# Effects of Grain Magnetic Properties and Grain Growth on Synthetic Dust
Polarization of MHD Simulations in Protostellar Environments

Nguyen Chau Giang,1,2 Thiem Hoang,1,2
1 Korea Astronomy and Space Science Institute, Daejeon 34055, Republic of
Korea
2Department of Astronomy and Space Science, University of Science and
Technology, 217 Gajeong-ro, Yuseong-gu, Daejeon, 34113, Republic of Korea

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Thermal dust polarization is a powerful tool to probe magnetic fields
($\boldsymbol{B}$) and grain properties. However, a systematic study of the
dependence of dust polarization on grain properties in protostellar
environments is not yet available. In this paper, we post-process a non-ideal
MHD simulation of a collapsing protostellar core with our updated POLARIS code
to study in detail the effects of iron inclusions and grain growth on thermal
dust polarization. We found that superparamagnetic (SPM) grains can produce
high polarization degree of $p\sim 10-40\%$ beyond $\sim 500$ au from the
protostar because of their efficient alignment by magnetically enhanced
Radiative Torque mechanism. The magnetic field turbulence in the envelope
causes the decrease in $p$ with increasing emission intensity $I$ as $p\propto
I^{\alpha}$ with the slope $\alpha\sim-0.3$. But within 500 au, SPM grains
tend to have inefficient internal alignment (IA) and be aligned with
$\boldsymbol{B}$ by RATs only, producing lower $p\sim 1\%$ and a steeper slope
of $\alpha\sim-0.6$. For paramagnetic (PM) grains, the alignment loss of
grains above $1\,{\mu\rm{m}}$ in the inner $\sim 200$ au produces $p<<1\%$ and
the polarization hole with $\alpha\sim-0.9$. Grain growth can increase $p$ in
the envelope for SPM grains, but cause stronger depolarization for SPM grains
in the inner $\sim 500$ au and for PM grains in the entire protostellar core.
Finally, we found the increase of polarization angle dispersion function $S$
with iron inclusions and grain growth, implying the dependence of B-field
strength measured using the DCF technique on grain alignment and grain
properties.

###### keywords:

stars: formation, magnetic fields, low-mass stars, dust extinction,
polarization

††pubyear: 2023††pagerange: Effects of Grain Magnetic Properties and Grain
Growth on Synthetic Dust Polarization of MHD Simulations in Protostellar
Environments–LABEL:lastpage

##  1 Introduction

Magnetic fields ($\boldsymbol{B}$) are thought to play an essential role in
the collapse of dense cores and the formation of protostar, protostellar disks
and planetary systems (Shu et al. 1987, McKee & Ostriker 2007, Allen et al.
2003, Tsukamoto et al. 2022). The dominant effect of magnetic fields over
gravity produces strong magnetic pressure, which prevents the collapse of the
core in the early stage (Nakano & Nakamura 1978, Krumholz & Federrath 2019),
but helps to maintain the cloud structure against the destruction from
radiation and mechanical feedback from protostars in later period (Krumholz et
al. 2014, Pabst et al. 2019). The dominant magnetic energy over turbulence
energy also helps to guide the infall gas motion inside the self-collapsing
cloud, which shapes the structure of molecular clouds and filaments
(Hennebelle & Ciardi 2009, Pattle et al. 2018), protostellar cores and disks,
and controls the coherence of magnetic field morphology across different
scales in star-forming regions (Fiedler & Mouschovias 1993, Galli & Shu 993a,
Galli & Shu 993b, Allen et al. 2003, Kataoka et al. 2012, (Seifried et al.
2015)).

During the Class 0/I stage of Young Stellar Objects (YSOs), magnetic fields
are predicted to help transport the disk angular momentum outward via the
magnetic braking effect (Mestel & Spitzer Jr 1956, Mouschovias & Paleologou
1979a, Mouschovias & Paleologou 1979b, Basu & Mouschovias 1994, Allen et al.
2003, Mellon & Li 2008a). Basically, the coupling between the fast-rotating
disk and the poloidal component of magnetic fields produces the magneto-
centrifugal force that transports the matter from the disk away along the disk
rotation axis, forming a protostellar outflow (Pudritz & Norman 1983, Bally
2016). The removal of matter and angular momentum via the outflow saves the
life of protostellar disks from being fragmented by the strong centrifugal
force and allows the remaining matter to continue feeding the growth of the
central protostar (Galli et al. 2006, Li et al. 2014). However, if the
magnetic field is strong and aligned with the disk rotation axis, the magnetic
braking can become too efficient in removing the disk angular momentum. As a
result, the remaining matter in the disk will be quickly accreted onto the
protostar that stops the protostellar disk formation, named as the magnetic
braking catastrophe (Allen et al. 2003, Galli et al. 2006, Mellon & Li 2008b,
Hennebelle & Fromang 2008, Machida et al. 2011). One of the possible scenarios
to solve the above problem is to introduce the initial misalignment between
the rotational axis and the magnetic field orientation (Hennebelle & Ciardi
2009, Joos et al. 2012, Krumholz et al. 2013). Besides, the effect of non-
ideal magnetohydrodynamic (MHD), i.e., ambipolar diffusion (Duffin & Pudritz
2009, Mellon & Li 2009), Ohmic dissipation (Dapp et al. 2012, Machida et al.
2014, Tomida et al. 2015, Lam et al. 2019), the Hall effect (Tsukamoto et al.
2015, Wurster et al. 2018), turbulence-induced reconnection inside the
Keplerian disk (Santos-Lima et al. 2012, Santos-Lima et al. 2013, Li et al.
2014, Seifried et al. 2015), are also suggested to be able to solve the
magnetic braking catastrophe. However, different scenarios lead to different
outcomes (in terms of disk size, outflow strength, and magnetic field
morphology). Consequently, accurate measurements of both the morphology and
strength of magnetic fields in all spatial scales of protostellar environments
become a key for accurately understanding the role of magnetic fields in
regulating the core collapsing and guiding the star, disk, and outflow
formation.

The most popular technique to measure magnetic fields in protostellar
environments is to use polarized thermal dust emission (see, e.g., Hull &
Zhang 2019, Pattle et al. 2022 for recent reviews). Given the perpendicular
alignment between the grain longest axis and $\boldsymbol{B}$ that produces
polarization vectors $\boldsymbol{P}\perp\boldsymbol{B}$ (Lazarian & Hoang,
2007; Andersson et al., 2015; Lazarian et al., 2015), measurements of dust
polarization from protostellar cores of 0.1 pc down to protostellar disks of
100 au are widely conducted. This is done using both single-dish telescopes
such as the James Clerk Maxwell Telescope (JCMT, (Matthews et al., 2009) and
interferometric arrays such as the Jansky Very Large Array (JVLA),
Submillimeter Array (SMA, (Ho et al., 2004), and the Atacama Large
Millimeter/submillimeter Array (ALMA). Especially, the operation of ALMA,
which can resolve up to $10$ au around the protostar, currently opens a golden
era for using dust polarization to study the role of magnetic fields in star
and planet formation. Many low/intermediate-mass class 0/I Young Stellar
Objects (YSOs) reveal a consistent picture of magnetic fields from the core to
envelope scale (Ching et al., 2017; Hull et al., 2014; Davidson et al., 2014;
Hull et al., 2017b), with well-ordered hourglass-shaped $\boldsymbol{B}$
fields preserved from 1000 au to 100 au (Girart et al., 1999, 2006; Gonçalves
et al., 2008; Rao et al., 2009; Stephens et al., 2013; Maury et al., 2018;
Kwon et al., 2019; Sadavoy et al., 2019). These features match well with
theoretical predictions of the magnetically-regulated collapsing core (i.e.,
(Allen et al., 2003), revealing the important dynamics of $\boldsymbol{B}$ in
the formation of these objects. Some other sources, in contrast, do not
clearly show the correlation of magnetic fields between core, envelope, and
disk scales (i.e., Serpens SMMI1 studied by Hull et al. 2017a), revealing the
dynamic importance of turbulence and gas kinematics over magnetic energy in
guiding core and disk formation (see review by (Hull & Zhang, 2019).

Focusing on observations probed by ALMA, the situation becomes much more
complicated since magnetic fields significantly deviate from the inner
envelope of $\sim 500$ au to the disk scale within $\sim 100$ au (Hull et al.
2014, Cox et al. 2015, Cox et al. 2018, Takahashi et al. 2019, Sadavoy et al.
2019). For example, Class 0 Protobinary System L1448 IRS 2 studied by Kwon et
al. (2019) clearly reveals the change from the hourglass-shaped
$\boldsymbol{B}$-fields in the envelope to the pinched fields along the major
disk axis within $\sim 100$ au. Similarly, magnetic fields from Class 0
OMC-3/MMS 6 studied by Takahashi et al. (2019) show the significant transition
from the spiral pattern within $\sim 800$ au to the radial field at $\sim 200$
au, and to the pinched field in the disk scale within $\sim 100$ au. These
authors suggest that the pinched or circular field inside the protostellar
disk could be induced by the formation of the toroidal $\boldsymbol{B}$-fields
wrapping by the fast rotation disk. But it also could be explained by self-
scattering of thermal emission from sub-millimeter grains (Yang et al. 2016,
Cox et al. 2018, Takahashi et al. 2019, Sadavoy et al. 2018, Sadavoy et al.
2019, Lam et al. 2021), whose polarization signal does not contain any
magnetic field information. Furthermore, observations of dust polarization
toward the inner 100 au region of NGC 1333 IRAS4A (Ko et al. (2020)) and
OMC-3/MMS 6 (Liu 2021) even reveal the $90^{\circ}$ flipping of the
polarization pattern between millimeter and submillimeter wavelengths, which
challenges our understanding about origin of polarization signal obtained by
ALMA. The uncertainty of the polarization origin inside the inner $\sim 100$
au region raises the serious question of whether we could use dust
polarization to study the role of $\boldsymbol{B}$ fields in building the star
and disk formation in this scale.

Besides the complex polarization pattern, interpreting the properties of dust
polarization measured toward $100-1000$ au of Class 0/I YSOs by ALMA poses
challenges. The most prominent feature detected in this class is the reduction
of the polarization degree toward the center, named as the polarization hole,
from a rather high $p>5-40\%$ in the envelope to $p\sim 1\%$ inside the disk
(Hull et al. 2014, Cox et al. 2015, Cox et al. 2018, Maury et al. 2018,
Sadavoy et al. 2018, Kwon et al. 2019, Takahashi et al. 2019, Gouellec et al.
2019, Le Gouellec et al. 2023a). The depolarization effect was first
attributed to the low resolution of single-dish telescopes, as they could not
resolve the complicated magnetic field in the center. However, the
polarization hole remains clearly detectable with the generations of SMA,
JVLA, and ALMA at all observed wavelengths. Thus, the distortion of
$\boldsymbol{B}$ fields by turbulence and the weak alignment efficiency of
dust grains with magnetic fields (Lazarian 2007, Yang 2021) are usually
referenced to explain the low polarization degree in the center. However, the
exact contribution of each effect to depolarization and the reasons for the
poor grain alignment around the protostar remain unclear. Furthermore, as
grains are expected to be inefficiently aligned with $\boldsymbol{B}$ in
protostellar environments due to strong gas randomization, the detection of
high $10-40\%$ polarization in the envelope by ALMA, and the subsequent
significant reduction of $p(\%)$ from such high values to $1\%$ inside the
disk challenge our understanding of grain alignment in these regions.

