm. Later, detailed numerical modeling by Tram et al. (2021a, b) using
the DustPOL code successfully reproduced the $P-T\rm{{}_{d}}$ trend observed
in the OMC-1 and $\rho$ Ophiuchus A molecular clouds, which showed the initial
success of the joint effect of RAT-A and RAT-D.

However, the effects of the B-field fluctuations on the polarization hole and
the variation of $P-T_{\rm d}$ have not been considered yet in the previous
testings of the RAT paradigm using polarization data (see Tram & Hoang 2022
for a review). It is, therefore, necessary to take into account the effects of
B-field fluctuations to achieve a complete testing of the RAT paradigm.

Filaments are ubiquitous and form a complex network within most interstellar
clouds, as revealed by Herschel. Filaments are considered the earliest stage
of the star formation process in which the gas density and dust temperature
vary significantly across the filaments (e.g., André et al. 2014), which makes
them ideal targets for testing the RAT paradigm. Therefore, the two main
objectives of our study are to (1) test the RAT paradigm and (2) study the
effect of the B-field fluctuations on the net thermal dust polarization of
aligned dust in filaments. To achieve these objectives, we use dust
polarization data observed toward Musca by Planck and OMC-1 by SOFIA/HAWC+ and
perform synthetic modeling of the dust polarization with our DustPOL code. Our
selection of these filaments is intended to account for the different local
properties, e.g., the gas density and radiation field (with or without
embedded protostars). While Musca is one of the simplest filaments without
embedded protostars, Orion is the closest high-mass star-forming region. We
use our DustPOL code to perform pixel-by-pixel polarization modeling of the
observed polarization data with the key local physical parameters inferred
from observations.

This paper is the second one in our series aimed at characterizing the
properties of B-fields And dust in interstelLar fiLAments using Dust
POLarization (BALLAD-POL). Our first paper focuses on the B-fields and dust in
a massive filament, G11.11-0.12 (Ngoc et al., 2023). We structure this paper
as follows. In Section 2, we describe the observational data on which we work,
and our data analysis techniques are described in Section 3. Section 4 recalls
the basis of our model and shows the comparison with observational data. We
discuss the results in Section 5, and summarize the main findings in Section
6.

##  2 Observations

###  2.1 Polarization Data

Table 1: Properties of Musca and OMC-1 and their observations. Objects | Distance | Facility | Wavelength | Beam size | Physical scale | Reference
---|---|---|---|---|---|---
Musca |  $170$ (pc) | Planck |  $850$ (${\,\mu\rm{m}}$) | $5\arcmin$ |  $0.24$ (pc) | Planck Collaboration XII (2020)
OMC-1 |  $388$ (pc) | SOFIA |  $214$ (${\,\mu\rm{m}}$) | $18\arcsec.2$ |  $0.03$ (pc) | Chuss et al. (2019)

We use the thermal dust polarization data observed by Planck and SOFIA/HAWC+
from Musca and OMC-1, respectively. The information on these filaments and
their observations are provided in Table 1.

For linear polarization, the polarization states are defined by the Stokes
parameters $I$, $Q$, and $U$. Because of the presence of noise in $Q$ and $U$,
the polarized emission intensity, $PI$, is debiased by (Vaillancourt, 2006):

$\displaystyle PI=\sqrt{Q^{2}+U^{2}-0.5(\delta Q^{2}+\delta U^{2})},$ (1)

where $\delta Q$ and $\delta U$ are the uncertainties on $Q$ and $U$,
respectively. Assuming that $\delta Q$ and $\delta U$ are uncorrelated, the
uncertainty on $PI$ is

$\displaystyle\delta PI=\sqrt{\frac{Q^{2}\delta Q^{2}+U^{2}\delta
U^{2}}{Q^{2}+U^{2}}}.$ (2)

The polarization degree, $P$, and its uncertainty, $\delta P$, are calculated
as

$\displaystyle P(\%)=100\times\frac{PI}{I},$ (3)

and

$\displaystyle\delta P(\%)=100\times\sqrt{\frac{\delta
PI^{2}}{I^{2}}+\frac{\delta I^{2}(Q^{2}+U^{2})}{I^{4}}}{\rm,}$ (4)

where $\delta I$ is the uncertainty on $I$ which is provided together with $I$
from observations.

The polarization angle, $\theta$, and its uncertainty, $\delta\theta$, are
calculated as follows:

$\displaystyle\theta=\frac{1}{2}\mathrm{atan2}\left(\frac{U}{Q}\right)$ (5)
$\displaystyle\delta\theta=0.5\times\frac{\sqrt{U^{2}\delta Q^{2}+Q^{2}\delta
U^{2}}}{(Q^{2}+U^{2})}.$ (6)

For the Planck data, $\theta$ is calculated by
$\theta=\frac{1}{2}\mathrm{atan2}(-U/Q)$ because of the different convention
used for the Stokes parameters (Planck Collaboration XII, 2020).

The polarization direction of thermal dust emission is perpendicular to the
projection of B-fields in the POS; therefore, the B-field orientation angles
on the POS are obtained by rotating $90^{\circ}$ from the polarization angle
$\theta$. Following the IAU convention, the angles of the B-field lines are
east of north, ranging from 0∘ to 180∘.

###  2.2 Polarization Angle Dispersion Function

To quantify the tangling of B-fields at small scales, we compute the
polarization angle dispersion function, denoted as $\mathcal{S}$ (see Section
3.3 in Planck Collaboration XII 2020). For each pixel at location $\bm{x}$,
$\mathcal{S}(\bm{x},\delta)$ is determined as the root mean square of the
polarization angle difference between the reference pixel $\bm{x}$ and pixel
$i$, $\mathcal{S}_{xi}=\theta(\bm{x})-\theta(\bm{x}+{\delta})$. Pixel $i$ is
situated on a circle with $\bm{x}$ as the center and a radius of $\delta$:

$\mathcal{\mathcal{S}}^{2}(\bm{x},\delta)=\frac{1}{N}\sum_{i=1}^{N}\mathcal{S}_{xi}^{2},$
(7)

with $N$ is the number of pixels lying on the circle. For the current study,
we calculate $\mathcal{S}(\bm{x},\delta)$ for $\delta$ taking a value of one
beam size of the observations. Due to noise on the Stokes parameters $Q$ and
$U$, $\mathcal{S}$ is biased. The variance of angle dispersion function,
$\sigma_{\mathcal{S}}$, and the debiased $\mathcal{S}_{\rm db}$ for a certain
pixel $\bm{x}$ are calculated as (see Section 3.5 of Planck Collaboration XII
2020):

$\begin{split}\sigma^{2}_{\mathcal{S}}&=\frac{\delta\theta^{2}(\bm{x})}{N^{2}\mathcal{S}^{2}}\left(\sum_{i=1}^{N}\mathcal{S}_{xi}\right)^{2}\\\
&+\frac{1}{N^{2}\mathcal{S}^{2}}\sum_{i=1}^{N}(\mathcal{S}_{xi})^{2}\delta\theta^{2}(\bm{x}+\bm{\delta}),\end{split}$
(8)

and

$\mathcal{S}_{\rm db}^{2}=\mathcal{S}^{2}-\sigma^{2}_{\mathcal{S}}.$ (9)

Here, $\mathcal{S}_{\rm db}$ can be positive or negative depending on whether
the true value is smaller or larger than the random polarization angle of
$52^{\circ}$ (Alina et al., 2016). The data points with
$\mathcal{S}\leq\sigma_{\mathcal{S}}$ are removed. Hereafter, we refer to
$\mathcal{S}_{\rm db}$ as $\mathcal{S}$ for convenience.

###  2.3 Dust Temperature and Column Density Maps

For further studies, we adopt the dust temperature and column density maps of
Musca and OMC-1 from previous studies. These maps are obtained by fitting a
modified black body function to the spectral energy distributions (SED).

Musca: The Planck Collaboration uses Planck at 850, 550, 350 ${\,\mu\rm{m}}$
and IRAS at 100 ${\,\mu\rm{m}}$ to fit with a modified black-body function to
obtain all-sky maps of dust temperature, optical depth, and spectral index
(Planck Collaboration XI, 2014). The resolution of the resulting maps is
$5\arcmin$. The Musca region extracted from the Planck all-sky maps is shown
in Figure 1.

OMC-1: The dust temperature and column density maps of OMC-1 are adopted from
Chuss et al. (2019). These two maps were obtained by fitting SED from emission
in the range of 53 ${\,\mu\rm{m}}$ to 35 mm, including data from SOFIA at 53,
89, 154, and 214 ${\,\mu\rm{m}}$; Herschel at 70, 100, 160, and 250
${\,\mu\rm{m}}$; James Clerk Maxwell Telescope (JCMT) at 850 ${\,\mu\rm{m}}$.
The longest wavelength of the combined Green Bank Telescope and Very Large
Array data at 3.5 and 35 mm was used to subtract the free–free emission caused
by UV in the region. The final temperature and density maps have a resolution
of $18.\arcsec 2$.

##  3 Data Analysis

Figure 1: Musca observed by Planck. Left: B-field map. The length of the line
segment is proportional to the polarization fraction $P$ with a 10% reference.
Center: Column density. Right: Dust temperature. The half-power beam width
(HPBW) of the maps $5\arcmin$ is shown by the red circle in the lower left
corner. The gray contours are $3.7\times 10^{21}\,{\rm cm}^{-2}$. The black
rectangle enclosing the southern section of the filament is the area of
interest for further analysis.

