# Recent progress in theory and observational study of dust grain alignment
and rotational disruption in star-forming regions

Le Ngoc Tram 1 and Thiem Hoang 2,3

###### Abstract

Modern understanding of dust astrophysics reveals that RAdiative Torques
(RATs) arising from the radiation-dust interaction can induce two fundamental
effects, including grain alignment and rotational disruption. Here we review
the recent progress in the theoretical development and observational testing
of these effects using dust polarization observed toward star-forming regions
(SFRs). We first review the basic theory of the RAT alignment and RAT
disruption, which are referred to as RAT-A and RAT-D effects, respectively. We
then briefly describe the numerical method we use to model polarized thermal
dust emission by accounting for both RAT-A and RAT-D and theoretical
predictions of dust polarization for observations. Next, we review our
observational efforts to search for observational evidence of the RAT-A and
RAT-D effects using thermal dust polarization toward SFRs. Finally, we discuss
magnetic fields inferred from dust polarization observed toward these SFRs and
implications of the RAT paradigm for different astrophysical conditions,
including protostellar environments, dust evolution, and time-domain
astrophysics.

## 1

##  2 Keywords:

Interstellar medium; Astrophysical dust processes; Interstellar dust;
Interstellar dust extinction; Dust continuum emission; Starlight polarization;
Interstellar magnetic fields; Star formation.

##  3 Introduction

Dust is an essential component of the interstellar medium (ISM) and plays an
important role in many astrophysical phenomena, from the ISM evolution, star
formation, planet formation to surface chemistry. The dust properties,
including the size, shape, and composition, determine the extinction of
starlight and reemission in infrared emission, which controls the appearance
of the Universe (Draine 2011). Thermal dust emission is at the foundation of
infrared, submm/mm astronomy, a central topic of which is the formation
process of stars and planets. In particular, the discovery of starlight
polarization (Hall 1949; Hiltner 1949) revealed that dust grains are non-
spherical and aligned with a preferred direction in space, which demonstrated
the existence of the interstellar magnetic field.

Magnetic field is believed to play an important role in various astrophysical
phenomena, including the evolution of the ISM and star formation (Crutcher,
2012), cosmic ray acceleration and transport. Therefore, observations of
magnetic field have important implications for understanding astrophysical
phenomena. Motivated by the discovery of starlight polarization and grain
alignment, Davis (1951) and Chandrasekhar and Fermi (1953) introduced a
technique to measure the field strength using dust polarization orientations,
which is based on the energy balance between turbulent energy and magnetic
energy (known as DCF technique). Hildebrand (1988) realized that polarization
of far-IR thermal emission from dust grains aligned with the magnetic field
can be used to trace the field morphology in very cold regions where young
stars are being formed because long-wavelength photons can propagate a long
distance. These enabled the rapid development of far-IR/submm polarimetric
capabilities in the last decades, including SMA (Marrone and Rao 2008),
CSO/SHARP (Li et al. 2008), Planck (Planck Collaboration et al. 2014), BLAST-
Pol (Galitzki et al. 2014); PILOT (Misawa et al. 2014), APEX/POLKA (Wiesemeyer
et al. 2014), CARMA (Hull and Plambeck 2015), ALMA (Nagai et al. 2016),
JCMT/POL2 (Friberg et al. 2018), and SOFIA/HAWC+ (Harper et al. 2018). As a
result, significant progress in understanding the role of magnetic fields in
the interstellar process have been made, from the diffuse interstellar clouds
(e.g., Heiles and Crutcher 2005;Planck Collaboration: et al. 2015), star-
forming regions (e.g., Curran and Chrysostomou 2007; Chuss et al. 2019; Doi et
al. 2020; Arzoumanian et al. 2021; Devaraj et al. 2021; Guerra et al. 2021;
Kwon et al. 2022, see also Pattle et al. 2022 for reviews), to extra-galactic
scales (e.g., Lopez-Rodriguez et al. 2020; Lopez-Rodriguez 2021;Jones et al.
2020; Borlaff et al. 2021; Pattle et al. 2021a).

However, the question of how dust grains get aligned and whether dust
polarization is a reliable tracer of magnetic field is a long-standing
question in astrophysics (see Lazarian 2007 for a review). In principle, the
process of grain alignment includes two stages, (1) the alignment of the grain
axis of maximum inertia moment (i.e., the shortest axis) with its angular
momentum (the so-called internal alignment), and (2) the alignment of the
grain angular momentum with a preferred axis in the space (i.e., axis of grain
alignment), which includes the external magnetic field and radiation direction
(the so-called external alignment). The leading process for the internal
alignment of interstellar grains is the Barnett relaxation effect (Barnett
1909; Purcell 1979). The Barnett effect is the reverse of the Einstein-de-Haas
effect (Einstein and de Haas 1915), arising from the dissipation of grain
rotational energy into heat due to rotating magnetization within a
paramagnetic grain rotating around a non-principal axis.

For the ISM (e.g., the diffuse ISM and molecular clouds (MCs)) where grains
are essentially small (of radius of $a<1\,\rm\mu m$), the Barnett relaxation
is usually much faster than the randomization of grain orientation by gas
collisions, resulting in the perfect internal grain alignment (i.e., the grain
axis of maximum inertia moment is parallel to its angular momentum; see e.g.,
Hoang 2022). For external alignment, the leading theory of grain alignment is
based on RAdiative Torques (hereafter RATs, Dolginov and Mitrofanov 1976;
Draine and Weingartner 1996; Lazarian and Hoang 2007a) and/or mechanical
torques (METs; Lazarian and Hoang 2007b; Hoang et al. 2018. Many predictions
of the RAT alignment (or RAT-A) theory were successfully tested with
observations (Andersson et al., 2015). The fast Larmor precession establishes
the magnetic field as a preferred axis of the external alignment (see e.g.,
Hoang 2022). Therefore, dust polarization is an established, reliable tracer
of magnetic fields in such conditions, which is the focus of this review.

Yet, for special environments where the radiation intensity is strong, while
magnetic field is weak or grain susceptibility is negligible (e.g.,
carbonaceous dust), or the gas is very dense (e.g., protostellar cores/disks
and protoplanetary disks, in which grains could grow to very large size of
$>10\,\mu$m), the physics of grain alignment becomes more complex. For
example, large grains may be aligned with the radiation direction instead of
magnetic field, as originally suggested in Lazarian and Hoang (2007a) and
modeled in Tazaki et al. (2017) and Pattle et al. (2021b). As a result, it is
not clear whether dust polarization can still trace magnetic fields in such
very dense environments. A detailed study of grain alignment for large grains
in protostellar environments is presented in Hoang et al. (2022a).

While dust polarization orientations become a powerful method to measure
magnetic field, the dust polarization degree ($p$) provides a unique probe
into dust properties and dust fundamental physics. Indeed, the degree of dust
polarization depends on the grain alignment efficiency, grain properties, the
local density and magnetic field structure (e.g., Draine and Fraisse 2009;
Guillet et al. 2018). According to the RAT-A theory, the alignment efficiency
of dust grains depends on the local conditions (gas properties and radiation,
Hoang et al. 2021b), grain geometric properties (size and shape) and magnetic
properties (Lazarian and Hoang, 2007a; Hoang and Lazarian, 2008, 2016a;
Herranen et al., 2021). Therefore, $p$ allows to probe physics of grain
alignment and physical properties of dust grains such as grain sizes and
composition. The variation of $p$ with the total emission intensity ($I$)
permits a popular analysis of polarimetric data to study the variation of
grain alignment and the magnetic field geometry across the observed region.
The general trend observed toward molecular clouds is $p\propto I^{-\alpha}$
with the power-index $\alpha$ between $0$ and $1$ (e.g., Coudé et al. 2019;
Pattle et al. 2019; Ngoc et al. 2021). If the magnetic field is uniform,
$\alpha=0$ indicating that grain alignment is uniform, while $\alpha=1$
implying that grain alignment only occurs in the outer layer of the cloud and
becomes completely lost in the inner region (Whittet et al. 2008). If magnetic
fields are completely random, the uniform grain alignment results in a slope
of $\alpha\approx 0.5$ (see Jones et al. 1992, 2015).

Nevertheless, $p-I$ relation is not a perfect tracer of grain alignment
because of the complex dependence of the emission intensity on the gas density
and dust properties. Indeed, the total emission intensity observed at
frequency $\nu$ is $I_{\nu}\propto B(T_{\rm d})\tau_{\nu}\sim f_{d/g}N_{\rm
H}\times B(T_{\rm d})\times\kappa_{\nu}$ with $f_{d/g}$ the dust-to-gas mass
ratio and $\kappa_{\nu}$ the dust opacity, which is the product of the gas
column density ($N_{\rm H}$), dust temperature ($T_{\rm d}$), and dust opacity
(i.e., grain size distribution and composition) along the line of sight.
Therefore, the $p-I$ relation reflects the overall dependence of $p$ on the
product of three key parameters (gas density, radiation field, and grain size
distribution). To disentangle the effect of the radiation field from the gas
density, it is required to analyze the variation of the polarization degree
with the dust temperature and the gas column density. The first analysis
directly reveals the grain alignment by RATs, while the second one reflects
the effect of grain randomization induced by gas collisions. The dust
polarization is predicted to increase with radiation emission intensity (or
$T_{\rm d}$) by theory (Hoang et al. 2021b), mumerical modeling (Lee et al.
2020) and numerical simulations (Reissl et al. 2016). A detailed analysis of
the polarization data for various clouds by Planck Collaboration et al. (2020)
shows that the polarization degree at 850 $\mu$m decreases with dust
temperature for $T_{\rm d}\geq 19\,\rm K$ (see their Figure 18). This observed
feature challenges the classical picture of the RAT-A theory.

Recently, a new fundamental effect induced by RATs, so-called Radiative Torque
Disruption (or RAT-D), has been realized by Hoang et al. (2019); Hoang (2019).
The effect is based on the fact that an irregular dust grain exposed to a
strong radiation field could be disrupted into small fragments when the
centrifugal stress induced by suprathermal rotation by RATs exceeds the
maximum binding energy of the grain material (i.e., tensile strength). Since
RATs are stronger for larger grains (Lazarian and Hoang 2007a; Hoang and
Lazarian 2008), RAT-D is effective for the large grains, which then determines
the upper cutoff of the grain size distribution (GSD) and causes the variation
of GSD with the radiation field. The modified GSD by RAT-D in turn, affects
dust optical-UV extinction, thermal emission from infrared to radio
wavelengths, as well as polarization (see Hoang 2020 for a review). The RAT-D
effect has been studied in many astrophysical environments, including
extinction curves toward starburst galaxies (Hirashita and Hoang 2020), light-
curve of type Ia supernovae (Giang et al. 2020) and of Gamma-Ray Burst (GRB)
Afterglows (Hoang et al. 2020), evolution of ice in protoplanetary disks (Tung
and Hoang 2020) and cometary comae (Hoang and Tung 2020), circumsolar dust
(Hoang et al. 2021a), rotational desorption of interstellar organic molecules
(Hoang and Tram 2020), spinning dust nanoparticles in circumstellar envelopes
of evolved stars (Tram et al. 2020), radiation pressure feedback from massive
protostars (Hoang, 2021b) and dust evolution across cosmic time (Hoang,
2021a). In particular, Lee et al. (2020) performed numerical modeling of
multi-wavelength polarized thermal dust emission from aligned grains and
disruption by RATs in MCs. They showed that the polarization at 850$\,\mu$m
first increases and then decreases with $T_{\rm d}$, which could reproduce the
$p-T_{\rm d}$ trend observed by Planck (Planck Collaboration et al. 2020).
Tram et al. (2021a) carried out a detailed modelling to interpret the
$p-T_{\rm d}$ of Ophiuchus-A cloud observed by SOFIA/HAWC+ with higher
resolution than Planck. The authors demonstrated that the combination of the
RAT-A and RAT-D effects could successfully reproduce the observational data.
Since then, more observations have shown evidences for the RAT-D mechanism
(e.g., Ngoc et al. 2021; Tram et al. 2021b; Hoang et al. 2022b).

High-resolution polarimetric observations toward starless cores (e.g.,Alves et
al. 2014; Jones et al. 2015) and protostars (e.g., Hull et al. 2014; Cox et
al. 2018; Pattle and Fissel 2019) usually report the decrease of the
polarization degree with increasing column density or intensity of dust
emission, a.k.a “polarization hole”. Two popular reasons proposed to explain
the polarization hole includes the decrease of grain alignment efficiency
toward denser gas/lower radiation field and the tangling of the magnetic field
(Goodman et al. 1992, 1995; Hull et al. 2014). In the case of protostars, one
expects, however, an increase of grain alignment efficiency toward the
protostar due to the increasing incident radiation flux with accordance to the
RAT-A theory. Hence, an increase in the induced polarization degree is
expected in proximity to the peak intensity. Therefore, the underlying origin
of the polarization hole toward protostars is difficult to understand and
reconcile in terms of grain alignment theory. The miss-match between theory
and observations has been reported in Pillai et al. (2020) and Zielinski et
al. (2021). Hoang et al. (2021b) showed that the rotational disruption by RATs
could occur in the vicinity of protostars and suggested the combination of
grain alignment and disruption by RATs as a plausible origin of the
polarization hole. For MCs without embedded sources or line of sights far away
from the source, the polarization hole may be dominated by the effect of the
field tangling and grain alignment loss (see Pattle and Fissel 2019).

In this review, we summarize the fundamentals of grain alignment in the
context of suprathermal rotation and disruption by RATs, internal and external
alignment processes within conditions of molecular cloud in Section 4. In
Section 5, we summarize the numerical modelling that predicts the polarized
thermal dust emission of molecular cloud irradiated by a nearby luminosity
source in the framework of the RAT paradigm (RAT-A and RAT-D effects). We
review some representative observational interpretations in Section 6. In
Section 7, we discuss grain alignment to trace magnetic fields in molecular
clouds and the very dense conditions in protostellar cores/disks, including
polarization hole within the RAT paradigm, effects of RAT paradigm in time-
domain astrophysics and on grain growth and evolution, and suprathermal
rotation induced by mechanical torques.