The key to answering the above questions is to perform synthetic modeling of
polarized dust emission based on a realistic model of grain alignment in Class
0/I YSOs. Grain alignment involves two main stages: 1) internal alignment,
which aligns the grain angular momentum $\boldsymbol{J}$ with the axis of
maximum inertia moment ${\boldsymbol{a}}_{\rm 1}$ (Lazarian 2007), and 2)
external alignment, which results in the alignment between $\boldsymbol{J}$
and $\boldsymbol{B}$ (see Andersson et al. 2015, Lazarian et al. 2015, Hoang
et al. 2022 for a review). For grains containing iron atoms (i.e., silicate
grains), which are paramagnetic material (PM grains), they can be magnetized
via the Barnett effect (Barnett 1915) and have the internal alignment through
Barnett relaxation (Purcell 1979). This mechanism brings grains back to the
minimum rotational energy level with
$\boldsymbol{J}\parallel{\boldsymbol{a}}_{\rm 1}$ by dissipating the grain
rotational energy via the precession of the grain magnetic moment around
${\boldsymbol{a}}_{\rm 1}$ within the grain’s inertia frame. In addition to
Barnett relaxation, nuclear relaxation (Lazarian & Draine 1999, Lazarian &
Hoang 2008), which shares similar physics to the former ones except the origin
of the magnetic moment comes from nuclear spins, also can lead
$\boldsymbol{J}$ to align with ${\boldsymbol{a}}_{\rm 1}$. Inelastic
relaxation (Purcell 1979, Lazarian & Efroimsky 1999, Efroimsky 2000) which
dissipates grain rotational energy via the deformation of spinning inelastic
grains can also induce grain internal alignment. While Barnett relaxation is
found to work effectively for sub-micron and micron-sized grains of $\sim
1\,{\mu\rm{m}}$, nuclear relaxation and inelastic relaxation are efficient for
larger grains, from a few microns to very large grains (VLGs) above
$>10\,{\mu\rm{m}}$ (Lazarian & Hoang 2008, Lazarian & Hoang 2019). Given the
presence of external magnetic fields, grains can couple with $\boldsymbol{B}$
owning to the Larmor precession formed by the interaction between the grain
magnetic moment with the ambient magnetic field. Then, RAdiative Torque (RATs)
(Dolginov & Mitrofanov 1976, Draine & Weingartner 1996 Lazarian & Hoang 2007,
Hoang & Lazarian 2008), or magnetic relaxation (Davis Jr & Greenstein 1951)
plays a role in driving the alignment between $\boldsymbol{J}$ and
$\boldsymbol{B}$. Magnetic relaxation leads grains to align with
$\boldsymbol{B}$ by dissipating the grain rotational energy via the rotating
induced magnetic moment in magnetized environments (Davis Jr & Greenstein
1951). In contrast, RAT aligns grains with $\boldsymbol{B}$ via their
alignment torque component produced from the differential scattering and
absorption between the ambient anisotropic radiation fields and the left- and
right-hand of irregular dust grains (Dolginov & Mitrofanov 1976, Draine 1996,
Lazarian & Hoang 2007, Hoang & Lazarian 2008). However, RAT can lift up grains
to suprathermal rotation (Hoang & Lazarian 2008), which stabilizes their
orientation in space regardless of gas randomization. Consequently, RAT is the
most effective mechanism driving the grain alignment with $\boldsymbol{B}$ in
the diffuse interstellar medium (ISM) and molecular clouds (MCs, Reissl et al.
2017, Seifried et al. 2018, Reissl et al. 2020a). Indeed, RAT only can drive a
part of grains, which is parameterized by $f_{\rm high-J}$, to have perfect
magnetic alignment at their suprathermal rotation (or called high-J
attractors, Hoang & Lazarian 2008). The value of $f_{\rm high-J}$ for PM
grains varies between 0.2 - 0.7 (Lazarian & Hoang 2008, Hoang & Lazarian
2016a, Herranen et al. 2021, Lazarian & Hoang 2021), depending complexly on
grain properties (size, grain shape, and composition) and their orientation
with radiation fields and magnetic fields.

Reissl et al. 2016 first combined three-dimensional (3D) Monte-Carlo Radiative
Transfer with the RAT alignment physics into the POLArized RadIation Simulator
(POLARIS) code, enabling users to simulate polarized dust emission from
aligned dust grains by RATs. Using the POLARIS code, Brauer et al. (2016)
conducted numerical modeling of dust polarization by aligned grains from a Bok
globule. They suggested that the existence of aligned VLGs of $\sim
10-100\,{\mu\rm{m}}$ inside the protostellar disk can cause depolarization at
submillimeter wavelengths, as detected in many Class 0/I YSOs (Hull et al.
2014, Cox et al. 2015, Cox et al. 2018, Maury et al. 2018 Kwon et al. 2019,
Sadavoy et al. 2018). VLGs also can lead to the $90^{\circ}$ flipping of the
polarization pattern as found in the protostellar disk of NGC 1333 IRA4A (Ko
et al. 2020) and OMC3 MMS6 (Liu 2021) due to the change in the polarization
mechanism from dichroic emission at optically thin wavelengths to dichroic
extinction at optically thick wavelengths. Regarding the observed level of
dust polarization in Class 0/I YSOs, Valdivia et al. (2019) and Le Gouellec et
al. (2023b) found that the high polarization of $p\geq 5\%$ observed in the
envelope (Hull et al. 2014, Cox et al. 2015, Sadavoy et al. 2018, Maury et al.
2018) only can be explained by the presence of aligned VLGs above
$10\,{\mu\rm{m}}$ beyond 100 au by RATs. The alignment of VLGs in the envelope
is also crucial to validate the usage of dust polarization in probing
$\boldsymbol{B}$ fields on this scale (Valdivia et al. 2022). However, Le
Gouellec et al. (2020) and Le Gouellec et al. (2023b) recently found the RAT
alignment mechanism alone cannot explain the high grain alignment efficiency
required in Class 0/I YSOs by ALMA. But the perfect alignment model of VLGs
does address this issue. Furthermore, the model proposed by Brauer et al.
(2016) is only effective inside the optically thick disk at submillimeter
wavelengths, which contain a high amount of aligned VLGs (Zielinski et al.
2021). The question of whether VLGs can exist and have perfect alignment with
magnetic fields by RATs in both the protostellar envelope and disk during
Class 0/I stage remains unclear.

Back to the RAT alignment paradigm, Hoang & Lazarian (2016b) found that for
grains with embedded iron clusters (specifically, superparamagnetic (SPM)
grains), the enhanced magnetic susceptibility (Jones & Spitzer 1967) can
increase the magnetic relaxation over gas randomization. Consequently, grains
can quickly align with $\boldsymbol{B}$ with higher $f_{\rm high-J}$ through
the joint action of RATs and superparamagnetic relaxation, named the
Magnetically enhanced RAdiative Torque (MRAT) alignment mechanism.
Additionally, they found that iron inclusions can also strengthen both the
Larmor precession and Barnett relaxation over gas randomization (Hoang &
Lazarian 2008). This enhancement enables more large grains in dense
environments to be aligned with $\boldsymbol{B}$ and achieve efficient
internal alignment.

Recently, Hoang (2022) and Hoang et al. (2022) revisited the grain alignment
theory of both PM and SPM grains in protostellar environments. They showed
that grains larger than $1\,{\mu\rm{m}}$ must be SPM to have the magnetic
alignment inside protostellar disks and envelopes owing to enhanced Larmor
precession caused by iron clusters. Iron inclusions are also necessary to
facilitate fast internal relaxation for micron-sized and VLGs at both high and
low-J attractors through enhanced Barnett relaxation (Lazarian & Hoang 2019).
Furthermore, they pointed out that VLGs containing a high amount of iron
clusters may achieve perfect magnetic alignment in protostellar envelopes by
MRAT mechanism (Le Gouellec et al. 2020, Le Gouellec et al. 2023b).

Given the crucial importance of iron inclusions on grain alignment and
resulting dust polarization, Giang et al. (2023) upgraded the POLARIS code by
incorporating grain magnetic properties and detailed grain alignment physics.
The significant improvement in this updated version enables us to model both
the internal and external alignment of grains consistently based on the level
of embedded iron inclusions, instead of fixing the RAT alignment efficiency as
in previous studies using the original POLARIS version (Brauer et al. 2016,
Valdivia et al. 2019, Le Gouellec et al. 2020, Le Gouellec et al. 2023b). Our
recent synthetic modeling of polarized dust emission for the Bok globule in
Giang et al. (2023) using this updated POLARIS code confirmed the numerical
predictions from Hoang et al. (2022). We found that SPM grains of size
$100\,{\mu\rm{m}}$ with $10^{3}-10^{4}$ iron atoms per cluster can be
perfectly aligned with $\boldsymbol{B}$ in the envelope by MRAT mechanism, as
implied from Le Gouellec et al. 2020 and Le Gouellec et al. 2023b and
predicted in Hoang et al. (2022). These grains also can produce high $p\sim
10-40\%$, which is consistent with observations of many YSOs by ALMA (Cox et
al. 2015, Sadavoy et al. 2018, Maury et al. 2018, Kwon et al. 2019, Takahashi
et al. 2019). . On the other hand, we found that inside the disk, PM grains
are not aligned with $\boldsymbol{B}$, while SPM grains above $1\,{\mu\rm{m}}$
always have slow internal relaxation at both high and low-J attractors.
Furthermore, these SPM grains are only aligned with $\boldsymbol{B}$ by RATs
(in contrast to MRAT alignment in the envelope) regardless of the help of iron
inclusions. Consequently, we propose that the misalignment of PM grains, or
the reduction of the internal and external alignment degree of SPM grains is
the major origin causing depolarization in protostellar cores.

However, our previous study (Giang et al., 2023) used the idealized Bok
globule model with uniform $\boldsymbol{B}$ fields and the Bonnor-Ebert gas
density distribution. These assumptions may overestimate the impact of iron
inclusions, and eliminate the turbulence and geometrical effect of magnetic
fields on polarized dust emission. In addition, we found that VLGs tend to
have inefficient internal and external alignment due to strong gas
randomization. Therefore, in contrast to the positive correlation between
grain growth activities and the observed degree of polarization (i.e., larger
$a_{\rm max}$ induces larger $p$), found in Valdivia et al. (2019) and Le
Gouellec et al. (2023b), we predict grain growth will suppress polarized dust
emission. However, this effect was not addressed in Giang et al. (2023).
Therefore, our following paper aims to comprehensively understand the effects
of grain magnetic properties and grain growth on polarized thermal dust
emission in the realistic magnetohydrodynamic (MHD) simulation of the
protostellar core. In this study, we focus on the protostellar envelope of
$\sim 1000$ au and the protostellar disk of $\sim 100$ au. A study of grain
alignment and polarized dust emission inside the bipolar outflow will be
presented in our upcoming paper (Giang et al. in prep). In addition, since
this paper focuses on the relationship between iron inclusions and polarized
dust emission, we consider only Barnett relaxation as the major internal
alignment mechanism of dust grains. The effects of nuclear relaxation and
inelastic relaxation on the net degree of grain alignment (Lazarian & Hoang
2019, Hoang et al. 2022) will be discussed and studied in detail in future
work. The structure of our paper is organized as follows. We first describe
the MHD datacube in Section 2, our model setups, and how to post-process the
MHD simulation with POLARIS in Section 3. Results for grain alignment is
presented in Section 4. The effects of iron inclusions and maximum grain size
on the polarization map, the variation of $p(\%)$, polarization angle
dispersion function $S$, and the alignment efficiency characterized by
quantity $p\times S$ with intensity are shown in Sections 5, 6 and 7,
respectively. The influence of iron inclusions on dichroic extinction at
submillimeter wavelengths is studied in Section 8. Further discussions and
conclusions of our study are provided in Sections 9 and 10, respectively.