To investigate the physics of dust alignment, the analysis of polarization
fraction and total intensity is frequently employed for simplification.
However, total intensity is a function of column density and dust temperature,
making it challenging to disentangle the effects of these two parameters from
the polarization effect. Therefore, this study examines the relationship
between the polarization fraction, $P$, gas column density, $N(\rm H_{2})$,
and dust temperature, $T\rm{{}_{d}}$. Additionally, we analyze the variation
of the angle dispersion function, $\mathcal{S}$, to explore the impact of
B-field tangling on dust polarization.

In the following subsections, we will first introduce Musca and OMC-1.
Secondly, we will study the dependence of $P$ on physical conditions, e.g.,
column density and dust temperature. This will reveal the basic properties of
grain alignment and disruption in the regions. Then, we will present the
dependence of $P$ on the polarization angle dispersion function $\mathcal{S}$
to investigate the effects of the B-field tangling.

###  3.1 Musca

Figure 2: Musca. Variations of $P$, $\mathcal{S}$, $P\times\mathcal{S}$, and
$T_{\rm d}$ as a function of the radial distance ($r$) constructed using
RadFil by sampling radial cuts every 0.01 pc along the spine (similar to
Figure A.1). The radial distance is then defined as the projected distance;
the negative distance is to the left, and the positive is to the right. Thin
gray lines show the radial cuts, and the mean values are shown as dot makers
per bin size of 0.1 pc.

Musca (Figure 1) is located at a distance of $170\,{\rm pc}$ (Zucker et al.,
2021). The filament is in its early evolutionary stage and lacks active star
formation (Cox et al., 2016). One T Tauri star candidate is located at the end
of its north (Vilas-Boas et al., 1994) (in the galactic coordinates). Musca is
known for being an isothermal filament in hydrostatic equilibrium and already
fragmented; new stars can be formed in the near future (Kainulainen et al.,
2016; Bonne et al., 2020). Polarization measurements from starlight (Pereyra &
Magalhães, 2004) and thermal dust emission (Planck Collaboration XXXIII 2016;
Cox et al. 2016) show that local B-fields are generally perpendicular to the
filament’s spine. Using Herschel data, Cox et al. (2016) found that B-fields
follow the low-column density striations. It suggested that Musca has been
accreting interstellar material through these low-column density striations.
Recently, B-fields in a small part of the filament obtained by SOFIA/HAWC+ at
higher spatial resolution also indicate that the B-field orientation is
generally perpendicular to the filament’s spine (Kaminsky et al., 2023).

We use the polarization map from Planck at $\sim$$850\;{\,\mu\rm{m}}$ (353
GHz). This is the highest frequency and most sensitive channel of Planck to
dust polarization (Planck Collaboration XXXIII, 2016). The Stokes $I$, $Q$,
and $U$ are mapped at the resolution of $5\arcmin$ corresponding to a physical
scale of $\sim$$0.24\,{\rm pc}$. The pixel size of the maps is $112\arcsec$.
Musca is one of the high signal-to-noise ratio (S/N) regions of the Planck
full-sky polarization map (Planck Collaboration XXXIII, 2016). Figure 1 (left)
displays the B-field map of Musca from Planck with the lengths of line segment
proportional to the polarization fraction ($P$). The map is extracted from the
Planck map and then reprojected to the galactic coordinates (using the
reproject Python package of astropy).

Figure 1 (center and right) show the $N(\rm H_{2})$ and $T\rm{{}_{d}}$ maps
obtained from the SED fitting by Planck Collaboration XI (2014). The density
map reveals a nice filamentary structure with several fragments along the
filament, and the column density increases from the filament’s outer to inner
parts. The dust temperature map shows two trends in the filament’s spine: high
temperature in the northern part $\sim$$21\;{\,\rm K}$ and low temperature in
the southern one $<17\;{\,\rm K}$. However, we note that the dust temperature
of the filament from Planck data, which can go up to $\sim$$22\;{\,\rm K}$, is
higher than that obtained by Herschel with a maximum $T\rm{{}_{d}}$ of
$\sim$$18\;{\,\rm K}$. Moreover, Bonne et al. (2020) reported using Herschel
data that there is no high-temperature core as seen with the Planck
temperature map (Figure 1 right). Be noted that the Planck dust temperature is
obtained based on six channels ranging from 20 to 3000 $\mu$m with a
representative spatial resolution of $4\arcmin.9$ at 850 $\mu$m while for
Herschel it is five channels ranging from 70 to 500 $\mu$m with much higher
spatial resolution such as $36\arcsec$ at 500 $\mu$m. The differences of the
resulting $T\rm{{}_{d}}$ maps may come from the different dust components
probed by Planck and Herschel. Moreover, the high dust temperatures from
Planck may be due to the confusion between the cold and warm components of the
SED fitting (Lars Bonne in private communication). With Musca, we aim to study
the non-embedded radiation source regions; for further studies, we only work
with the southern part of the filament encompassed by the black rectangle in
Figure 1 (right).

Planck Collaboration XXXIII (2016) conducted a detailed study of Musca using
polarization data obtained by Planck. Their findings revealed a depolarization
effect within the filament, manifesting as a drop in polarization fraction in
the filament’s spine and a decrease in the polarization fraction with
increasing gas column density. To understand the cause of the depolarization,
we will examine the filament’s density, temperature, and B-field tangling. Due
to its well-defined filamentary structure, we will explore the radial profiles
of those quantities and their relations.

####  3.1.1 Gas Density Profile

We fit the radial density profile of Musca with a Plummer-like function using
a Python package RadFil applied for a column density map (Zucker & Chen, 2018)
(more details can be found in Appendix A.2). Assuming that the filament is in
the POS, the Plummer-like function is as follows (Arzoumanian et al., 2011;
Cox et al., 2016; Zucker & Chen, 2018):

$\displaystyle N({\rm H_{2}})(r)=\frac{N({\rm H_{2}})(0)}{[1+(r/R_{\rm
flat})^{2}]^{\frac{q-1}{2}}}$ (10)

where $r$ is the radial distance from the filament’s spine; $R_{\rm flat}$ is
the radius of the inner, flat region of the density profile; $N({\rm
H_{2}})(0)$ is the peak column density, and $q$ is the power-law density
exponent.

With the best-fit model given by RadFil, we find $q=1.94$ and $R_{\rm
flat}=0.28\,{\rm pc}$. Cox et al. (2016) fitted a Plummer-like function for
Musca using Herschel data and obtained $q=2.2$ and $R_{\rm flat}=0.08\,{\rm
pc}$. Because Planck’s beam size of $\sim$$5\arcmin$ ($\sim$$0.24\,{\rm pc}$)
is much larger than that of Herschel of $\sim$$36\arcsec$ ($\sim$$0.025\,{\rm
pc}$), our fitted $R_{\rm flat}$ is much larger than that of Herschel.

####  3.1.2 Filament Radial Profiles

Figure 2 shows the physical parameters across the filament obtained by RadFil,
including polarization fraction (a), polarization angle dispersion function
(b), grain alignment efficiency $P\times\mathcal{S}$ (c), and dust temperature
(d).

It is seen from the $P$ profile (a) that $P$ is the lowest in the filament’s
spine ($\sim$8%). $P$ increases to 15% at about $0.7\;\,{\rm pc}$ from the
spine and then slightly decreases to 10% when going to the ISM. This trend is
also found by Planck Collaboration XXXIII (2016) (see their Figure 10).

To unravel the effects of B-field tangling on the decrease of $P$ with
decreasing $r$ (depolarization), we analyze the polarization angle dispersion
function, $\mathcal{S}$, and the product $P\times\mathcal{S}$ shown in Figure
2 (b and c). $\mathcal{S}$ is generally low in Musca, and its mean value is
smaller than $10^{\circ}$. Planck Collaboration XXXIII (2016) claim that this
depolarization is not attributed to the B-field fluctuations because
$\mathcal{S}$ is small ($\sim$$10^{\circ}$), significantly smaller than
$52^{\circ}$ for random fields.

In addition to B-field tangling, the polarization fraction is influenced by
environmental factors and alignment efficiency. Removing the effect of the
B-field tangling, $P\times\mathcal{S}$ represents the alignment efficiency of
grains along the LOS (Planck Collaboration XII, 2020). The
$P\times\mathcal{S}$ radial profile reveals that the alignment efficiency is
the lowest in the filament spine and increases when moving from the inner to
the outer filament.

The temperature profile (d) shows that the temperature and column density
(Figure A.2) behave in the opposite manner; the temperature is low in the
filament’s spine, which has the highest density ($\sim$$5\times 10^{21}\,{\rm
cm}^{-2}$). The $N(\rm H_{2})-$$T_{\rm d}$ anti-correlation was also found
with Herschel data by Bonne et al. (2020). The temperatures in the spine are
below $18\;{\,\rm K}$ and can go down to $16\;{\,\rm K}$. The mean
temperatures increase from $17\;{\,\rm K}$ to $20\;{\,\rm K}$ from the spine
to the outer areas.