##  4 Grain Alignment and Disruption by Radiative Torques: the RAT paradigm

###  4.1 Radiative torques and its fundamental effects

Dolginov and Mitrofanov (1976) first suggested that the interaction of an
anisotropic radiation with a helical grain can induce RATs due to the
differential scattering/absorption of left- and right-handed circularly
polarized photons. Latter, RATs were numerically demonstrated in Draine and
Weingartner (1996) for three irregular grain shapes. Lazarian and Hoang
(2007a) introduced an Analytical Model (AMO) of RATs which is based on a
helical grain consisting of an oblate spheroid and a weightless mirror. The
AMO is shown to reproduce the basic properties of RATs obtained from numerical
calculations for realistically irregular grain shapes (Lazarian and Hoang,
2007a; Hoang and Lazarian, 2008; Herranen et al., 2021), and enables us to
make quantitative predictions for various conditions (Hoang and Lazarian,
2014) and dust compositions (Lazarian and Hoang, 2008; Hoang and Lazarian,
2009, 2016a). Many predictions from the RAT alignment (hereafter RAT-A) theory
were observationally tested (see Andersson et al. 2015 Lazarian et al. 2015
for recent reviews).

As shown in previous studies (Draine and Weingartner, 1997; Lazarian and
Hoang, 2007a; Hoang and Lazarian, 2008), RATs in general can induce three
fundamental effects on grain rotational dynamics, including (1) the grain
precession around the radiation direction, (2) spin-up the grain to
suprathermal rotation as well as spin-down to thermal rotation, and (3) align
the grain with $\boldsymbol{J}$ along the radiation $\boldsymbol{k}$.

###  4.2 Grain model and magnetic properties

####  4.2.1 Grain model

Let us define an irregular grain of size $a$ and mass density $\rho$ (Figure
1). In the fixed grain’s frame, this irregular shape is described by there
principal axes $\hat{a}_{1},\hat{a}_{2},\hat{a}_{3}$ with the inertia moments
of $I_{1}>I_{2}\geq I_{3}$, respectively. For the sake of simplicity, we
assume an oblate spheroidal shape (i.e., $I_{1}>I_{2}=I_{3}$) with the length
of the semi-minor axis of $c$ and the lengths of the semi-major axes of $a$.
The corresponding inertial moments are $I_{\|}\equiv I_{1}=\frac{8\pi}{15}\rho
a^{4}c=\frac{8\pi}{15}\rho sa^{5}$ and $I_{\perp}\equiv
I_{2}=I_{3}=\frac{4\pi}{15}\rho
a^{2}c\left(a^{2}+c^{2}\right)=\frac{4\pi}{15}\rho
sa^{5}\left(1+s^{2}\right)$, where $s=c/a$ is the axial ratio of oblate
grains. The volume of this grain is $V=4/3\pi sa^{3}$.

Figure 1: (a): illustration of internal alignment between the angular momentum
($\boldsymbol{J}$) and the short axis ($\boldsymbol{a_{1}}$). (b):
illustration of external alignment between $\boldsymbol{J}$ and the external
magnetic field ($\boldsymbol{B}$) or radiation direction ($\boldsymbol{k}$) or
gas flow ($\boldsymbol{v}$). $\boldsymbol{\Omega}$ is the angular velocity,
and $\omega$ is the angular frequency. Figures taken from Hoang et al.
(2022a).

For an isolated grain, the angular momentum $\boldsymbol{J}$ is conserved, and
the grain axis of maximum inertia in general makes an angle with
$\boldsymbol{J}$. Figure 1(a) describes the torque-free motion of the oblate
grain, which includes the precession of angular velocity of
$\boldsymbol{\Omega}$ or momentum $\boldsymbol{J}$ with a frequency of
$\omega$ along the grain shortest axis. In the presence of an external
magnetic field, radiation field or a gas flow, the angular momentum
$\boldsymbol{J}$ precesses around the magnetic field, radiation field or
incident gas flow, Figure 1(b) (Lazarian and Hoang, 2007a, b).

####  4.2.2 Grain magnetic properties

Magnetic properties of dust are essential for their interaction with the
ambient magnetic field and grain alignment. We consider grains having embedded
iron atoms (such as silicate grains). The presence of iron atoms with unpaired
electrons makes interstellar dust a natural paramagnetic material. When iron
atoms are diffusely distributed, the grain is called paramagnetic (PM). When
iron atoms are distributed as iron clusters, dust grains become
superparamagnetic material (SPM, see Hoang and Lazarian 2016a).

The magnetic susceptibility of a paramagnetic grain at rest is given by

$\chi_{\rm PM}(0)=0.03n_{23}f_{p}\hat{p}^{2}\left(\frac{20\,\rm K}{T_{\rm
d}}\right)~{}{,}$ (1)

with $\hat{p}=p/5.5$ ($p\simeq 5.5$ for silicate grains; see Hoang and
Lazarian 2016a, and references therein). This equation shows the dependence on
$T_{\rm d}$. The magnetic susceptibility reduces for higher $T_{\rm d}$
because of the halt in magnetization due to the thermal fluctuation of free-
electron spins.

For a superparamagnetic grain, we get

$\chi_{\rm SPM}(0)=0.026N_{\rm cl}\phi_{\rm sp}\hat{p}^{2}\left(\frac{20\,\rm
K}{T_{\rm d}}\right)~{}{.}$ (2)

where $N_{\rm cl}$ is the number of Fe atoms per cluster, $\phi_{sp}$ is the
volume filling factor of iron cluster in grain. $N_{\rm cl}$ could vary from
20 to $10^{5}$ (Jones and Spitzer 1967). One can see that iron inclusion
increases significantly the magnetic property of grains.

For a grain rotating at a frequency $\omega$, the magnetic susceptibility is a
complex number with the imaginary part that describes the absorption
properties of magnetic energy given by

$\chi_{2}(\omega)=\frac{\chi(0)\tau_{\rm el}\omega}{[1+(\omega\tau_{\rm
el}/2)^{2}]^{2}}~{}{,}$ (3)

where $\tau_{\rm el}=2.9\times 10^{-12}/f_{p}n_{23}$ is the spin-spin
relaxation time with $n_{23}=n/10^{23}\,\rm cm^{-3}$ the atomic density of
material, $f_{p}$ the fraction of Fe atoms in the grain (e.g., $f_{p}=1/7$ for
MgFeSiO4 grains, see Hoang et al. 2014).

Let us define a quantity $K(\omega)=\chi_{2}(\omega)/\omega$, which yields

$K_{\rm PM}(\omega)\simeq 8.7\times 10^{-14}\hat{p}\left(\frac{20\,\rm
K}{T_{\rm d}}\right)\left(\frac{1}{[1+(\omega\tau_{\rm
el}/2)^{2}]^{2}}\right)~{}{,}$ (4)

and

$K_{\rm SPM}(\omega)\simeq 2.6\times 10^{-11}N_{\rm cl}\phi_{\rm
sp}\hat{p}^{2}\left(\frac{20\,\rm K}{T_{\rm d}}\right)k_{\rm
SPM}(\omega)~{}{,}$ (5)

with

$k_{\rm SPM}(\omega)=\exp\left(\frac{{\rm 0.011\,K}\times N_{\rm cl}}{T_{\rm
d}}\right)\left[1+\left(\frac{\omega\tau_{\rm sp}}{2}\right)^{2}\right]^{-2},$
(6)

where $\tau_{\rm sp}$ is the timescale of thermally activated remagnetization
given by

$\tau_{\rm sp}\simeq 10^{-9}\exp\left(\frac{0.011\,{\rm K}\times N_{\rm
cl}}{T_{\rm d}}\right)~{}{\rm s.}$ (7)

The supermagnetic grain has enormously increased magnetic susceptibility
(Jones and Spitzer 1967), and the coupling with the magnetic field is stronger
than in an ordinary paramagnetic grain. Note that a superparamagnetic grain
behaves like a paramagnetic grain in the absence of an external magnetic
field.

###  4.3 Grain suprathermal rotation by RATs and rotational damping

Let us consider a grain exposed to a radiation field of the energy density of
$u_{\rm rad}\,(\rm erg\,cm^{-3})$, the mean wavelength of $\bar{\lambda}$ and
an anisotropy degree of $\gamma$111$\gamma=0$ for a isotropic radiation field,
and $\gamma=1$ for an unidirectional radiation field. For a typical diffuse
interstellar radiation field: $\gamma\simeq 0.1$. For molecular cloud:
$\gamma\simeq 0.7$ (Draine and Weingartner 1996). Its strength is defined by a
dimensionless $U=u_{\rm rad}/u_{\rm ISRF}$, where $u_{\rm ISRF}=8.64\times
10^{-13}\,\rm erg\,cm^{-3}$ is the radiation energy density of the
interstellar radiation field (ISRF) in the solar neighborhood (Mathis et al.
1983).

The rotation of an irregular grain of size $a$ is described by the
suprathermal number $St=\Omega/\Omega_{\rm T}$ with $\Omega_{\rm T}=(k_{\rm
B}T_{\rm gas}/I_{\|})^{0.5}$ the thermal angular velocity. Grain rotation is
called ”suprathermal” for $St>1$ and ”subthermal” for $St<1$.

One of the important effects of RATs is to spin-up the dust grains of
irregular shapes. On the another hand, gas collisions and IR re-emission spin-
down the grain (see Hoang 2019). The timescale characterizing the grain
rotational damping by gas collisions and IR emission is

$\tau_{\rm damp}=\frac{\tau_{\rm gas}}{1+F_{\rm IR}}$ (8)

where $\tau_{\rm gas}$ and $F_{\rm IR}$ are the gas collision damping
timescale and the ratio of the IR re-emission to collisional damping times,
which are

$\tau_{\rm gas}=\frac{3}{4\sqrt{\pi}}\frac{I_{\|}}{1.2n_{\rm H}m_{\rm H}v_{\rm
T}a^{4}\Gamma_{\|}}\simeq 8.3\times
10^{3}\hat{\rho}\left(\frac{sa_{-5}}{n_{3}T_{1}^{1/2}\Gamma_{\|}}\right)~{}{\rm
yr},$ (9) $F_{\rm IR}\simeq 3.8\times
10^{-2}\left(\frac{U^{2/3}}{a_{-5}}\right)\left(\frac{10^{3}\,\rm{cm^{-3}}}{n_{\rm
H}}\right)\left(\frac{10\,\rm K}{T_{\rm gas}}\right)^{1/2}.$ (10)

with $n_{3}=n_{\rm H}/(10^{3}\,\rm cm^{-3})$ the gas volume density,
$T_{1}=T_{\rm gas}/10\,\rm K$ the gas temperature, $a_{-5}=a/(10^{-5}\,\rm
cm)$ the grain size, $\hat{\rho}=\rho/(3\,\rm g\,cm^{-3})$ the mass density of
the grain, and $\Gamma_{\|}$ the geometrical parameter ($\Gamma_{\|}=1$ for a
spherical grain and $0.82$ for a grain of axial ratio $s=1/2$; see Hoang et
al. 2022a).

The balance between the spin-up by RATs and spin-down by gas collisions and IR
emission establishes the maximum rotation rate of grains due to RATs. For a
simple case of a constant luminosity source222If the radiation source varies
in luminosity, the angular velociy $\Omega_{\rm RAT}$ is obtained by solving
the equation of motion; see Equation 12 in Hoang et al. (2019), the maximum
angular velocity that an irregular grain acquired by RATs is

$\Omega_{\rm RAT}=\frac{\bar{\Gamma}_{\rm RAT}\tau_{\rm damp}}{I_{\parallel}}$
(11)

where $\bar{\Gamma}_{\rm RAT}$ is the average RAT acting on grain, which
depends on the RAT efficiency ($Q_{\Gamma}$), grain size ($a_{\rm eff}$) and
local radiation ($u_{\rm rad}$, $\bar{\lambda}$, $\gamma$) as

$\bar{\Gamma}_{\rm RAT}=\pi a_{\rm
eff}^{2}\left(\frac{\bar{\lambda}}{2\pi}\right)\gamma u_{\rm
rad}\bar{Q}_{\Gamma}$ (12)

The RAT efficiency strongly depends on the ratio of radiation wavelength and
grain size as $\overline{Q}_{\Gamma}\propto(\lambda/a)^{-\eta}$333the power
index $\eta$ describes the changes in RAT efficiency with respect to grain
size; see Figure 1 in Hoang et al. 2021b (see e.g., Hoang et al. 2019, 2021b),
indicating that RATs could be effective at longer wavelengths for large grains
(e.g., optical-IR photons) rather than only UV photons. Thus, the RAT effect
is effective in more extended regions than other dust effects induced by UV
photons, such as photo-electron emission.

Because the RAT efficiency changes its slope at $a\simeq\bar{\lambda}/2.5$
(see e.g., Hoang et al. 2021b), the suprathermal number of grains spun-up by
RATs is described separately as (Hoang et al. (2022a))

$St_{\rm RAT}\simeq
180\hat{\rho}^{1/2}s^{5/6}a_{-5}^{7/2}\left(\frac{\bar{\lambda}}{1.2\,\rm\mu
m}\right)^{-2}\left(\frac{\gamma U}{n_{3}T_{1}}\right)\left(\frac{1}{1+F_{\rm
IR}}\right),$ (13)

for $a\lesssim\bar{\lambda}/2.5$, and

$St_{\rm RAT}\simeq 2.1\times
10^{4}\hat{\rho}^{1/2}s^{-1/6}a_{-5}^{1/2}\left(\frac{\bar{\lambda}}{1.2\,\rm\mu
m}\right)\left(\frac{\gamma U}{n_{3}T_{1}}\right)\left(\frac{1}{1+F_{\rm
IR}}\right).$ (14)

for $a>\bar{\lambda}/2.5$.

For regions where the gas density is very high and the radiation strength is
rather small (i.e., $F_{\rm IR}\ll 1$), such as in pretostellar cores and the
mid-plane of protoplanetary disks, one can see that $St_{\rm RAT}\propto
U/n_{3}T_{1}$. For regions with very strong radiation fields but a low gas
density (i.e., $F_{\rm IR}\gg 1$), such as in the vicinity of a radiation
source, one has $St_{\rm RAT}\propto U^{1/3}$. Note that $St_{\rm RAT}$ also
strongly depends on the size of the grains, the bigger grain size, the faster
rotation induced by RATs. Therefore, large grains are easily to be spun-up
suprathermally in strong radiation fields.

###  4.4 Internal alignment

The internal alignment describes the alignment of the angular momentum
$\boldsymbol{J}$ with the short axis $\boldsymbol{a_{1}}$ of the grain as
illustrated in Figure 1(a). Here we discuss three relaxation processes that
induce internal alignment, including Barnett relaxation, nuclear relaxation,
and inelastic relaxation, as well as show the critical grain sizes for each
internal alignment.

####  4.4.1 Barnett relaxation

The most important process relies on the Barnett relaxation (Purcell 1979). A
rotating (super-)paramagnetic grain will be magnetized due to the Barnett
effect, which results in the dissipation of rotational energy due to the
rotating component perpendicular to $\boldsymbol{a_{1}}$. The dissipation will
eventually bring $\boldsymbol{\Omega}$ and $\boldsymbol{J}$ aligned with
$\boldsymbol{a_{1}}$, which is the minimum energy state. The stage of
$\boldsymbol{J}\perp\boldsymbol{a_{1}}$ is called ”right internal alignment”.