Figure 1: 2D map of the mean gas density obtained on the x$-$z plane (left
panel) and x$-$y plane (right panel). The material in the envelope is
accumulated into the equatorial plane under the contraction from gravity,
forming the protostellar disk of radius $\sim 100$ au around the protostar
with the mean gas density of $n_{\rm H}\sim 10^{8}-5\times
10^{9}\,{\rm{cm}}^{-3}$. The gas density in the envelope is low of $n_{\rm
H}\sim 10^{5}-10^{6}\,{\rm{cm}}^{-3}$.

##  2 MHD Simulation of Collapsing Cores

###  2.1 Simulation setup and gas density distribution

We use a datacube that shows the clearest formation of a protostellar disk
from the series of non-ideal MHD simulations of the self-collapsing low-mass
core from Lam et al. (2019) (Model M1.0A10.0, Table 1). The simulation starts
with a cubic box of 5000 au, containing a spherical core of total gas mass
$M=0.5M_{\odot}$ and a radius of 2000 au. The core’s gas density follows the
Bonor-Ebert density profile, and it is assumed to rotate around the z
direction with an angular velocity of $\Omega=6\times
10^{-14}\,{\rm{s}}^{-1}$. Initial magnetic fields are assumed to be uniform
along the z direction. The core is supercritical with the dimensionless mass-
to-flux ratio of $\lambda\sim 2.6$, and turbulence is transonic with the sonic
Mach number $M_{s}=1$. The ambipolar diffusion coefficient of the simulation
is 10 times higher than the standard values of $Q_{\rm A,0}=95.2\rm
g^{1/2}\,{\rm{cm}}^{-3/2}$, assuming the iron-neutral drag coefficient of
$\gamma=3.5\times 10^{13}\,{\rm{cm}}^{3}\rm g^{-1}\rm s^{-1}$ and the cosmic
ray ionization rate of $10^{-17}\rm s^{-1}$ (Shu 1992). The core is assumed to
collapse isothermally with the gas temperature of $T_{\rm g}=10$ K.

Lam et al. (2019) used the ATHENA code and the uniform cubic grids of
$512^{3}$, which corresponds to the spatial resolution of $\Delta x=20$ au, to
simulate the collapse of the dense core and the formation of the protostellar
disk. After a sink particle forms, they rebin the cubic box of $256^{3}$
around the sink particle to the new size of $512^{3}$ to better capture the
disk formation with a higher resolution of $\Delta x=5$ au.

To perform synthetic observations of dust polarization in both the
protostellar core and disk, we take the same snapshot used in Lam et al.
(2021) when the cloud evolves 0.158 Myrs. In this snapshot, the protostellar
disk with a radius of $\sim 100$ au is clearly formed around the sink particle
whose mass is about $M_{\rm star}=0.22M_{\odot}$. The mean gas density
distribution on the x$-$z and x$-$y plane taken from the snapshot is shown in
Figure 1. One can see the existence of the disk on the equatorial plane (at
$z\sim 0$) when looking along the edge-on direction (left panel) and the clear
accretion of material into the disk scale of 100 au in the anti-clockwise
direction when looking along the face-on direction (right panel). The gas
density inside the disk is about $n_{\rm H}\sim 10^{8}-10^{9}\,{\rm{cm}}^{-3}$
and it decreases outward to $n_{\rm H}\sim 10^{6}-10^{4}\,{\rm{cm}}^{-3}$ at
$500$ au and $\sim 1000$ au, respectively.

###  2.2 Mean magnetic field orientation

Figure 2 shows the mean orientation of magnetic fields integrated along the y-
and z-direction overplotted with the density-weighed magnetic field strength
$\langle B\rangle$ on x$-$z and x$-$y plane, respectively. The values of
$\langle B\rangle$ is given by:

$\displaystyle\langle B\rangle=\frac{\int_{\rm los}Bn_{\rm H}dl}{\int_{\rm
los}n_{\rm H}dl},$ (1)

where $B$ and $n_{\rm H}$ are the magnetic field strength and the gas density
in each cell along the observed direction. We weigh magnetic field strength
(and later other quantities, Section 4) with gas density to better visualize
the amplification of $\boldsymbol{B}$-fields in dense regions due to the
magnetic flux freezing effect. For the mean orientation of magnetic fields on
the plane of the sky (POS), we use the same approach proposed by Valdivia et
al. (2022) to emphasize the magnetic field structure in dense regions and to
later compare with the inferred magnetic fields from dust polarization in
Section 9.2.2. We first compute the density-weighted sinus and cosines of
$2\phi_{\rm B}$, $\langle\sin(2\phi_{\rm B})\rangle$ and
$\langle\cos(2\phi_{\rm B})\rangle$, along the y- and z-direction by using
Equation (1), where $\phi_{\rm B}$ is the angle of projected magnetic field on
the POS to the North direction in each cell. The mean angle of
$\boldsymbol{B}$ to the North direction from MHD simulation $\langle\phi_{\rm
B}\rangle$ is then given by:

$\displaystyle\langle\phi_{\rm
B}\rangle=\frac{1}{2}\arctan\frac{\langle\sin(2\phi_{\rm
B})\rangle}{\langle\cos(2\phi_{\rm B})\rangle}.$ (2)

Due to the magnetic flux freezing, the initial uniform magnetic fields along
the z-direction are dragged by infall gas toward the center, forming the
hourglass-shaped magnetic fields seen in the x$-$z plane (left panel) and the
spiral pattern seen in the x$-$y plane (right panel). The magnetic field along
the outflow (left panel) is slanted toward the right, and the large-scale
hourglass shape magnetic field is bent to be pinched along the disk within
$\sim 200$ au due to the formation of the toroidal $\boldsymbol{B}$ component
wrapped by the fast rotation of the disk. The interaction between the outflow
and infalling gas develops the small-scale toroidal pattern seen on the x$-$y
plane.

##  3 Synthetic Polarization Observations of MHD simulations with the updated
POLARIS

We perform synthetic multi-wavelength observations of polarized dust emission
from the protostellar core and disk by post-processing the above MHD
simulation with the updated POLARIS code introduced by Giang et al. (2023).
Here, we first introduce our setup for the radiation source and the dust model
in Section 3.1, then we briefly explain our working flow in POLARIS from
Sections 3.2 to 3.4.

Figure 2: The mean orientation of magnetic fields on the x$-$z (left panel)
plane and x$-$y (right panel) (black segments) overplotted over the density-
weighted magnetic field strength (color code) integrated along the y and
z-direction, respectively. In the x$-$z plane, the magnetic field changes from
the hourglass-shaped field along the z direction to the pinch field
perpendicular to the outflow in the center induced by the development of the
toroidal field from the fast rotation of the disk. On the x$-$y plane, the
magnetic field follows the spiral pattern driven by the infall and accretion
motion of material from the envelope into the central protostar. The magnetic
field strength increases continuously from $B\sim 100\mu G$ in the envelope to
$B\sim 5\times 10^{3}\mu G$ in the disk due to the magnetic flux freezing.

###  3.1 Radiation source and dust model

We consider the stellar radiation from the sink particle to be the unique
heating source in our modeling. The contribution from the interstellar
radiation field is neglected due to their inefficient effect on heating dust
grains at $\sim 2500$ au scale of the protostellar core. For the stellar
radiation source, we assume the sink particle is a black body with the radius
of $R_{\rm star}=2.5R_{\odot}$ and temperature of $T_{\rm star}=9707$ K, which
corresponds to the stellar luminosity $L_{\rm star}=50L_{\odot}$ (or low-mass
protostar). The choice of $L_{\rm star}$ is motivated by the study of Le
Gouellec et al. (2023b) who found that the high $p\sim 20\%$ in the outflow
cavity of Class 0/I low-mass YSOs only can be reproduced if the protostar has
$L_{\rm star}\sim 50-100L_{\odot}$. We consider the wide radiation spectrum
with the lower cutoff at $\lambda_{\rm min}=0.1\,{\mu\rm{m}}$ at which UV
photons are mainly absorbed by Hydrogen atoms, and the upper cutoff at
$\lambda_{\rm max}=2$ mm in which the dust-radiation interactions become
insignificant. We assume the source emits $N_{\rm photon}=10^{7}$ photons per
each wavelength.

For the dust model, we assume the uniform distribution of dust grains in the
entire protostellar core with the typical gas-to-dust mass ratio $\eta=0.01$
found in ISM. Dust grains are assumed to be in the composite form with
$67.5\%$ of silicate and $32.5\%$ of graphite, and follow the standard Mathis
Rumpl Nordsieck (MRN) distribution of $dn/da=n_{\rm H}Ca^{-3.5}$ (Mathis et
al. 1977) with $C$ the normalization constant determined from the value of
$\eta=0.01$ (see Giang et al. 2023 for detail). We assume the composite dust
model because sub-micron grains dense regions such as in protostellar
environments have higher probability to collide and stick together, forming
larger grains (Okuzumi et al. 2012,Kataoka et al. 2013). Under this
assumption, composite grains containing a mixture of silicate with iron
inclusions and carbonaceous material become magnetic and are aligned by the
similar alignment physics as silicate grains. Thus, both silicate and
carbonaceous material within composite dust grains will produce polarized dust
emission. We consider the wide grain size distribution from the minimum size
of $a_{\rm min}=3.5$ Å to the maximum size varying from $a_{\rm
max}=1\,{\mu\rm{m}}$ to $a_{\rm max}=100\,{\mu\rm{m}}$. The choice of VLGs
within the $2500$ au scale stems from the detection of large grains via the
SED fitting (see, e.g., Kwon et al. 2009; Miotello et al. 2014; Galametz et
al. 2019; Liu 2021) as well as the synthetic observations of dust polarization
(see, e.g., Valdivia et al. 2019, Le Gouellec et al. 2023b).

###  3.2 Monte-Carlo Radiative Transfer and Dust temperature calculation

The first step of our post-processing is to perform the radiative transfer of
photons emitting from the sink particle using the Monte-Carlo technique
introduced by Lucy (1999). POLARIS simulates the scattering, absorption, and
dust emission processes for all dust grains in the MHD simulation box. For
each absorption event, they manage the energy conservation between the stellar
absorption and thermal dust emission by immediately correcting the grain
temperature and sending one lower energy photon into the grid space. The
energy density distribution inside each cell is updated continuously when
photons enter, interact with dust grains, and leave the cell until the end of
the simulation. POLARIS stores the direction of each photon inside the cell to
calculate the anisotropic degree $\gamma_{\rm rad}$ and the radiative torques
required to model the grain alignment by RATs in the latter step.