Figure 3: OMC-1. a) B-field orientation map observed at $214{\,\mu\rm{m}}$
wavelength by SOFIA/HAWC+. The black lines show the orientation of the B-field
vector and are overplotted on the intensity map. The vector spacing equals the
beam size, shown by a solid red circle in the lower-left corner. Three
regions, North, BN-KL, and South, are labeled. b) the column density map, and
c) the dust temperature map. The light green, dark blue, and violet contours
correspond to $P=0.6,1.0,1.4\%$. The star indicates the location of BN/KL, and
the triangle shows the location of the Trapezium cluster.

From those filament profiles, we see that the polarization fraction is lowest
in the filament bone in which the temperature (or radiation field) is lowest,
density is highest, and alignment efficiency is the lowest. The polarization
fraction increases when density decreases, dust temperature increases, and
alignment efficiency increases.

Figure 4: OMC-1. Variations of the polarization fraction, $P$, with the column
density $N(\rm H_{2})$ (a), dust temperature $T\rm{{}_{d}}$ (b), and angle
dispersion function $\mathcal{S}$ (c). The blue, red, green, and violet dots
represent the North, BN-KL, South, and the H II region and the lines are the
results of fits to power-law models, respectively.

Figure 5: OMC-1. The dependence of column density on dust temperature (a), the
angle dispersion function $\mathcal{S}$ on the column density $N(\rm H_{2})$
(b) and dust temperature $T\rm{{}_{d}}$ (c), and the dependence of the
$P\times\mathcal{S}$ on the dust temperature $T\rm{{}_{d}}$ (d). The blue,
red, green, and violet dots represent the data in the North, BN-KL, South, and
the H II regions, respectively, and the solid curves show the running mean
values of the data.

###  3.2 OMC-1

Located at a distance of $388\pm 5\,{\rm pc}$ (Kounkel et al., 2017), the
Orion Nebula is the nearest high-mass star formation region (O’dell et al.,
2008). OMC-1 or Orion A is located behind an H II region ionized by O-B stars
from the Trapezium cluster. OMC-1 consists of two principal clumps, the
northern Becklin-Neugebauer-Kleinmann-Low (BN/KL) clump (Becklin & Neugebauer,
1967; Kleinmann & Low, 1967) and the southern Orion S clump (Batria et al.,
1983). BN/KL hosts an extremely explosive molecular outflow with a wide
opening angle and multiple ejecta. B-fields of OMC-1 have an hourglass
morphology and strengths of the order of mG (e.g., Pattle et al. 2017; Chuss
et al. 2019; Guerra et al. 2021).

We use thermal dust polarization data observed toward the Orion BN/KL by
SOFIA/HAWC+ at $214{\,\mu\rm{m}}$ and beam size of $18\arcsec.2$, which was
firstly reported by Chuss et al. (2019). Analyzing this data set, Chuss et al.
(2019) and later Guerra et al. (2021) mainly focused on the studies of
B-fields. Chuss et al. (2019) found the depolarization effect and Tram et al.
(2021c) found evidence for RAT-D. For the present work, we only use the data
having signal-to-noise ratios ($S/N$s) satisfying: $S/N(I)>250$ and
$S/N(P)>3$.

Figure 3 displays the maps of B-fields (a), column density (b), and dust
temperature (c) observed toward OMC-1 by SOFIA/HAWC+ at 214 ${\,\mu\rm{m}}$.
We only consider the OMC-1 filament and H II region encompassed by the gray
contours. The H II region has low column density of $N(\rm H_{2})<2\times
10^{22}\,{\rm cm}^{-2}$, and high dust temperature $T_{\rm d}>45{\,\rm K}$.
While the filament has high column density $N(\rm H_{2})>2\times 10^{22}\,{\rm
cm}^{-2}$ and strong emission $I>15$ $\rm mJy/arcsec^{-2}$; for a detailed
study, we divided the OMC-1 filament into three sub-regions: North, BN-KL, and
South.

The $N(\rm H_{2})$ map (Figure 3 b) shows that the column density is highest
in the filament’s spine and divided into three clumps residing in the North,
BN-KL, and South regions. BN-KL exhibits the highest temperatures, reaching
100 K (Figure 3 c). The temperatures of the eastern part of OMC-1 and the H II
region are higher than that of the western one. This is due to the effect of
ionization from the Trapezium cluster. The H II region has the lowest
densities compared to other regions.

The lengths of the line segments in Figure 3 (a) are proportional to the
polarization fraction, revealing high polarization fractions in the H II
region and particularly low polarization fractions in certain parts of the
filament. The color contours in Figure 3 (b) and (c) highlight three
polarization holes in the central part of each sub-region, with polarization
fractions smaller than 1%. In Figure 3 (b) and (c), the polarization holes in
the North and South regions show lower dust temperatures compared to the
polarization hole in BN-KL, despite all regions having high column density.
This suggests that the depolarization in the low and high dust temperatures
may be caused by different mechanisms.

To explicitly see how the polarization fraction, $P$, changes with the column
density, $N(\rm H_{2})$, dust temperature, $T\rm{{}_{d}}$, and B-field
tangling, $\mathcal{S}$, we show the variations of $P$ with $N(\rm H_{2})$,
$T\rm{{}_{d}}$, and $\mathcal{S}$ in Figure 4. The $P-N(\rm H_{2})$ relation
(Figure 4 a) shows an anti-correlation of the polarization fraction and the
gas column density (i.e., polarization hole). We fit the data with a power-law
model and obtain the power indices $\alpha=-0.33\pm 0.03$ for North,
$\alpha=-0.50\pm 0.01$ for BN-KL, $\alpha=-0.45\pm 0.03$ for South, and the
smallest slope value $\alpha=-0.08\pm 0.01$ for the H II region. In the OMC-1
filament, the $P-T\rm{{}_{d}}$ relation reveals that the polarization fraction
increases at low $T\rm{{}_{d}}$ and dramatically decreases as the dust
temperature increases to $T\rm{{}_{d}}>70{\,\rm K}$. In detail, the
polarization fraction remarkably increases with increasing $T\rm{{}_{d}}$ with
a power index $\alpha=2.82\pm 0.11$ in the South, while in the North with the
low temperature ($T\rm{{}_{d}}<40{\,\rm K}$), the polarization fraction
slightly increases with $\alpha=0.09\pm 0.08$. In BN-KL, having massive
protostars and a high dust temperature region, the polarization fraction only
decreases as the dust temperature increases with $\alpha=-0.57\pm 0.06$. Tram
et al. (2021c) studied this relation for BN-KL using data collected by
JCMT/POL-2 and SOFIA/HAWC+ and found this increase-decrease feature. Although
dust temperature can go up to 80 K, the polarization fraction gradually
increases with increasing $T\rm{{}_{d}}$ in the H II region with
$\alpha=0.41\pm 0.05$.

We also fit a power law function to the $P-\mathcal{S}$ relation and find the
anti-correlation of $P$ and $\mathcal{S}$ with $\alpha=-0.61\pm 0.02,-0.48\pm
0.02$, $-0.69\pm 0.02$, and $-0.17\pm 0.01$ for North, BN-KL, South, and H II
region, respectively. The H II region seems to have weak B-field fluctuations
as shown by the small angle dispersion function of $\mathcal{S}<10^{\circ}$.
However, the South region appears to have stronger B-field fluctuations due to
the high angle dispersion function of $\mathcal{S}>10^{\circ}$.

From the $N(\rm H_{2})$$-T_{\rm d}$ relation shown in Figure 5 (a), we see
that $N(\rm H_{2})$$-T_{\rm d}$ is anti-correlated in the North, South, and H
II regions, while, it is correlated in the BN-KL region with the presence of
massive protostars. To show the effect of B-field tangling on the
depolarization, we plot the relations of $\mathcal{S}-N(\rm H_{2})$ and
$\mathcal{S}-T\rm{{}_{d}}$ in Figure 5 (b) and (c). The $\mathcal{S}-N(\rm
H_{2})$ relation shows the correlation of the polarization angle dispersion
function $\mathcal{S}$ and the column density in general. This clearly shows
the increasing importance of B-field tangling in dense regions. The
$\mathcal{S}-T\rm{{}_{d}}$ relation shows that the polarization angle
dispersion function $\mathcal{S}$ has anti-correlation with the dust
temperature in the South and H II regions, but a correlation in the North and
BN-KL regions. The relationship between $P\times\mathcal{S}$ and
$T\rm{{}_{d}}$ (Figure 5 d) shows that the average grain alignment efficiency
firstly slightly increases with dust temperature for $T\rm{{}_{d}}<35{\,\rm
K}$, remains constant in the range of 35-70 K, and then decreases for
$T\rm{{}_{d}}>70{\,\rm K}$.

##  4 Numerical Modeling and Results

This section presents the numerical modeling method of polarized thermal dust
emission from aligned grains using the RAT theory for Musca and OMC-1 to test
the RAT paradigm. We first use the DustPOL
code111https://github.com/lengoctram/DustPOL, which was first presented in Lee
et al. (2020) and improved by Tram et al. (2021a) to obtain the dust
polarization for the ideal model in which the B-fields are assumed to lie in
the POS. Then, we incorporate the depolarization effects caused by the B-field
inclination angle and their fluctuations into our ideal model to obtain a
realistic polarisation model and confront it with the observational data.