The relaxation times by the Barnett effect for a paramagnetic (so-called
Barnett relaxation) and a supermagnetic grain (so-called super-Barnett
relaxation) are given by

$\tau_{\rm BR,PM}=\frac{\gamma_{e}^{2}I_{\|}^{3}}{VK_{\rm
PM}(\omega)h^{2}(h-1)J^{2}}\simeq
0.5\hat{\rho}^{2}a^{7}_{-5}f(\hat{s})\left(\frac{J_{\rm
d}}{J}\right)^{2}\left[1+\left(\frac{\omega\tau_{\rm
el}}{2}\right)^{2}\right]^{2}~{}{\rm yr,}$ (15) $\tau_{\rm
BR,SPM}=\frac{\gamma_{e}^{2}I_{\|}^{3}}{VK_{\rm
SPM}(\omega)h^{2}(h-1)J^{2}}\approx
0.16\hat{\rho}^{2}f(\hat{s})a_{-5}^{7}\left(\frac{1}{N_{\rm cl}\phi_{\rm
sp,-2}\hat{p}^{2}}\right)\left(\frac{J_{\rm
d}}{J}\right)^{2}\left(\frac{T_{d,1}}{k_{\rm SPM}(\omega)}\right)~{}\rm yr,\\\
$ (16)

where $V=4\pi/3a^{3}$ is the grain volume, $\gamma_{e}=-g_{e}\mu_{B}/\hbar$ is
gyromagnetic ratio of an electron ($g_{e}\simeq 2$ and $\mu_{B}\simeq
9.26\times 10^{-21}\,\rm erg\,G^{-1}$),
$\hat{s}=s/0.5,f(\hat{s})=\hat{s}[(1+\hat{s}^{2})/2]^{2}$, and $J_{\rm
d}=\sqrt{I_{\|}k_{\rm B}T_{\rm d}/(h-1)}$ is the dust thermal angular momentum
(see also Hoang and Lazarian 2014) with $h=I_{\|}/I_{\perp}=2/(1+s^{2})$ and
$\phi_{\rm sp,-2}=\phi_{\rm sp}/0.01$.

However, gas collisions can strongly randomize the internal alignment if the
Barnett relaxation is slower than the grain randomization by gas collisions.
Therefore, grains have efficient internal alignment when the timescale for the
Barnett relaxation is shorter than that for gas rotational damping. Thus, the
maximum size that a paramagnetic and superparamagnetic grain has the internal
alignment with $\boldsymbol{J}$ along the short axis $\boldsymbol{a_{1}}$ can
be determined by the balance between these two timescales, which yields (Hoang
et al. (2022a))

$a^{\rm PM}_{\rm max,aJ}\simeq 0.39\times
h^{1/3}St^{1/3}\left(\frac{\hat{p}^{2}}{\hat{\rho}n_{3}T^{1/2}_{1}}\right)^{1/6}\times\left(\frac{1}{1+(\omega\tau_{\rm
tel}/2)^{2}}\right)^{1/3}\left(\frac{(h-1)T_{\rm gas}}{T_{\rm
d}}\right)^{1/6}~{}{\rm\mu m},$ (17)

and

$a^{\rm SPM}_{\rm max,aJ}\simeq 2.18h^{1/3}St^{1/3}\left(\frac{N_{\rm
cl,4}\phi_{\rm
sp,-2}\hat{p}^{2}}{\hat{\rho}n_{3}T_{1}^{1/2}\Gamma_{\|}}\right)^{1/6}\times\left(\frac{1}{k_{\rm
SPM}(\omega)}\right)^{1/6}\\\ \times\left(\frac{(h-1)T_{\rm gas}}{T_{\rm
d}}\right)^{1/6}~{}{\rm\mu m},$

with $N_{\rm cl,4}=N_{\rm cl}/10^{4}$ (Hoang et al., 2022a). One can see that
$a_{\rm max,aJ}$ increases with grain rotation as $St^{1/3}$ and decreases
with gas density as $\sim n^{-1/6}_{\rm H}$. With iron inclusion,
superparamagnetic grains could achieve efficient internal alignment for larger
grain sizes.

####  4.4.2 Nuclear relaxation

Atomic nuclei within the grain can also have nuclear paramagnetism due to
unpaired protons and nucleons. The attachment of H atoms to the grain or water
ice mantle induce nuclear magnetism due to proton spin (Purcell 1979). Nuclear
paramagnetism can also induce internal relaxation as Barnett effect for
electron spins (Lazarian and Draine 1999). The nuclear relaxation (NR) time is
given by

$\displaystyle\tau_{\rm NR}$ $\displaystyle=$
$\displaystyle\frac{\gamma_{n}^{2}I_{\|}^{3}}{VK_{n}(\omega)h^{2}(h-1)J^{2}},$
(18) $\displaystyle\simeq$ $\displaystyle
125\hat{\rho}^{2}a_{-5}^{7}f(\hat{s})\left(\frac{n_{e}}{n_{n}}\right)\left(\frac{J_{d}}{J}\right)^{2}\left(\frac{g_{n}}{3.1}\right)^{2}\left(\frac{2.79\mu_{N}}{\mu_{n}}\right)^{2}\times\left[1+\left(\frac{\omega\tau_{n}}{2}\right)^{2}\right]^{2}{\rm
s},$

where $\gamma_{n}$ is nuclear gyromagnetic ratio, $n_{e}$ is the number
density of unpaired electrons in the dust, $n_{n}$ is the number density of
nuclei that has the magnetic moment $\mu_{n}$, and
$\mu_{N}=e\hbar/2m_{p}c=5.05\times 10^{-24}$ erg G-1 with $m_{p}$ the proton
mass is the nuclear magneton, $\tau_{n}\sim 10^{-5}(3.1/g_{n})^{2}(10^{22}{\rm
cm}^{-3}/n_{e})$s is the nuclear spin-spin relaxation time, and
$K_{n}=\chi_{n}/\omega$ with $\chi_{n}$ the nuclear magnetic susceptibility
(see Hoang 2022).

The maximum size for internal alignment induced by nuclear relaxation is given
by Hoang (2022),

$\displaystyle a_{\rm max,aJ}({\rm
NR})<3.03h^{1/3}St^{1/3}\left(\frac{(n_{e}/n_{n})(g_{n}/3.1)^{2}}{n_{3}T_{1}^{1/2}\Gamma_{\|}}\right)^{1/6}\left(\frac{1}{1+(\omega\tau_{\rm
n}/2)^{2}}\right)^{1/3}\times\left(\frac{(h-1)T_{\rm gas}}{T_{\rm
d}}\right)^{1/6}~{}{\rm\mu m.}$ (19)

which increases with the suprathermal rotation as $St^{1/3}$ for
$\omega<2/\tau_{n}$, but decreases as $St^{-1/3}$ for $\omega>2/\tau_{n}$ due
to suppression of nuclear magnetism at fast rotation. For the typical density
of $n=10^{3}\,\rm cm^{-3}$, large grains of size upto $a\sim 3\,\mu$m can have
efficient internal alignment by nuclear relaxation.

####  4.4.3 Inelastic relaxation

Atoms and molecules in a rotating grain experience a centrifugal force that
stress them out from the grain central of mass. The chemical attractive
forces, on the other hand, tend to tight them together. These effects cause
the grain deformation. If grains are made up inelastic materials, the
deformation results in the dissipation of rotational energy, which eventually
brings the angular momentum $\boldsymbol{J}$ to align with the grain’s
shortest axis $\boldsymbol{a_{1}}$ (see Purcell 1979; Lazarian and Efroimsky
1999). This internal alignment mechanism is named as inelastic relaxation.

The characteristic time of inelastic relaxation for an oblate spheroidal grain
can be estimated as (see Hoang et al. 2022a)

$\tau_{\rm iER}=\frac{\mu Q}{\rho
a^{2}\Omega_{0}^{3}}g(s)=\frac{\rho^{1/2}a^{11/2}\mu Q}{(kT_{\rm
gas})^{3/2}St^{3}}g^{\prime}(s)=0.034\hat{\rho}^{1/2}a_{-5}^{11/2}\frac{\mu_{8}Q_{3}}{T_{1}^{3/2}}\frac{g^{\prime}(s)}{St^{3}}~{}{\rm
yr},$ (20)

where $Q$ is the quality factor of grain material (Q=100 for a silicate rocks,
Efroimsky and Lazarian 2000), $\mu$ is the modulus of rigidity ($\mu\sim
10^{7}\,\rm erg\,cm^{-3}$ for cometary, Knapmeyer et al. 2018).
$\Omega_{0}=St\Omega_{T}$, $\mu_{8}=\mu/(10^{8}\,\rm erg\,cm^{--3})$,
$Q_{3}=Q/10^{3}$, and $g^{\prime}=2.2s^{3/2}g(s)$ with $g(s)$ the geometrical
factor as

$g(s)=\frac{2^{3/2}7}{8}\frac{(1+s^{2})^{4}}{s^{4}+1/(1+\sigma)},$ (21)

which corresponds to $g(s)=7.0$ and $4.6$ for $s=1/2$ and $1/3$, respectively.

Following Hoang et al. (2022a), the maximum grain size for efficient inelastic
relaxation is defined by $\tau_{\rm iER}=\tau_{\rm gas}$ and given by

$a_{\rm max,aJ}({\rm iER})\simeq
1.55\left(\frac{\mu_{8}Q_{3}}{\sqrt{\hat{\rho}}}\right)^{-2/9}\left(\frac{T_{1}}{n_{3}}\right)^{2/9}\times\left(\frac{s}{g^{\prime}(s)\Gamma_{\|}}\right)^{2/9}St^{2/3}~{}{\rm\mu
m,}$ (22)

which implies that large grains of $a\sim 1.55\,\rm\mu m$ can be efficiently
aligned via inelastic relaxation even at thermal rotation of $St=1$, and large
grains can be aligned for $St>1$.

Figure 2: Comparison of minimum (dashed line) and maximum (solid line) size of
grains having internal alignment by Barnett relaxation (BR) for
superparamagnetic grain (SPM, color lines) and inelastic mechanism (iER, black
lines) with respect to a large range of gas density (Figure adopted from Hoang
et al. 2022a). The grains are spun-up by RATs with $U=10^{2}$ (panel (a)) and
$U=10^{4}$ (panel (b)).

Figure 2 shows the comparison of grains having internal alignment by Barnett
relaxation for superparamagnetic grains (color lines) and inelastic relaxation
(black lines) as functions of the gas density. Grains are assumed to be spun-
up by RATs. First, the Barnett relaxation is not effective in very dense
regions (e.g., $n_{\rm H}>10^{9}\,\rm cm^{-3}$) because of the gas rotational
damping and lower $St_{\rm RAT}$. For a given gas density, one can see that
Barnett relaxation is more efficient, but the inelastic relaxation is able to
align larger grains (panel (a)). The reason is that the maximum grain size for
inelastic relaxation scales as $\sim St^{2/3}$, while it scales as $\sim
St^{1/3}$ for Barnett relaxation. For stronger radiation fields, the inelastic
relaxation is efficient in denser gas (panel (b)), which is due to the fact
that inelastic timescale reduces as $\sim St^{-1/3}$.

For a typical density of the ISM ($n_{\rm H}\simeq 10^{3}\,\rm cm^{-3}$), one
can see that the Barnett relaxation for PM/SPM grains are much more effective
than the inelastic relaxation. However, unlike the PM/SPM, diamagnetic grains
(e.g., carbonaceous dust) does not experience the Barnett relaxation effect,
but they could still have efficient internal alignment for size $a<a_{\rm
max,aJ}(\rm iER)$ due to inelastic relaxation, because this relaxation is
independent from the grain magnetic properties.

###  4.5 External alignment

The external alignment describes the alignment of the grain angular momentum
$\boldsymbol{J}$ with a preferred direction in space, which can be the ambient
magnetic field or the radiation direction (see Figure 1(b)).

####  4.5.1 Alignment with magnetic fields (magnetic alignment): Larmor
precession and magnetic relaxation

Larmor precession: The interaction between the grain magnetic moment due to
the Barnett effect with the magnetic field induces the Larmor precession of
$\boldsymbol{J}$ around $\boldsymbol{B}$, resulting the first coupling of the
grain with the magnetic field (see e.g., Hoang 2022).

The rate of the Larmor precession for (super)paramagnetic grains are
respectively given by

$\tau_{B,\rm PM}=\frac{2\pi}{|d\phi/dt|}=\frac{2\pi I_{\|}\Omega}{|\mu_{\rm
Bar}|B}=\frac{2\pi|\gamma_{e}|I_{\|}}{\chi_{\rm PM}(0)VB}\simeq 3.5\times
10^{-4}\frac{\hat{\rho}T_{d,1}a^{2}_{-5}}{n_{23}f_{p}\hat{p}^{2}}\left(\frac{B}{20\,\rm\mu
G}\right)^{-1}~{}{\rm yr,}$ (23)

and

$\tau_{B,\rm SPM}=\frac{2\pi|\gamma_{e}|I_{\|}}{\chi_{\rm SPM}(0)VB}\simeq
4.05\times 10^{-2}\frac{\hat{\rho}T_{d,1}a_{-5}^{2}}{N_{\rm
cl}\phi_{sp,-2}\hat{p}^{2}}\left(\frac{B}{20\,\rm\mu G}\right)^{-1}\rm
yr.~{}~{}~{}~{}$ (24)

When the Larmor precession is faster than the gas randomization, the magnetic
field becomes an axis of grain alignment, which is known as magnetic
alignment. The maximum size for the fast Larmor precession around the B-field
is defined by $\tau_{B}=\tau_{\rm gas}$, which is

$a^{\rm Lar,PM}_{\rm max,JB}\simeq 1.2\times
10^{6}\hat{s}n^{-1}_{3}T^{-1/2}_{1}T^{-1}_{d,1}\left(\frac{B}{20\,\rm\mu
G}\right)\left(\frac{n_{23}f_{p}\hat{p}^{2}}{\Gamma_{\|}}\right)~{}{\rm\mu
m,}$ (25)

and

$a^{\rm Lar,SPM}_{\rm max,JB}\simeq 1.02\times
10^{9}\hat{s}n^{-1}_{3}T_{1}^{-1/2}T^{-1}_{d,1}\left(\frac{B}{20\,\rm\mu
G}\right)\left(\frac{N_{\rm cl,4}\phi_{\rm
sp,-2}\hat{p}^{2}}{\Gamma_{\|}}\right)~{}{\rm\mu m.}~{}~{}~{}$ (26)

for PM and SPM grains, respectively.