After the radiative transfer simulation, we calculate the absorption rate for
each grain size using the energy density distribution stored in each cell. The
grain temperature is determined by solving the energy conservation between
radiative heating and cooling of dust grains. The average dust temperature
$T_{\rm d}$ inside the cell is calculated by integrating the grain temperature
over the grain size distribution, and the gas temperature is given by
multiplying $T_{\rm d}$ by a correlation factor. We consider $T_{\rm g}=T_{\rm
d}$ in our simulation because gas-grain collision is the main heating source
of gas in protostellar environments.

###  3.3 Grain alignment physics in the updated POLARIS

Taking into account the complexity of grain alignment physics in protostellar
environments (Hoang 2022, Hoang et al. 2022), we use the updated version of
POLARIS introduced by Giang et al. (2023) to model in detail the alignment of
grains with $\boldsymbol{B}$ in the MHD model of the protostellar core and
disk. To understand the role of grain magnetic properties, we consider three
types of grains: paramagnetic (PM) grains with the iron fraction $f_{\rm
p}=0.1$, superparamagnetic (SPM) grains with a moderate value of $N_{\rm
cl}=100$ iron atoms/cluster, and SPM grains with the high value of $N_{\rm
cl}=10^{4}$ iron atoms/cluster. We assume the same volume filling factor of
iron clusters $\phi_{\rm sp}=0.1$ for two types of SPM grains corresponding to
$\sim 30\%$ of iron abundance locked inside dust grains in the form of iron
clusters.

Following the RAT theory (Lazarian & Hoang 2007, Hoang & Lazarian 2008), we
first calculate the radiative torques acting on each grain size based on the
radiation field obtained from the Monte-Carlo radiative transfer simulation
(Section 3.2, Reissl et al. 2016, Giang et al. 2023). Then we calculate the
maximum rotational rate that grains gained by RATs (Hoang & Lazarian 2014) and
select the grain size which rotates at least three times faster than their
thermal rotation as the minimum alignment size $a_{\rm align}$ (Hoang &
Lazarian 2008). The maximum alignment size $a_{\rm max,JB}^{\rm Lar}$ is
determined based on the competition between the Larmor precession and the gas
randomization. In detail, grains rotating suprathermally by RATs only be
considered to receive $\boldsymbol{B}$ as their alignment direction if their
Larmor precession timescale is at least ten times shorter than the gas damping
timescale (Yang 2021, Giang et al. 2023). For grains within the alignment
range, their internal alignment is considered to have fast internal relaxation
if their Barnett relaxation timescale is shorter than the gas damping
timescale, constrained by the range $a_{\rm min,aJ}-a_{\rm max,aJ}$ (Hoang
2022, Hoang et al. 2022). Grains beyond this limit will have inefficient IA by
slow internal relaxation. Taking into account the dependence of the Barnett
relaxation with the grain rotational rate, we determine in detail the internal
alignment of each grain size in their alignment state at both high-J and low-J
attractors (see Giang et al. 2023 for detailed calculations).

Figure 3: Illustration of the variation of $f_{\rm high-J}$ with grain sizes
within the alignment range $[a_{\rm align}-a_{\rm max,JB}^{\rm Lar}]$. The
green area determines the size of grains being aligned with $\boldsymbol{B}$
by the MRAT mechanism. We consider $f_{\rm high-J}=1$ for grains having high
magnetic relaxation ratio $\delta_{\rm m}\geq 10$, and $f_{\rm high-J}=0.5$
for grains having $1\geq\delta_{\rm m}\leq 10$. The red area determines the
size range for RAT alignment with $f_{\rm high-J}=0.25$. The maximum size for
grains having $f_{\rm high-J}=1$ and $f_{\rm high-J}=0.5$ by MRAT alignment is
denoted by $a_{\rm max,JB}^{\rm DG,1}$ and $a_{\rm max,JB}^{\rm DG,0.5}$,
respectively.

Following the RAT paradigm, grains can have the external alignment with
$\boldsymbol{B}$ by RATs or MRAT alignment depending on their magnetic
properties (Hoang & Lazarian 2016b, Hoang et al. 2022). Giang et al. (2023)
determine the external alignment mechanism for each grain size based on the
magnetic relaxation parameter $\delta_{\rm m}$, which characterizes the
magnetic relaxation efficiency over the gas randomization. As shown in Figure
3, MRAT will be the major alignment mechanism for grains having $\delta_{\rm
m}>1$, and RATs will play the main role in driving the alignment of small
grains having inefficient magnetic relaxation, i.e., $\delta_{\rm m}<1$. For
RAT alignment, we assume about $\sim 25\%$ of aligned dust grains to be
efficiently aligned with $\boldsymbol{B}$ at high-J attractors (or $f_{\rm
high-J}=0.25$). For MRAT alignment, we consider higher $f_{\rm high-J}=0.5$ if
grains have $1<\delta_{\rm m}<10$, and $f_{\rm high-J}=1$ for grains having
very efficient magnetic relaxation with $\delta_{\rm max}\geq 10$.

POLARIS determines the overall alignment degree for each grain size with
magnetic fields (Hoang & Lazarian 2014) by using the Rayleigh reduction factor
$R$ (Greenberg 1968) given by:

$\displaystyle R=f_{\rm high-J}Q_{\rm X}^{\rm high-J}Q_{\rm J}^{\rm
high-J}+(1-f_{\rm high-J})Q_{\rm x}^{\rm low-J}Q_{\rm J}^{\rm low-J},$ (3)

where $Q_{\rm X}$ and $Q_{\rm J}$ characterize the internal and external
alignment degree.

For grains having fast internal relaxation, i.e., grains within $a_{\rm
min,aJ}-a_{\rm max,aJ}$, their internal alignment is perfect if grains
aligning with $\boldsymbol{B}$ at high-J attractors, but will be imperfect for
grains at low-J attractors due to their internal thermal fluctuation (Purcell
1979)). We consider $Q_{\rm X}^{\rm high-J}=1$ for the former case and
describe $Q_{\rm X}^{\rm low-J}$ for the latter case by the local thermal
equilibrium (TE) Boltzmann distribution (Lazarian & Roberge 1997).

For grains having slow internal relaxation, i.e., grains beyond the range
$a_{\rm min,aJ}-a_{\rm max,aJ}$, they still can have right IA if they are
aligned with $\boldsymbol{B}$ at high-J attractors, but grains can have right
IA or wrong IA (i.e., grains rotate around their longest axis and thus the
grain longest axis will be aligned parallel with the magnetic field direction)
if they have the magnetic alignment at low-J attractors (Hoang & Lazarian
2009). We characterize the right IA by using the positive value of $Q_{\rm
X}^{\rm high-J}$ for grains at high-J and positive $Q_{\rm X}^{\rm low-J}$ for
grains at low-J, and characterize the wrong IA by using the negative $Q_{\rm
X}^{\rm low-J}$.

The value of $Q_{\rm X}$ for grains having slow internal relaxation is
controlled by hand in POLARIS due to the difficulty of studying the dynamic of
grains without internal relaxation (Hoang & Lazarian 2009). However, we expect
this value should be close to zero due to their weak internal alignment
degree. We use the similar setup in Giang et al. (2023) with $Q_{\rm X}^{\rm
high-J}=0.15$ for grains at high-J, and $Q_{\rm X}^{\rm low-J}=0.05$ for the
case of right IA and $Q_{\rm X}^{\rm high-J}=-0.1$ for the case of wrong IA at
low-J attractors to study the effect of grains with slow internal relaxation
on the observed polarization signal, named as model rIA and wIA.

For the external alignment between $\boldsymbol{J}$ and $\boldsymbol{B}$, we
consider perfect alignment for grains at both high and low-J attractors, or
$Q_{\rm J}^{\rm high-J}=Q_{\rm J}^{\rm low-J}=1$. The adoption of $Q_{\rm
J}^{\rm high-J}=1$ for high-J state is reasonable due to the grain
suprathermal rotation. In terms of $Q_{\rm J}^{\rm low-J}=1$, this adoption
may overestimate the realistic external alignment efficiency of grains at
low-J. However, since the net polarization signal is dominated by the emission
from grains at high-J, the overestimated external alignment efficiency of
grains at low-J may not significantly affect our final results. Therefore, it
is acceptable to adopt $Q_{\rm J}^{\rm low-J}=1$.

Finally, knowing the internal and external alignment degrees of all grain
sizes within the alignment range, we thus can determine the overall alignment
degree of grains with $\boldsymbol{B}$ using Equation (3). The Rayleigh
reduction factor $R$ will be zero for grains beyond the range of alignment.
The summary of parameters used in our model is given in Table
LABEL:tab:parameter.

Besides two models of grain alignment rIA and wIA described in the above
paragraph, we also consider the ideal case that all grains above $a>a_{\rm
align}$ have perfect alignment with $\boldsymbol{B}$, named as model PA. By
comparing model PA with models rIA and wIA, one thus can separate the effect
of turbulence and magnetic field geometry from grain alignment efficiency,
which helps to clarify the contribution of each factor on the dust
polarization. The summary of our model names and their setups is shown in
Table LABEL:tab:model_table.

###  3.4 Polarized Radiative Transfer of Stokes Parameters

Given the alignment degree and alignment direction of all grain sizes with the
ambient magnetic field, we finally perform the synthetic observation of dust
polarization by solving the polarized radiative transfer of Stokes parameters
(Reissl et al. 2016). We put the plane detector with $256\times 256$ pixels at
100 pc to the source to observe the envelope of $2500$ au and then the disk of
$500$ au scale. Each pixel on the detection will cover the area with a
sidelength of $\Delta x=9.4$ au when observing the full map of 2500au scale
and $\Delta x=1.95$ au when zooming onto the protostellar disk. We perform the
multi-wavelength observations with $\lambda=2$ mm, $870\,{\mu\rm{m}}$,
$450\,{\mu\rm{m}}$, and $250\,{\mu\rm{m}}$ with different inclination angles
from the face-on to the edge-on direction to understand behaviors of dust
polarization at different wavelengths with different light of sight (LOS). The
polarization degree in terms of percentage $p(\%)$ in each pixel is given by:

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

and the polarization angle $\psi$ in the unit of radian is:

$\displaystyle\psi=\frac{1}{2}\arctan\frac{U}{Q}$ (5)

where $I$ is the first Stokes parameter describing the thermal intensity, and
$Q$ and $U$ describe the linear polarization states of dust emission.

We calculate the polarization angle dispersion function $S$ to study the
effect of B-field tangling on the observed polarization map. For each cell at
position $x$, we calculate the difference in $\psi$ between this cell and
their neighborhoods within the sphere of lag $\delta$ given by (Planck
Collaboration et al., 2015):

$\displaystyle S(x,\delta)=\sqrt{\frac{1}{N}\sum_{\rm i=1}^{\rm
N}[\psi(x+\delta_{\rm i})-\psi(x)]^{2}},$ (6)

where $N$ is the total number of cells inside the spherical window. We choose
$\delta=4\Delta_{\rm x}$ with $\Delta_{\rm x}$ the region covered by one pixel
inside the detector plane to understand the effect of B-field tangling in
cases of very high resolution.