Below, we briefly describe the main features of DustPOL and its key input
parameters, including the physical parameters of the gas and dust, the local
radiation field, and the alignment of dust grains by RATs. We then present the
main modeling results and compare them to the data.

###  4.1 Methods and Assumptions

####  4.1.1 Gas Properties and Radiation Fields

The gas density ($n_{\rm H}=2n_{\rm H_{2}}$) and temperature ($T_{\rm gas}$)
are the two important physical parameters in the RAT alignment theory because
they determine the randomization and alignment of dust grains (see e.g., Hoang
et al. 2021a). For our modeling, these parameters are inferred from
observational data and used as the input parameters for DustPOL.

For radiation fields, the radiation strength, $U$ (effectively equivalent to
the dust temperature $T\rm{{}_{d}}$) and the anisotropy degree of the
radiation, $\gamma$, are the key parameters of the RAT theory because they
determine the alignment degree of dust grains (Hoang et al., 2021a). The
anisotropy degree varies in a range from 1 to 0; $\gamma=1$ represents a
unidirectional radiation field from a nearby star, while $\gamma=0.1$ for the
diffuse ISM (Draine & Weingartner, 1996). The mean wavelength,
$\bar{\lambda}$, depends on specific regions.

####  4.1.2 Grain Size Distribution

We adopt a power-law grain size distribution, $dn/da\propto a^{-\beta}$, where
$\beta$ is in the range of $3.5-4.5$. The grain size distribution in the ISM
follows the MRN distribution with a power-law index $\beta=3.5$ (Mathis et
al., 1977). The smallest grain size, $a_{\rm min}$, is 10$\,{\rm\AA}$, while
the initial maximum grain size, $a_{\rm max}$, can be varied in the range of
$0.2\;{\,\mu\rm{m}}$ to several microns.

When considering the RAT-D effect, the value of $a_{\rm max}$ varies with the
local conditions of the gas densities, radiation fields, and tensile
strengths, as given by Equation A4. Their tensile strengths determine the
internal structure of grains, and the disruption temperature depends on this
property. For composite grains, the tensile strengths vary around $S_{\rm
max}=10^{7}\,{\rm erg}\,{\rm cm}^{-3}$ (Hoang, 2019).

####  4.1.3 Grain Alignment Function by RATs

In the RAT-A theory, grains are first spun up to suprathermal rotation and
then driven to be aligned with the ambient B-fields by RATs (Lazarian & Hoang,
2007; Hoang & Lazarian, 2008). For grains with iron inclusions, which are
superparamagnetic material, the joint effect of enhanced magnetic relaxation
and RATs could make grains achieve perfect alignment with $\bf B$, a mechanism
known as magnetically enhanced RAT (MRAT) (Hoang & Lazarian, 2016). Therefore,
grains are only efficiently aligned when they can rotate suprathermally. The
minimum size for grain alignment, $a_{\rm align}$ (also called critical
alignment size), is determined by the suprathermal rotation criterion of
$\omega_{\rm RAT}(a_{\rm align})=3\omega_{\rm T}$ (Hoang & Lazarian, 2008),
where $\omega_{\rm RAT}(a_{\rm align})$ is the angular velocity of grains
spun-up by RATs and $\omega_{\rm T}$ is the grain’s thermal angular velocity.
From Equation A1, we have:

$\displaystyle a_{\rm align}$ $\displaystyle\simeq$ $\displaystyle
0.024\hat{\rho}^{-5/32}\gamma^{-5/16}U^{-5/16}\left(\frac{10^{3}\,{\rm
cm}^{-3}}{n_{\rm H}}\right)^{-5/16}$ (11)
$\displaystyle\times\left(\frac{\bar{\lambda}}{0.5{\,\mu\rm{m}}}\right)^{17/32}\left(\frac{20\;{\,\rm
K}}{T_{\rm gas}}\right)^{-5/16}\left(\frac{1}{1+F_{\rm
IR}}\right)^{-5/16}{\,\mu\rm{m}},$

where $U$ is the radiation strength of the local radiation field.

The degree of grain alignment as a function of the grain size can be described
by:

$\displaystyle f_{\rm align}=f_{\rm max}\left[1-\exp\left(-a/2a_{\rm
align}\right)^{-3}\right],$ (12)

where $a_{\rm align}$ is given by Equation 11, $f_{\rm max}$ describes the
maximum degree of grain alignment, and the negligible alignment of small
grains of $a\ll a_{\rm align}$ is disregarded (Hoang & Lazarian, 2016).

According to the MRAT mechanism, the exact value of $f_{\rm max}$ depends on
the ratio of the magnetic relaxation to the gas damping rate, which is a
function of the grain size, magnetic susceptibility, and local gas density, as
shown in Hoang & Lazarian (2016). They also numerically demonstrated that
grains with embedded iron clusters (i.e., superparamagnetic grains) can
achieve perfect alignment of $f_{\rm max}=1$. Moreover, inverse modeling of
thermal dust polarization observed by Planck in Hensley & Draine (2023) found
that perfect alignment of large grains is required to reproduce the
observational data. Therefore, in this paper, we assume $f_{\rm max=1}$, which
is expected from the MRAT theory for composite grains containing embedded iron
inclusions.

####  4.1.4 Ideal Model of Thermal Dust Polarization with DustPOL

Dust grains are heated by starlight and subsequently re-emit radiation in
infrared wavelengths. In the assumption of a dust environment containing both
carbonaceous and silicate grains, the total emission intensity is calculated
as:

$\displaystyle\frac{I_{\rm em}(\lambda)}{N_{\rm H}}=$
$\displaystyle\sum_{j=sil,cal}\int_{a_{\rm min}}^{a_{\rm max}}Q_{\rm abs}\pi
a^{2}$ (13) $\displaystyle\times\int dTB_{\lambda}(T_{\rm
d})\frac{dP}{dT}\frac{1}{n_{\rm H}}\frac{dn_{j}}{da}da,$

where $Q_{\rm abs}$ is the absorption efficiency, $dP/dT$ is the temperature
distribution function, and $B_{\lambda}(T_{\rm d})$ is the Planck function.
Here, we disregard the minor effect of grain alignment on total emission
intensity (cf. Hoang & Truong 2023).

If carbonaceous and silicate grains exist in two separate populations, only
silicate grains can be aligned with B-fields due to their paramagnetic nature,
while carbonaceous grains (e.g., pure graphite) cannot be efficiently aligned
because of their diamagnetic nature Hoang & Lazarian (2016).222Although ideal
carbon grains being diamagnetic cannot be aligned with B-fields, hydrogenated
carbon grains can be aligned with the B-fields because those grains contain
unpaired electrons and become paramagnetic material (Hoang et al., 2023).
However, for a composite dust model (a mixture of silicate and carbon in a
dust grain), the composite dust can be aligned with B-fields. Note that
observational constraints favor the composite model for interstellar dust,
such as Astrodust (Draine & Hensley, 2021) and THEMIS model (Ysard et al.,
2024). Hence, for this study, we assume the composite dust model, so that
carbonaceous and silicate grains have the same temperature.

We consider first the ideal model in which B-fields lie in the POS and the
impact of B-field fluctuations is disregarded. The polarized emission
intensity from aligned grains is then given by:

$\displaystyle\frac{I_{\rm pol}(\lambda)}{N_{\rm H}}=$
$\displaystyle\sum_{j=sil,cal}\int_{a_{\rm min}}^{a_{\rm max}}f_{\rm
align}(a)Q_{\rm pol}\pi a^{2}$ (14) $\displaystyle\times\int
dTB_{\lambda}(T_{\rm d})\frac{dP}{dT}\frac{1}{n_{\rm H}}\frac{dn_{j}}{da}da,$

where $Q_{\rm pol}$ is the polarization efficiency (Lee et al., 2020).

The polarization degree of thermal dust emission is given by

$\displaystyle P_{\rm mod}^{\rm ideal}(\%)=100\%\times\frac{I_{\rm
pol}}{I_{\rm em}},$ (15)

where the subscript “mod” is for the values calculated from the model and
superscript “ideal” for the ideal polarization degree achievable by grain
alignment, assuming the optimal case of B-fields in the POS with the absence
of the B-field tangling.