Para- and superpara-magnetic relaxation: The angle between $\boldsymbol{J}$
and the magnetic field can be reduced by paramagnetic relaxation, resulting in
the magnetic alignment of $\boldsymbol{J}$ with $\boldsymbol{B}$. This
mechanism is known as the DG relaxation (Davis and Greenstein 1951). As
numerically shown in Hoang and Lazarian (2016a, b), magnetic relaxation alone
is not enough to produce efficient grain alignment when grains have thermal
rotation (i.e., $St\leq 1$) due to strong internal thermal fluctuations and
gas randomization. Yet, the joint action of suprathermal rotation by RATs and
magnetic relaxation however can make grains to achieve high degree of
alignment.

The characteristic times of the DG relaxation for the PM and SPM grains are
given by (see e.g., Hoang and Lazarian 2016a)

$\tau_{\rm mag,PM}=\frac{I_{\|}}{VK_{\rm PM}(\Omega)B^{2}}=\frac{2\rho
a^{2}}{5K_{\rm PM}(\Omega)B^{2}}\simeq 8.6\times
10^{4}\frac{\hat{\rho}a^{2}_{-5}T_{d,1}}{\hat{p}}\left(\frac{B}{20\,\rm\mu
G}\right)^{-2}\left(1+(\omega\tau_{\rm el}/2)^{2}\right)^{2}~{}{\rm
yr,}~{}~{}~{}$ (27)

and

$\tau_{\rm mag,SPM}=\frac{2\rho a^{2}}{5K_{\rm SPM}(\Omega)B^{2}}\simeq
3.75\times 10^{2}\frac{\hat{\rho}a_{-5}^{2}}{N_{\rm cl}\phi_{\rm
sp}\hat{p}^{2}}\frac{T_{d,1}}{k_{\rm SPM}(\Omega)}\left(\frac{B}{20\,\rm\mu
G}\right)^{-2}~{}{\rm yr.}~{}~{}~{}$ (28)

The respective maximum of size that grain alignment can still be affected by
the DG magnetic relaxation is defined by $\tau_{\rm mag}=\tau_{\rm gas}$ and
is equal to (Hoang et al., 2022a)

$a^{DG,\rm PM}_{\rm max,JB}\simeq 0.0097\times
n^{-1}_{3}T^{-1/2}_{1}T^{-1}_{d,1}\left(\frac{B}{20\,\rm\mu
G}\right)^{2}\left(1+(\omega\tau_{el}/2)^{2}\right)^{-2}~{}{\rm\mu m,}$ (29)

and

$a_{\rm max,JB}^{DG,\rm SPM}\approx 224\times
n^{-1}_{3}T^{-1/2}_{1}T^{-1}_{d,1}\left(\frac{B}{20\,\rm\mu
G}\right)^{2}N_{\rm cl,4}\phi_{\rm sp,-2}\hat{p}^{2}k_{\rm
SPM}(\omega)~{}{\rm\mu m.}$ (30)

One can see that the gas density and B-field strength determine which grains
are aligned with B-field as $a^{DG}_{\rm max,JB}\sim n^{-1}_{\rm H}B^{2}$. The
DG efficiency drops dramatically with increasing the gas density. Paramagnetic
relaxation is negligible for large grains, but superparamagnetic relaxation
can be efficient even for large grains up to $\sim 20\,\rm\mu m$ for MCs with
$n_{\rm H}\sim 10^{3}\,\rm cm^{-3}$.

####  4.5.2 The RAT Alignment Paradigm

Here we describe the basic elements of the RAT alignment paradigm, including
radiative precession, grain alignemnt at low-$J$ and high-$J$ attractors, and
the minimum size for RAT alignment.

Radiative precession: In the presence of an anisotropic radiation field,
irregular grains experience radiative precession due to RATs, with the grain
angular momentum precessing around the radiation direction ($\boldsymbol{k}$).
For a grain with angular momentum $J$, the radiative precession time is given
by (Lazarian and Hoang 2007a; Hoang and Lazarian 2014),

$\tau_{k}=\frac{2\pi}{|d\phi/dt|}\approx\frac{2\pi J}{\gamma u_{\rm
rad}\lambda a_{\rm eff}^{2}Q_{e3}}\simeq
56.8\hat{\rho}^{1/2}T_{1}^{1/2}\hat{s}^{-1/6}a_{-5}^{1/2}St\left(\frac{1.2\,\rm\mu
m}{\gamma\bar{\lambda}\hat{Q}_{e3}U}\right)\rm yr,~{}~{}~{}$ (31)

where $\hat{Q}_{e3}=Q_{e3}/10^{-2}$ with $Q_{e3}$ the third component of RATs
that induces the grain precession around $\boldsymbol{k}$ and the
normalization is done using the typical value of $Q_{e3}$ (see Lazarian and
Hoang 2007a).

In the absence of magnetic field, radiative precession can be much faster than
gas damping time for a thermal rotation of $St\sim 1$ (Eq. 9). In this case,
the anisotropic radiation direction is the axis of grain alignment, which is
known as k-RAT alignment.

Low$-J$ and high$-J$ Attractor Alignment by RATs: In addition to radiative
precession, RATs can align grains at high$-J$ attractors and low$-J$
attractors. In general, a fraction of grains ($f_{\rm high-J}$) can be aligned
at a high$-J$ attractor due to the aligning and spin-up effects of RATs
(Lazarian and Hoang 2007a; Hoang and Lazarian 2008). Grains aligned at the
high$-J$ attractor have the suprathermal rotation with $St_{\rm high-J}>1$. On
the other hand, due to the aligning and spin-down effects of RATs, the
$1-f_{\rm high-J}$ fraction of grains, are aligned at the low$-J$ attractor
with $St_{\rm low-J}\sim 1$ (Hoang and Lazarian, 2014). The RAT alignment
process can occur on a timescale smaller than the gas damping time, which is
called fast alignment (Lazarian and Hoang, 2007a, 2021) with a strong
radiation field. Grains can be stably aligned at low$-J$ attractors when the
grain randomization by gas collisions is not considered. Otherwise, the gas
randomization could randomize grains at low$-J$ attractors and eventually
transport them to the high$-J$ attractor after a timescale greater than the
gas damping time, which is usually called slow alignment (Hoang and Lazarian,
2008; Lazarian and Hoang, 2021). The fast RAT alignment occurs in the local
environment of intense transient radiation sources such as supernova (Giang et
al., 2020), kilonova, and gamma-ray bursts (Hoang et al., 2020), whereas the
slow RAT alignment occurs in most astrophysical environments, from the ISM,
MCs, to SFRs. A detailed discussion of fast and slow alignment by RATs is
discussed in detail in Lazarian and Hoang (2021).

The exact value of $f_{\rm high-J}$ depends on the grain properties (shape and
size) and magnetic susceptibility. For grains with ordinary paramagnetic
material (e.g., silicate), Herranen et al. (2021) found that $f_{\rm high-J}$
can be about $10-70\%$ based on calculations of RATs for an ensemble of
Gaussian random shapes. The presence of iron inclusions embedded in the grains
increases grain magnetic suceptibility and superparamagnetic relaxation, which
can produce in the universal high$-J$ attractors(i.e., $f_{\rm high-J}\sim
100\%$) (Hoang and Lazarian, 2016a; Lazarian and Hoang, 2021).

Minimum size of RAT alignment: Due to gas collisions, grains can be
efficiently aligned only when they rotate suprathermally at a high$-J$
attractor. Hoang and Lazarian (2008) and Hoang and Lazarian (2014)
demonstrated that grain alignment is stablized under the gas randomization
when $\Omega_{\rm RAT}\equiv 3\times\Omega_{\rm T}$ (or $St_{\rm RAT}=3$).
This criteria yields the minimum size for grain alignment by RATs as

$a_{\rm align}\simeq
0.055\hat{\rho}^{-1/7}\left(\frac{n_{3}T_{1}}{\gamma_{-1}U}\right)^{2/7}\times\left(\frac{\bar{\lambda}}{1.2\,\rm\mu
m}\right)^{4/7}\left(\frac{1}{1+F_{\rm IR}}\right)^{-2/7}~{}\rm{\mu m},$ (32)

where $\gamma_{-1}=\gamma/10^{-1}$. One can see that for a typical MC with
$n_{\rm H}=10^{3}\,\rm cm^{-3}$, grains larger than $a_{\rm
align}=0.055\,\mu$m can be aligned by RATs. However, the radiation strength is
not an observable quantity, thus we could use the dust temperature $T_{\rm d}$
as a proxy for this strength. If we adopt $U\simeq(T_{\rm d}/16.4\,\rm
K)^{6}(a/10^{-5}\,\rm cm)^{6/15}$ (Draine 2011), $a_{\rm align}\sim
n^{2/7}_{\rm H}T^{-12/7}_{\rm dust}$ that illustrates that higher $T_{\rm d}$
is able to bring smaller grains to align. Vice-versa, only larger grains are
able to align in higher $n_{\rm H}$ (denser gas). Note that the power-index in
Equation 32 slightly differs from Hoang et al. (2021b) and Tram et al.
(2021b). The reason is due to the slightly difference in the scaling of the
average RAT efficiency, but this difference results in a negligible
discrepancy of $a_{\rm align}$ (see Figure 1 in Hoang et al. 2021b).

k-RAT vs. B-RAT Alignment: Using Equation (31) one calculates the radiative
precession (k-RAT) timescale for grains aligned at low$-J$ attractors with
$St=1$ as

$\displaystyle\tau_{k}^{\rm
low-J}=56.4\hat{\rho}^{1/2}T_{1}^{1/2}\hat{s}^{-1/6}a_{-5}^{1/2}\left(\frac{1.2\mu
m}{\gamma\bar{\lambda}\hat{Q}_{e3}U}\right){\rm yr},~{}~{}~{}~{}~{}$ (33)

and for grains aligned at high$-J$ attractors with $St=St_{\rm RAT}$ from
Equation (14), one has

$\displaystyle\tau_{k}^{\rm high-J}$ $\displaystyle=$ $\displaystyle
1.17\times
10^{6}\hat{s}^{-1/3}\left(\frac{\hat{\rho}a_{-5}}{\hat{Q}_{e3}}\right)\left(\frac{1}{n_{3}T_{1}^{1/2}}\right)\left(\frac{1}{1+F_{\rm
IR}}\right){\rm yr}.~{}~{}~{}~{}~{}$ (34)

In the presence of a magnetic field, the grain experiences simultaneous Larmor
precession and radiative precession. If the Larmor precession is faster than
the radiative precession, i.e., $\tau_{B}<\tau_{k}$, the axis of grain
alignment by RATs changes from with $\boldsymbol{J}$ along $\boldsymbol{k}$ to
the magnetic field, $\boldsymbol{B}$, which is known as B-RAT. The minimum
grain size that grain alignment still occurs via k-RAT, which is also the
maximum size grains can be aligned via B-RAT, is defined by
$\tau_{k}=\tau_{B}$ and is derived in Hoang et al. (2022a). For grains aligned
at a low$-J$ attractor, one has

$a^{\rm low-J,PM}_{\rm min,Jk}\equiv a_{\rm max,JB}^{\rm low-J,PM}\simeq
2.96\hat{\rho}^{-1/2}\hat{s}^{-1/9}\left(\frac{\bar{\lambda}}{1.2\,\rm\mu
m}\right)^{-2/3}\left(\frac{n_{23}f_{\rm
p}\hat{p}^{2}T^{1/2}_{1}}{T_{d,1}\gamma
U_{3}\hat{Q}_{e3}}\right)^{2/3}\left(\frac{B}{20\,\rm\mu
G}\right)^{2/3}~{}{\rm\mu m,}$ (35)

and

$\displaystyle a^{\rm low-J,SPM}_{\rm min,Jk}\equiv a_{\rm max,JB}^{\rm
low-J,SPM}$ $\displaystyle=$ $\displaystyle
57.8\hat{\rho}^{-1/2}\hat{s}^{-1/9}\left(\frac{\bar{\lambda}}{1.2\,\rm\mu
m}\right)^{-2/3}\left(\frac{\hat{p}^{2}T^{1/2}_{1}}{T_{d,1}\gamma_{\rm
rad}U_{3}\hat{Q}_{e3}}\right)^{2/3}$ $\displaystyle\times$
$\displaystyle\left(\frac{B}{20\,\rm\mu G}\right)^{2/3}\left(N_{\rm
cl,4}\phi_{sp,-2}\right)^{2/3}~{}{\rm\mu m.}~{}~{}~{}~{}~{}$

for PM and SPM grains, respectively. The above equations imply that the
minimum size for k-RAT when grains aligned at low$-J$ attractors is smaller
for higher radiation strength as $U^{-2/3}$, but larger for stronger magnetic
field as $B^{2/3}$. It is harder to bring SPM grains to align with the
radiation direction because of their larger magnetic susceptibility and then
faster Larmor precession.

Similarly, for grains aligned at a high$-J$ attractor (Hoang et al., 2022a)

$a^{\rm high-J,PM}_{\rm min,Jk}\equiv a_{\rm max,JB}^{\rm high-J,PM}\simeq
3.3\times 10^{8}\hat{s}^{-1/3}\left(\frac{n_{23}f_{\rm
p}\hat{p}^{2}}{n_{3}T_{d,1}T^{1/2}_{1}\hat{Q}_{e3}}\right)\left(\frac{B}{20\,\rm\mu
G}\right)\left(\frac{1}{1+F_{\rm IR}}\right)~{}{\rm\mu m,}$ (37)

and

$a^{\rm high-J,SPM}_{\rm min,Jk}\equiv a_{\rm max,JB}^{\rm high-J,SPM}\simeq
2.8\times
10^{10}\hat{s}^{-1/3}\left(\frac{\hat{p}^{2}}{n_{3}T_{d,1}T_{1}^{1/2}\hat{Q}_{e3}}\right)\left(\frac{B}{20\,\rm\mu
G}\right)\left(\frac{N_{\rm cl,4}\phi_{sp,-2}}{1+F_{\rm IR}}\right)~{}{\rm\mu
m.}~{}~{}~{}$ (38)

One realizes that the minimum size for k-RAT alignment is very large for
typical molecular clouds, especially when grains are aligned at high$-J$
attractors. Therefore, in these conditions, grains are aligned with
$\boldsymbol{J}$ along the magnetic field via B-RAT. Yet, for very strong
radiation fields of strength of $U\gg 1$, small grains at low$-J$ attractors
can have k-RAT alignment.

