# SOFIA observations of 30 Doradus: I - Far-Infrared dust polarization and
implications for grain alignment and disruption by radiative torques

Le Ngoc Tram  Stratospheric Observatory for Infrared Astronomy, Universities
Space Research Association, NASA Ames Research Center, MS 232-11, Moffett
Field, 94035 CA, USA Le Ngoc Tram<EMAIL_ADDRESS>Thiem Hoang  Korea
Astronomy and Space Science Institute, Daejeon 34055, South Korea  Korea
University of Science and Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon,
34113, South Korea  Enrique Lopez-Rodriguez  Kavli Institute for Particle
Astrophysics and Cosmology (KIPAC), Stanford University, Stanford, CA 94305,
USA  Simon Coudé  Stratospheric Observatory for Infrared Astronomy,
Universities Space Research Association, NASA Ames Research Center, MS 232-11,
Moffett Field, 94035 CA, USA  Archana Soam  Stratospheric Observatory for
Infrared Astronomy, Universities Space Research Association, NASA Ames
Research Center, MS 232-11, Moffett Field, 94035 CA, USA  B-G Andersson
Stratospheric Observatory for Infrared Astronomy, Universities Space Research
Association, NASA Ames Research Center, MS 232-11, Moffett Field, 94035 CA,
USA  Min-Young Lee  Korea Astronomy and Space Science Institute, Daejeon
34055, South Korea  Lars Bonne  Stratospheric Observatory for Infrared
Astronomy, Universities Space Research Association, NASA Ames Research Center,
MS 232-11, Moffett Field, 94035 CA, USA  William D. Vacca  Stratospheric
Observatory for Infrared Astronomy, Universities Space Research Association,
NASA Ames Research Center, MS 232-11, Moffett Field, 94035 CA, USA  Hyeseung
Lee  Korea Astronomy and Space Science Institute, Daejeon 34055, South Korea

###### Abstract

Located in the Large Magellanic cloud and mostly irradiated by a massive-star
cluster R 136, 30 Doradus is an ideal target to test the popular theory of the
grain alignment and rotational disruption by Radiative Torques (RATs). Here,
we use publicly available polarized thermal dust emission observations of 30
Doradus at 89, 154, and 214$\,\mu$m using SOFIA/HAWC+. We analyse the
variation of the dust polarization degree ($p$) with the total emission
intensity ($I$), the dust temperature ($T_{{\rm d}}$), and the gas column
density ($N_{\rm H}$) constructed from Herschel data. The 30 Doradus complex
is divided into two main regions relative to R 136, namely North and South. In
the North, we find that the polarization degree first decreases and then
increases before decreasing again when the dust temperature increases toward
the irradiating cluster R 136. The first depolarization likely arises from the
decrease of grain alignment efficiency toward the dense medium due to
attenuated interstellar radiation field. The second trend (the increase of $p$
with $T_{\rm d}$) is consistent with the RAT alignment theory. The final trend
(the decrease of $p$ with $T_{\rm d}$) is inconsistent with the RAT alignment
theory, but can be explained by rotational disruption of large grains by RATs.
In the South, we find that the polarization degree is nearly independent of
the dust temperature, while the grain alignment efficiency is higher around
the peak of the gas column density and decreases toward the radiation source.
The latter feature is also consistent with the prediction of the rotational
disruption by RATs.

ISM: dust, extinction $-$ ISM: clouds $-$ ISM: individual objects (30 Doradus,
LMC) $-$ ISM: polarization

††journal: ApJ

##  1 Introduction

Grain alignment induces the polarization of background starlight as well as
the polarization of thermal dust emission. Dust polarization induced by
aligned dust grains is widely used to map magnetic fields (see e.g., Lazarian
2007). A popular theory describing grain alignment is based on RAdiative
Torques (hereafter RATs), which arises from the interaction of an anisotropic
radiation field on irregular grains (Draine & Weingartner 1996; Lazarian &
Hoang 2007; see Lazarian et al. 2015 and Andersson et al. 2015 for recent
reviews). According to the RAT alignment theory, the alignment efficiency of
dust grains depends on the radiation field and the local gas properties (e.g.,
Hoang et al. 2021). Gas collisions tend to damp the grain rotation and
randomize the grain orientation, whereas RATs act to spin-up and align dust
grains. Therefore, toward the center of a dense cloud, the degree of grain
alignment by RATs decreases due to the attenuation of the interstellar
radiation field (ISRF) and the increase of gas density. This results in the
decrease of the dust polarization degree ($p$) toward the center of molecular
clouds (MCs), i.e., higher gas column density $N_{{\rm H}}$ (see e.g., Hoang
et al. 2021). Numerous observations toward MCs report the decrease of the
polarization degree with $N_{{\rm H}}$, which favors the RAT theory (Whittet
et al. 2008; Alves et al. 2014). Note that the tangling of magnetic fields is
suggested to produce the decrease of the polarization with increasing $N_{{\rm
H}}$.

With the advance of high resolution polarimetric facilities (JCMT/POL2, ALMA,
SOFIA/HAWC+), one can now observe the polarization induced by dust grains in
proximity of an embedded source (e.g., a protostar) where the effect of
stellar radiation becomes dominant over that of the ISRF. As a result, toward
the proximity of an embedded source, the RAT alignment theory predicts the
increase of the dust polarization with increasing the radiation emission
intensity (or dust temperature $T_{{\rm d}}$, Hoang et al. 2021). Numerical
modeling (Lee et al. 2020) and numerical simulations (Reissl et al. 2016) of
dust polarization using the RAT alignment theory report a monotonic increase
of the polarization degree of thermal dust emission with increasing the
radiation intensity (or $T_{{\rm d}}$). Such a correlation of $p$ with
$T_{{\rm d}}$ is an important feature predicted by the RAT theory. As a
result, observing dust polarization around a strong radiation source is
crucial to test the RAT alignment theory (see e.g., Medan & Andersson 2019;
Soam et al. 2021b).

Interestingly, polarimetric observations reveal that the thermal dust
polarization does not always increase with the radiation intensity (or dust
temperature, $T_{\rm d}$) and decrease with increasing $N_{{\rm H}}$ as
implied by the RAT theory. For example, Planck observations at 850 $\mu$m
toward four molecular clouds, including Aquila Rift, Cham-Musca, Orion, and
Ophiuchus in the Gould Belt cloud, showed that the polarization degree
decreases for $T_{\rm d}>19~{}\,{\rm K}$ (see Planck Collaboration et al.
2020). Shorter wavelength polarimetric measurements at 500 $\mu$m from the
air-balloon BLASTPol toward the Vela C also show a similar decrease at
$T_{{\rm d}}\geq 19\,{\rm K}$ (see Fissel et al. 2016). Further shorter
wavelength observations by the High-resolution Airborne Wide band Camera Plus
(HAWC+) instrument (Harper et al. 2018) aboard the Stratospheric Observatory
for Infrared Astronomy (SOFIA) toward the molecular cloud Ophiuchus A at 89
and 154 $\mu$m also reported the decrease of the polarization degree for
$T_{\rm d}\geq 25-32\,{\rm K}$ and $N_{{\rm H}}\leq 10^{22}\,{\rm cm}^{-2}$
(see Santos et al. 2019 and Tram et al. 2021a). For the case of $\rho$
Ophiuchus A, the peak of the dust temperature is close to the central source
(a B-association star), while the gas column density peaks further away. Thus,
the decrease of the dust polarization degree toward high dust temperatures and
low gas column density (toward the source) could not be explained by the loss
of grain alignment due to weak radiation intensity or enhanced collisional
randomization as in the case of starless cores (Whittet et al. 2008; Alves et
al. 2014).

Hoang et al. (2019) discovered that large grains cannot survive once being
exposed to a strong radiation field owing to the Radiative Torque Disruption
(RATD) mechanism (see Hoang 2020 for a review). The RATD mechanism causes the
fragmentation of large grains into many smaller species when the centrifugal
stress induced by suprathermal rotation due to RATs exceeds the tensile
strength of the grain material, which is determined by the binding energy that
hold their constituents together. RATD is more efficient for large grains
because the RAT efficiency increases with the grain size (Lazarian & Hoang
2007; Hoang & Lazarian 2008). The depletion of large grains due to RATD is
expected to result in the decrease of the polarization degree of the thermal
dust emission because large grains dominate the polarized emission at long
wavelengths.