####  4.1.5 Realistic Polarization Model: Accounting for B-field Tangling

For DustPOL, B-fields are assumed to lie in the POS, so that the degree of
thermal dust polarization could achieve the maximum value, as given by
Equation 15. In a realistic situation, the inclination angle of B-fields and
LOS, denoted by $\psi$, contributes to the polarization degree due to the
projection effect. Moreover, the fluctuations of B-fields along the LOS cause
the depolarization of thermal dust emission, which is described by an
additional depolarization parameter $F_{\rm turb}$. $F_{\rm turb}$ is a
function of the angle between the local B-field and the mean B-field and is
anti-correlated with the polarization angle dispersion function $\mathcal{S}$
(Hoang & Truong, 2023). Therefore, the realistic dust polarization degree
becomes

$\displaystyle P_{\rm mod}=P_{\rm mod}^{\rm ideal}\sin^{2}\psi F_{\rm turb}.$
(16)

Numerical simulations in Hoang & Truong (2023) using MHD simulations and
polarized radiative transfer in POLARIS found that the polarization model
given by Equation 16 is in excellent agreement with synthetic polarization.
They also found that $F_{\rm turb}$ decreases with $\mathcal{S}$ as $F_{\rm
turb}\propto\mathcal{S}^{-\eta_{1}}$, where $\eta_{1}$ depends on the
inclination angle. Numerical simulations in Planck Collaboration et al. (2015)
showed that there is a correlation between the average of the B-field
inclination angle and $\mathcal{S}$, which can be described as
$\langle\sin^{2}\psi\rangle\propto\mathcal{S}^{-\eta_{2}}$ (see e.g., Hensley
et al. 2019). Furthermore, the B-field tangling within the beam size (i.e.,
B-field fluctuations in the POS) can cause an additional depolarization
described by the factor $F_{\rm beam}$ (Hoang & Truong, 2023). Accounting for
all these depolarization effects, the net polarization degree of thermal dust
emission can be described by

$\displaystyle P_{\rm mod}=\Phi P_{\rm mod}^{\rm
ideal}\left(\frac{\mathcal{S}}{1^{\circ}}\right)^{-\eta},$ (17)

where $\Phi$ is a coefficient that describes the depolarization by the
B-field’s inclination angle, and $\eta>0$ is the power index describing the
depolarization effect caused by B-field fluctuations along the LOS and in the
POS, which is expected to vary with the local conditions. The value $\Phi$ is
obtained by fitting the polarization fraction from the model to the maximum
polarization observed in each filament. The best-fit values of $\Phi$ are
shown in Table 2.

###  4.2 Results

Table 2: Physical parameters for model | Parameter | Musca | OMC-1
---|---|---|---
Anisotropic degree | $\gamma$ | 0.3 | 1
Mean wavelength | $\bar{\lambda}\;[{\,\mu\rm{m}}]$ | 1.2 | 0.3
Gas temperature |  $T_{\rm gas}$ [K] | $T_{\rm dust}$ |  $T_{\rm dust}$ (filament), 5000 K(H II region)
Grain composition | - | Sil+Car | Sil+Car
Dust mass density | $\rho\;[\,{\rm g}\,{\rm cm}^{-3}]$ | 3 | 3
Minimum grain size | $a_{\rm min}\;[{\,\mu\rm{m}}]$ | $10^{-3}$ | $10^{-3}$
Maximum grain size | $a_{\rm max}\;[{\,\mu\rm{m}}]$ | $0.25-0.5$ | $0.2-2$
Power-index | $\beta$ | $-3.5$ | $-3.5$
Maximum tensile strength |  $S_{\rm max}$ [erg cm-3] | $10^{7}$ | $10^{7}-10^{8}$
B-field’s inclination effect | $\Phi$ | $0.53$ | $0.28$

Table 2 summarizes the model parameters for Musca and OMC-1, including the gas
number density, $n_{\rm H}$, gas temperature, $T_{\rm gas}$, ($a_{\rm min}$,
$a_{\rm max}$), shape (grain aspect ratio = minor axis/major axis taken to be
equal to 1/3), size distribution power index, $\beta$, tensile strength,
$S_{\rm max}$, radiation field strength, $U$, the mean wavelength,
$\bar{\lambda}$, and the anisotropy degree, $\gamma$.

####  4.2.1 Musca

In Musca, where a filamentary structure is evident and the physical conditions
do not vary significantly along the filament, we will model the variation of
polarization faction $P$ across the filament, as the function of the radial
distance, $r$, from the filament’s spine to the outer regions.

Figure 6: Musca. Upper: Minimum alignment size by RATs calculated from our
model for Musca. Lower: Comparison of the polarization fraction from our model
(thick solid lines) with observation data. The gray curves show $P$ vs. $r$
for radial cuts along the filament’s spine from observational data (see
Section 3.1.2). Green dots represent the mean values of the observational $P$
at a certain value of $r$. The color curves show polarization fractions from
our model with different $a_{\rm max}$. The rapid increase of alignment size
toward the filament center causes the decrease of the model polarization
fraction and successfully reproduces the observed polarization hole.

To model the dust polarization as a function of the radial distance, $r$, we
first need to know how the local gas volume density and dust temperature
(radiation field) vary with $r$. The dust temperature profile is taken to be
the mean temperature obtained from Planck (Figure 2 d).

For the regions of $|r|<1$ pc within the filament, the gas number density
varies with the radial distance $r$ as the Plummer-like function (e.g.,
Arzoumanian et al. 2011),

$\displaystyle n_{{\rm H}_{2}}(r)=\frac{n_{{\rm H}_{2}}(0)}{[1+(r/R_{\rm
flat})^{2}]^{p/2}},$ (18)

where $n_{\rm H_{2}}(0)$ is the number density at the filament center $r=0$,
given by

$\displaystyle n_{\rm H_{2}}(0)=\frac{N({{\rm H}_{2}})(0)}{A_{\rm p}R_{\rm
flat}},$ (19)

where

$\displaystyle A_{\rm p}=\frac{1}{\cos
i}\int_{-\infty}^{\infty}\frac{du}{(1+u^{2})^{p/2}},$ (20)

with $i$ is the inclination angle of the filament relative to the POS, and we
assumed $i=0$. Following Plummer-like fitting as described in Section 3.1.2,
we obtain $n_{\rm H_{2}}(0)\simeq 1300\,{\rm cm}^{-3}$.

For the outer regions of $|r|>1\,{\rm pc}$, we assume a constant gas density
of $n_{\rm H_{2}}=100\,{\rm cm}^{-3}$ and constant dust temperature of $T_{\rm
d}=18.3\;{\,\rm K}$ (Figure 2 d).

Figure 6 (upper panel) shows the variation of the minimum alignment size by
RATs ($a_{\rm align}$) computed for Musca. The alignment size increases
rapidly toward the filament spine due to the increase in the gas volume
density and reduction of the radiation field (dust temperature). For a fixed
maximum size, $a_{\rm max}$, the increase in $a_{\rm align}$ corresponds to
the increasing loss of grain alignment of grains having sizes smaller than
$a_{\rm align}$ toward the filament’s spine.

Figure 6 (lower panel) compares the polarization profile obtained from our
modeling, $P_{\rm mod}$, given by Equation 17, with the observational data for
the case without RAT-D. Musca has low temperatures $<19\;{\,\rm K}$ and high
densities of $10^{2}-10^{3}\,{\rm cm}^{-3}$, therefore, RAT-D cannot occur.
Models with only RAT-A can very well reproduce the data. We note that the
B-field tangling is expected to have a minor effect on the polarization hole
in Musca because the values of polarization angle dispersion function
$\mathcal{S}$ are small ($<10^{\circ}$) and do not change considerably across
the filament (Figure 2 b). In addition, there is little dependence of $P$ on
$\mathcal{S}$ with $\eta=0.03$.

Figure 6 shows model predictions of polarization fraction with different
values of maximum grain sizes varying from 0.25 to 0.5 ${\,\mu\rm{m}}$. Dust
grains can be aligned between $a_{\rm align}$ and $a_{\rm max}$; therefore,
higher $a_{\rm max}$ produces a higher polarization fraction. Moreover, as
$a_{\rm max}$ increases, the size distribution of aligned grains becomes
broader, narrowing the difference between the polarization fraction between
the outer and inner of Musca. As shown, our polarization models with $a_{\rm
max}=0.25-0.35{\,\mu\rm{m}}$ can successfully reproduce the strong decrease in
the polarization fraction toward the filament’s spine. Therefore, the loss of
grain alignment by RATs (i.e., increasing $a_{\rm align}$) toward the denser
and colder region of the filament (upper panel) is the main origin of the
polarization hole observed in Musca.

####  4.2.2 OMC-1

For OMC-1 that has the structure and physical conditions varying
significantly, such as between the east and west (with/without H II region)
and the presence/absence of an embedded protostar within the filament, we will
perform pixel-by-pixel modeling for the whole region to obtain maps of
polarization fraction to compare with the observational data.

Figure 7: OMC-1. The polarization fraction maps, (a) from observation and (b)-(e) from numerical modeling: (b) $P_{\rm mod}^{\rm ideal}$ without RAT-D, (c) $P_{\rm mod}^{\rm ideal}$ with RAT-D, (d) $P_{\rm mod}$ without RAT-D, and (e) $P_{\rm mod}$ with RAT-D. The gray contour corresponds with 4 sub-regions defined in Figure 3. The star indicates the location of BN/KL, and the triangle shows the location of the Trapezium cluster. Figure 8: OMC-1. Comparison of the polarization degree vs. dust temperature relation from our models (color lines) with observational data (black line). The shaded area represents the 1-$\sigma$ deviation of each bin of data points. The ideal polarization model overestimates the observational data (blue and orange lines), but the realistic polarization model (red line) considering RAT-A, RAT-D, and B-field tangling ($P_{\rm mod}$) can best reproduce the observed variation of $P-T_{\rm d}$. Figure 9: OMC-1. Same as Figure 8 but for $P-N({\rm H_{2}})$ relation. The ideal polarization model (blue and orange lines) cannot reproduce the data, but the realistic polarization model (red line) considering RAT-A, RAT-D, and B-field tangling ($P_{\rm mod}$) can best reproduce the observed polarization hole. Table 3: The comparisons between models and observations for different regions in OMC-1: North, BN-KL, South, H II, and all. The statistical evaluation is shown by $\chi^{2}=\frac{1}{N}\sum_{i=1}^{N}\frac{(P^{\rm i}_{\rm mod}-P^{\rm i}_{\rm obs})^{2}}{P^{\rm i}_{\rm obs}}$ with $N$ the number of pixels within each region. As shown, the values of $\chi^{2}$ are the highest for $P_{\rm mod}^{\rm ideal}$. $P_{\rm mod}$ with B-field tangling and RAT-D produces the smallest $\chi^{2}$, which implies that both grain disruption and B-field tangling play an important role in the observed polarization. Moreover, the model with the rotational disruption effect matches much better with observations in BN-KL and HII regions. Model |  |  | $\chi^{2}$ |  |
---|---|---|---|---|---
| North | BN-KL | South | H II | All
$P_{\rm mod}^{\rm ideal}$ without RAT-D | 21.65 | 17.34 | 46.77 | 1.39 | 18.76
$P_{\rm mod}^{\rm ideal}$ with RAT-D | 21.65 | 12.24 | 45.88 | 0.65 | 17.20
$P_{\rm mod}$ without RAT-D | 0.32 | 0.83 | 0.30 | 0.32 | 0.43
$P_{\rm mod}$ with RAT-D | 0.32 | 0.55 | 0.4 | 0.17 | 0.34