###  4.6 Radiative Torque Disruption (RAT-D)

####  4.6.1 The RAT-D mechanism

A dust grain rotating with an angular velocity $\Omega$ develops a tensile
stress $S=\rho\Omega^{2}a^{2}/4$ on grain material. This stress points outward
from the grain center of mass and tends to tear the grain apart. When the
tensile stress exceeds the maximum tensile strength of the grain material,
$S_{\rm max}$, the grain is instantaneously disrupted in fragments. The
maximum angular velocity beyond which the grain is disrupted is defined by
$S=S_{\rm max}$ and reads

$\Omega_{\rm disr}=\frac{2}{a}\left(\frac{S_{\rm
max}}{\rho}\right)^{1/2}\simeq\frac{3.65\times 10^{8}}{a_{-5}}S^{1/2}_{\rm
max,7}\hat{\rho}^{-1/2}{~{}\rm rad/s},$ (39)

where $S_{\rm max,7}=S_{\rm max}/(10^{7}\rm erg\,cm^{-3})$. The exact value of
$S_{\rm max}$ depends on the grain internal structure (see Hoang 2020, and
references therein). For example, composite grains have $S_{\rm max}\sim
10^{7}\,\rm erg\,cm^{-3}$, whereas compact grains have a higher value.

Subject to a strong radiation field, dust grains will be spontaneously
fragmented into many smaller species when the grain angular velocity spun-up
by RATs, $\Omega_{\rm RAT}$, exceeds the critical value $\Omega_{\rm disr}$.
This is called the radiative torque disruption (RAT-D) mechanism, which was
first introduced in Hoang et al. (2019). Due to the dependence of $\Omega_{\rm
RAT}$ on the grain size and local conditions, there is a range of grain sizes
that can be disrupted, as illustrated by the shaded areas in Figure 3. The
minimum size of grains that can be disrupted by RAT-D is

$\displaystyle a_{\rm disr}$ $\displaystyle=$
$\displaystyle\left(\frac{0.8n_{\rm H}\sqrt{2\pi m_{\rm H}kT_{\rm
gas}}}{\gamma u_{\rm rad}\bar{\lambda}^{-2}}\right)^{1/2}\left(\frac{S_{\rm
max}}{\rho}\right)^{1/4}(1+F_{\rm IR})^{1/2}$ (40) $\displaystyle\simeq$
$\displaystyle
1.7\left(\frac{\gamma_{-1}U}{n_{3}T_{1}^{1/2}}\right)^{-1/2}\left(\frac{\bar{\lambda}}{1.2\,\rm\mu
m}\right)\hat{\rho}^{-1/4}S_{\rm max,7}^{1/4}(1+F_{\rm IR})^{1/2}\,\rm\mu m,$

and the maximum size of grains that can still be disrupted by RAT-D is

$\displaystyle a_{\rm disr,max}$ $\displaystyle=$ $\displaystyle\frac{\gamma
u_{\rm rad}\bar{\lambda}}{16n_{\rm H}\sqrt{2\pi m_{\rm H}kT_{\rm
gas}}}\left(\frac{S_{\rm max}}{\rho}\right)^{-1/2}(1+F_{\rm IR})^{-1}$ (41)
$\displaystyle\simeq$ $\displaystyle
0.03\left(\frac{\gamma_{-1}U}{n_{3}T_{1}^{1/2}}\right)\left(\frac{\bar{\lambda}}{1.2\,\rm\mu
m}\right)\hat{\rho}^{1/2}S_{\rm max,7}^{-1/2}(1+F_{\rm IR})^{-1}\,\rm\mu m,$

which depend on the local gas properties, radiation field, and the grain
tensile strength (Hoang and Tram, 2020; Hoang et al., 2021b).

As shown, the RAT-D efficiency strongly depends on the local conditions, i.e.,
more efficient for higher radiation strength (U) or $T_{\rm d}$, but less
efficient for higher $n_{\rm H}$.

Figure 3: Illustration of the rotation rate by RATs and disruption limit for
different tensile strength. The lower and upper intersections define the lower
and upper disruption sizes. Figure adopted from Hoang (2020).

####  4.6.2 Rotational desorption of ice mantles

Detection of interstellar complex organic molecules (COMs) are extensively
reported toward low-/high-mass protostars and protoplanetary disks (see e.g.,
van Dishoeck 2014, and references therein). As a popular scenario, COMs are
believed to first form on the icy mantle of dust grains (e.g., Garrod et al.
2008; Jiménez-Serra et al. 2016), and subsequently desorbed to the gas phase
during the warm and hot phase, where icy grain mantles can be heated to high
temperatures $T_{\rm d}\sim 100–300\,\rm K$ (see e.g., Blake et al. 1987;
Brown et al. 1988; Bisschop et al. 2007). However, icy mantles might not be
survived under an intense radiation field before the critical temperature for
thermal sublimation is reached (Hoang and Tram 2020). The centrifugal force
can break the icy mantel from the bare core, then the evaporation and
sublimation processes are enhanced owing to smaller size of the icy fragments,
enabling COMs to release (see Figure 4(a)). This non-thermal process is called
the rotational desorption, and found to be effective at lower dust
temperatures than the classical thermal sublimation, which is able to explain
the detection of COMs in cold regions (see Hoang and Tram 2020; Tram et al.
2021c for further details).

Figure 4: (a):Illustration of the rotational desorption process (Figure
adopted from Hoang and Tram 2020). There are two stages: (1) disruption of icy
mantles into small fragments by the RAT-D mechanism, and (2) rapid evaporation
due to thermal spikes for very small fragments or increased sublimation for
larger fragments. The rotational desorption is thus able to desorb complex
organic molecules (COMs) formed on the icy mantle. (b): Desorption timescales
of some COMs having very high binding energy in Orion BN/KL (Figure adopted
from Tram et al. 2021c). The rotational desorption and sublimation are shown
by the colored and black lines, while the UV photodesorption is displayed by
the gray dashed lines. The rotational desorption is effective for $T_{\rm
d}\simeq 80-100\,\rm K$ at which these COMs are detected (cyan area). The gray
area is where the sublimation is dominant.

Figure 4(b) shows an evidence for rotation desorption of COMs from Orion BN/KL
region. The timescales of rotational desorption (color lines) and thermal
sublimation (dashed dotted black lines) for three COMs with very high binding
energy, including ethyl formate ($\rm C_{2}H_{5}OCHO$), methoxymethanol ($\rm
CH_{3}OCH_{2}OH$) and ethylene glycol ($\rm OHCH_{2}CH_{2}OH$) that are
reported in Tercero et al. (2018), are shown for comparison. One can see that
for given low dust temperatures constrained by SOFIA/FORCAST observations
($T_{\rm d}\simeq 80-100\,\rm K$; De Buizer et al. 2012), rotational
desorption is more effective than thermal sublimation in desorbing COMs from
the grain surface. With regard to the UV photo-desorption (dashed horizontal
lines), the rotational desorption could be as efficient as (or more effective
than) the UV photo-desorption regarding to the photo-desorption yield
uncertainty. However, UV photons are expected to be quickly attenuated by dust
absorption, so that the volume in which photo-desorption is important is less
extended than the rotational desorption which can work with optical-IR
photons.

Figure 5: (a): Alignment of paramagnetic grains for a typical molecular cloud
in proximity to an ionization source with $\gamma=1$ and
$\bar{\lambda}=0.5\,\rm\mu m$. The internal alignment of most grains is
efficient (i.e., $\boldsymbol{a_{1}}\|\boldsymbol{J}$). For the external
alignment, the paramagnetic relaxation (DG mechanism) is much less efficient
than Larmor precession (B-RAT) to align $\boldsymbol{J}$ with
$\boldsymbol{B}$. Grains with micro-size are aligned with magnetic fields.
Larger grains are expected to align with radiation field (k-RAT). (b): Similar
but for superparamagnetic grains with $N_{\rm cl}=10^{3}$. Due to the
enhancement of the grain susceptibility, (1) all grains are internally aligned
with $\boldsymbol{J}$ perpendicular to $\boldsymbol{a_{1}}$, (2) DG mechanism
is effective to larger grain size, (3) all grains are externally aligned with
magnetic fields. Note that the B-RAT is much more effective than DG
relaxation.

###  4.7 Summary of grain alignment and disruption in molecular clouds

In this section, we discuss the alignment of magnetic grains in a typical
molecular cloud, which is irradiated by an ionization source (e.g., OB stars).
We adopted the following representative parameters: $n_{\rm H}=10^{3}\,\rm
cm^{-3}$, $T_{\rm gas}=T_{\rm d}=30\,\rm K$, $U\simeq a^{6/15}_{-5}(T_{\rm
d}/16.4\,\rm K)^{6}$, $\bar{\lambda}=0.5\,\mu$m, $\gamma=1.0$, and
$B=20\,\rm\mu G$. The maximum grain size is limited to $100\,\mu$m.

Figure 5(a) illustrates the relative importance of different alignment
mechanisms. For the internal alignment, most grains are aligned with the
angular momentum parallel to the short axis (i.e.,
$\boldsymbol{J}\parallel\boldsymbol{a_{1}}$). The paramagnetic relaxation is
much less efficient than the Larmor precession. Grains can be aligned by RATs
along B-fields (B-RAT) up to sizes of 1.1$\,\mu$m. Above that size, the
radiative precession is faster than the Larmor precession such that grains are
turned to align with the radiation direction (k-RAT). Hence, B-RAT is the most
important alignment mechanism in the diffuse ISM and typical molecular clouds
where dust grains could not grow to beyond $1\mu$m. Furthermore, if the
grain’s structure is not very compact (e.g., with a low tensile strength of
$S_{\rm max}=10^{8}\,\rm erg\,cm^{-3}$), the RAT-D mechanism can disrupt large
grains with size of 0.1-7.5$\,\mu$m. The rotational fragmentation enhances the
abundance of smaller grains, which, in turn, can allow more grains to align
with the magnetic field.

Similarly, Figure 5(b) shows alignment for superparamagnetic grains with
$N_{cl}=10^{3}$ and $\phi_{\rm sp}=0.01$. Due to the enhancement of
susceptibility, all superparamagnetic grains are aligned with
$\boldsymbol{J}\parallel\boldsymbol{a_{1}}$ for the internal alignment. For
the external alignment, B-RAT alignment is dominant over k-RAT. Even the DG
mechanism is more effective for larger grains up to 6.5$\,\mu$m compared to
ordinary paramagnetic grains, yet the B-RAT is much more efficient than the
superparamagnetic relaxation. Thus, superparamagnetic grains are precessed
along the external magnetic fields.

Therefore, grain alignment is able to probe the magnetic field in typical
molecular clouds. However, as shown in this section, the gas damping, the
B-field strength and the radiation strength are as key to characterize which
alignments take place. Hence, all of alignment sizes are subjected to change
with the local conditions, and the picture of grain alignment become very
complicated in very dense regions, such as protostellar cores/disks (see
Section 7.2).

##  5 Modelling thermal dust polarization within the RAT paradigm

Polarized thermal dust emission depends on the grain properties (size, shape,
and composition), grain alignment degree, and the magnetic field geometry (see
e.g., Draine and Fraisse 2009). Within the RAT paradigm, RATs can induce the
simultaneous effects of RAT-A and RAT-D on dust grains. The RAT-A describes
the alignment degree of grains of different sizes with the magnetic field,
while RAT-D affects the grain size distribution. Therefore, an accurate
physical model of polarized thermal dust emission requires the treatment of
both RAT-A and RAT-D effects.

###  5.1 Assumptions and basic equations

To account for both the RAT-A and RAT-D effects on dust polarization, Lee et
al. (2020) adopted a simple cloud model444The state-of-the-art version is
publicly available at https://github.com/lengoctram/DustPOL consisting of a
homogeneous molecular cloud irradiated by a central radiation source, as
illustrated in Figure 6(a). The radiation strength ($U$) decreases with
distance from the central star and varies along the line of sight (solid and
dashed lines). For simplicity, all grains along each line of sight in the
cloud are assumed to be illuminated by the constant radiation field of
strength $U$.

Figure 6: (a): Schematic illustration of the model. A molecular cloud is
irradiated by the central star, so that the radiation strength decreases from
$U_{0}$ to $U_{\rm ISRF}$. The color gradient indicates the increase of the
gas density from the central star. Magnetic fields are assumed to be uniform.
(b): An example of polarization degree vs. radiation strength (dust
temperature) at $850\,\mu$m predicted by the numerical model (Lee et al.
2020). If we consider only the RAT-A effect, the polarization degree
monotonically increases with increasing the radiation strength (dust
temperature) as expected (black line). If we take into account the RAT-D
mechanism, rotational disruption occurs when the radiation strength (dust
temperature) is sufficiently high. Once this happens, the depletion of largest
grains results in the decrease of the polarization degree (color lines).
Figure taken from Lee et al. (2020) and Tram et al. (2021a).

As shown in Section 4.4 (also Section 4.7), both Barnett and inelastic
relaxation effects are very efficient for standard grains in the ISM and MCs,
leading to the perfect internal alignment of dust grains (see Figures 5).
Therefore, grain alignment is mostly determined by external alignment by RATs.
For the rotational damping, we consider only collisions with gas and re-
emission of thermal dust as two main sources to damp the grain rotation
against the spinup by RATs. The variation of B-fields along the line of sight
is neglected by assuming the uniform magnetic field (the field tangling effect
is discussed in Section 6.5). The detailed description of this model was
presented in Lee et al. (2020) and Tram et al. (2021a).

Dust grains are heated by absorption of the stellar radiation and cool down by
re-emission in infrared radiation. Assuming a dust environment containing
carbonaceous and silicate grains, the total dust emission intensity is given
by

$\frac{I_{\rm em}(\lambda)}{N_{\rm H}}=\sum_{j=\rm sil,carb}\int^{a_{\rm
max}}_{a_{\rm min}}Q_{\rm ext}(a,\lambda)\pi a^{2}\times\int
dTB_{\lambda}(T_{\rm d})\frac{dP}{dT}\frac{1}{n_{\rm
H}}\frac{dn_{j}}{da}da.~{}~{}~{}$ (42)

where $a_{\rm min}$ and $a_{\rm max}$ are the minimum and maximum grain sizes,
$N_{\rm H}$ is the hydrogen column density, $B_{\lambda}(T_{\rm d})$ is the
Planck function at dust temperature $T_{\rm d}$, $dP/dT$ is the grain
temperature distribution function, $dn_{j}/da$ is the grain-size distribution
of dust population $j$, and Qext is the extinction coefficient. The dust
temperature distribution depends on the grain size, optical constant, and the
radiation field ($U$), which is computed by the DustEM code (Compiègne et al.
2011, see e.g., Figure 8 in Lee et al. 2020).