The first numerical modeling of dust polarization that considers both grain
alignment and rotational disruption by RATs was performed in Lee et al.
(2020). They found that the dust polarization increases monotonically with
$T_{{\rm d}}$ if grains are aligned by RATs. In the presence of RATD, the
polarization first increases and then decreases when the dust temperature
becomes sufficiently large to activate the RATD. The critical temperature for
the RATD depends on the local gas density and grain tensile strength that
depends on the grain structure. Their modeling results could successfully
reproduce the anti-correlation $p-T_{{\rm d}}$ trend at 850$\,\mu$m observed
by Planck (Planck Collaboration et al. 2020). As a result, the variation of
$p$ with $T_{{\rm d}}$ observed toward an intense radiation source becomes an
important test for the RAT alignment and RATD theory. The first detailed
analysis of the $p-T_{{\rm d}}$ relation is carried out in Tram et al. (2021a)
for the SOFIA/HAWC+ observations from Ophiuchus A cloud at 89 and 154 $\mu$m.
The authors found the anti-correlation of $p-T_{{\rm d}}$ for sufficiently
large $T_{{\rm d}}$ and show that it could be reproduced by the RAT alignment
and RATD. Ngoc et al. (2021) also found the anti-correlation of $p-T_{{\rm
d}}$ toward the proximity of the LkH$\alpha$ 101 star in the Auriga-California
cloud. These studies reveal the importance of RATD that needs to be taken into
account together with RAT alignment to interpret dust polarization data
observed toward an intense radiation source. Moreover, observational data also
allow us to constrain the physical properties of dust grains, such as shape,
internal structure, and size distribution.

In this paper, we will use the thermal dust polarization observed by
SOFIA/HAWC+ at 89, 154, and 214 $\mu$m toward a massive star-forming cloud, 30
Doradus (hereafter 30 Dor), to test the RAT alignment and RATD mechanisms. 30
Dor is located in the Large Magellanic Cloud (LMC) with the distance of $\sim$
50 kpc from us (e.g., De Marchi et al. 2011) and powered by a massive star
cluster R 136, with mass of $\simeq 5\times 10^{4}M_{\odot}$ (see Indebetouw
et al. 2009 and references therein) and the bolometric luminosity of
$7.8\times 10^{7}L_{\odot}$ (see Lopez et al. 2011 and references therein).
Note that LMC is a low-metallicity galaxy ($Z\simeq 0.5Z_{\odot}$, Dufour et
al. 1982; Galliano et al. 2008b; Chevance et al. 2016), with a low dust-to-gas
ratio (e.g., Galliano et al. 2008a; Roman-Duval et al. 2014), which makes the
dust-shielding relatively low such that dust grains can be affected by the
radiation field over a large scale (see Figure 2). Thus, the 30 Dor cloud
offers a valuable environment to test the physics of grain alignment and
disruption by RATs. Toward this end, we will concentrate on the correlation of
the polarization degree with the emission intensity, the dust temperature, and
the gas column density.

Figure 1: Background is a RGB composite image with R: 8$\,\mu$m, G:
4.5$\,\mu$m, and B: 3.6$\,\mu$m. The black contours show an example of the FIR
observations at 154$\,\mu$m with SOFIA/HAWC+. There are two main regions in 30
Dor: North and South in relative to the massive star cluster, R 136. The data
analyzed in this work locate in the strong radiation field.

Figure 2: Polarization maps (E-vectors) of 30 Dor observed using SOFIA/HAWC+
at 89 (left), 154 (middle), and 214 $\mu$m (right). The background color is
the total intensity (Stokes I). The vectors are selected within the thresholds
of $I/\sigma_{I}\geq 100$ and $p/\sigma_{p}\geq 3$ (see Section 2.1). The beam
size (red circle) at the respective wavelengths and a 5% polarization
measurement (top-right) are shown. The location of R136 (white cross) is shown
in every panel.

This paper is structured as follows. We first present the FIR polarimetric
observations of 30 Dor observed by SOFIA/HAWC+ in Section 2. We then present
(1) the distribution of the polarization angle and the polarization degree,
(2) the relation of the polarization degree with the total intensity, the gas
column density, and with dust temperature in Section 3. The implications of
the observational data for grain alignment and rotational disruption by RATs
are presented in Section 4. The summary of our main findings is presented in
Section 5.

##  2 Observations of 30 Doradus

###  2.1 SOFIA/HAWC+ multi-wavelength polarization

Polarization of thermal dust emission from 30 Dor was observed by SOFIA/HAWC+
at three bands: C at 89$\,\mu$m, D at 154$\,\mu$m, and E at 214$\,\mu$m. The
beams sizes (Full-Width-at-Half-Maximum, FWHM) are 7.8′′, 13.6′′, and 18.2′′
at 89$\,\mu$m, 154$\,\mu$m, and 214$\,\mu$m, respectively. Data are publicly
available under the Strategic Director’s Discretionary Time (S-DDT) program
(PI: Yorke, H., ID: 76$\\_$0001), which observations were taken during the
SOFIA New Zealand deployment in July 2018. Data have been presented by Gordon
et al. 2018 and no further data reduction has been performed.

For the linear polarization, the polarized intensity is defined as (Gordon et
al. 2018)

$\displaystyle I_{p}=\sqrt{Q^{2}+U^{2}}$ (1)

with $Q$ the Stokes parameter Q, $U$ the Stokes parameter U. The association
error is then

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

with $\sigma_{Q}$ the error of Stokes Q, and $\sigma_{U}$ the error of Stokes
U.

The debiased polarization degree is calculated as

$\displaystyle
p=\frac{100}{I}\sqrt{Q^{2}+U^{2}-\sigma^{2}_{Ip}}=100\frac{I_{p}}{I}~{}~{}~{}(\%)$
(3)

with $I$ the Stokes I parameter. The associated error is then

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

with $\sigma_{I}$ the error of Stokes I.

Finally, the polarization angle and its error are given by

$\displaystyle\theta=\frac{1}{2}\arctan{\left(\frac{U}{Q}\right)}$ (5)

and

$\displaystyle\sigma_{\theta}=\frac{1}{2}\frac{\sqrt{(Q\sigma_{U})^{2}+(U\sigma_{Q})^{2}}}{Q^{2}+U^{2}}.$
(6)

Figure 3: Panels(a,c): Maps of the total intensity and the polarization degree
at 214$\,\mu$m after masking from the common sky positions in which all 3
bands in Figure 2 detected the signal. Panels(b,d): Maps of the gas column
density and the dust temperature. The black cross locates the massive star
cluster R 136. The total intensity correlates well to the gas column density.
The gas column density peak is offset from the dust temperature, whose peak is
close to R 136.

We then performed quality cuts to obtain statistically significant
polarization measurements. The quality cuts on the polarization map are very
conservative. In this work, we mask our HAWC+ maps by using two common
thresholds. The first quality cut depends on the signal-to-noise ratio of the
Stokes-I ($I/\sigma_{I}$), we adopt $I/\sigma_{I}\geq 100$, which allows
$\sigma_{p}\simeq\sqrt{2}\times(I/\sigma_{I})^{-1}=1.4\%$111This approximation
is derived from Equation 4 by assuming that $\sigma_{Q}=\sigma_{U}$. The
second threshold is the signal-to-noise ratio of the fractional polarization
$p/\sigma_{p}\geq 3$. Figure 2 show the final map of the polarization vectors
(E-vectors, not B-vectors). The length of the polarization measurements is
referenced to $5\%$ of polarization degree. Data for band C (89$\,\mu$m), D
(154$\,\mu$m) and E (214$\,\mu$m) is show from left to right, and the
background color is the original total intensity (Stokes I). Figure 3(a,c)
show the final maps of the stokes I and the polarization degree at 214$\,\mu$m
as an example. The correlations to the gas column density are visualized by
the black contours.

###  2.2 Gas column density and dust temperature

We constructed the maps of dust temperature, $T_{{\rm d}}$, and the gas column
density, $N_{{\rm H}}\simeq N({\rm H}+{\rm H}_{2})$, using Herschel data at
100, 160, 250, 350 and 500 $\mu$m from HERITAGE (Meixner et al. 2013). The
observations were registered and sampled to the 160 $\mu$m with a pixelscale
of 3′′ and FWHM = 11.4′′, then a modified black-body function was used to fit
to each of the line-of-sigh (LOS) SED with an assume spectral index $\beta$ =
1.62. This value was taken as the median from the range of $\beta\sim 1.5-1.8$
by Gordon et al. (2014), and $\beta=2.0$ used by Bernard et al. (2008).