For the modeling of the OMC-1 molecular cloud, we assume the mean radiation
wavelength to be $\bar{\lambda}\simeq 0.53\,{\rm cm}{\,\rm K}/T_{\star}$
(Hoang et al., 2021a), which corresponds to $\bar{\lambda}=0.3{\,\mu\rm{m}}$
for the presence of the nearby massive OB star cluster of $T_{\star}>2\times
10^{4}$ K. The pixel-by-pixel maps of the number density, $n_{\rm H_{2}}$, and
dust temperature, $T\rm{{}_{d}}$, are used for modeling grain alignment and
polarization fraction. To derive the number density map, OMC-1 is assumed to
have the same depth $d_{\rm OMC}=0.15\,{\rm pc}$, as obtained by Chuss et al.
(2019). Therefore, the volume density is accordingly calculated by $n_{\rm
H_{2}}=N({\rm H_{2}})/d_{\rm OMC}$. To reduce measurement noise and optimize
model computation time, we increase the pixel size by a factor of $\sim$4 to
be $14\arcsec$, almost equal to the spatial resolution of the input data maps.

Since grain growth is expected to be efficient in dense star-forming regions,
we assume the maximum grain size of $a_{\rm max}=2{\,\mu\rm{m}}$ for $N(\rm
H_{2})>2\times 10^{22}$ (cm-2) in OMC-1. In the remaining regions of low
density, such as the outer region of the filament structure, and the H II
region, we assume the maximum grain size of $a_{\rm max}=0.2{\,\mu\rm{m}}$,
similar to that of the diffuse ISM. When modeling the RAT-D effect expected to
be important in OMC-1 due to strong radiation fields, the tensile strength of
dust grains is required. Because the tensile strength of dust grains depends
on the grain internal structure (e.g., porosity) which varies with the grain
size due to grain coagulation (Tatsuuma et al., 2019), we assume $S_{\rm
max}=10^{7}\,{\rm erg}\,{\rm cm}^{-3}$ for large grains having $a_{\rm
max}=2{\,\mu\rm{m}}$, and a higher tensile strength of $S_{\rm
max}=10^{8}\,{\rm erg}\,{\rm cm}^{-3}$ for grains having $a_{\rm
max}=0.2{\,\mu\rm{m}}$.

As shown in Figure 4 (c), the angle dispersion function in OMC-1 can go up to
50∘, and the slopes of $P-\mathcal{S}$ relation are 0.47–0.69 in the filament,
which means that B-field tangling in OMC-1 is stronger than Musca and varies
significantly across the object. Therefore, one cannot take the dust
polarization results directly obtained by DustPOL as in Musca. We use the
polarization model, which takes into account the effect of B-field
fluctuations given by Equation 17 for different subregions in OMC-1 with
different $\eta$ (see Section 4).

Figure 7 (b) - (e) shows the polarization fraction maps obtained by using our
numerical modeling. Figure 7 (a) shows the observational data for comparison.
From Figure 7 (b), one can see that $P_{\rm mod}^{\rm ideal}$ without RAT-D
can reproduce a part of the South and North polarized holes; however, the
model cannot reproduce the polarization hole in BN-KL. Figure 7 (c) shows the
result from $P_{\rm mod}^{\rm ideal}$ with RAT-D; the polarization fraction
decreases in the BN-KL region, but $P$ from the model is significantly higher
than the observed data. The models in Figure 7 (d) and (e) incorporate B-field
tangling effects, $P_{\rm mod}$ without/with RAT-D with exponents
$\eta=0.61,0.48,0.69,$ and $0.17$ for the North, BN-KL, South, and H II
regions, respectively. Here, the values of $\eta$ are chosen based on the
slopes of the $P-\mathcal{S}$ relation in Figure 4 (c). It is evident that
$P_{\rm mod}$ without RAT-D is limited to reproducing observations in the
southern and northern areas. However, when RAT-D is incorporated, our model
aligns more closely with observations, including those in the BN-KL and H II
regions. In the ensuing analysis, we conducted a thorough comparison utilizing
two distinct diagnostic methods for the $P-T_{\rm d}$ relationship (which
directly reflect the impact of alignment and disruption) and $P-N({\rm
H_{2}})$ (which directly reflects the gas damping effect).

Figure 8 shows the comparison of the $P-T_{\rm d}$ relation produced by our
model and from observations for the whole OMC-1. There is a noticeable
correlation between $P$ and $T\rm{{}_{d}}$ at low temperatures ($<70{\,\rm
K}$), while an anti-correlation is observed at high temperatures ($>70{\,\rm
K}$). It is clear that models that do not take into account the fluctuations
of B-fields fail to reproduce the observations, while those incorporating both
the RAT-D mechanism and B-field fluctuations provide the most satisfactory
explanation for these two trends compared to other models.

The left panels of Figure A.2 show the comparisons for the individual regions,
including the North, South, BK-KL, and H II regions. In the regions with lower
dust temperatures (North and South), models with only RAT-A and a combination
with RAT-D are identical because the disruption effect does not occur. For the
regions of high dust temperatures (BN-KL and H II), models with RAT-D and
B-field fluctuations best reproduce the decrease of polarization degree at
$T_{\rm d}>50$ K.

Figure 9 compares the $P-N({\rm H_{2}})$ relations of our models with
observations. One can see that models without B-field tangling, $P_{\rm
mod}^{\rm ideal}$, produce a rather high polarization and cannot reproduce the
polarization hole. On the other hand, models with the B-field tangling,
$P_{\rm mod}$, better reproduce the data. Therefore, the main cause of the
depolarization in $P-N({\rm H_{2}})$ is due to the B-field tangling, and the
grain alignment is still efficient at large column density due to the effect
of embedded radiation sources. This is different from the case of Musca, where
the main cause of the polarization hole is the loss of grain alignment toward
the denser and weaker radiation field.

We refer to Table 3 for a quantitative $\chi^{2}$ comparison. As shown, the
values of $\chi^{2}$ are larger for $P_{\rm mod}^{\rm ideal}$ and smaller for
$P_{\rm mod}^{\rm ideal}$ with the RAT-D effect. It is improved for the BN-KN
and H II regions where RAT-D happens. $P_{\rm mod}$ with the B-field tangling
produces the smaller $\chi^{2}$, which implies that both grain alignment and
B-field tangling play a role in the observed polarization. Therefore, we see
that the best model is $P_{\rm mod}$ with both RAT-D and B-field tangling
taken into account.

##  5 Discussions

In this section, we will discuss the evidence of grain alignment and
disruption by RATs obtained from comparing modeling results with observational
data for Musca and OMC-1.

###  5.1 Evidence of RAT-A Mechanism

####  5.1.1 Decrease of polarization fraction with column density: Origin of
polarization hole

The anti-correlation between $P$ and $N(\rm H_{2})$ (or $Av$), aka.
polarization hole, is usually found in various environments, from the diffuse
ISM to molecular clouds, filaments, and star-formation regions (e.g., Pattle &
Fissel, 2019). Yet, the exact origin of the polarization hole remains unclear.
The loss of grain alignment toward denser regions by RATs is a leading
explanation for the polarization hole (Hoang et al., 2021a). Qualitatively,
according to the RAT-A theory, grains are only efficiently aligned with
B-fields if the grain size is larger than the minimum alignment size of
$a_{\rm align}\propto n^{2/7}_{\rm H}T\rm{{}_{d}}^{-12/7}$ (see Equation 11).
The alignment size increases with increasing gas density and decreasing
$T_{\rm d}$, leading to a narrower size distribution of aligned grains and the
lower dust polarization fraction (Lee et al., 2020).

In this paper, to quantitatively test the RAT-A theory, we analyzed the
thermal dust polarization data toward one of the simplest filaments, Musca,
that exhibited a clear polarization hole. We performed a detailed polarization
modeling for Musca using the local volume density and radiation fields (dust
temperatures) from observational data as the input parameters for the DustPOL
code. Our modeling results showed that the alignment size ($a_{\rm align}$)
increases rapidly from the outer to inner filament when the density increases
and dust temperature decreases (see upper panel of Figure 6). As a result, the
polarization degree ($P$) decreases rapidly toward the filament spine,
successfully reproducing the polarization hole observed toward Musca (see the
lower panel of Figure 6).