If silicate and carbon dust populations are separated, then as paramagnetic
grains, silicates can align with the ambient magnetic field, while
carbonaceous grains cannot. Thus, the polarized emission intensity resulting
from aligned grains is given by (Hoang and Lazarian 2016a)

$\frac{I_{\rm pol}(\lambda)}{N_{\rm H}}=\int^{a_{\rm max}}_{a_{\rm
min}}f(a)Q_{\rm pol}\pi a^{2}\times\int dTB_{\lambda}(T_{\rm
d})\frac{dP}{dT}\frac{1}{n_{\rm H}}\frac{dn_{sil}}{da}da,~{}~{}~{}$ (43)

where $f(a)=1-\exp[-(a/2a_{\rm align})^{3}]$ is the alignment function, and
Qpol is the polarization efficiency coefficient.

If silicate and carbon grains are mixed together (e.g., Jones et al. 2013;
Draine and Hensley 2021), which may be the case in dense clouds due to many
cycles of photo-processing, coagulation, shattering, accretion, and erosion,
carbon grains could be aligned with the ambient magnetic field and their
thermal emission could be polarized. In the simplest case, assuming these
grain populations have the same alignment parameters (i.e., $a_{\rm align}$
and $f(a)$), the total polarized intensity is

$\frac{I_{\rm pol}(\lambda)}{N_{\rm H}}=\sum_{j=\rm sil,carb}\int^{a_{\rm
max}}_{a_{\rm min}}f(a)Q_{\rm pol}\pi a^{2}\times\int dTB_{\lambda}(T_{\rm
d})\frac{dP}{dT}\frac{1}{n_{\rm H}}\frac{dn_{j}}{da}da.$ (44)

The extinction ($Q_{\rm ext}$) and polarization coefficients ($Q_{\rm pol}$)
are computed by the DDSCAT model (Draine and Flatau 1994, 2008; Flatau and
Draine 2012). The maximum grain size is constrained by the RAT-D mechanism,
$a_{\rm max}\equiv a_{\rm disr}$.

The fractional polarization of the thermal dust emission is the ratio of the
polarized intensity ($I_{\rm pol}$) to the total emission intensity ($I_{\rm
em}$), which yields

$P(\%)=100\times\frac{I_{\rm pol}}{I_{\rm em}}.$ (45)

###  5.2 Model parameters and numerical prediction

Our model parameters include the gas properties (gas number density, $n_{\rm
H}$) and gas temperature, $T_{\rm gas}$), the dust properties (size $a$,
shape, internal structure described by the tensile strength $S_{\rm max}$, and
size distribution power index $\beta$), and the ambient properties (radiation
field strength $U$, mean wavelength $\bar{\lambda}$, and an anisotropy degree
$\gamma$ of the radiation field).

Radiation strength cannot be constrained from observations, thus, the
equilibrium temperature of dust grains is used as a proxy of the radiation
strength, as $U\simeq a^{6/15}_{-5}(T_{\rm d}/16.4\,\rm K)^{6}$. Here, the
dust temperature is derived from fitting a modified black-body function to the
continuum spectral energy distribution. The mean wavelength ($\bar{\lambda}$)
of the stellar radiation field from a central source with $T_{\ast}$ is given
as $\bar{\lambda}\simeq 0.53\,\rm cm\,K/T_{\ast}$ (see Eq. 8 in Hoang et al.
2021b). The internal structure of grains is determined via their tensile
strength ($S_{\rm max}$). Unfortunately, this quantity is poorly constrained
in astrophysical environments. However, compact grains in general are expected
to have higher $S_{\rm max}$ than composite and fluffy grains. For instance,
one expects $S_{\rm max}\sim 10^{9}-10^{10}\,\rm erg\,cm^{-3}$ for compact
grains, while $S_{\rm max}\sim 10^{6}-10^{8}\,\rm erg\,cm^{-3}$ for composite
grains (see e.g., Hoang 2020). In the model, $S_{\rm max}$ can be set as a
free parameter.

Figure 6(b) shows the predicted dependence of the polarization degree on
radiation strength (or dust temperature) at a specific wavelength of
$850\,\mu$m for $n_{\rm H}=10^{4}\,\rm cm^{-3}$. In the case of only RAT-A
effect (black line), the polarization degree increases as radiation strength
(dust temperature) increases. In the case of both RAT-A and RAT-D effects, the
polarization degree first increases and turns to decrease toward higher dust
temperature. The second trend is caused by the rotational disruption which
occurs above a critical temperature. For dust temperatures lower than the
critical temperature, RATs are not enough to spin-up grains to the disruption
limit $\Omega_{\rm disr}$, so that the RAT-D cannot occur. Thus, the results
are the same as in the case of only RAT-A effect. For dust temperatures higher
than the critical value, RATs become stronger to trigger the RAT-D effect,
resulting in rotational disruption of large grains into small fragments. This
leads to a drop in the polarization degree due to the depletion of large
grains. This critical temperature and the level of the decline depend on the
internal structure of the grains controlled by $S_{\rm max}$.

Figure 7: Understanding of thermal dust polarization. For a fixed maximum
grain size ($a_{\rm max}$), higher radiation strength/dust temperature can
bring smaller grains to align (smaller size of $a_{\rm min,align}$) so that
the aligned grain size distribution becomes broader. This broad size
distribution increases the degree of thermal dust polarization with radiation
strength/dust temperature. If the maximum grain size is determined by the
RAT-D mechanism and grains are composite ($a_{\rm max}\rightarrow a_{\rm
disr}$), higher radiation/dust temperature results in smaller $a_{\rm disr}$.
The consequent narrower aligned grain size distribution drops the polarization
degree.

Figure 7 shows our understanding of the trend of polarization degree versus
radiation strength/dust temperature shown in Figure 6(b). For a given gas
density, the polarization degree is mainly determined by the grain size
distribution. If the only RAT-A effect is considered, the maximum grain size
is fixed. The minimum size that is aligned by RATs is smaller for higher
radiation strength (more small grains are able to be aligned). Hence, the size
distribution of aligned grains becomes broader, which deduces the increment of
the polarization degree. If the rotational disruption is taken into account
along with the RAT-A effect, the maximum grain size could vary. In this case,
if the radiation is insufficient strong the polarization degree is observed to
increase with radiation strength as previous case. Once the radiation strength
is sufficient to trigger disruption, the maximum grain size reduces from
$a_{\rm max}$ to $a_{\rm disr}$. The narrower size distribution of aligned
grains causes the polarization degree to drop.

As we mentioned, radiation strength is not an observable quantity, thus dust
temperature could be used as a proxy. Therefore, the relation between
polarization degree and dust temperature ($p-T_{\rm d}$) becomes a unique tool
to study the joint effect of grain alignment and disruption by RATs.

##  6 Observational Study of the RAT effects in star-forming regions

In this section, we review our observational results of thermal dust
polarization observed toward star-forming molecular clouds, including $\rho$
Ophiuchus-A, 30 Doradus, M 17, and Orion BN/KL. These star-forming MCs
containing intense radiation sources are the ideal targets to test the RAT-A
and RAT-D effects.

Figure 8: Top: Gas column density (panel (a)) and dust temperature (panel (b))
constructed from Herschel data (Figure adopted from Santos et al. 2019 and
Tram et al. 2021a). A star symbol locates the radiation source Oph-S1. Close
to this source, dust is hotter while gas is more diffuse. Middle: 2D-histogram
of polarization degree observed with SOFIA/HAWC+ at 89$\,\mu$m (panel (c)) and
154$\,\mu$m (panel (d)) and dust temperature (background). Dashed line is the
prediction by only the RAT-A effect, which could not explain the decrease of
$p$ for $T_{\rm d}>34\,\rm K$. The solid lines are the predictions made by
both the RAT-A and RAT-D effects for only silicate grains with the grain-size
distribution of $dn/da\sim a^{\beta}$. Bottom: Similar to the middle panel but
for and a mixture of silicate-carbonaceous grains at 89$\,\mu$m (panel(e)) and
154$\,\mu$m (panel(f)). The RAT paradigm (RAT-A and RAT-D) successfully
reproduces both the increase and decrease part of the data. Figures adopted
from Tram et al. (2021a).

###  6.1 $\rho$ Ophiuchus-A

$\rho$ Ophiuchus-A ($\rho$ Oph-A) is a molecular cloud in one of the closest
dark cloud complex and star-forming region $\rho$ Ophiuchi. The distance to
this complex is reported to be $\sim$ 120–160 pc (Chini, 1981; de Geus et al.,
1989; Ortiz-León et al., 2017). This cloud is irradiated by a high-mass star,
namely Oph-S1, which is a young B3-type star (Vrba 1977; Hamaguchi et al.
2003).

Figure 8 shows the observational data of $\rho$ Oph-A. The top panels show the
maps of gas column density (panel (a)) and dust temperature (panel (b))
constructed from Herschel. Along the radial direction toward Oph-S1 (marked by
the white star), the dust is hotter while the gas is more diffuse. The middle
and bottom panels show the $p-T_{\rm d}$ relations. Background shows the
observational data at 89 (left) and 154$\,\mu$m (right) observed by
SOFIA/HAWC+. The polarization degree tends to first increase and then decrease
as the dust temperature increases. One can see that the latter decrement
occurs in vicinity of Oph-S1 where the gas density is lower. Therefore, the
drop of $p$ toward higher $T_{\rm d}$ and lower $n_{\rm H}$ is opposite to the
fundamental prediction of the RAT-A effect.

The lines in Figure 8 show the results from our numerical modelling
overplotted with the observational data at two wavelengths, assuming only
silicate (panels (c,d)) or a combination of silicate and carbonaceous grains
(panels (e,f)) with different slopes ($\beta$) of the size distribution. We
considered two cases: only the RAT-A effect (no RAT-D, dashed line) and both
RAT-A and RAT-D effects (solid lines).

  * •

For the former case, the polarization degree monotonically increases with the
dust temperature. This rising $p-T_{\rm d}$ trend is caused by the increase in
the alignment efficiency (i.e., decrease in the alignment size $a_{\rm
align}$) toward the Oph-S1 due to stronger radiation intensity and lower gas
density. Obviously, the RAT-A effect is able to reproduce the first increasing
part of the observational data, but not the second decreasing part.

  * •

For the latter case, the polarization degree is similarly expected to
increases for $T_{\rm d}\simeq 34\,\rm K$. At higher dust temperatures, the
RAT-D happens for grains larger than $a_{\rm disr}$ owing to the higher
radiation intensity and the lower gas density, which reduces the polarization
degree. The joint effect of RAT-A and RAT-D mechanisms is therefore able to
reproduce the entire observational data.

###  6.2 30 Doradus

30 Doradus (30 Dor), the Tarantula Nebula, is an Hii region located in the
Large Magellanic Cloud (LMC) at the distance of $\sim 50\,\rm kpc$ from the
Sun (e.g., De Marchi et al. 2011). It is powered by the massive star cluster R
136 that has a bolometric luminosity of $7.8\times 10^{7}L_{\odot}$ (see Lopez
et al. 2011, and references therein). Since the LMC is a low-metallicity
galaxy ($Z\simeq 0.5Z_{\odot}$, see e.g., Dufour et al. 1982; Galliano et al.
2008) that has a low dust-to-gas ratio (e.g., $\sim 2–5\times 10^{-3}$; see
Roman-Duval et al. 2014), the radiative feedback could be effective deeper
inside the surrounding molecular cloud. Thus, the 30 Dor cloud offers a
valuable environment to test the physics of grain alignment and disruption by
RATs.

Figure 9: (a): SOFIA/HAWC+ observations of 30 Dor at 214$\,\mu$m. Contours are
the map of gas column density. The stellar cluster R 136 is located by a red
cross. We selected two sub-regions, namely North and South, to further
analysis. (b,c): 2D-histogram of gas column density vs. dust temperature in
two selected sub-regions: North and South. The black line shows the weighted
mean in each bin associated with $1-\sigma$ uncertainty. The blue line is a
double power-law fitting. In the North, we fit to an upper envelope of the
distribution by a dashed red line. In the South, the plot is more scatter.
However, these quantities are positively correlated upto $T_{\rm d}\simeq
37\,\rm K$, and negatively correlated to higher $T_{\rm d}$. Figure adopted
from Tram et al. 2021b.

Figure 9(a) shows the polarization degree map observed with SOFIA/HAWC+ at
214$\,\mu$m. The location of ionization source R 136 is denoted by a red
cross. For comparison, the contours show the map of gas column density. For
further analysis, we selected two sub-regions, namely North and South as
highlighted respectively by a red box and a red ellipse. Figure 9(b,c) show
the relation of the gas column density to the dust temperature in these two
sub-regions, respectively. In the North, the gas density unambiguously
increases and then decreases with increasing dust temperature. The transition
is at $T_{\rm d}\sim 37-40\,\rm K$. In the South, the relation is less clear.
Yet, it is likely to behave similarly as in the North, in which the gas
density positively correlates with the dust temperature upto $T_{\rm d}\sim
37-39\,\rm K$, and negatively correlates with higher $T_{\rm d}$.

Figure 10: Relations between polarization degree and dust temperature in three
bands (from top to bottom) in North (left panel) and in South (right panel)
regions highlighted in the left panel of Figure 9. The error bars connected by
a line shows the weighted-mean in each bin. The black dashed line shows the
piecewise line fit to the data. In the North, a complex $p-T_{\rm d}$ relation
is found, but $p$ decreases with higher $T_{\rm d}$. In the South, a weak
$p-T_{\rm d}$ relation is seen. Figure adopted from Tram et al. 2021b.

Figure 10 shows the $p-T_{\rm d}$ relations in the North (left panel) and in
the South (right panel) for three SOFIA/HAWC+ wavelengths (from top to
bottom).

  * 1-

In the North, there is a complex relation for all wavelength. Note that
$T_{\rm d}$ increases toward R 136.

    * –

For $T_{\rm d}\leq 37–40\,\rm K$ (region A), both dust temperature and gas
column density increase. The polarization degree first rapidly drops, then
rises, and finally slowly changes with increasing dust temperature. This trend
could be explained by the RAT-A effect. Increasing gas density causes
increased rotational damping by gas collisions, which is sufficiently
efficient for larger $a_{\rm align}$. As we discussed in Section 5, a higher
value of $a_{\rm align}$ causes $p$ to decrease. At some point in the cloud
where $T_{\rm d}$ is sufficient high, the variation of $a_{\rm align}$ might
be small because of opposite effects of $T_{\rm d}$ and $N_{\rm H}$. That
makes $p$ to slowly change.