Figure 4: Spatial distribution of the density-weighted radiation field strength $U$ (left panel), the mean anisotropic degree $\gamma_{\rm rad}$ (right panel), and the mean dust temperature $T_{\rm d}$ (right panel) on the x$-$y plane. The radiation field strength decreases continuously outward due to the extinction of surrounding dust grains, inducing the decrease of dust temperature from $T_{\rm d}\geq 100$K around the protostar to $T_{\rm d}\sim 20$ K at 2500 au. The radiation field is highly isotropic with $\gamma_{\rm rad}\sim 0.1-0.4$ inside the disk resulting from the strong interaction between dust grains and stellar radiation fields, but it becomes more anisotropic when moving outward, i.e., $\gamma_{\rm rad}\sim 0.9$, due the weak interaction between thermal dust emission at infrared wavelengths with cold dust grains there. Table 1: Setup for the radiation field and dust model in POLARIS Quantity | Symbol | Value
---|---|---
Radiation sources
Stellar radius | $R_{\rm star}$ | $2.5R_{\odot}$
Effective temperature | $T_{\rm star}$ | 9707 K
Stellar luminosity | $L_{\rm star}$ | $50L_{\odot}$
Dust model
Grain axial ratio | $s$ | 0.5
Dust-to-gas mass ratio | $\eta$ | 0.01
Grain size distribution | $\rm dn/da$ | C$a^{-3.5}$
Minimum grain size | $a_{\rm min}$ | 3.5Å
Maximum grain size | $a_{\rm max}$ | $1,5,10,20,50,100\mu m$
Fraction of silicate |  | $67.5\%$
Fraction of graphite |  | $32.5\%$
Grain magnetic properties
Iron fraction | $f_{\rm p}$ | 0.1
Iron atom/cluster | $N_{\rm cl}$ | 100 $\&$ $10^{4}$
Volume filling factor | $\phi_{\rm sp}$ | 0.1
of iron clusters |  |
Internal alignment degree
Grains with fast internal relaxation
High-$J$ attractors | $Q_{\rm X}^{\rm high-J}$ | 1
Low-$J$ attractors | $Q_{\rm X}^{\rm low-J}$ | TE Boltzmann distribution
External alignment degree
High-$J$ attractors | $Q_{\rm J}^{\rm high-J}$ | 1
Low-$J$ attractors | $Q_{\rm J}^{\rm low-J}$ | 1
Table 2: Parameters of the grain alignment model

Model name | Alignment range | Slow internal relaxation | $Q_{\rm X}^{\rm high-J}$ | $Q_{\rm X}^{\rm low-J}$ | $f_{\rm high-J}$
---|---|---|---|---|---
PA | $a_{\rm align}-a_{\rm max}$ | No | – | – | 1
rIA | $a_{\rm align}-a_{\rm max,JB}^{\rm Lar}$ | Yes | 0.15 | 0.05 | Depend on grain magnetic properties
wIA | $a_{\rm align}-a_{\rm max,JB}^{\rm Lar}$ | Yes | 0.15 | -0.1 | Depend on grain magnetic properties

Note: Value of $Q_{\rm X}^{\rm low-J}$ and $Q_{\rm X}^{\rm high-J}$ here are
for the internal alignment degree of grains with slow internal relaxation.

##  4 Numerical Results for Grain Alignment

In this section, we first show the distribution of the radiation field
strength and dust temperature on the x$-$y plane. We then show results about
grain alignment with $a_{\rm max}=100\,{\mu\rm{m}}$ for PM grains and SPM
grains from Section 4.2 to Section 4.4. Results of grain alignment with
different maximum grain sizes from $a_{\rm max}=1\,{\mu\rm{m}}$ to $a_{\rm
max}=100\,{\mu\rm{m}}$ are shown in Appendix A.

###  4.1 Radiation field and dust temperature

Figure 4 shows the spatial distribution of the radiation field strength $U$
(left panel), the anisotropic degree $\gamma_{\rm rad}$ (central panel), and
the dust temperature $T_{\rm d}$ (right panel) on the x$-$y plane, in which
each quantity is weighted with gas density along the z-direction. The
radiation field strength is strongest in the center where the protostar forms
and decreases outward due to the increasing extinction from surrounding dust
grains. The radiation field is highly isotropic, $\gamma_{\rm rad}\leq 0.2$,
within the inner 100 au due to the strong scattering between stellar radiation
and dust grains inside the dense disk. Then it will become highly anisotropic
when moving outward, i.e., $\gamma_{\rm rad}\sim 0.8$, due to weak
interactions of thermal emission from warm grains in the center with $T_{\rm
d}\geq 100$ K with colder grains in the envelope with $T_{\rm d}\sim 40$ K.

Figure 5: The density-weighted minimum alignment size $a_{\rm align}$ on the
x$-$y plane obtained in the full 2500au scale (left panel) and in the inner
500 au region (right panel), assuming $a_{\rm max}=100\,{\mu\rm{m}}$. The
contour line shows the boundary of $a_{\rm align}=0.16\,{\mu\rm{m}}$ in the
envelope, $0.45\,{\mu\rm{m}}$ in the inner part of the envelope, and
$2.7\,{\mu\rm{m}}$ found in the disk. Sub-micron size of $a_{\rm align}\sim
0.2\,{\mu\rm{m}}$ around the protostar can be aligned with $\boldsymbol{B}$
due to the efficient RATs in the strong stellar radiation field. However, this
value increases quickly to $a_{\rm align}\sim 8\,{\mu\rm{m}}$ inside the disk
due to the inefficient RATs induced by the high gas damping, then it decreases
again to $a_{\rm align}\sim 0.7\,{\mu\rm{m}}$ in the envelope because of the
reduced gas density there.

###  4.2 Grain Alignment Size Range

Figure 6: Distribution of the maximum alignment size $a_{\rm max,JB}^{\rm
Lar}$ weighed with gas density on the x$-$y plane for PM grains (left panel)
and SPM grains with $N_{\rm cl}=10^{4}$ (right panel). The maximum alignment
size generally reduces toward the center due to the significant increase in
gas randomization. Large PM grains of $a>10\,{\mu\rm{m}}$ basically cannot be
aligned with $\boldsymbol{B}$ in the entire protostellar core due to the weak
Larmor precession, but this problem can come over if grains are SPM with a
high amount of iron inclusions.

Figure 5 shows the map of the weight-density minimum alignment size $a_{\rm
align}$ on the x$-$y plane obtained in the full 2500 au scale (left panel) and
in the inner 500 au region (right panel). Around the protostar, sub-micron
grains of $a_{\rm align}\sim 0.2\,{\mu\rm{m}}$ (right panel) can be aligned
with $\boldsymbol{B}$ due to the efficient RATs in the strong stellar
radiation field. The alignment size then increases quickly to $a_{\rm
align}\sim 1-8\,{\mu\rm{m}}$ inside the disk (right panel) and decreases again
to $a_{\rm align}\sim 0.1-0.8\,{\mu\rm{m}}$ in the envelope scale (left
panel). The reduced RAT efficiency in the disk is caused by the attenuation of
the radiation field strength, the increase in the field isotropic degree, and
the enhanced gas damping by the high gas density. In the protostellar
envelope, more sub-micron grains can rotate suprathermally and be able to have
the magnetic alignment by RATs due to the increased anisotropic degree of the
radiation field and decreased gas damping.

Figure 6 shows the density-weight maximum alignment size $a_{\rm max,JB}^{\rm
Lar}$ calculated on the x$-$y plane for PM grains (left panel) and SPM grains
with $N_{\rm cl}=10^{4}$ (right panel). For both PM and SPM grains, one can
clearly see that large grains inside the disk are hard to be aligned with
$\boldsymbol{B}$ than the other due to the strong gas randomization on the
Larmor precession. However, grains with higher embedded iron inclusions have
higher possibilities to align with $\boldsymbol{B}$ due to the enhanced Larmor
precession by their larger magnetic susceptibility. For example, PM grains are
not aligned with $\boldsymbol{B}$ in the disk, i.e., $a_{\rm max,JB}<a_{\rm
align}$ (Figure 5), and only grains below $10\,{\mu\rm{m}}$ can have the
magnetic alignment in the envelope. In contrast, all SPM grains with $N_{\rm
cl}=10^{4}$ up to $a_{\rm max}=100\,{\mu\rm{m}}$ beyond the disk can be
aligned with $\boldsymbol{B}$. Inside the inner 100 au, grains up to
$30\,{\mu\rm{m}}$ can have the magnetic alignment, but larger grains have
random orientation because of their slow Larmor precession driven by their
higher inertia moment.

###  4.3 Grain sizes with fast internal relaxation

Figure 7: Spatial distribution of the density-weighed maximum size that grains
have fast internal relaxation at high-J attractors $a_{\rm max,aJ}^{\rm
high-J}$ for PM grains (left panel) and SPM grains with $N_{\rm cl}=10^{4}$
(right panel) on x$-$y plane. Generally, large grains inside the disk tend to
have slow internal relaxation, i.e., smaller $a_{\rm max,aJ}^{\rm high-J}$,
due to the strong gas randomization. The size range of grains having fast
internal relaxation is extended to larger sizes if grains have higher magnetic
susceptibility. However, it is insufficient to cause large SPM grains above
$20\,{\mu\rm{m}}$ inside the disk to have fast internal relaxation.

Figure 8:  Similar results as Figure 8 but for grains aligning with
$\boldsymbol{B}$ at low-J attractors $a_{\rm max,aJ}^{\rm low-J}$. The size
range of grains having fast internal relaxation reduces toward the center
region and can extend to larger sizes with increasing the amount of iron
clusters locked inside dust grains. However, all micron-sized grains above
$1\,{\mu\rm{m}}$ tend to have slow internal relaxation at low-J attractors in
the entire protostellar core regardless of the grain magnetic properties due
to their slow rotation in space.

Figures 8 and 8 show the density-weighted maximum size that grains have fast
internal relaxation for PM grains (left panel) and SPM grains (right panel).
The first figure is for grains at high-J attractors $a_{\rm max,aJ}^{\rm
high-J}$ and the second is for grains at low-J attractors $a_{\rm max,aJ}^{\rm
low-J}$. One can see that in both high and low-J attractors, large grains in
the protostellar disk are hard to have fast internal relaxation compared to
grains in the envelope due to the stronger gas randomization during their
internal alignment stage. The maximum size for fast internal relaxation
increases for grains having higher embedded iron inclusions due to the
enhanced Barnett relaxation by higher magnetic susceptibility. Furthermore,
this size significantly increases if grains are aligned with $\boldsymbol{B}$
at high-J attractors. For example, beyond $\sim 200$ au for SPM grains with
$N_{\rm cl}=10^{4}$, only small grains below $1-3\,{\mu\rm{m}}$ can have fast
internal relaxation at low-J attractors (Figure 8, right panel), while this
value can excess to $100\,{\mu\rm{m}}$ for grains at high-J attractors (Figure
8, right panel). The huge difference in the value of $a_{\rm max,aJ}$ between
high and low-J is caused by the difference in the rotation rate. In
particular, grains rotating suprathermally experience faster Barnett
relaxation and thus have higher possibilities of having fast internal
relaxation in dense environments. That also explains why we see the slight
increase of $a_{\rm max,aJ}^{\rm high-J}$ toward the strong stellar radiation
source in the case of high-J attractors (Figure 8).