####  5.1.2 Increase of polarization fraction with dust temperature

Figure 10: Musca: Comparison of the polarization degree vs. dust temperature
relation from our models (blue line) with observational data (black line). The
shaded area represents the 1-$\sigma$ deviation of each bin of data points.
The observed increase of $P$ with $T_{d}$ can be successfully reproduced by
the polarization model based on the RAT-A theory.

The second prediction of the RAT-A mechanism is the increase of the
polarization degree with increasing dust temperature because the alignment
size decreases with $T_{\rm d}$ (see Equation 11) and the size distribution of
aligned grains becomes broader (Lee et al., 2020).

For both filaments, the polarization data indeed show the increase of $P$ with
$T\rm{{}_{d}}$ for low temperatures (see Section 3). However, for OMC-1 with
strong radiation fields, the polarization degree reverses its trend at
sufficiently high temperatures, which suggests the signature of RAT-D. Our
detailed modeling for Musca and OMC-1 successfully reproduce the rising trend
of $P-T_{\rm d}$ (see Figures 10 and 11).

####  5.1.3 Role of B-field tangling in producing polarization hole: The anti-
correlations of $P\;vs.\;\mathcal{S}$ and $P\;vs.\;N({\rm H_{2}})$

Together with grain alignment and properties, polarization fraction, $P$, also
depends on the fluctuations of B-fields along a light of sight and within the
beam (Hoang & Truong, 2023). The polarization angle in the POS is the average
of the polarization patterns, which are aligned with the local B-fields along
a LOS. Therefore, polarization angle dispersion function $\mathcal{S}$ is
usually used to describe the B-field fluctuations.

In Musca, where the B-field fluctuations are small (see Figure 2, our
polarization model based on the RAT paradigm could reproduce the observed
polarization data (see Figure 6). Therefore, the loss of RAT alignment is the
main cause of the polarization hole, and the effect of B-field tangling is
negligible.

The situation becomes different for OMC-1, where B-field fluctuations are
strong and spatially dependent (see Figure 5). Our detailed modeling results
show that the model without B-field tangling, $P_{\rm mod}^{\rm ideal}$,
produces too high polarization and cannot reproduce the data. Only the
polarization models including B-field tangling effects, $P_{\rm mod}$, could
reproduce the observational data (see Figure 8). This reveals the important
role of B-field tangling in producing the polarization hole in OMC-1.

###  5.2 Evidence of RAT-D Mechanism in OMC-1

The correlation of $P$ and $T\rm{{}_{d}}$ at low dust temperature and anti-
correlation at high dust temperature is found in the OMC-1 filament. The
decreasing trend at high dust temperature is the opposite of what would be
expected from the RAT-A theory, but a signature of RAT-D.

To quantitatively study whether RAT-D could reproduce the declining trend in
OMC-1, we performed detailed modeling of dust polarization using the maps of
the gas volume density and dust temperature for each pixel obtained from
observations. Our modeling results could reproduce the $P\;{\rm
vs.}\;T\rm{{}_{d}}$ relation (see Figure 8).

We note that the B-field tangling ($P_{\rm mod}$) alone cannot reproduce the
decrease of $P$ with $T\rm{{}_{d}}$ at high temperatures. Indeed, as shown in
Figure 5, the polarization angle dispersion function $\mathcal{S}$ tends to
decrease or change slowly at high $T\rm{{}_{d}}$. Our models that take into
account both the B-field tangling and RAT-D shown in Section 4 could produce
the observed $P-T\rm{{}_{d}}$ relation at high $T\rm{{}_{d}}$. Therefore, the
RAT-D effect can explain the decreasing trend of $P-T_{\rm d}$ in OMC-1.

In previous simple modeling and data analyses, tentative evidence of RAT-D in
some regions of OMC-1 was revealed, e.g., in BN-KL region (Tram et al., 2021c)
and in Orion Bar (Le Gouellec et al., 2023), using SOFIA/HAWC+. It is worth
noting that evidence of RAT-D was previously found in several star-forming
regions: Auriga (Ngoc et al., 2021), Oph A (Tram et al., 2021a), 30 Doradus
(Tram et al., 2021b), and M17 (Hoang et al., 2022).

Figure 11: OMC-1. Comparison of the polarization fraction between our models
and observations for different initial maximum sizes of $a_{\rm
max}=0.25{\,\mu\rm{m}}$ (left panel) and $a_{\rm max}=1{\,\mu\rm{m}}$ (right
panel). The latter realistic models (red line) can reproduce better the data
for the region of high density but low temperatures of $T_{d}<50{\,\rm K}$,
revealing the significant grain growth in this dense region.

###  5.3 Implication for Probing Grain Growth

Observational studies suggest that the upper limit of the grain size
distribution in the ISM is $\sim$$0.25{\,\mu\rm{m}}$ (Mathis et al., 1977).
However, grain growth is expected to occur within dense molecular clouds or
filaments due to gas accretion and grain coagulation. Vaillancourt et al.
(2020) reveal grain growth in dense clouds using observations of starlight
polarization combined with numerical modeling using the RAT theory. Using
polarization data, Ngoc et al. (2023) gives evidence that grain growth is
moderate with $a_{max}>0.3{\,\mu\rm{m}}$ in the G11 filament when the volume
density is higher than $10^{4}\,{\rm cm}^{-3}$.

In this study, our numerical modeling for Musca shown in Figure 10 reveals
that the maximum grain size is $\sim 0.3-0.35{\,\mu\rm{m}}$. This implies
grain growth is effective but not significant in this not-dense filament, with
the highest density of $n_{{\rm H}}\sim 10^{3}\,{\rm cm}^{-3}$. On the other
hand, the density of the OMC-1 filament varies significantly and can reach a
high value of $10^{4}-10^{6}\;\,{\rm cm}^{-3}$. To probe grain growth in this
dense filament, we varied $a_{\rm max}$ from $0.2$ to $2\;{\,\mu\rm{m}}$ and
found that the model with $a_{\rm max}>1\;{\,\mu\rm{m}}$ can well reproduce
the $P-T\rm{{}_{d}}$ relation, in particular the low $T\rm{{}_{d}}$ region in
the lower dust temperature, $T\rm{{}_{d}}<50{\,\rm K}$ (see Figure 11 left).
This suggests that grain growth is significant in the dense regions of OMC-1.

###  5.4 Limitations of Our Modeling and Future Works

In this paper, we have used the hybrid approach to perform modeling of thermal
dust polarization and test the RAT paradigm. We first used the DustPOL code to
calculate the polarization fraction predicted by the RAT theory for the ideal
model with B-fields in the POS. To account for the effects of the inclination
and the tangling of B-fields, we then incorporated a coefficient $\Phi$ to
describe the inclination angle and a depolarization factor
$\mathcal{S}^{-\eta}$ to describe the effect of B-field fluctuations (Equation
17). The final realistic dust polarization model is based on the optically
thin assumption, which is successfully tested with synthetic observations of
MHD simulations using POLARIS in Hoang & Truong (2023). Although our numerical
modeling can successfully reproduce the observational data and support the RAT
paradigm, there are several limitations. First, we have assumed that grains
are aligned via the MRAT mechanism with $f_{\rm max}=1$, which is expected for
grains with embedded iron clusters. Second, the coefficient $\Phi$ obtained by
fitting the polarization model to the maximum observed polarization is assumed
to be constant across the filament. Such an assumption is likely valid for
Musca which has well-ordered B-fields and a small slope $\eta\sim 0.03$.
However, it may not be realistic for a complex star-forming region like OMC-1,
which has strong B-field fluctuations, and the slope $\eta$ is found to vary
considerably from region to region in OMC-1.

In our follow-up studies, we will perform synthetic dust polarization using
MHD simulations with our updated POLARIS code (Giang et al., 2023), which
includes both the RAT paradigm and the radiative transfer process. A detailed
comparison of synthetic simulations with observational data would help
disentangle the different origins of the polarization hole in filaments,
including grain alignment loss, disruption, and B-field tangling.

##  6 Conclusions

In this study, we utilize dust polarization data observed toward Musca and
Orion to test the grain alignment and rotational disruption mechanisms by
radiative torques (i.e., RAT paradigm). We investigate the relation of the
polarization degree, $P$, with gas column density, $N(\rm H_{2})$, and dust
temperature, $T\rm{{}_{d}}$, and compare them with theoretical predictions
using the RAT paradigm. Additionally, the variation of $P$ with polarization
angle function, $\mathcal{S}$, is explored to study the effects of B-field
fluctuations. Our results are as follows:

  1. 1.

Observational data show the existence of a polarization hole, i.e., the
decrease of the polarization degree with increasing the gas column density in
both Musca and OMC-1. The analysis of the observed polarization angle
dispersion reveals that B-fields fluctuations are rather weak in Musca but
strong in OMC-1.

  2. 2.

For Musca with weak B-field fluctuations, our detailed modeling of dust
polarization using the RAT theory with small B-field tangling effects succeeds
in reproducing the polarization hole, suggesting the loss of RAT alignment as
the main cause of the polarization hole in this filament. Moreover, our
modeling results show that grain growth is effective but not significant in
Musca.