Figure 4: Distribution of the polarization angle (upper panels) and the
polarization degree (lower panels) in 3 bands (from left to right). The cyan,
red and blue are the distributions of the polarization measurement in the
entire region, in the North and in the South regions (see Figure 2),
respectively. The vertical dashed red and blue lines are the weighted mean of
the North and the South distributions. The corresponding thinner lines
represent for the $1\sigma$ uncertainties.

Panels (b,d) in Figure 3 show the maps of the gas column density and dust
temperature in 30 Dor. Toward R 136, the dust temperature increases, while the
gas column density first increases and then decreases after its peak, which is
further away from the center. The correlations are visualized by the black
contours in these maps. In addition, the dust temperature (gas column density)
in the North is hotter (denser) than in the South.

##  3 Results

We now show the results analyzing the polarization degree of thermal dust
emission in 30 Dor in terms of the total emission intensity, dust temperature
and gas column density.

Bands | $\theta^{o}(\pm 1\sigma)$ | $p(\%)(\pm 1\sigma)$
---|---|---
| North | South | North | South
89$\,\mu\rm m$ | 15.4(27.32) | -12.43(50.48) | 4.46(2.39) | 4.29(2.27)
154$\,\mu\rm m$ | 13.21(24.41) | -11.55(49.61) | 4.29(2.05) | 3.19(1.94)
214$\,\mu\rm m$ | 13.53(22.31) | -21.27(52.28) | 5.08(2.14) | 4.25(2.14)
Table 1: Mean value and uncertainties of the polarization angle ($\theta$) and
the polarization degree ($p$) in the North and South regions within 3 bands.

###  3.1 Variation of polarization in 30 Dor

The upper panels in Figure 4 show the distribution of the polarization angle
of the thermal dust polarization in 30 Dor at 3 bands. The positive angle
means that the polarization angle is counterclockwise rotation with respect to
the North direction. The variation of the polarization angle is very similar
for the three bands. The distributions of all vectors are shown by the cyan
area, which shows that the polarization vectors vary quite significantly,
spanning from $-90^{\circ}$ to $90^{\circ}$, and peaks at around $20^{\circ}$
and $-60^{\circ}$. The first peak is characterized in the North (see the red
distribution). The second peak is responsible in the South (see the blue
distribution), which is more statistically random. These distributions have
the same bin width, which is chosen following the Freedman-Diaconis rule in
python-
package222https://docs.astropy.org/en/latest/visualization/histogram.html from
the all vectors. Then, the mean-weighted values and the uncertainties have
been calculated as shown by the dashed vertical thick and corresponding thin
lines, whose values are listed in Table 1.

The lower panels in Figure 4 show the distribution of the polarization degree
in 3 bands. The distribution peaks at between $2\%$ and $4\%$. The red and
blue histograms show the distributions in the North and South, respectively.
The choice of the bin width is the same as above. The weighted mean and the
associated $1\sigma$ uncertainties are showed by the dashed vertical thick and
thin lines, whose values are also listed in Table 1. Notably, the $\sigma$ is
larger than the maximum associated uncertainty of $1.4\%$, thus the $1\sigma$
is intrinsically associated to changes in the source.

###  3.2 Polarization degree versus total intensity

Figure 5: Relation of the polarization degree and the total intensity in 3
bands in the North region. The gray error bars are the mean-weighted within
the bins of each data. A single power-law fitting is shown by the orange line,
while the blue line is for a double power-law. In the latter case, the
transitions are shown by the dashed vertical lines. As a comparison, the
power-law with $\alpha=1$ is shown by the dashed black line.

Figure 6: Similar to Figure 6 but for the South. The data points likely group
separately, hence we classify them into two different clusters by using
BayesianGaussianMixture library in python-package. These clusters are colored
by the gray and orange points in panels (a,b,c), which are followed by the
power-law fittings. The slope differs from one to another cluster, which
indicates a distinguished grain alignment efficiency between the two regions.
Panels(d,e) visualize the spatial distribution of these clusters. The slope is
shallower at around the peaked intensity and gas column density, whereas the
polarization decreases steeply to higher intensity.

Figure 6 shows the relation of the polarization degree to the total intensity
for all 3 bands in the North region. We first fit the observational data with
a single power-law function (solid orange line) using the lmfit python-package
(Newville et al. 2014), and compare it with the limiting case of $\alpha=1$
(black dashed line). The best fit indicates the slope of $\alpha=0.6\pm 0.01$
at 89$\,\mu$m is steeper than 0.5 and the estimated slopes at 154$\,\mu$m
($\alpha=0.34\pm 0.02$) and 214$\,\mu$m ($\alpha=0.46\pm 0.02$). We then fit
the data with a double power-law function (solid and dashed blue line) using
the piecewise linear least square fit pwlf (Jekel & Venter 2019). The best
fits illustrate two distinct slopes, whose transitions are marked by the
dashed dotted vertical lines. The transitions occur at 0.28$\pm 0.02\,\rm
Jy/arcsec^{2}$, 0.05$\pm 0.01\,\rm Jy/arcsec^{2}$, and 0.02$\pm 0.003\,\rm
Jy/arcsec^{2}$ at 89$\,\mu$m, 154$\,\mu$m, and 214$\,\mu$m, respectively. In
the latter case, as the intensity increases, the polarization degree first
declines rapidly with a steep slope of $\alpha>0.5$ (solid blue line), and
then varies with a shallow slope with $\alpha<0.5$ (dashed blue line). The
first steep slope is still shallower than $\alpha=1$ predicted by the model in
which grains are only aligned in the outer envelope of the cloud (see e.g.,
Hoang et al. 2021). We note that the slope difference gets smaller for longer
wavelengths.

Figure 6 shows the relation of the polarization degree to the total intensity
for all 3 bands in the South region (panels a,b,c). The relations are complex,
but the data appear to group into two different clusters. We then used the
BayesianGaussianMixture library in scikit-sklearn python-package (Pedregosa et
al. 2011) to classify the data points into 2 groups. Since the separation at
89$\,\mu$m is less obvious than at 154 and 214$\,\mu$m, the classification
just works well at these longer wavelengths and fails at 89$\,\mu$m. Thus, to
make it consistent, we classify this case into 3 groups then merge two of them
together. Each cluster is plotted in a different color and is fitted by a
power law, as shown in the legend. The polarization degree is higher, and the
slope is steeper from one cluster to another.

To better understand the observed features, we show in panels (e) and (d) of
Figure 6 the spatial positions of these two clusters at 214$\,\mu$m as an
example. The central region (colored area in panel e) seems to be shielded
partially from the irradiation of R 136, whereas the outer region (colored
area in panel d) contains a near-side region which is directly irradiated by
the source R 136. Surprisingly (also differing from the North), the high
polarization degree and shallower slope of the $p-I$ diagram is seen for the
central region, located around the intensity peak (or the gas column density
peak, see Figure 3a). Moreover, the polarization degree in the near-side
region (close to R 136) is much lower than that in the far-side region (lower
and upper right corners; see color bars). This feature is unexpected from the
RAT alignment theory.

We note that the best-fits to the $p-I$ data only reflect the general trend
and cannot describe accurately the underlying physics across the different
regions. Indeed, as shown in Figure 6, there are many data points that appear
to follow the steep slope of $\alpha=1$ toward high intensity. This can also
be seen from Figure 3c that the polarization degree is minimum around the
central radiation source, while it becomes higher at the farthest side from
this source where the dust temperature is relatively low (see Figure 3b,c,d).
Therefore, in the next section, we will perform additional analysis of the
polarization with the gas column density and dust temperature.

Figure 7: Sub-region selections. Background is the polarization degree at band
E of the common space position which all SOFIA/HAWC+ detected data. The
contours show the gas density map. The red rectangle (origin at 5h38m51.041s
-69∘04′25.084′′, width=17.12s, height=2.5′) and ellipse (origin at 5h38m32.4s
-69∘06${}^{{}^{\prime}}$42${}^{{}^{\prime\prime}}$, major axis = 6.05s, minor
axis = 0.81${}^{{}^{\prime}}$) show the selected regions, where we are
interested in.

Figure 8: Relation of the gas column density and the dust temperature in the
North region (upper panel) and in the South region (lower panel). They are
positively correlated for $T_{\rm d}<37\,{\rm K}$, while they are negatively
correlated for $T_{\rm d}>37\,{\rm K}$.