Figure 9: Spatial distribution of the maximum size that $50\%$ of grains can
be aligned with $\boldsymbol{B}$ by MRAT alignment $a_{\rm max,JB}^{\rm
DG,0.5}$ for PM grains (left panel) and SPM grains with $N_{\rm cl}=10^{4}$
(right panel). Visually, PM grains are aligned with $\boldsymbol{B}$ by RATs
due to the weak magnetic relaxation, i.e., $a_{\rm max,JB}^{\rm DG,0.5}\sim
0.005\,{\mu\rm{m}}$, but SPM grains can be aligned via the MRAT alignment due
to the enhanced magnetic relaxation by iron inclusions. However, the perfect
alignment of VLGs cannot happen in the disk even if grains contain a high
amount of iron clusters due to the high gas randomization there. Figure 10:
The inferred magnetic field map obtained in the entire protostellar core from
dust polarization at 2mm with the inclination angle of $45^{\circ}$. The color
code shows the polarization degree $p(\%)$ and white segments show magnetic
field orientation obtained by rotating the polarization vector
$\boldsymbol{P}$ by $90^{\circ}$. The first row shows results for model PA,
while the second to fourth rows show results for model rIA of PM grains and
SPM grains with $N_{\rm cl}=100$ and $N_{\rm cl}=10^{4}$, respectively. With
each dust model, we show results with different maximum grain sizes from
$a_{\rm max}=1\,{\mu\rm{m}}$ in the left to $a_{\rm max}=100\,{\mu\rm{m}}$ in
the right. Dust polarization from both model PA and rIA reveals the spiral
magnetic field pattern driven by the gas infalling motion inside the
rotational-collapsing core. However, in terms of polarization degree, model
rIA for PM grains reveals much lower $p(\%)$ compared with model PA and model
rIA for SPM grains due to their weak alignment with $\boldsymbol{B}$. Besides,
for grains with low iron inclusions, their polarization degree will decrease
with increasing $a_{\rm max}$ owing to the increased amount of grains with
inefficient alignment. But for grains are SPM with $N_{\rm cl}=10^{4}$,
$p(\%)$ can increase with $a_{\rm max}$ as model PA due to the extension of
efficient alignment range to larger sizes.

###  4.4 Effective grain sizes of MRAT alignment

Figure 9 shows the density-weighed maximum size that $50\%$ of grains will be
aligned with $\boldsymbol{B}$ at high-J attractors by MRAT alignment, $a_{\rm
max,JB}^{\rm DG,0.5}$. The left panel shows the result obtained for PM grains
and the right panel if for SPM grains with $N_{\rm cl}=10^{4}$. Obviously, RAT
is the major mechanism driving the alignment of PM grains in protostellar
environments, i.e., $a_{\rm max,JB}^{\rm DG,0.5}<<a_{\rm align}$ (Figure 5),
owing to their weak magnetic relaxation strength. On the contrary, all SPM
grains with $N_{\rm cl}=10^{4}$ can be efficiently aligned with
$\boldsymbol{B}$ by MRAT alignment in the envelope as a result of the enhanced
magnetic relaxation by iron inclusions. However, within the inner $\sim 200$
au region, MRAT alignment is only the major alignment mechanism for micron-
sized grains below $\sim 20\,{\mu\rm{m}}$. For larger grains, RATs play a main
role in driving the magnetic alignment due to the reduced magnetic relaxation
driven by the higher inertia moment of VLGs.

##  5 Effects of Iron Inclusions and Grain Growth on Synthetic Polarization
Map

We next move to analyze the effect of iron inclusions and maximum grain size
on the inferred magnetic field map from dust polarization. We place the
detector at the inclination angle $\Theta=45^{\circ}$ (to the North direction)
and observe the object at an optically thin wavelength $\lambda=2$ mm. We
first analyze results from the full-scale observation of 2500 au in Section
5.1, then from the zoom-in scale of 500 au around the protostar in Section
5.2.

Figure 11: Effect of maximum grain size on the dependence of the polarization
degree map and inferred magnetic field orientation from dust polarization
obtained in the inner 500 au region from model PA (upper panels) and rIA for
PM and SPM grains (second to fourth rows). In model PA, for $a_{\rm max}\geq
5\,{\mu\rm{m}}$, dust polarization reveals the complex magnetic field map with
$\boldsymbol{B}$ vectors change from the spiral pattern to be concentric along
the disk minor axis due to the projection effect of $\boldsymbol{B}$-fields on
the POS. In model rIA, only SPM grains with $N_{\rm cl}=10^{4}$ can reveal a
similar magnetic field map as model PA because only they can be aligned with
$\boldsymbol{B}$ in the disk. Grains with lower levels of iron inclusions do
not reveal the magnetic field information within 200 au around the protostar
because they cannot be aligned with $\boldsymbol{B}$ there.  Figure 12: Same
as Figure 11 but for model wIA. Similar to model rIA, only SPM grains with
$N_{\rm cl}=10^{4}$ can reveal the complex orientation of magnetic fields
driven by the gas dynamic in the central 500 au region. However, their
inferred magnetic field within 200 au is pinched along the disk major axis,
which is $90^{\circ}$ difference from one obtained in model rIA and model PA
(Figure 11). The wrong inferred magnetic fields in the disk from model wIA is
caused by the wrong interpretation of polarized dust emission emitting from
wrong aligned dust grains wholes polarization vectors already tell the
magnetic field direction, i.e., $\boldsymbol{P}\|\boldsymbol{B}$.

###  5.1 Protostellar envelope

Figure 10 shows the effects of iron inclusions and maximum grain size on the
polarization degree and the inferred magnetic field direction obtained in the
entire protostellar core. In each column, we fix the maximum grain size and
show the comparison between model PA (upper row) and model rIA for PM grains
(second row), SPM grains with $N_{\rm cl}=100$ (third row), and SPM grains
with $N_{\rm cl}=10^{4}$ (fourth column). For each dust model, we show results
for different maximum grain sizes of $a_{\rm max}=1\,{\mu\rm{m}}$,
$5\,{\mu\rm{m}}$, $10\,{\mu\rm{m}}$, $50\,{\mu\rm{m}}$, and $a_{\rm
max}=100\,{\mu\rm{m}}$, from left to right, respectively. The color code shows
the polarization degree in the unit of percentage $p(\%)$, and white segments
show the inferred magnetic field direction obtained by rotating the
polarization vector $\boldsymbol{P}$ by $90^{\circ}$.

####  5.1.1 Effects of grain magnetic properties

For $a_{\rm max}=1\,{\mu\rm{m}}$ (first column), dust polarization from model
PA reveals the spiral magnetic field pattern driven by the gas infall inside
the rotating collapsing core. The map of polarization degree also shows the
spiral structure with $p\sim 5\%$ inside the arm along the South-West and
North-East direction and $p\sim 10-15\%$ in the remaining parts. The complex
map of polarization degree in the protostellar envelope is a result of the
projection effect of hourglass-shaped magnetic fields on the POS with the
inclination angle $\Theta=45^{\circ}$.

In model rIA (second to fourth panels, first column), polarized dust emission
from both PM and SPM grains generally can infer again the spiral
$\boldsymbol{B}$-fields pattern because all grains up to $1\,{\mu\rm{m}}$ can
be aligned with $\boldsymbol{B}$ in the envelope scale (Figure 25, upper left
panel). However, PM grains produce the uniform low $p\sim 1-5\%$ in the entire
envelope because most of them have inefficient magnetic alignment, i.e.,
aligned dust grains above $0.5\,{\mu\rm{m}}$ have inefficient IA by slow
internal relaxation and only $25\%$ of them will be aligned with
$\boldsymbol{B}$ at high-J attractors by RATs (Figures 27, 27, 28, upper left
panel). In contrast, SPM grains can produce higher $p\geq 10\%$ and reflect
well the spiral pattern in the polarization degree map as model PA owing to
their efficient magnetic alignment by MRAT alignment (Figures 27, 27, and 28,
center and lower left panels).

####  5.1.2 Effects of maximum grain size

In model PA (first row), one can see the clear increase of the polarization
fraction from $p\sim 10-15\%$ to $p\sim 15-40\%$ when the maximum grain size
increases from $a_{\rm max}=1\,{\mu\rm{m}}$ to $a_{\rm max}=100\,{\mu\rm{m}}$.
The positive correlation between $p$ and $a_{\rm max}$ for model PA is owed to
the extension of the alignment range to larger sizes. However, when we take
into account the realistic alignment of PM grains (second row), the rise of
$p$ with $a_{\rm max}$ only happens when grains grow from $a_{\rm
max}=1\,{\mu\rm{m}}$ to $a_{\rm max}=10\,{\mu\rm{m}}$. The further growth in
grain size, in contrast, weakens the observed polarization fraction due to the
misalignment of VLGs above $10\,{\mu\rm{m}}$ with magnetic fields in the
envelope (Figure 25, upper panels). For SPM grains with low $N_{\rm cl}=100$,
$p$ is higher than the case of PM grains because VLGs can be aligned with
$\boldsymbol{B}$ and achieve efficient IA at high-J attractors (Figures 25 and
27, central row), but the dependence of $p$ on $a_{\rm max}$ is still the same
as the case of PM grains because VLGs cannot achieve efficient external
alignment by MRAT alignment as smaller grains (Figures 28 and 29, central
row). For SPM grains with high $N_{\rm cl}=10^{4}$, one can obtain the
continuous increase of $p$ with $a_{\rm max}$ as model PA because all grains
above $a_{\rm align}$ can achieve perfect magnetic alignment by efficient MRAT
alignment there (Figure 27, 28, 29, lower panels).

The effect of iron inclusions and maximum grain size on the inferred magnetic
fields within thousands au scale from model wIA are similar to model rIA (see
the map in Appendix B). It is because polarized dust emission from wrong-
aligned dust grains in the envelope is subdominant to polarized dust emission
from grains with the right IA there.

Figure 13: Left panel: variation of the mean polarization degree $p(\%)$ with
normalized intensity $I/I_{\rm max}$ at 2mm for model PA (black line) and
model rIA with different magnetic properties of grains, assuming $a_{\rm
max}=10\,{\mu\rm{m}}$ and $\Theta=45^{\circ}$. Right panel: similar results as
the left panel but for results from model wIA. The rough position of $1000$,
$400$, $200$, and $100$ au to the protostar is marked on the upper x-axis. In
general, the polarization degree obtained from all models tends to decrease
with increasing intensity toward the central region. However, for model PA,
the depolarization effect is only caused by the narrower of alignment range
toward the disk (Figure 5). For model rIA and wIA, the reduction of $p$ with
$I/I_{\rm max}$ is faster than model PA due to the reduced grain alignment
efficiency in the central region. The value of $p(\%)$ is smaller and the
depolarization will appear more prominent with decreasing amount of iron
locked inside dust grains. In addition, model wIA produces slightly lower
$p(\%)$ compared with model rIA due to the additional suppression of polarized
dust emission arising from the coexistence of grains with the right and wrong
IA.

Figure 14: Effect of maximum grain size on the variation of $p(2\rm mm)(\%)$
with $I/I_{\rm max}$, assuming $\Theta=45^{\circ}$. The upper left panel shows
results for model PA, the upper right panel is for model rIA with SPM grains
having high $N_{\rm cl}=10^{4}$, the lower left panel is for SPM grains with
low $N_{\rm cl}=100$, and the lower right panel is for PM grains. In model PA,
$p$ increases with increasing $a_{\rm max}$ due to the extension of the
alignment range. However, in model rIA, the grain growth in dense environments
suppresses the observed degree of polarization due to the extension of the
amount of large grains with inefficient magnetic alignment. The decrease of
$p$ with increasing $a_{\rm max}$ appears clearer for grains containing lower
levels of iron inclusions.