    * –

For $T_{\rm d}>37–40\,\rm K$, the gas column density decreases, the
polarization degree first increases slightly (regions B) and then drops again
(region C). The negative slopes are steeper for longer wavelengths, namely
$-0.52$, $-0.72$ and $-0.83$ at 89, 154 and 214$\,\mu$m, respectively. As gas
damping become less effective due to lower $N_{\rm H}$ and higher $T_{\rm d}$,
the RAT-A effect could explain the increase in $p$ in region B (i.e., smaller
$a_{\rm align}$), but fails to explain the drop of $p$ in region C. The latter
is evidence for the RAT-D mechanism at work in proximity to radiation source
where $N_{\rm H}$ continues to be small while $T_{\rm d}$ is highest. As the
largest grains get disrupted, $p$ decreases faster at longer wavelength. Thus,
the observed drops of $p$ in region C are expected by RAT-A and RAT-D.

  * 2-

In the South, the $p-T_{\rm d}$ relation has a high degree of scatter and is
ambiguous. There might be three reasons for the slow variation of $p$ with
$T_{\rm d}$. First, the saturation of RAT-D, which occurs at very high
temperatures (i.e., $T_{\rm d}$ is much larger than the disruption threshold
temperature; see Figure 14 in Lee et al. 2020). Small grains resulting from
the disruption do not experience RAT-D due to lower RATs. Second, when dust
grains are very hot, IR damping is so strong and dominates the gas damping,
resulting in the slow variation of $a_{\rm disr}$ with $T_{\rm d}$. Third,
effective gas damping due to the high gas density in the selected region (see
Figure 9).

###  6.3 M17

M17, also known as the Omega Nebula or the Horseshoe Nebula is prominent Hii
region with and associated molecular cloud and star-forming region (Povich et
al. 2009; Lim et al. 2020). It is located in the constellation of Sagittarius
at a distance of 1.98 kpc (Xu et al. 2011). M17 hosts the stellar cluster NGC
6618 and its relative closeness makes it an excellent target to test the RAT
paradigm.

Figure 11: (a): SOFIA/HAWC+ observations of M17. The color background shows
the total continuum intensity (Stokes-I). The black vectors represent the
magnetic field morphology. We devided M17 into two sub-regions: M17-N and
M17-S. In the M17-S region, IR sources are denoted by the white crosses and
numbered. Star cluster NGC 6618 is represented by star symbols. (b): The
relation of gas column density vs. dust temperature in the two sub-regions.
(c): The relation of polarization degree vs. dust temperature. Error bars
connected by a line shows the weighted-mean in each bin and illustrates that
the lower density gas has a higher dust temperature. Color lines are the
fitting to all data point (the grey line), to only M17-S (the red line), and
to only M17-N (the orange line). The polarization increases with increasing
dust temperature in M17-S, while it drops to higher $T_{\rm d}$ and lower
$N(\rm H_{2})$ in M17-N. Figure adopted from Hoang et al. (2022b).

Figure 11(a) shows the SOFIA/HAWC+ observations of M17. Generally, the stellar
cluster NGC 6618 (the most massive stars in this cluster are indicated by the
white star symbols) are responsible for ionizing and creating the Hii region.
We divided M17 into two sub-regions, namely M17-N and M17-S. M17-S contains
several compact sources as marked by white crosses. Figure 11(b,c)
respectively show the relation between gas column density and dust
temperature, and the relation of polarization degree with dust temperature.
Lower density gas is associated with higher dust temperature.

  * •

In M17-S, where the density is relatively high, and the dust temperature is
relatively low, the observed polarization degree increases with increasing
dust temperature. In this condition of high gas density and low temperatures,
gas damping is more efficient so that RAT-D cannot occur. Therefore, the
increase of $p-Td$ is consistent with the prediction of the RAT-A effect.

  * •

In M17-N, where the gas density is rather low and the dust dust temperature is
relatively high ($T_{\rm d}\geq 60\,\rm K$), the observed polarization tends
to decrease with $T_{d}$. In this condition of lower densities and high
temperatures, RAT-D can easily occur due to weak gas damping and strong RATs.
As a result, the decrease in $p$ with increasing $T_{\rm d}$ is consistent
with the joint effect of RAT-A and RAT-D.

###  6.4 OMC-1 or Orion BN/KL

At a distance of $\simeq 400\,$pc (Menten et al. 2007; Kounkel et al. 2017),
the Orion nebula is the closest and the most extensive studied region of
massive star-forming region. Due to its proximity, the arcsecond resolution
observations of polarized thermal dust that have been carried out sample
scales of a few thousand AU (Pattle et al. 2017; Chuss et al. 2019;Ajeddig et
al. 2022). These observations have provided a wealth of information of dust
physics in the different physical environments of the Orion region. Here we
focus on only OMC-1 region. This region can be divided into: (1) the Becklin-
Neugebauer-Kleinmann-Low Nebula (BN/KL; Becklin and Neugebauer 1967; Kleinmann
and Low 1967) which hosts on-going star formation; (2) the Southern clump
(OMC-S; Batria et al. 1983) that is quiescent. These sub-regions are labelled
in Figure 12(a). We concentrated on the variation of the polarization degree
along the main filament in Orion, in which we selected one more sub-region
(OMC-N) north of BN/KL.

Figure 12: (a): SOFIA/HAWC+ observations of OMC-1 at 214$\,\mu$m. The
background color is polarization degree. The black vectors illustrate the
B-field morphology inferred from thermal dust polarization (Ngoc et al. in
preparation). Three sub-regions, namely BN-KL, North and South are selected
for further analysis. (b): The relation of polarization degree at 214$\,\mu$m
to dust temperature at three sub-regions in OMC-1 (Ngoc et al. in
preparation). In OMC-N and OMC-S, $p$ is seen to monotonically increases with
$T_{\rm d}$. The increment in the first sub-region is faster than the latter
one. It differs from OMC-BN/KL, for which $p$ slightly increases before deeply
decreasing to higher $T_{\rm d}$.

Figure 12(b) shows the variation of the polarization degree with the dust
temperature for these three sub-regions. In two quiescent OMC-N and OMC-S, $p$
increases with increasing $T_{\rm d}$, which is predicted by the RAT-A effect.
Since $T_{\rm d}$ is higher in OMC-S, the grain alignment is more efficient
resulting in more significant increment in this region compared to OMC-N,
which is understandable in the RAT-A framework. However, the $p-T_{\rm d}$
relation in OMC-BN/KL is different from other regions. The polarization degree
slightly increases and then decreases for $T_{\rm d}\geq 70\,\rm K$. At such
high $T_{\rm d}$, the rotational disruption is effective, leading to the
decline of $p$. The RAT paradigm is therefore able to reproduce the $p-T_{\rm
d}$ relations in these three regions of OMC-1.

###  6.5 Can the anti-correlation of $p$ vs. $T_{\rm d}$ arise from B-field
tangling?

We showed that the anti-correlation of $p$ vs. $T_{\rm d}$ is produced by
RAT-D mechanism, assuming uniform magnetic fields. This disregard of the
magnetic field tangling on variation of $p$ could lead to biased conclusions.
To check whether or not the field tangling could cause the decrease of $p$, we
analyze the variation of the polarization angle dispersion function $S$ as a
function of $T_{\rm d}$. The biased dispersion at position $\boldsymbol{r}$ on
the sight-line is given as

$S_{\rm
biased}(\boldsymbol{r})=\sqrt{\frac{1}{N}\sum^{N}_{i=1}[\psi(\boldsymbol{r}+\boldsymbol{\delta}_{i})-\psi(\boldsymbol{r})]^{2}}$
(46)

where $\psi(\boldsymbol{r})$ and $\psi(\boldsymbol{r}+\boldsymbol{\delta}$)
are the polarization angles at position $\boldsymbol{r}$ and
$\boldsymbol{r}+\boldsymbol{\delta}$, respectively. In practice, for a given
position $\boldsymbol{r}$ we select all data points ($N$) within a circular
aperture centered at this position with a diameter of two beam sizes.

Then, the unbiased dispersion function is

$S(\boldsymbol{r})=\sqrt{S^{2}_{\rm
biased}(\boldsymbol{r})-\sigma^{2}_{S}(\boldsymbol{r})}$ (47)

whose an associated error is (see Eq. 8 in Planck Collaboration et al. 2020)

$\displaystyle\sigma_{S}^{2}(\boldsymbol{r})$ $\displaystyle=$
$\displaystyle\frac{1}{N^{2}S^{2}}\sum_{i=1}^{N}\sigma^{2}_{\psi}(\boldsymbol{r}+\boldsymbol{\delta_{i}})[\psi(\boldsymbol{r}+\boldsymbol{\delta}_{i})-\psi(\boldsymbol{r})]^{2}$
(48)
$\displaystyle+\frac{\sigma^{2}_{\psi}(\boldsymbol{r})}{N^{2}S^{2}}\left[\sum_{i=1}^{N}\psi(\boldsymbol{r}+\boldsymbol{\delta}_{i})-\psi(\boldsymbol{r})\right]^{2}$

with $\sigma(\boldsymbol{r})$ and $\sigma(\boldsymbol{r}+\boldsymbol{\delta})$
the errors of polarization angle at the position $\boldsymbol{r}$ and
$\boldsymbol{r}+\boldsymbol{\delta}$.

Figure 13: (a): 2D-histogram of $S$ and $T_{\rm d}$ in 30 Dor at 214$\,\mu$m
(Figure adopted from Tram et al. in prep.). They have a loose relation with a
correlation coefficient of $r=0.06$. Thus, the variation of magnetic field
could not be a main reason for a decrease in $p$ with $T_{\rm d}>43\,\rm K$ as
shown in Figure 10. (b): Scatter plot of $S$ vs. $T_{\rm d}$ for M17 at
154$\,\mu$m (Figure adopted from Hoang et al. 2022b) with a correlation
coefficient of $r=0.01$. In particular, $S$ is weakly correlated with $T_{\rm
d}>90\,\rm K$ (dashed vertical line) in M17-N, and could not explain the drop
of $p$ for such high $T_{\rm d}$ as shown in Figure 11.

Figure 13 shows the variation of the dispersion angle $S$ with $T_{\rm d}$ in
30 Dor (panel (a)) and in M17 (panel (b)). For 30 Dor, $S$ is poorly
correlated with $T_{\rm d}$, indicating that the field tangling could not be
the main explanation for the anti-correlation of $p$ vs. $T_{\rm d}$ as shown
in Figure 10. In the case of M17, $S$ slowly varies for $T_{\rm d}>90\,\rm K$.
Similarly, the variation of the magnetic field ambiguously results in the
$p-T_{\rm d}$ relation in Figure 11). Via these two examples, one can see that
the disruption of large grains in the vicinity of an intense radiation source
is the major responsible for the decrease in $p$, instead of owing to the loss
of alignment efficiency by gas collision and magnetic field tangling.

##  7 Discussion

###  7.1 Dust polarization as a reliable tracer of magnetic fields in star-
forming regions

Magnetic fields are thought to play an important role in the evolution of the
ISM and star formation. However, measurement of interstellar magnetic fields
is rather difficult. One of the leading applications of dust polarization is
tracing magnetic fields. Thus, the question of to what extent dust
polarization can reliably trace the magnetic field is crucially important.

As reviewed in Section 4, grain alignment in the ISM and MCs is dominated by
the RAT mechanism. The (super)paramagnetic material (e.g., silicate grains) is
magnetized through the Barnett effect, which causes the grain to have rapid
Larmor precession around the magnetic field. Moreover, Barnett and inelastic
relaxation effects induce efficient internal alignment of grain shortest axis
with the angular momentum. The external alignment of the grain angular
momentum along the ambient magnetic fields is dominated by $B-$RAT alignment
because the Larmor precession is much faster than the gas randomization as
well as the radiative precession. Thus, grains are aligned with their longest
axes perpendicular to the magnetic field. The polarization vector of thermal
dust emission is perpendicular to, whereas the polarization of starlight by
dust extinction is parallel to the local magnetic field. As a result, dust
polarization is a reliable tracer of magnetic fields in the diffuse ISM to
molecular clouds to star-forming regions.

Dust polarization has been used widely to measure magnetic fields in star-
forming regions when combined with DCF method. This enables us to quantify the
role of magnetic fields in the star formation process (see Pattle and Fissel
2019 for a recent review). Here, we discuss our recent study using dust
polarization to measure magnetic fields in two star-forming regions discussed
in Section 6 with SOFIA/HAWC+.

#### (a) Mean magnetic field strength in M17

Knowing the variation of the magnetic fields (i.e., characterized by the
polarization angle $\theta$) induced by magnetically aligned dust grains, we
can estimate the field strength using the DCF method as

$B_{\rm mean}=f_{\rm DCF}\sqrt{4\pi\rho}\frac{\delta_{v}}{\delta_{\theta}}$
(49)

where $f_{\rm DCF}$ is a factor adjusting for line-of-sight and beam-
integration effects, $\rho$ is the gas mass density, $\delta_{v}$ is the
dispersion in velocity of non-thermal gas, and $\delta_{\theta}$ is the one
for polarization angle. The dispersion of non-thermal gas is derived from
spectral lines of gas tracer, while we followed an modification made by
Hildebrand et al. (2009) and Houde et al. (2009) that one replaces
$\delta_{\theta}\rightarrow D^{1/2}_{\theta}$ with $D_{\theta}$ the structure
function of $\theta$ to estimate the polarization angle dispersion. We
constrained the field strengths are around $980\pm 230\,\mu$G in M17-N and
$1665\pm 885\,\mu$G in M17-S. The mean value of the Alfvénic Mach number
($\mathcal{M}_{A}$) and the mass-to-flux ratio ($\lambda$) are consequently
lower than 1, indicating that M17 is sub-Alfvénic and sub-critical or the
magnetic fields are stronger than the turbulence and able to resist against
gravitational collapse. Our constraints are consistent with the low star-
formation efficiency in M17 (see Hoang et al. 2022b for details computations).

#### (b) Variation of magnetic field strength in 30 Dor

Figure 14: Maps of magnetic strength (panel a), of mass-to-flux ratio (panel
b), of turbulence-to-magnetic pressure (panel c), and of Alfvénic Mach number
(panel d) in 30 Dor (Figure adopted from Tram et al. 2022). Stream lines are
the magnetic field morphology inferred from thermal dust polarization. Star
symbol locates the massive stellar cluster R 136. Black contours enclose the
area where we trust the data. Circles indicate where the gas velocity gradient
parallels to the field lines (red for CII, and white for CO), whereas gas
velocity gradient perpendiculars to the field lines. Magenta crosses are the
YSOs candidates.