###  3.3 Polarization degree versus dust temperature and gas column density

For the optically thin case, the total dust emission intensity is
$I_{\nu}\propto B(T_{{\rm d}})\tau_{\nu}=\kappa_{\nu}N_{{\rm H}}B(T_{{\rm
d}})$ with $\kappa_{\nu}$ the dust opacity, which is the product of the gas
column density and dust temperature along the line of sight. Therefore, the
$p-I$ relation reflects the overall effect of both the radiation field and the
gas density on grain alignment. To disentangle the effect of the radiation
field from the gas density, we analyze the variation of the polarization
degree with the dust temperature and the gas column density. The first
analysis directly reveals the grain alignment and disruption by RATs, while
the second one reflects the effect of grain randomization induced by gas
collisions.

To make comparison among 3 bands, we first smooth the maps of the dust
temperature, gas column density, and SOFIA/HAWC+ bands C and D to the
$18.7^{\prime\prime}$ FWHM, which is equivalent to the SOFIA/HAWC+ band E
beam-size. Then, we selected the common space position where three SOFIA/HAWC+
bands all detect data. The final map is shown in Figure 7. From this final
map, we analyze the correlations in two sub-regions: within a rectangle in the
North, and within an ellipse in the South. These extraction areas cover the
representative features of both the gas density and the dust temperature in 30
Dor, which are shown in Figure 7.

Figure 9: Left panel: correlation of the polarization degree to the dust
temperature. Middle and right panels: correlation of the polarization to the
gas column density within $T<37\,{\rm K}$ and $T_{\rm d}\geq 37\,{\rm K}$,
respectively. Solid black line is the weighted-mean within $1-\sigma$
uncertainty in each bin. Dashed black line is the piecewise line fitting. On
the left column, the dashed vertical lines indicate the corresponding $T^{\rm
trans}_{{\rm d}}$. The $p-T_{{\rm d}}$ relation shows a three features: $p$
decreases (region A), slightly increases (region B), and decreases (region C)
as $T_{{\rm d}}$ increases. The mean value of the slopes and their associated
95$\%$ level of confidences are given in region C. Figure 10: Similar to
Figure 9 but for the South region. The polarization degree tends to decrease
slowly with $T_{{\rm d}}$ (left panel). The polarization degree decreases with
$N_{{\rm H}}$ for $T_{{\rm d}}<37\,{\rm K}$ (middle panel), but it increases
and decreases with $N_{{\rm H}}$ for $T_{{\rm d}}>37\,{\rm K}$ (right panels).

Figure 8 shows the two-dimensional histogram of the gas column density and the
dust temperature relation for these two representative North (upper panel) and
South (panel) regions. In the North, a pwlf fitting (Jekel & Venter 2019) to
all data points shows the gas column density is positively proportional and
then negatively proportional to the dust temperature, for which the dust is
below and above $T^{\rm trans}_{\rm d}\simeq 37\,{\rm K}$. However, since the
error bar is significantly large, we fit an upper envelope of the distribution
and obtain a slightly higher value of $T^{\rm trans}_{\rm d}\simeq 40\,{\rm
K}$. In the South, the transition temperature occurs around $37-39\,{\rm K}$.
However, the plot is much more scattered then in the North.

In Figure 9, we show the relations of the polarization degree in 3 bands to
the dust temperature (left panels from top to bottom). The black line shows
the weighted mean fit (filled dots), and the black dashed line shows the
piecewise line fit to the data. The separation owing to the $T_{{\rm
d}}-N_{{\rm H}}$ is shown by two vertical dashed lines. The corresponding
relations to the gas column density are shown in the middle and right panels,
respectively. Notably, we only show the separation for $T^{\rm trans}_{\rm
d}\simeq 37\,{\rm K}$ here because the behavior is the same for any values in
between $37\,{\rm K}$ and $40\,{\rm K}$.

For $T_{{\rm d}}\leq 37\,{\rm K}-40\,{\rm K}$, in which the dust temperature
and the gas column density both increase toward the R 136, the polarization
degree rapidly increases, then decreases and finally gets nearly plateau with
increasing dust temperature (region A in left panels). Meanwhile, the
polarization degree tends to decrease up to $N_{{\rm H}}\sim 10^{21.7}\,\,{\rm
cm}^{-2}$ and gets plateau with increasing the gas column density (middle
panels). For $T_{{\rm d}}>37\,{\rm K}-40\,{\rm K}$, in which the dust
temperature and the gas column density are anti-correlated toward the R 136,
the polarization degree first increases slightly and then decreases with
increasing both the dust temperature (region B and C in left panels) and the
gas column density (right panels). Furthermore, the declining slope of
$p-T_{{\rm d}}$ is steeper for the longer wavelength, i.e., the mean slope of
$-0.52$ at $89\,\mu$m, $-0.72$ at $154\,\mu$m and $-0.83$ at $214\,\mu$m.

Similarly, Figure 10 shows the correlations of the polarization degree to the
dust temperature and the gas column density in the South region. Although
these plots are scattered, one can see a weak correlation between the
polarization degree with the dust temperature. The relation to the gas column
density is also separated for the different range of dust temperature
according to Figure 8. The relation to the gas column density looks somehow
similar to the middle and right panels of Figure 9. For $T_{\rm d}<37-39\,{\rm
K}$ (middle panel), the polarization degree shows a monotonic decrease with
increasing the column density. However, for $T_{\rm d}\geq 37-39\,{\rm K}$
(right panel), the polarization degree first increases and then decreases with
increasing the column density, as seen in the North region.

In summary, the $p-T_{{\rm d}}$ and $p-N_{{\rm H}}$ relations are not
monotonic as in $p-I$. The most interesting feature is the anti-correlation of
$p$ vs. $T_{{\rm d}}$ and the correlation of $p$ with $N_{{\rm H}}$, which
cannot be explained by the RAT alignment theory.

##  4 Discussion

In this section, we will discuss the implications of the observed $p-I$,
$p-T_{{\rm d}}$, and $p-N_{{\rm H}}$ relations for the theory of grain
alignment and disruption based on RATs.

###  4.1 On the $p-I$ relation and grain alignment

The variation of the polarization degree of thermal dust emission with the
total emission intensity ($I$) is a popular analysis of polarimetric data,
which provides information on grain alignment and magnetic fields. Numerous
studies show that the polarization decreases with $I$ as $p\propto
I^{-\alpha}$ with the slope $\alpha\simeq 0-1$. If grain alignment and the
magnetic field are uniform throughout the cloud, one expects $\alpha=0$. If
grain alignment only occurs in the outer layer and becomes completely lost in
the inner region, one expects $\alpha=1$ (Whittet et al. 2008). The latter
slope is previously reported in the case of starless cores (see e.g., Crutcher
et al. 2004; Alves et al. 2014). Theoretically, for uniform grain alignment in
the cloud, the stochastic magnetic field is found to induce a slope of
$\alpha=0.5$ (see Jones et al. 2015).

Figure 11: The alignment size (minimum size of aligned grains by RATs, $a_{\rm
align}$) as functions of the dust temperature ($T_{{\rm d}}$) and the gas
volume density ($n_{{\rm H}}$) computed with $\bar{\lambda}=0.3$, $\gamma=1$,
and $T_{\rm gas}=T_{{\rm d}}$. The alignment size is larger (smaller) for
higher (lower) gas density and lower (higher) dust temperature. The white
region shows the range of the transition temperature ($T^{\rm trans}_{{\rm
d}}$).

Figure 12: Similar to Figure 12 but for the grain disruption size ($a_{\rm
align}$) for different values of the tensile strength ($S_{\rm max}$). $a_{\rm
disr}$ is smaller (larger) for higher (lower) $T_{{\rm d}}$ and lower (higher)
$n_{{\rm H}}$. For the same physical condition, $a_{\rm disr}$ is larger for
higher $S_{\rm max}$. In each panel, the white dashed horizontal line
represents to $T_{{\rm d}}=43\,{\rm K}$.