###  5.2 Protostellar disk

####  5.2.1 Model rIA

Figure 11 shows similar results as Figure 10, but zooming in the central
region of 500 au, which reveals dust polarization in the protostellar disk. In
model PA with $a_{\rm max}=1\,{\mu\rm{m}}$, the inferred magnetic field from
dust polarization continues to follow the spiral pattern driven by the
accretion of material from the inner part of the envelope onto the disk. The
obtained polarization fraction is quite low of $p<5\%$ because of the narrow
alignment range ($a_{\rm align}\sim 0.95\,{\mu\rm{m}}$, Figure 24, right
panel). When grains grow to above $a_{\rm max}\geq 5\,{\mu\rm{m}}$, $p$
increases due to the extension of the alignment range to large sizes, from
$p\sim 15\%$ for $a_{\rm max}=5\,{\mu\rm{m}}$ to $p\sim 30-35\%$ for $a_{\rm
max}=100\,{\mu\rm{m}}$. The inferred $\boldsymbol{B}$-fields are similar to
the case of $a_{\rm max}=1\,{\mu\rm{m}}$. But moving toward the disk scale,
$\boldsymbol{B}$ vectors change to be pinched along the disk minor axis, which
could be explained by the projection effect of $\boldsymbol{B}$-fields on the
POS. The reason why we do not obtain this feature in the case of $a_{\rm
max}=1\,{\mu\rm{m}}$ is because we do not have enough tracers inside the disk
need to reflect well the change of $\boldsymbol{B}$-fields around the
protostar.

Taking into account the realistic model of grain alignment, one can see that
for PM (second row) and SPM grains with low $N_{\rm cl}=100$ (third row), the
dust polarization from all models of $a_{\rm max}$ only can reflect the spiral
$\boldsymbol{B}$-field pattern beyond $\sim 200$ au. The deviation of
$\boldsymbol{B}$-fields inside the disk is not revealed because of the
alignment loss of micron-sized grains within $\sim 200$ au around the
protostar (see Figure 24, right panel and Figure 25, upper and central rows).
The polarization degree obtained in $\sim 500$ au region is low of $p<3\%$ and
$p$ clearly decreases with increasing maximum grain sizes as a result of
increasing amount of misaligned dust grains (Figure 25) and grains with
inefficient IA by slow internal relaxation (Figures 27, 27, and 28 ).

For SPM grains with high $N_{\rm cl}=10^{4}$, polarized dust emission from all
models of $a_{\rm max}$ clearly reveals the change from the spiral
$\boldsymbol{B}$-fields pattern beyond $\sim 200$ au to the pinched field
along the disk minor axis as found in model PA. It is because micron-sized
grains having high amounts of iron inclusions can have magnetic alignment in
the disk scale by their fast Larmor precession (Figure 25, third row).
However, their polarization degree is lower than model PA, i.e., $p\sim
2.5-20\%$, because dust grains are not able to have perfect alignment with
$\boldsymbol{B}$ inside the disk owning to the high gas randomization here
(see Appendix A, Figures 28 and 29). In addition, the increase in maximum
grain size from $1\,{\mu\rm{m}}$ to $10\,{\mu\rm{m}}$ can help to increase $p$
due to the extension of the alignment range. However, beyond $a_{\rm
max}=10\,{\mu\rm{m}}$, the further growing of grains will suppress polarized
dust emission due to the alignment loss of VLGs and the presence of micron-
sized grains with inefficient internal and external alignment with magnetic
fields.

####  5.2.2 Model wIA

Figure 12 shows the effect of iron inclusions (from top to bottom), and
maximum grain size (from left to right) on the inferred magnetic field
morphology in the inner 500 au region for model wIA. Similar to model rIA,
only SPM grains with high $N_{\rm cl}=10^{4}$ can produce the detected
polarization degree above $1\%$ in this area and are able to trace the change
of magnetic fields inside the protostellar disk as model PA. The grain growth
activities inside the inner 500 au region help to increase $p$ for grains
having high amount of iron inclusions but decrease $p$ for grains with low
magnetic susceptibility.

However, one can see that for model wIA, the inferred magnetic field from SPM
grains with low $N_{\rm cl}=100$ is more distorted than results found in model
rIA, with some areas showing $\boldsymbol{B}$ vectors being perpendicular to
the large-scale spiral pattern, i.e., the edge of the disk in the West
direction (third row). Furthermore, magnetic fields within 100 au inferred
from dust polarization of SPM grains with high $N_{\rm cl}=10^{4}$ are pinched
along the disk major axis (fourth row), which is $90^{\circ}$ difference from
one obtained from model PA and rIA. The change in the inferred magnetic fields
in the above areas results from the wrong interpretation of the polarization
signal originating from grains with the wrong IA, whose polarization vectors
$\boldsymbol{P}$ already show the magnetic field direction. That explains why
rotating $\boldsymbol{P}$ in these regions by $90^{\circ}$ shows $90^{\circ}$
difference in $\boldsymbol{B}$ compared with results from model PA and model
rIA. The effect of wrong aligned dust grains on dust polarization becomes
prominent when grains just grow to above $5\,{\mu\rm{m}}$ because micron-sized
grains always tend to have slow internal relaxation and be aligned with
$\boldsymbol{B}$ at low-J attractors inside the disk owning to the strong gas
randomization around the protostar.

##  6 Effects of Iron Inclusions and Grain Growth on Synthetic Dust
Polarization degree

We next move to analyze the variation of the mean polarization degree $p(\%)$
with intensity. The first section is on the effect of iron inclusions on the
$p-I/I_{\rm max}$ with $I_{\rm max}$ the maximum intensity of thermal dust
emission, the second section is about the effect of maximum grain size on
$p-I/I_{\rm max}$ , and the last section is the effect of iron inclusions and
grain growth on the slope of $p-I/I_{\rm max}$.

###  6.1 Effects of grain magnetic properties

The left panel of Figure 14 shows the variation of the mean polarization
degree $p(\%)$ (color lines) with $I/I_{\rm max}$ obtained in the entire
protostellar core at $\lambda=2$ mm for model PA (black line) and model rIA,
assuming different grain magnetic properties and $a_{\rm
max}=10\,{\mu\rm{m}}$. The boundary of the inner 100 au, 200 au, 400 au, and
1000 au from the protostar are roughly marked in the upper x-axis to
distinguish the variation of $p$ with $I/I_{\rm max}$ in the envelope (beyond
400au) and in the disk scale. In model PA, the polarization degree beyond 400
au slightly decreases with increasing intensity, from $\sim 30\%$ to $\sim
10\%$, due to the effect of turbulence and the disorganization of
$\boldsymbol{B}$-fields along the LOS. Moving toward the inner region, $p$
continues to reduce to $p\sim 8\%$ at $\sim 100$ au then increases again to
$\sim 20\%$ owing to the narrower of the alignment range in the dense
protostellar disk and the enhanced alignment of small grains by efficient RATs
near the protostar, respectively (Figures 5).

Taking the realistic model of grain alignment (model rIA) into account, one
obtains a similar reduction of $p$ with increasing $I/I_{\rm max}$, but the
slope of $p-I$ is much steeper compared with model PA due to the significant
reduction of the grain alignment efficiency toward the center. The obtained
polarization degree depends on the magnetic susceptibility of dust grains. For
instance, PM grains (red line) show $p\sim 4\%$ in the envelope because they
are only aligned with $\boldsymbol{B}$ by RATs and almost all of them have
inefficient IA by slow internal relaxation. Moving toward the inner region,
their polarization degree decreases significantly to negligible values of
$p<<0.1\%$ owing to the alignment loss around the protostar (Figures 24, 25
and 27, upper left panels). In contrast, SPM grains with low $N_{\rm cl}=100$
can produce higher $p\sim 20\%$ in the envelope because of their enhanced
internal and external alignment by iron inclusions. But they still produce
$p\sim 0.1\%$ in the inner 100 au due to the misalignment of grains above
$1\,{\mu\rm{m}}$ there. For SPM with higher $N_{\rm cl}=10^{4}$, MRAT can
drive all grains of $a\geq a_{\rm align}$ to be perfectly aligned with
$\boldsymbol{B}$ in the envelope, producing the similar high $p\sim 10-40\%$
as model PA. But within 400 au, $p$ quickly decreases from $\sim 10\%$ to
$\sim 1\%$ because of the change of the alignment mechanism from MRAT to RATs
and the presence of VLGs with slow internal relaxation in this area.

The right panel of Figure 14 shows similar results as the left panel but for
model wIA. In general, the polarization degree tends to decline from the
envelope to the disk scale following the significant reduction of grain
alignment efficiency with increasing gas density. $p$ produced from grains
with higher embedded iron inclusions is higher owing to the enhanced magnetic
alignment of dust grains. However, comparing to model rIA, one can see that
reduction of $p(\%)$ versus $I/I_{\rm max}$ in model wIA is slightly stronger,
which is caused by the self-suppression of dust polarization signal radiating
from grains with right and wrong IA in the protostellar core (Figure 12).

Figure 15: Variation of the slope of $p-I$ relation obtained from model rIA at
2mm: $\alpha_{\geq 500}$ (left panel), $\alpha_{\rm 200-500}$ (central panel),
and $\alpha_{\leq 200}$ (right panel) as the function of $a_{\rm max}$ for
different grain magnetic properties, assuming $\Theta=45^{\circ}$. In general,
$\alpha_{\geq 500}$, $\alpha_{\rm 200-500}$, and $\alpha_{\leq 200}$ increases
with increasing amount of iron inclusions inside grains but tend to decreases
with increasing maximum sizes. The reduction of $p-I$ is steeper in the
central region due to the higher significant effect of gas randomization on
both internal and external alignment of dust grains in the innermost region of
the protostellar core.

###  6.2 Effects of grain growth

The upper left panel of Figure 14 shows the effect of maximum grain size on
the variation of $p-I/I_{\rm max}$ for model PA. As the maximum grain size
increases, the overall polarization degree obtained in thousands au scale
around the protostar increases due to the extension of the alignment range
toward larger sizes. Besides, the grain growth eliminates the effect of
alignment range on the values of $p$ obtained in the protostellar disk. In
detail, in a model with $a_{\rm max}=1\,{\mu\rm{m}}$, the increase of
alignment size toward the disk by increasing gas density (Figure 24, right
panel) induces the decrease of $p$ from $\sim 8\%$ at $\sim 400$ au to $p\sim
0.7\%$ at $\sim 100$ au. Then, the decrease of $a_{\rm align}$ toward the
protostar by increasing RATs efficiency (Figure 24) induces the increase of
$p\sim 0.7\%$ at $\sim 100$ au to $p\sim 1\%$ at the peak of dust emission.
However, as grains grow to $\sim 10\,{\mu\rm{m}}$, the above decrease and
increase of $p$ with $I/I_{\rm max}$ becomes less prominent. The obtained
polarization degree then becomes a constant in the entire $\sim 400$ au region
if the maximum size exceeds $\sim 20\,{\mu\rm{m}}$, in which the alignment
range is too large to feel the change of $a_{\rm align}$ with gas density and
radiation field strength.