  3. 3.

For the high-mass star-forming region, OMC-1, we find that, in general,
polarization fraction ($P$) and polarization angle dispersion function
($\mathcal{S}$) are anti-correlated but $N(\rm H_{2})$ and $\mathcal{S}$ are
correlated, especially at low temperatures. This trend suggests that the
B-field fluctuations in dense regions dominantly contribute to the
depolarization effect.

  4. 4.

Observational data exhibited a general correlation of $P$ with $T\rm{{}_{d}}$
at low temperatures, such as in the northern and southern. In the Orion BN-KL
region of high temperatures, $P$ reverses the trend and decreases with
$T\rm{{}_{d}}$.

  5. 5.

Using the pixel-by-pixel polarization modeling method with local physical
parameters (gas density and radiation fields) inferred from observations as
the input, we demonstrate that the observed $P-T_{\rm d}$ and $P-N({\rm
H_{2}})$ (polarization hole) relations can be simultaneously reproduced by our
polarization models incorporating both RAT-A and RAT-D, and the contribution
of B-field fluctuations. The decreasing trend of $P-T_{\rm d}$ when taking
into account the RAT-D effect from models is in agreement with observations
from the BN-KL region.

  6. 6.

The successful testing of the RAT paradigm through the combination of pixel-
by-pixel polarization modeling using DustPOL with observational data to two
different filaments of quiescent one (Musca) and an active star-forming region
(OMC-1) opens the avenue for testing the RAT paradigm and constraining dust
properties in other astrophysical environments.

Stratospheric Observatory for Infrared Astronomy (SOFIA), Planck

##  7 Acknowledgement

This research is supported by a grant from the Simons Foundation to IFIRSE,
ICISE (916424, N.H.) and by Vingroup Innovation Foundation (VINIF) under
project code VINIF.2023.DA.057. We sincerely thank Dr. Hyeseung Lee for her
help with DustPol in the initial stage of this work and Lars Bonne for his
comments on Planck data. N.B.N. was funded by the Master, Ph.D. Scholarship
Programme of Vingroup Innovation Foundation (VINIF), code VINIF.2023.TS077.
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. Based on observations obtained with
Planck (http://www.esa.int/Planck), an ESA science mission with instruments
and contributions directly funded by ESA Member States, NASA, and Canada.

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##  Appendix A Appendix

###  A.1 Method

The polarization model of the thermal dust emission is described in detail in
(Lee et al., 2020) and (Tram et al., 2021a).

####  A.1.1 Grain Alignment by Radiative Torques

We consider a radiation field with a total energy density $u_{\rm rad}=\int
u_{\lambda}d\lambda\;(\,{\rm erg}\,{\rm cm}^{-3})$. $U=u_{\rm rad}/u_{\rm
ISRF}$ is radiation strength, where $u_{\rm ISRF}=8.64\times 10^{-13}$ (erg
cm-3) is the radiation energy density of the interstellar radiation field
(ISRF) in the solar neighborhood (Mathis et al., 1983).

One important effect of RATs is to spin up dust grains of irregular shapes. On
the other hand, gas collisions and IR re-emission spin down grains.

The maximum angular velocity spun up by RAT,

$\displaystyle\frac{\omega_{\rm RAT}}{\omega_{\rm T}}$ $\displaystyle\simeq$
$\displaystyle
48.7\hat{\rho}a_{-5}^{3.2}U\left(\frac{\gamma}{0.1}\right)\left(\frac{30\,{\rm
cm}^{-3}}{n_{\rm
H}}\right)\left(\frac{\bar{\lambda}}{1.2\;{\,\mu\rm{m}}}\right)$ (A1)
$\displaystyle\times\left(\frac{100\;{\,\rm K}}{T_{\rm gas}}\right)(1+F_{\rm
IR})$

where $a$ is the grain size, $a_{-5}=a/(10^{-5}\,{\rm cm})$, $\rho$ is grain
mass density, $\hat{\rho}=\rho/(3\,{\rm g}\,{\rm cm}^{-3)}$, and
$\bar{\lambda}=1.2{\,\mu\rm{m}}$ is a mean wavelength of the ISRF (Draine &
Weingartner, 1997). $F_{\rm IR}$ is a dimensionless coefficient of rotational
damping by IR emission and is calculated by:

$\displaystyle F_{\rm IR}\simeq
0.91\left(\frac{U^{2/3}}{a_{-5}}\right)\left(\frac{30\,{\rm cm}^{-3}}{n_{\rm
H}}\right)\left(\frac{100\;{\,\rm K}}{T_{\rm gas}}\right)^{1/2}.$ (A2)

####  A.1.2 Grain size distribution and RAT-D

The tensile stress of a spinning grain having angular velocity $\omega$ is
calculated by $S=\rho\omega^{2}a^{2/4}$ where $\rho$ is the mass density of
dust. For large grains in the strong radiation field, the angular velocity by
RAT can be sufficiently large such that $\mathcal{S}$ exceeds the maximum
tensile strength $S_{\rm max}$ of the grain material, resulting in rotational
disruption (Hoang et al., 2019):

$\displaystyle\omega_{\rm crit}=\frac{2}{a}\left(\frac{S_{\rm
max}}{\rho}\right)\simeq\frac{3.6\times 10^{8}}{a_{-5}}S_{\rm
max,7}^{1/2}\hat{\rho}^{-1/2},$ (A3)

with, $S_{\rm max,7}=S_{\rm max}/(10^{7}\,{\rm erg}\,{\rm cm}^{-3})$. We can
obtain the critical size $a_{\rm disr}$ above which grains are disrupted
following this equation:

$\displaystyle\left(\frac{a_{\rm disr}}{0.1{\,\mu\rm{m}}}\right)^{2.7}\simeq
5.1\gamma_{0.1}^{-1}U^{-1/3}\bar{\lambda}_{0.5}^{1.7}S_{\rm max,7}^{1/2},$
(A4)

with, $\bar{\lambda}_{0.5}=\bar{\lambda}/(0.5{\,\mu\rm{m}})$. For strong
fields of $U\ll 1$ and $a_{\rm max}\gg\bar{\lambda}/1.8$.

The maximum size that grains are still disrupted by RAT-D is given by (Hoang
et al. (2019)):

$\displaystyle a_{\rm disr,max}\simeq
1.7\gamma\bar{\lambda}_{0.5}\left(\frac{U}{\bar{n}\bar{T}_{{\rm
gas}}^{1/2}}\right)^{1/2}\left(\frac{1}{1+F_{\rm IR}}\right),$ (A5)

where $\bar{n}=n_{\rm H}/30\,{\rm cm}^{-3}$ and $\bar{T}=T_{\rm gas}/100{\,\rm
K}$.

In this work, we adopt a power-law grain size distribution assumption for both
the original large grains and the smaller grains produced by disruption by a
power-low index $\beta$ (Mathis et al., 1977):

$\displaystyle\frac{1}{n_{\rm H}}\frac{dn_{\rm sil,car}}{da}=C_{\rm
sil,car}a^{\beta}(a_{\rm min}\leq a\leq a_{\rm max})$ (A6)

where $dn$ is the number density of grains of material between $a$ and $a+da$.
$C_{\rm sil}$ and $C_{\rm car}$ are the normalization constants for silicate
and carbonaceous grains, respectively. The smallest grain size is chosen as
$a_{\rm min}=10\,{\rm\AA}$, while the $a_{\rm max}$ is constrained by the
RAT-D mechanism.

###  A.2 Musca

We build the density profile and the spine of Musca using a Python package
RadFil applied for a column density map (Zucker & Chen, 2018). The obtained
spine is shown by the thick red line over-plotted on the density map
(grayscale background) in Figure A.1 (left). Subsequently, RadFil generates
radial cuts (red thin lines) perpendicular to the spine; from that, we have
the radial profiles shown in black dotted curves in Figure A.1 (right). We
estimated the background by subtracting a first-order polynomial from all
profiles. The background fitting is performed within the radii $2<|r|<6$ pc
(green area in the upper panel of Figure A.1) right. Then, the fitted profile
with a Plummer-like function again using RadFil is shown by the thick curve in
Figure A.1 (lower right).

Figure A.1: Musca. Left: Column density map is used to build the radial
profile of Musca by sampling radial cuts (red thin lines) every 0.1 pc along
the spine (thick red line). The spine is determined by Radfil using the
provided mask using the B-spline smoothness algorithm (Zucker & Chen, 2018).
The radial distance is defined as the projected distance from the considered
pixel to the peak emission (blue dot) on a given cut. Right: Background fit
(upper) and Plummer-like fit (lower). The results of the fits are shown in the
inserts. Green and blue shaded areas are the regions used for the background
and Plummer-like fits, respectively.

###  A.3 OMC-1

Figure A.2: OMC-1. Comparison of the polarization degree vs. dust temperature
relation from our models (color lines) with observational data (gray line).
The shaded area represents the 1-$\sigma$ deviation of each bin of data
points. Left and center-left panels: $P\;vs.\;T\rm{{}_{d}}$. Right and center-
right panels: $P\;vs.\;N(\rm H_{2})$. The color lines represent the same
models as in Figure 8.