In Section 3.2, we showed the $p-I$ diagrams in all 3 bands in both North and
South regions. In the North, a single power-law fitting shows that the diagram
at 89$\,\mu$m has a steep slope of $\alpha>0.5$, while the ones at longer
wavelengths show a shallower slope of $\alpha<0.5$. The slope of $p-I$
relations seem rather low compared to other regions, e.g., Auriga-California
cloud ($\alpha\simeq 0.82$, Ngoc et al. 2021), Perseus B1 ($\alpha\simeq 0.8$,
Coudé et al. 2019), Ophiuchus B, C ($\alpha\simeq 0.6-0.7$, Pattle et al.
2019), Serpens South ($\alpha\simeq 0.55$, Pillai et al. 2020). A double
power-law fitting indicates the diagrams change from the first, steep slope of
$\alpha>0.5$ to the second, shallower slope of $\alpha<0.5$ toward the peak
dust emission intensity. This transition feature seems similar to the case of
NGC 6334 (Arzoumanian et al. 2020). In both cases, the shallow slope of the
$p-I$ diagram is shown, which can be explained by the high grain alignment
efficiency by RATs owing to the intense radiation flux from the luminous
source R 136.

Indeed, according to the RAT alignment theory, the degree of dust polarization
is determined by the minimum size of aligned grains ($a_{\rm align}$). A
smaller value of $a_{\rm align}$ results in a higher $p$ because the size
distribution of aligned grains (i.e., from $a_{\rm align}$ to the maximum
grain size ($a_{\max}$)) is broader. Similarly, a larger value of $a_{\rm
align}$ causes the lower polarization due to a narrower size distribution of
aligned grains, assuming that $a_{\max}$ is fixed (without RATD). The
alignment size depends on the local physical conditions as $a_{\rm align}\sim
n^{2/7}_{{\rm H}}T^{-12/7}_{{\rm d}}$ (see Equation A).

Since 30 Dor is a complex region, it is very difficult to accurately constrain
the local physical properties such as $n_{{\rm H}}$ and $T_{{\rm d}}$ (see
Okada et al. 2019). Therefore, in Figure 12, we show the map of $a_{\rm
align}$ with respect to $T_{\rm d}$ and $n_{{\rm H}}$. The values of $n_{{\rm
H}}$ span broadly from $10^{3}$ to $10^{6}\,\,{\rm cm}^{-3}$ (see Chevance et
al. 2016 and Lee et al. 2019), while that of $T_{{\rm d}}$ is adopted from
observations (Figure 3d). The alignment size increases with increasing gas
density and decreasing dust temperature (from upper left corner to lower right
corner). For a given dust temperature, $a_{\rm align}$ becomes larger for
higher gas density (from left panel to right panel).

In the case of two steps, when approaching the luminous source of R 136, the
local dust temperature (i.e., radiation strength) increases, which
significantly decreases $a_{\rm align}$ (see Figure 12). As a result the
polarization degree of thermal dust emission increases (Hoang et al. 2021).
This can reproduce the shallow slope of $\alpha\sim 0.3-0.4$ observed toward
the peak intensity. For the first steep slope of $\alpha>0.5$, the alignment
size increases due to the decrease of the interstellar radiation field ($U$),
and one expects the slope of $\alpha=1$ when grain alignment is completely
lost. However, if grain growth occurs, the size distribution of aligned grains
($a\sim a_{\rm align}-a_{\rm max}$) is still finite, which can produce the
slope shallower than $\alpha\sim 1$ (see Hoang et al. 2021). Incidentally, the
optical to near-infrared observations of 30 Dor by Hubble Space Telescope (De
Marchi & Panagia 2014) reported a high value of the ratio of the total and
selective extinction $R_{\rm V}$ as 4.4, which implies the grain growth in 30
Dor. Other effects, including the tangling of the magnetic fields, can also
reproduce the first slope.

In the South, the $p-I$ diagrams illustrate two clusters with different
properties. The first group shows a quite shallow slope with $\alpha\simeq
0.2-0.5$ at around the peak of the intensity and gas column density, which is
opposite to the North. The second group otherwise shows a very deep slope as
$\alpha\simeq 1$. Naturally, the contradiction of the polarization degree
reflects a difference in the radiation flux and the grain size distribution.
However, the dust temperature is not the highest at the dense gas (Figure 3d).
Thus, it is likely that grains are larger in the dense region and responsible
for a shallow slope of the $p-I$ diagram.

The North and South regions share a common feature that the polarization
degree seems to be larger at greater distances from the central source with a
low gas density and low dust temperature. An interesting feature is that the
polarization degree appears to be low in the vicinity of R$\,136$, in which
the radiation is strong and the gas density is not the densest (Figure 3c).
This is evidence in support of the RATD mechanism, which will be discussed in
detail in the next section.

Figure 13: Understanding the thermal dust polarization in 30 Dor in the light
of RAT alignment and RATD. According to the RAT theory, the polarization
degree is determined by the grain alignment size ($a_{\rm align}$), which
depends on the dust temperature ($T_{{\rm d}}$) and the gas volume density
($n_{{\rm H}}$). Therefore, the observed $p-T_{{\rm d}}$ and $p-N_{{\rm H}}$
trends could be explained by three distinct stages denoted by $①$, $②$, and
$③$. For ① with $T_{{\rm d}}<T^{\rm trans}_{{\rm d}}$, $T_{{\rm d}}$ is
positively proportional to $n_{{\rm H}}$ (solid white arrow), the grain
alignment size increases along this direction only if $n_{{\rm H}}$ increases
faster than the radiation strength $U\propto T_{{\rm d}}^{6}$, which results
in the decrease of the polarization degree ($p$) with both $T_{{\rm d}}$ and
$n_{{\rm H}}$. If $T_{{\rm d}}$ increases such that $U$ increases faster than
$n_{{\rm H}}$, one expects $p$ to go up. For $②$ with $T_{{\rm d}}>T^{\rm
trans}_{{\rm d}}$, $T_{{\rm d}}$ is negatively proportional to $n_{{\rm H}}$
(dashed white arrow), $p$ increases with increasing $T_{{\rm d}}$ and
decreasing $n_{{\rm H}}$ as $a_{\rm align}$ becomes smaller along this
direction. Finally, for $③$ with $T_{\rm d}>T^{\rm disr}_{{\rm d}}$, the RATD
effect becomes active, resulting in the drop of $p$ toward higher $T_{{\rm
d}}$ and lower $n_{{\rm H}}$ (orange lines), which explains the observed
trend. Without RATD, $p$ is expected to continue increasing or be constant.
Therefore, the joint effect of grain alignment and disruption by RATs can
successfully explain the observed polarized thermal dust emission of 30 Dor.

###  4.2 On the $p-T_{{\rm d}}$ and $p-N_{{\rm H}}$ relations and grain
alignment

As shown in Figure 9, the variation of $p$ with $T_{{\rm d}}$ and $N_{{\rm
H}}$ in the North region is more complex than the $p-I$ relation. For $T_{{\rm
d}}\leq 37\,{\rm K}$, $p$ decreases with increasing $T_{{\rm d}}$ (left
panels) and $N_{{\rm H}}$ (middle panels). For $T_{{\rm d}}>37\,{\rm K}$, $p$
first increases with increasing $T_{{\rm d}}$ and $N_{{\rm H}}$, then
decreases when $T_{{\rm d}}$ becomes sufficiently large (see B and C regions
in the left panels). The increase of $p$ with $T_{{\rm d}}$ and the decrease
of $p$ with $N_{{\rm H}}$ are expected from the RAT alignment theory as the
alignment size decreases with increasing $U$ and decreasing $n_{{\rm H}}$ (see
Figure 12, also Equation A). However, the decrease of $p$ with $T_{{\rm d}}$
is unexpected from the RAT alignment theory because the alignment size $a_{\rm
align}$ is smaller for larger $T_{{\rm d}}$ ($U$). Similarly, the increase of
$p$ with $N_{{\rm H}}$ is unexpected because $a_{\rm align}$ increases with
the gas density.

In the South, the relations of $p$ to $T_{\rm d}$ and $N_{{\rm H}}$ are more
scattered and not very obvious as shown in Figure 10. However, the South is
likely closer to R 136 than the North, and the selected area covers mostly the
region where the grain alignment efficiency is high. However, the polarization
degree seems to not vary with $T_{{\rm d}}$ and decreases with decreasing the
gas column density for $T_{{\rm d}}>T^{\rm trans}_{{\rm d}}$. Same as in the
North, the latter trend is opposite to the expectation by the RAT alignment
theory.