The DCF method expressed in Equation 49 can be used to make maps of magnetic
field strength (see Guerra et al. 2021). The maps of $\rho$, $\delta_{v}$ and
$D_{v}$ are naturally required. The map of $\rho$ is due to the width of
cloud, which could be determined by the half-width at half-maximum of the
polarized flux autocorrelation function (see e.g., Figure 1 in Houde et al.
2009). The map of $\delta_{v}$ is done by using a position-position-velocity
datacube, while that of $D_{\theta}$ is acquired within a circular kernel of
radius $w$ (see Section 3.2 in Guerra et al. 2021 for a critical selection of
a best kernel size, $w_{\rm opt}$).

Figure 14 shows the case of 30 Dor as an example. The magnetic field
morphology (stream lines) is inferred from SOFIA/HAWC+ observations (Tram et
al. 2021b, 2022), which illustrates a complex but ordered structure. We
constructed a map of the magnetic strength using the DCF method (panel (a)),
and found a quite strong field with a maximum of few hundred micro-Gauss.
Consequently, most of the material associated with 30 Dor is (1) sub-critical
(i.e., $\lambda<1$, panel (b)), meaning that the magnetic field is stronger
than gravity; and (2) sub-Alfvénic (i.e, $\mathcal{M}_{A}<1$, panel (d)),
meaning that magnetic field is stronger than turbulence, which is illustrated
by the turbulence-to-magnetic pressure (panel (c)). Our estimation shows that
magnetic field is strong enough to support the cloud against the feedback from
the intense ionization source R 136. We compared the field morphology inferred
from thermal dust polarization with that inferred from the velocity gradient
technique (or VGT, see Hu et al. 2019, and references therein), in which the
velocity gradient is perpendicular to the ambient magnetic field. These two
different techniques deliver different results at certain locations (located
by the red and white circles in panel (d)) in 30 Dor. This disagreement
implies that the velocity gradient is parallel to the field lines (or gas
moves along the field) and gives evidence for local gravitational collapse (Hu
et al. 2021). Because gas moves along the field, the gas motion is not
affected by the magnetic pressure (which preferentially acts perpendicular to
the field lines), and thus able to accumulate mass to trigger new stars to
form in such a strong magnetic field. The locations of Young Stellar Objects
are indicated by the magenta crosses for comparison.

###  7.2 Can thermal dust polarization trace magnetic fields in
protostellar/protoplanetary disks?

In very dense regions such as protostellar cores and disks, the process of
grain alignment is far more complicated than in molecular clouds, as studied
in detail in Hoang et al. (2022a). A detailed modelling of grain alignment in
a protoplanetary disk by Tazaki et al. (2017) shows that very large grains
tend to align along the radiation direction (k-RAT) instead of the magnetic
field (B-RAT). Their study, however, evaluated the radiative precession rate
for grains at low$-J$ attractors, which overestimates the efficiency of k-RAT.
They assumed that all grains have right internal alignment. This assumption is
likely too simplistic knowing that grain alignment can be very complicated due
to a complexity of gas density. Therefore, understanding both internal and
external alignments of grains becomes the key to interpret dust polarization
in protostellar environments.

(Sub)millimeter polarimetry oberservations of protostars (e.g., Zhang et al.
2014; Cox et al. 2018; Sadavoy et al. 2019) showed that the pattern of
polarized thermal dust is complicated and diverse. Thus, we need to establish
a robust understanding of grain alignment in dense regions in order to
interpret observations in a reliable way (whether or not the observed patterns
actually trace magnetic fields). Detailed theoretical studies (Hoang et al.
2022a) and numerical modeling (Giang et al. in preparation) indicate that
thermal dust polarization does not always trace magnetic fields under such
conditions. A diversity of polarization patterns can be the result of multiple
alignment mechanisms, and inferring the magnetic field from rotating
polarization vectors might lead to inaccurate conclusions (Hoang et al.
2022a).

###  7.3 Polarization hole within the RAT paradigm

In this section, we review the polarization hole effect that is frequently
reported in starless cores and protostars (e.g.,Alves et al. 2014; Jones et
al. 2015; Hull et al. 2014; Cox et al. 2018; Pattle and Fissel 2019) in the
context of the RAT paradigm. As shown in Section 5, the alignment size $a_{\rm
min,align}$ and maximum size $a_{\rm max}$ are keys in determining the
polarization degree of thermal dust. These sizes are, in turn, determined by
the properties of radiation field and of environment (gas temperature and gas
density). In the following we consider sources with different physical
conditions, including starless cores and protostellar cores.

For simplicity’s sake, we also assume all grains have efficient internal
alignment. Note that grain alignment under such conditions is much more
complicated due to the complex gas density profile and interplay between
different alignment mechanisms as discussed in Section 7.2.

Figure 15: (a): Variation of the polarization fraction (arbitrary unit) with
the visual extinction expected from the RAT paradigm, described by $p\propto
A^{-\xi}_{V}$. The blue lines are for a starless core with two values of the
maximum grain size (solid and dashed lines). The slope changes from $\xi<1$ to
$\xi=1$ when grain alignment is completely lost, highlighted by two dotted
vertical lines. The orange line is for a protostellar core. Similarly, the
shaded region indicates the range of $A_{V}$ where grain alignment is
completely lost. Nevertheless, the internal radiation increases the alignment
of grains near the source, changing the slope from $\xi=1$ to $\xi=0$ in the
absence of RAT-D and to $\xi>0$ in the presence of RAT-D. Uniform magnetic
field geometry is assumed. (b): Variation of the $A_{V,\rm loss}$ above which
grain alignment by RATs is completely lost (i.e., $a_{\rm align}\geq a_{\rm
max}$), as a function of $a_{\rm max}$ for high-mass protostars. Solid lines
show the results for grain alignment induced by central protostars of
different luminosity. Black dashed line shows grain alignment induced by
attenuated ISRF. Shaded region indicates the parameter space for complete
alignment loss. Stellar temperature $T_{\ast}=10^{4}\,\rm K$ and gas density
$\propto r^{-3/2}$ are adopted. Figure adopted from Hoang et al. (2021b).

Figure 15(a) shows the expected polarization degree versus the visual
extinction from the cloud surface of a starless core (blue lines) and a
protostellar core (orange line) within the RAT paradigm.

  * •

For a starless core (a cloud is illuminated by the interstellar radiation
field (ISRF) and has no embedded sources), grain alignment is controlled by
this external field. The grain alignment becomes less efficient deeper inside
the cloud due to the attenuation of the ISRF and increase of gas damping. The
polarization degree gradually decreases with $A_{V}$. Once the minimum aligned
grain size is larger than the maximum grain size ($a_{\rm align}>a_{\rm
max}$), all grains completely lost their alignment, resulting at $p\sim
A^{-1}_{V}$. Thus, the location ($A_{V,\rm loss}$) where grain alignment is
lost depends on $a_{\rm max}$ (vertical lines).

  * •

For a protostellar core, there is an internal radiation source located at
within of the cloud, in addition to the external interstellar radiation field.
The polarization degree similarly gradually decreases as $p\sim A^{-\rm
slope}_{V}$ with a slope $<1$ before the slope becomes $1$ due to the loss of
grain alignment by the ISRF. In the vicinity of the protostar, owing to the
effect of the protostellar radiation, more small grains become aligned by
RATs. The polarization degree is hence expected to increase monotonically (or
at least stay constant when all grains are brought to align) toward the
protostar. If the RAT-D mechanism occurs near the central star, the
polarization degree is predicted to decrease instead of increase (dashed
orange line).

Figure 15(b) shows the region from the cloud surface where grains are
completely lost their alignment ($A_{V,\rm loss}$) in the case of a massive
protostar. It is determined by both the internal and external radiation
fields. For a given maximum grain size, the more luminous source, the narrower
$A_{v,\rm loss}$, because this intense radiation field could align grains
loser to the cloud surface by RATs. For a given luminosity, one can expect to
have a similar narrower in range of $A_{V,\rm loss}$ for larger grain size,
because the RAT alignment efficiency is proportional to grain size as shown in
Section 4.3. For a starless core, as mentioned, $A_{V,\rm loss}$ is only
dependent on the maximum grain size which implies that the observed
polarization hole could help to constrain grain growth in such a condition.

###  7.4 Effects of RAT-A and RAT-D on grain growth and evolution, and
observations

Grain growth is thought to occur in molecular clouds where grain alignment is
ubiquitous as revealed by dust polarization. Therefore, the process of grain
growth must involve the growth of aligned grains. Hoang (2022) suggested that
the alignment of dust grains with ambient magnetic fields has an important
effect on grain growth in molecular clouds. Due to grain alignment, the
process of grain growth via gas accretion and grain-grain collisions is
anisotropic instead of isotropic as induced by Brownian random motions. As a
result, large grains grown by gas accretion are expected to have elongation
increasing with it radius. Moreover, dust aggregates formed by grain-grain
collisions contain binary structures with aligned axes (Hoang 2022), which is
recently reported in primitive interplanetary dust grains (Hu and Winarski,
2021).

On the other hand, the RAT-D induces the change of the grain size
distribution, depending on the local radiation field and gas densities (Hoang,
2021b). This disruption affects both the upper cutoff size and the abundance
of smaller grains (including nanoparticles), and hence imprints on dust
extinction, emission and polarization. The RAT-D effect is also important for
grain surface chemistry, including rapid desorption of ice mantles,
enhancement of thermal desorption of water and COMs. Therefore, the effect of
RAT-A and RAT-D on the grain evolution can be accessible from observations in
a large window from optical-UV to infrared to radio wavelengths (see Hoang
2020 for a review).

###  7.5 Effects of RAT-A and RAT-D in the era of time-domain astrophysics

Dust grains in the local environment of transient sources (as supernovae,
kilonovae, and gamma-ray bursts) can be transiently illuminated by an intense
radiation flash from the source and are transiently aligned by RAT with both
high$-J$ and low$-J$ attractor (fast RAT alignment). The efficiency of fast
RAT alignment and disruption is determined by $f_{\rm high-J}^{\rm fast}$. The
exact value of $f_{\rm high-J}^{\rm fast}$ depends on the grain properties
(shape and size) and magnetic susceptibility. For grains consisting of
ordinary paramagnetic material (e.g., silicate), Herranen et al. (2021) found
that $f_{\rm high-J}$ can be about $10-70\%$ based on calculations of RATs for
an ensemble of Gaussian random shapes.

Therefore, a fraction $f_{\rm high-J}^{\rm fast}$ of grains with size
$a>a_{\rm disr}$ can be disrupted into fragments. However, the fraction
$1-f_{\rm high-J}$ grains aligned at low$-J$ attractors are not disrupted by
RAT-D. Therefore, in the presence of RAT-D, large grains at $a>a_{\rm disr}$
now can only rotate thermally at low$-J$ attractors. This selective disruption
has an important implication for grain alignment and dust polarization. This
results in a significant change in the grain size distribution and dust
properties in the local environment of transients (Giang et al., 2020; Hoang
et al., 2020). Since the RAT-D process is time-dependent, the optical and
polarization properties of transient sources would change with time just
because of the change in foreground dust. This is an important effect that
needs to be accounted for to infer the intrinsic properties of SNe before
their luminosity and colors can be used for cosmological studies of dark
energy (Giang et al., 2020).

We note that the RAT-A and RAT-D effects also work for dust around other
transient sources, including nova, episodic protostars, and AGN, which will be
quantified in our future works.

###  7.6 Mechanical Torque (MET) Alignment

Lazarian and Hoang (2007b) and Hoang et al. (2018) realized that a gas flow
could act to spin-up and align an irregular grain via the so-called Mechanical
Torques (or METs). Instead of the effect of photons, as in the RAT paradigm,
METs arises from the scattering of gas atoms by the grain surface that drifts
through the ambient gas with the speed $v_{\rm d}$. The suprathermal number of
a grain spun-up by METs is given by (Hoang et al. 2022a)

$St_{\rm MET}\simeq
0.86\hat{\rho}^{1/2}s_{d,-1}^{2}a_{-5}^{3/2}\left(\frac{s^{3/2}Q_{\rm
spinup,-3}}{\Gamma_{\|}}\right),$ (50)

where $s_{\rm d}=v_{\rm d}/v_{\rm T}$ with $v_{T}$ the thermal gas velocity
($s_{d,-1}=s_{\rm d}/0.1$), and $Q_{\rm spinup,-3}=Q_{\rm spinup}/10^{-3}$ is
the spin-up efficiency. Here $Q_{\rm spinup}=0$ for a spherical grain, while
$Q_{\rm spinup}=10^{-6}-10^{-3}$ for irregular shapes (Hoang et al., 2018).

Interestingly, one can realize that $St_{\rm MET}$ is independent to the gas
density, and only depends on the grain properties and drift velocity. It
differs from the corresponding value for the RATs, which is less efficient in
dense gas due to the gas damping. Comparing $St_{\rm MET}$ to $St_{\rm RAT}$
from Section 4.3, we see that METs are more important than RATs for only the
very small grains (e.g., $a\leq 0.02\,\mu$m for $U=10^{6}$ and $s_{\rm
d}=0.5$) for low density ($n_{\rm H}=10^{3}\,\rm cm^{-3}$). In higher density
($n_{\rm H}=10^{8}\,\rm cm^{-3}$), The effect of RATs is reduced significantly
so that METs become dominant mechanism to spin-up grains. Therefore, METs play
an important role in dense regions (e.g., protostellar cores or disks) as
shown in Hoang et al. (2022a). Similar to RATs, grains at low$-J$ attractors
induced by METs are likely to precess around the gas flow $\boldsymbol{v}$,
the so-called v-MET. Whereas grains at high$-J$ attractors are likely
experienced in precessing around $\boldsymbol{B}$, the so-called B-MET.
Therefore, the grain alignment and induced polarization due to METs needs to
be considered in modelling or simulation for a better interpretation of
observations, as well as understanding physical properties of dust grain in
such conditions.

## Conflict of Interest Statement

This review uses the results from the both published and unpublished works
with permissions, and the authors declare no conflict of interest.

## Author Contributions

T.H led the theory part and L.N.T led the observational data analysis part of
the manuscript. L.N.T wrote the first draft of the manuscript. T.H. outlined
and revised significantly the manuscript. Both authors contributed to
manuscript writing, revision, read, and approved the manuscript.

## Funding

T.H. is funded by National Research Foundation of Korea (NRF) grant funded by
the Korea government (MSIT) (No. 2019R1A2C1087045)

## Acknowledgments

We are grateful to the referees for helpful comments that improved our
manuscript. We thank Karl Menten and Helmut Wiesemeyer for careful proof
reading and comments, Nguyen Chau Giang and Nguyen Thi Phuong for useful
suggestions, and Nguyen Bich Ngoc for allowing us to use her preliminary
results.

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