###  4.3 A new evidence of the RATD mechanism

In Figure 9 and 10, we found that the polarization degree first increases and
then decreases as the dust temperature increases and the gas column density
decreases (region C in Figure 9) for $T_{\rm d}>43\,{\rm K}$. The decrease of
$p$ with $T_{{\rm d}}$ (higher radiation flux, closer R 136) and the increase
of $p$ with decreasing $N_{{\rm H}}$ cannot be explained by the RAT alignment
theory, as discussed in the previous subsection. Below, we suggest that the
RATD mechanism (Hoang et al. 2019) can reproduce this feature.

According to the RATD mechanism, dust grains subject to a strong radiation
field (i.e., close the source R 136) can be spun-up to extremely fast
rotation. The induced centrifugal stress due to grain suprathermal rotation
can exceed the tensile strength of the grain material, resulting in the
spontaneous fragmentation of large grains into smaller ones. The depletion of
large grains causes the decrease of the thermal dust polarization degree at
long wavelengths (see Lee et al. 2020; Tram et al. 2021a).

To study if RATD can reproduce the observed anti-correlation of $p-T_{{\rm
d}}$, let us estimate the critical size above which dust grains are disrupted,
$a_{\rm disr}$. The disruption size depends on the local conditions and the
grain structure as $a_{\rm disr}\sim n_{{\rm H}}^{1/2}T^{-3}_{\rm
d}S^{1/4}_{\rm max}$ (see Section B). As shown in Tram et al. (2021a, b), the
disruption occurs at the critical disruption temperature of $T^{\rm
disr}_{{\rm d}}\simeq 32\,{\rm K}$ for $n_{{\rm H}}\simeq 10^{5}\,\,{\rm
cm}^{-3}$, and $T^{\rm disr}_{{\rm d}}\simeq 71\,{\rm K}$ for $n_{{\rm H}}\sim
10^{6}\,\,{\rm cm}^{-3}$. Thus, the gas volume density in the region within
$43\leq T_{{\rm d}}<50\,{\rm K}$ could be in $(10^{5}-10^{6}\,\,{\rm
cm}^{-3})$ interval. Figure 12 shows the map of the grain disruption size as a
function of $T_{{\rm d}}$ and $n_{{\rm H}}$ for different value of $S_{\rm
max}$. Grains within composite structure ($S_{\rm max}\sim
10^{5}-10^{7}\,\,{\rm erg}\,{\rm cm}^{-3}$) are being disrupted at smaller
size than the more compact grains (higher value of $S_{\rm max}$). For
instance, for $n_{{\rm H}}=6\times 10^{5}\,\,{\rm cm}^{-3}$ and $T_{{\rm
d}}=43\,{\rm K}$, $a_{\rm dirs}$=0.2, 0.4, 0.65, 1.2$\,\mu$m for
$S_{\max}=10^{6},\,10^{7},\,10^{8},\,10^{9}\,\,{\rm erg}\,{\rm cm}^{-3}$,
respectively. Thus, the RATD mechanism can reasonably disrupt the composite
grains for $T_{{\rm d}}\geq 43\,{\rm K}$. The disruption size gets smaller for
higher $T_{{\rm d}}$.

Figure 13 sketches our understanding on the main feature of dust polarization
in 30 Dor expected from theoretical modeling of dust polarization based on
grain alignment and disruption by RATs (Lee et al. 2020). For an illustration,
a color map of $a_{\rm align}$ (in arbitrary units) is shown as functions of
$T_{{\rm d}}$ and $n_{{\rm H}}$. The feature of $p$ fundamentally depends on
the $T_{{\rm d}}-n_{{\rm H}}$ relation. If we assume $n_{{\rm H}}\sim N_{{\rm
H}}$ maps (e.g., the local variation of the depth of cloud is significantly
small or negligible), the observed polarization degree ($p$) profile could be
explained in three ranges of $T_{{\rm d}}$.

For $T_{{\rm d}}<T^{\rm trans}_{{\rm d}}$, $T_{{\rm d}}$ positively correlates
to $n_{{\rm H}}$ (see Figure 8). If the gas density increases faster than the
radiation strength ($U\propto T_{d}^{6}$), such that the rotational damping by
gas collisions is sufficiently strong to prevent the grains from being spun-up
to the disruption limit by RATs, $a_{\rm align}$ is larger with increasing
$T_{{\rm d}}$ and $n_{{\rm H}}$ (solid white arrow). This increment of $a_{\rm
align}$ consequently causes the polarization degree to decrease to both the
dust temperature and gas density as expected from the RAT alignment theory. In
this case, $a_{\rm align}$ might also become constant with a specific range of
$T_{{\rm d}}$ and $n_{{\rm H}}$ because of their opposite effect. These
features explain the observed region A of $p-T_{{\rm d}}$ in Figure 9 and
$p-N_{{\rm H}}$ in the middle panel of Figures 9, 10. In the case if the dust
temperature increases such that $U$ increases faster than the gas density, the
rotational damping by gas collisions is insufficiently strong to prevent the
grains from being spun-up by RATs, $a_{\rm align}$ is getting smaller as
$T_{{\rm d}}$ and $n_{{\rm H}}$ increase. Thus, the polarization degree is
increasing instead of decreasing, which contradicts to the observed trend of
30 Dor.

For $T_{{\rm d}}>T^{\rm trans}_{{\rm d}}$, $T_{{\rm d}}$ and $n_{{\rm H}}$ are
anti-correlated (see Figure 8). The gas collision damping is further weak, the
grain alignment size gets smaller as $T_{{\rm d}}$ increases (dashed white
arrow). Thus, the polarization degree gets increased toward higher $T_{{\rm
d}}$ and lower $n_{{\rm H}}$. This feature could explain the region B of the
observed $p-T_{{\rm d}}$ in Figure 9.

For $T_{{\rm d}}>T^{\rm disr}_{{\rm d}}$, the depletion of large grains due to
the RATD mechanism results in the decrease of the polarization degree toward
higher $T_{{\rm d}}$ and lower $n_{{\rm H}}$. This feature successfully
reproduces the observed $p-T_{{\rm d}}$ in region C of Figure 9 and $p-N_{{\rm
H}}$ in the right panels of Figures 9, 10. Otherwise, the polarization degree
is expected to continuously increase or at least flat accordingly to the RAT
alignment theory (see Lee et al. 2020).

However, there is a question remaining why the relation of $p$ to $T_{{\rm
d}}$ is weak in the South. This slow variation may be due to the saturation of
RATD, which occurs when $T_{{\rm d}}>>T_{\rm disr}$ (see Figure 14 in Lee et
al. 2020). Induced small grains likely have compact structures and RATD ceases
to act at some high $T_{{\rm d}}$. Furthermore, when $T_{{\rm d}}$ is large,
IR damping is so strong that $a_{\rm disr}$ becomes to slowly change with
$T_{{\rm d}}$. Both effects induce the slow variation of $p$ with $T_{{\rm
d}}$.

Previous studies have reported the anti-correlation of the dust polarization
with grain temperature for several molecular clouds (Planck Collaboration et
al. 2020), the $\rho$ Ophiuchus A which is irradiated by a nearby
B-association star (Santos et al. 2019; Tram et al. 2021a)333In the case of
$\rho$ Ophiuchus A, the observed trend of $p-T_{{\rm d}}$ is explained by the
processes ② and ③ in Figure 13., and at 89, 154, 214, and 850$\,\mu$m of a
cloud surrounding the massive star BN/KL (Tram et al. 2021b). Numerical
modeling studies (Lee et al. 2020) show that the RATD mechanism (Hoang 2020)
could successfully reproduce the observed anti-correlation of the polarization
with grain temperature at the same FIR and submm wavelength ranges. The
observed anti-correlation of $p-T_{{\rm d}}$ in 30 Dor is a new evidence for
the RATD mechanism.

###  4.4 Dust polarization mechanisms

Our analysis so far is focused on the polarization degree with the assumption
that far-IR dust polarization is induced by dust grains aligned with a
preferred direction. Modern theory of RAT alignment theory implies that such a
preferred axis could be the magnetic field (B-RAT) or the radiation direction
(k-RAT) (Lazarian & Hoang 2007). As a result, the polarization vectors of
polarized thermal emission are perpendicular to the magnetic field (radiation
direction). The B-RAT alignment is most popular in the diffuse ISM and
molecular clouds, whereas the k-RAT can occur near a strong radiation source
where the rate of grain precession around the radiation direction is larger
than the Larmor precession around the magnetic field (Hoang & Lazarian 2016;
Lazarian & Hoang 2019). In this case, the polarization vectors of thermal dust
emission are perpendicular to the radiation direction. Therefore, the
polarization vectors provide valuable constraints on dust magnetic properties
and alignment mechanisms (Lazarian & Hoang 2019; Pattle et al. 2021).

In the South region of Figure 2, the polarization vectors (the E-vectors)
appear to be concentric with respect to the intense source R 136. Such a
polarization pattern is different from the one in the North region which is
far from the source R 136 where grains are most likely aligned via B-RAT. This
indicates that the alignment mechanisms in the South may be due to the
alignment with the radiation direction. Evidence for k-RAT is previously
reported for some star-forming regions (Chuss et al. 2019; Pattle et al. 2021)
and protoplanetary disks (Kataoka et al. 2017; Stephens et al. 2017). We will
study in detail this issue in a follow up paper on 30 Dor.

##  5 Summary

In this work, we report our results analyzing the far-IR polarimetric
observations of 30 Doradus by SOFIA/HAWC+. 30 Dor cloud is located in the
Large Magellanic Cloud and mainly irradiated by an intense radiation source
from the massive R 136 cluster. Thus, this region is an ideal target to study
the physics of grain alignment and disruption driven by RATs. To infer the
alignment and disruption properties of dust grains, we have performed three
different analyses based on the polarization degree, including $p-I$,
$p-N_{{\rm H}}$, and $p-T_{{\rm d}}$. Our main findings are summarized as
bellows

  * 1

The thermal dust polarization varies across the 30 Dor cloud, in which there
are two main regions, namely North and South in relative to the massive star
cluster R 136. In the North region, the polarization patterns are likely
radial from R 136 within the polarization angle peaked at $\simeq 20^{\circ}$,
while they are concentric around R 136 within the peaked polarization angle of
$\simeq-60^{\circ}$ in the South region.

  * 2

For the North region, a single power-law fitting show a shallow slope of the
$p-I$ diagrams, which could be explained by an intense radiation field. A
double power-law fitting shows a change from a steeper slope of $\alpha>0.5$
to a shallower slope as $\simeq I^{-0.4}-I^{-0.3}$ toward higher intensity.
The latter shallow slope most likely arises from the enhanced alignment of
grains by RATs when approaching the source R 136.

  * 3

The variations of the polarization degree with the dust temperature and gas
column density in the North are complex. The polarization degree decreases
with increasing the dust temperature and the gas column density for $T_{{\rm
d}}<37\,{\rm K}$, which might result from the loss of grain alignment due to
strong collisional damping by denser gas. For $T_{{\rm d}}>37\,{\rm K}$, the
polarization degree increases as the dust temperature increases and the gas
column density decreases. However, the polarization degree decreases as the
dust temperature increases beyond $\sim 43\,{\rm K}$ and the gas column
density further decreases. Such an anti-correlation of $p-T_{{\rm d}}$ is
consistent with the prediction of the RATD theory that the dust polarization
decreases due to disruption of large grains toward the intense radiation
source R 136.

  * 4

In the South region, the polarization degree is higher, and the $p-I$ slope is
shallower at the peak of the intensity and gas column density. Whereas, the
polarization degree is the highest at further away (low intensity and less
dense gas) and decreases steeply as $p\propto I^{-1}$ backward to R 136
(higher intensity and denser gas). The first feature is likely caused by
enhancement of grain alignment by RATs due to the radiation from the source R
136. The second feature is unexpected from the RAT alignment theory, but is
consistent with the prediction by the RATD effect, as in the North.

  * 5

In the South, the stronger grain randomization by gas collisions in the
densest part results in a weak correlation of the polarization degree to the
dust temperature. Whereas, the polarization degree likely decreases toward low
and high gas column density. However, the relations are quite scattered and
show a large error bar.

The values of the dust temperature and gas column density using in this work
are derived from the modified black-body fitting to the Herschel data, and
they thus encompass the projection effect. Our conclusions are given without
taking the B-field tangling into account. The variation of the magnetic field
could result in the modification of the grain alignment and then the
polarization degree. Despite these aspects, our SOFIA/HAWC+ data show that the
grain alignment is complicate and significantly varied across the 30 Dor
cloud, and evidence a strong impact of the intense radiation field of a
massive star cluster R 136. The drop of the polarization degree as close to
the central source, in which the gas column density is not peaked, is contrary
to the RAT alignment theory. The lower abundance of large grains via the
disruption mechanism by radiative torque could be responsible for this
declination.

Acknowledgments: 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. Financial support for this
work was provided by NASA through award 4$\\_$0152 issued by USRA. T.H is
funded by the National Research Foundation of Korea (NRF) grant funded by the
Korea government (MSIT) through the Mid-career Research Program
(2019R1A2C1087045).

Facility: SOFIA

##  Appendix A Grain alignment size

Grains are effectively aligned only when they can rotate suprathermally, i.e.,
the grain angular velocity ($\omega$) is greater than the thermal velocity
($\omega_{\rm T}$) (Hoang & Lazarian 2008; Hoang & Lazarian 2016). The minimum
alignment size ($a_{\rm align}$) above which grains are aligned can be
computed as $\omega(a_{\rm align})\equiv 3\times\omega_{\rm T}$. As exposed to
a radiation source, the minimum $a_{\rm align}$ is determined analytically as
(see Equation 26 in Hoang et al. 2021)

$\displaystyle a_{\rm align}$ $\displaystyle\simeq$ $\displaystyle
0.055\hat{\rho}^{-1/7}\left(\frac{\gamma}{0.1}\right)\left(\frac{n_{{\rm
H}}}{10^{3}\rm cm^{-3}}\right)^{2/7}U^{-2/7}$ $\displaystyle\left(\frac{T_{\rm
gas}}{10\,{\rm K}}\right)^{2/7}\left(\frac{\bar{\lambda}}{1.2\rm\mu
m}\right)^{4/7}(1+F_{\rm IR})^{2/7}~{}~{}~{}{\rm\mu m},$

where $\gamma$ is the anisotropy degree, $\bar{\lambda}$ is the mean
wavelength, and $U$444$U=u_{\rm rad}/u_{\rm ISRF}$, with $u_{\rm rad}=\int
u_{\lambda}d\lambda$ the radiation energy density of the local radiation field
and $u_{\rm ISRF}\approx 8.64\times 10^{-13}\,{\rm erg}\,\,{\rm cm}^{-3}$ the
radiation energy density of the local interstellar radiation field (Mathis et
al. 1983). is the strength of the radiation field, respectively. $n_{{\rm H}}$
and $T_{\rm gas}$ are the number density and temperature of the gas. $F_{\rm
IR}$ is the ratio of the IR damping to the collisional damping rate. If we use
$U\simeq(T_{\rm d}/16.4\,{\rm K})^{6}$ (Draine 2011), one can see that the
alignment size depends on the local properties as $a_{\rm align}\sim
n^{2/7}_{{\rm H}}T^{-12/7}_{\rm d}$. The alignment size decreases (increases)
with decreasing (increasing) gas density and increasing (decreasing) dust
temperature as shown in Figure 12.

##  Appendix B Grain disruption size

Large grains exposed to an intense radiation field can be spun-up to extremely
fast due to RATs. The fast rotation induces a strong centrifugal stress which
opposes the binding energy that holds up the grain structure (i.e., tensile
strength). Once the centrifugal stress exceeds the tensile strength, the grain
is spontaneously fragmented. This fragmentation is called Radiative torque
disruption (RATD). The critical grain size above which grains will be
disrupted by the RATD mechanism is

$\displaystyle a_{\rm disr}$ $\displaystyle\simeq$ $\displaystyle
0.22\hat{\rho}^{-1/4}\gamma^{-1/2}\left(\frac{n_{{\rm H}}}{10^{3}\,{\rm
cm}^{-3}}\right)^{1/2}U^{-1/2}\left(\frac{T_{\rm gas}}{10\,{\rm
K}}\right)^{1/2}$ $\displaystyle\times\left(\frac{\bar{\lambda}}{0.5\rm\mu
m}\right)\left(\frac{S_{\rm max}}{10^{7}\,{\rm erg}\,{\rm
cm}^{-3}}\right)^{1/4}(1+F_{\rm IR})^{1/2}\,{\mu\rm{m}},$

where $S_{\rm max}$ is the tensile strength which is determined by the
internal structure of dust grains (see Equation 29 in Hoang et al. 2021). The
compact grains have a higher tensile strength and are stronger than the
composite ones (see Hoang 2019). One can see that $a_{\rm disr}\sim n_{{\rm
H}}^{1/2}T^{-3}_{\rm d}S^{1/4}_{\rm max}$, which illustrates that the
disruption size also decreases (increases) with decreasing (increasing) gas
density and increasing (decreasing) dust temperature. Moreover, at the same
local condition, grains within a weaker structure are easier to be disrupted
than the stronger one as shown in Figure 12.

## References

  * Alves et al. (2014) Alves, F. O., Frau, P., Girart, J. M., et al. 2014, A&A, 569, L1
  * Andersson et al. (2015) Andersson, B. G., Lazarian, A., & Vaillancourt, J. E. 2015, ARA& A, 53, 501
  * Arzoumanian et al. (2020) Arzoumanian, D., Furuya, R., Hasegawa, T., et al. 2020, arXiv e-prints, arXiv:2012.13060
  * Bernard et al. (2008) Bernard, J.-P., Reach, W. T., Paradis, D., et al. 2008, AJ, 136, 919
  * Chevance et al. (2016) Chevance, M., Madden, S. C., Lebouteiller, V., et al. 2016, A&A, 590, A36
  * Chuss et al. (2019) Chuss, D. T., Andersson, B.-G., Bally, J., et al. 2019, ApJ, 872, 187. doi:10.3847/1538-4357/aafd37
  * Coudé et al. (2019) Coudé, S., Bastien, P., Houde, M., et al. 2019, ApJ, 877, 88
  * Crutcher et al. (2004) Crutcher, R. M., Nutter, D. J., Ward-Thompson, D., & Kirk, J. M. 2004, ApJ, 600, 279
  * De Marchi & Panagia (2014) De Marchi, G., & Panagia, N. 2014, MNRAS, 445, 93
  * De Marchi et al. (2011) De Marchi, G., Paresce, F., Panagia, N., et al. 2011, ApJ, 739, 27
  * Draine & Weingartner (1996) Draine, B. T. & Weingartner, J. C. 1996, ApJ, 470, 551. doi:10.1086/177887
  * Draine (2011) —. 2011, Physics of the Interstellar and Intergalactic Medium
  * Dufour et al. (1982) Dufour, R. J., Shields, G. A., & Talbot, R. J., J. 1982, ApJ, 252, 461
  * Fissel et al. (2016) Fissel, L. M., Ade, P. A. R., Angilè, F. E., et al. 2016, ApJ, 824, 134
  * Galliano et al. (2008a) Galliano, F., Dwek, E., & Chanial, P. 2008a, ApJ, 672, 214
  * Galliano et al. (2008b) Galliano, F., Madden, S. C., Tielens, A. G. G. M., Peeters, E., & Jones, A. P. 2008b, ApJ, 679, 310
  * Gordon et al. (2014) Gordon, K. D., Roman-Duval, J., Bot, C., et al. 2014, ApJ, 797, 85
  * Gordon et al. (2018) Gordon, M. S., Lopez-Rodriguez, E., Andersson, B. G., et al. 2018, arXiv e-prints, arXiv:1811.03100
  * Harper et al. (2018) Harper, D. A., Runyan, M. C., Dowell, C. D., et al. 2018, Journal of Astronomical Instrumentation, 7, 1840008
  * Hoang (2019) Hoang, T. 2019, ApJ, 876, 13
  * Hoang (2020) Hoang, T. 2020, Galaxies, 8, 52
  * Hoang & Lazarian (2008) Hoang, T., & Lazarian, A. 2008, MNRAS, 388, 117
  * Hoang & Lazarian (2016) Hoang, T. & Lazarian, A. 2016, ApJ, 831, 159. doi:10.3847/0004-637X/831/2/159
  * Hoang et al. (2019) Hoang, T., Tram, L. N., Lee, H., & Ahn, S.-H. 2019, Nature Astronomy, 3, 766
  * Hoang et al. (2021) Hoang, T., Tram, L. N., Lee, H., Diep, P. N., & Ngoc, N. B. 2021, ApJ, 908, 218
  * Indebetouw et al. (2009) Indebetouw, R., de Messières, G. E., Madden, S., et al. 2009, ApJ, 694, 84
  * Jekel & Venter (2019) Jekel, C. F., & Venter, G. 2019, pwlf: A Python Library for Fitting 1D Continuous Piecewise Linear Functions
  * Jones et al. (2015) Jones, T. J., Bagley, M., Krejny, M., Andersson, B. G., & Bastien, P. 2015, AJ, 149, 31
  * Kataoka et al. (2017) Kataoka, A., Tsukagoshi, T., Pohl, A., et al. 2017, ApJL, 844, L5. doi:10.3847/2041-8213/aa7e33
  * Lazarian (2007) Lazarian, A. 2007, J. Quant. Spec. Radiat. Transf., 106, 225
  * Lazarian & Hoang (2007) Lazarian, A., & Hoang, T. 2007, MNRAS, 378, 910
  * Lazarian & Hoang (2019) Lazarian, A. & Hoang, T. 2019, ApJ, 883, 122. doi:10.3847/1538-4357/ab3d39
  * Lazarian et al. (2015) Lazarian, A., Andersson, B.-G., & Hoang, T. 2015, Polarimetry of Stars and Planetary Systems, 81
  * Lee et al. (2020) Lee, H., Hoang, T., Le, N., & Cho, J. 2020, ApJ, 896, 44
  * Lee et al. (2019) Lee, M. Y., Madden, S. C., Le Petit, F., et al. 2019, A&A, 628, A113
  * Lopez et al. (2011) Lopez, L. A., Krumholz, M. R., Bolatto, A. D., Prochaska, J. X., & Ramirez-Ruiz, E. 2011, ApJ, 731, 91
  * Mathis et al. (1983) Mathis, J. S., Mezger, P. G., & Panagia, N. 1983, A&A, 500, 259
  * Medan & Andersson (2019) Medan, I., & Andersson, B. G. 2019, ApJ, 873, 87
  * Meixner et al. (2013) Meixner, M., Panuzzo, P., Roman-Duval, J., et al. 2013, AJ, 146, 62
  * Newville et al. (2014) Newville, M., Stensitzki, T., Allen, D. B., & Ingargiola, A. 2014, LMFIT: Non-Linear Least-Square Minimization and Curve-Fitting for Python, v.0.8.0, Zenodo, doi:10.5281/zenodo.11813
  * Ngoc et al. (2021) Ngoc, N. B., Diep, P. N., Parsons, H., et al. 2021, ApJ, 908, 10
  * Okada et al. (2019) Okada, Y., Güsten, R., Requena-Torres, M. A., et al. 2019, A&A, 621, A62
  * Pattle et al. (2019) Pattle, K., Lai, S.-P., Hasegawa, T., et al. 2019, ApJ, 880, 27
  * Pattle et al. (2021) Pattle, K., Lai, S.-P., Wright, M., et al. 2021, MNRAS, 503, 3414. doi:10.1093/mnras/stab608
  * Pedregosa et al. (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., et al. 2011, Journal of Machine Learning Research, 12, 2825
  * Pillai et al. (2020) Pillai, T. G. S., Clemens, D. P., Reissl, S., et al. 2020, Nature Astronomy, 4, 1195
  * Planck Collaboration et al. (2020) Planck Collaboration, Aghanim, N., Akrami, Y., et al. 2020, A&A, 641, A12
  * Reissl et al. (2016) Reissl, S., Wolf, S., & Brauer, R. 2016, A&A, 593, A87
  * Roman-Duval et al. (2014) Roman-Duval, J., Gordon, K. D., Meixner, M., et al. 2014, ApJ, 797, 86
  * Santos et al. (2019) Santos, F. P., Chuss, D. T., Dowell, C. D., et al. 2019, ApJ, 882, 113
  * Stephens et al. (2017) Stephens, I. W., Yang, H., Li, Z.-Y., et al. 2017, ApJ, 851, 55. doi:10.3847/1538-4357/aa998b
  * Soam et al. (2021b) Soam, A., Andersson, B. G., Straižys, V., et al. 2021b, AJ, 161, 149
  * Tram et al. (2021a) Tram, L. N., Hoang, T., Lee, H., et al. 2021a, ApJ, 906, 115
  * Tram et al. (2021b) Tram, L. N., Lee, H., Hoang, T., et al. 2021b, ApJ, 908, 159
  * Whittet et al. (2008) Whittet, D. C. B., Hough, J. H., Lazarian, A., & Hoang, T. 2008, ApJ, 674, 304