The upper right panel of Figure 14 shows similar results as the upper left
panel but for the model rIA of SPM grains with high $N_{\rm cl}=10^{4}$.
Beyond $\sim 400$ au, larger $a_{\rm max}$ induces higher $p$ as model PA
because SPM grains with high embedded iron clusters can achieve perfect
magnetic alignment by efficient MRAT alignment. As increasing intensity toward
the center, the polarization degree obtained from all models of $a_{\rm max}$
decreases as a result of the reduced grain alignment efficiency (Figure 14).
The increase of $a_{\rm max}$ from $1\,{\mu\rm{m}}$ to $2\,{\mu\rm{m}}$ helps
to increase $p$ by about twice the order of magnitude due to the extension of
the alignment range. However, the further growing of $a_{\rm max}$ from
$2\,{\mu\rm{m}}$ to $100\,{\mu\rm{m}}$ does not clearly induce the increase of
$p$ as model PA because the alignment range now covers large grains with
inefficient IA by slow internal relaxation (Figure 27 and 27). Besides, as
grains grow to above $50\,{\mu\rm{m}}$, VLGs within $100$ au are not able to
be aligned with $\boldsymbol{B}$ (Figure 25), inducing the slight reduction of
$p$ with $a_{\rm max}$ (blue line).

The lower panels of Figure 14 show results for model rIA with SPM grains with
low $N_{\rm cl}=100$ (left panel) and with PM grains (right panel). For all
models of $a_{\rm max}$, $p$ always becomes smaller toward the protostar as a
result of reducing the grain alignment efficiency, and the depolarization
becomes more prominent for larger $a_{\rm max}$. The suppression of grain
growth on $p$ is stronger for SPM grains with lower $N_{\rm cl}$ because of
the dominance of large grains with slow internal relaxation. While for PM
grains, it is caused by the alignment loss of VLGs around the protostar. For
example, for SPM grains with low $N_{\rm cl}=100$, $p$ at $\sim 200$ au will
decrease $\sim 2$ times, from $p\sim 2\%$ for $a_{\rm max}=1\,{\mu\rm{m}}$ to
$p\sim 1\%$ for $a_{\rm max}=100\,{\mu\rm{m}}$. For PM grains, the reduction
is much more significant with the continuous reduction from $p\sim 0.3\%$ at
$\sim 200$ au for $a_{\rm max}=1\,{\mu\rm{m}}$ to $p\sim 0.01\%$ if grains
grow to $a_{\rm max}=100\,{\mu\rm{m}}$.

Figure 16: Effect of iron inclusions on the variation of the polarization
angle dispersion function $S$ in the unit of degrees with normalized intensity
$I/I_{\rm max}$ for model rIA (left panel) and model wIA (right panel),
assuming $a_{\rm max}=10\,{\mu\rm{m}}$ and $\Theta=45^{\circ}$. The result
from model PA is shown in the black line for comparison. In general, $S$
obtained from model PA, rIA, and wIA increases with increasing intensity due
to the increased turbulence toward the center, then it decreases toward the
protostar due to the formation of the well-ordered toroidal field inside the
fast rotating protostellar disk (Figure 2). But in model rIA and wIA, $S$
obtained from aligned dust grains with lower levels of embedded inclusions is
smaller than $S$ obtained from grains with higher embedded iron inclusions
owing to the smaller area where they can trace $\boldsymbol{B}$-fields.
Besides, the wrong interpretation of the polarization signal from wrong
aligned dust grains (model wIA) also can induce extra dispersion in the
polarization pattern compared with results in model PA.

Figure 17: Effect of maximum grain size on the variation of $S$ with $I/I_{\rm
max}$ obtained in model PA (upper left panel) and model rIA for SPM grains
with high $N_{\rm cl}=10^{4}$ (upper right panel), SPM grains with low $N_{\rm
cl}=100$ (lower left panel), and PM grains (lower right panel). In model rIA
for SPM grains, $S$ obtained in the envelope increases with increasing $a_{\rm
max}$ due to the increased amount of good tracer of magnetic fields. But in
the inner region, $S$ decreases with increasing $a_{\rm max}$ because large
grains now are not coupled well with $\boldsymbol{B}$fields, which reduces the
information of $\boldsymbol{B}$-fields and turbulence carried inside dust
polarization. The decrease of $S$ with grain growth is clearer for grains
having smaller magnetic susceptibility.

###  6.3 On the slope of $p-I$

The slope of the $p-I$ is an important quantity for characterizing the effects
of grain alignment, grain sizes, and B-fields on the observed polarization
fraction. To connect above factors with the slope of $p-I$, we separate the
$p-I$ relation at $\lambda=2$ mm found in Sections 6.1 and 6.2 into three
segments which characterize the area of $>500$ au (envelope scale), $200-500$
au, and $<100$ au. Each segment is fitted with the power law $p\sim
I^{\alpha}$ which $\alpha$ characterizes the slope of $p-I$ relation. The
fitting is done by using the lmfit function in Python (Newville et al. 2021).
We show the details of our fitting in Appendix F.

Figure 15 shows the variation of $\alpha$ with $N_{\rm cl}$ for different
maximum grain sizes from $a_{\rm max}=1\,{\mu\rm{m}}$ to $a_{\rm
max}=100\,{\mu\rm{m}}$, considering model rIA. The left panel is for the slope
of $p-I$ beyond $\sim 500$ au ($\alpha_{\rm>500}$), the center panel is for
the area within $\sim 200-500$ au ($\alpha_{\rm 200-500}$), and the right
panel is for the disk scale within $200$ au ($\alpha_{\rm<200}$). In general,
the reduction of $p$ versus intensity in the entire protostellar core is
shallower, i.e., lower negative $\alpha$, with increasing $N_{\rm cl}$ due to
the enhanced grain alignment with magnetic fields by iron inclusions.

Beyond 500 au (left panel), for grains with low $N_{\rm cl}<10$,
$\alpha_{\rm>500}$ is in the range of $[-0.4,-0.6]$ and it reduces
continuously from $-0.4$ for $a_{\rm max}=1\,{\mu\rm{m}}$ to $-0.6$ for
$a_{\rm max}=100\mu m$ causing by the misalignment of grains above
$10\,{\mu\rm{m}}$ in the envelope (Figure 25, upper panels). For grains with
$N_{\rm cl}>10$, $\alpha_{\rm>500}$ becomes less negative, i.e.,
$\alpha_{\rm>500}$ belongs to $[-0.25,-0.4]$, because of increasing the
alignment degree of grains with magnetic fields as increasing the magnetic
susceptibility (Figure 25, upper panels). The value of $\alpha_{\geq 500}$
increases with increasing maximum grain sizes, from $\sim-0.4$ for $a_{\rm
max}=1\,{\mu\rm{m}}$ to $\sim-0.25$ for $a_{\rm max}=100\,{\mu\rm{m}}$, owing
to the extension of alignment range to larger sizes. Since SPM grains are able
to have perfect alignment with $\boldsymbol{B}$ by MRAT alignment, turbulence
and the projection effect of $\boldsymbol{B}$-fields are the major factors
controlling the slope of $p-I$ in this envelope region (see the effect of iron
inclusions and the inclination angle in Appendix E).

Moving toward the inner $200-500$ au region (center panel), the value of
$\alpha_{\rm 200-500}$ for grains having low $N_{\rm cl}<10$ decreases to
$-0.8$ for $a_{\rm max}=1\,{\mu\rm{m}}$ and even $-1$ for $a_{\rm
max}=100\,{\mu\rm{m}}$ due to the misalignment of micron-sized grains there.
Grains with higher $N_{\rm cl}\geq 10$ show the similar range of $\alpha_{\rm
200-500}$ within $[0,-0.4]$ as found for $\alpha_{\rm>500}$. However,
$\alpha_{\rm 200-500}$ becomes more negative for larger maximum grain size
owing to the presence of VLGs with inefficient IA by slow internal relaxation
(Figures 27 and 27).

Inside the disk within $\sim 200$ au (right panel), $\alpha_{\rm<200}$ for all
values of $N_{\rm cl}$ and $a_{\rm max}$ decreases significantly to smaller
values due to the reduced grain alignment efficiency in dense environments.
For grains with low $N_{\rm cl}<10$, one gets very low $\alpha\sim-0.8$ up to
$-1$ due to the alignment loss of dust grains with magnetic fields around the
protostar. For grains with higher $N_{\rm cl}\geq 10$, $\alpha_{\leq 200}$
significantly reduces to $\sim[-0.5-0.7]$ and it decreases with increasing
$a_{\rm max}$ owing to the change of the external alignment mechanism from
MRAT to RATs and the increased amount of micron-sized grains with slow
internal relaxation. For example, for SPM grains with $N_{\rm cl}\sim
10-10^{3}$, the slope of their $p-I$ relation changes from $\alpha_{\leq
200}\sim-0.6$ for $a_{\rm max}=1\,{\mu\rm{m}}$ to $\alpha_{\leq 200}\sim-0.8$
if grains grow to $a_{\rm max}=100\,{\mu\rm{m}}$.

##  7 Effect of Iron Inclusions and Grain growth on $S$ and $p\times S$

Next, we move to study the influence of iron inclusions and grain growth on
the variation of the polarization angle dispersion function $S$ and the grain
alignment efficiency $p\times S$ with intensity.

###  7.1 $S-I/I_{\rm max}$ relation

The left panel of Figure 17 shows the variation of $S$ in unit of degrees with
intensity obtained at 2 mm for model PA (black curve) and model rIA, assuming
different grain magnetic properties and $a_{\rm max}=10\,{\mu\rm{m}}$. For
both model PA and model rIA, the angle dispersion function generally increases
from $S\sim 3$ degrees at $\sim 1000$ au to peak in the outer edge of the disk
of $\sim 100$ au then decreases toward the central region. The rise of $S$ in
the envelope is caused by the increased turbulence in the contact areas
between different infalling gas flows and between infalling and outflowing
material. And the decrease of $S$ in the disk could be understood by the
formation of the well-ordered toroidal $\boldsymbol{B}$-fields around the
protostar wrapped by the fast rotating protostellar disk. However, taking into
account the realistic model of grain alignment, $S$ seems not only to depend
on the distortion of magnetic fields by turbulence but also depends on the
grain magnetic properties. Particularly, $S$ for PM grains only slightly
increases from $S\sim 3$ degrees in the envelope to a constant $S\sim 5$
degrees inside the disk, but $S$ for SPM grains with $N_{\rm cl}=10^{4}$
increases significantly from $S\sim 3$ degrees in the envelope to the peak at
$S\sim 25$ degrees in the outer edge of the disk and decreases to $S\sim 10$
degrees near the protostar as model PA. The variation of $S$ with $N_{\rm cl}$
is caused by the difference in the area where dust polarization can trace
magnetic fields. In detail, PM grains only can trace $\boldsymbol{B}$-fields
beyond 500 au (Figure 11, second row), thus, their angle dispersion function
only can reflect the turbulence level inside the envelope, which explains why
$S$ only slightly increases by few degrees toward the center. In contrast,
polarized dust emission from SPM grains with higher $N_{\rm cl}$ carries more
information on magnetic fields and turbulence levels in the innermost region
of the core (Figure 11, third and fourth rows), inducing the clearer
reflection of the variation of $S$ with $I/I_{\rm max}$ as model PA.

The right panel of Figure 17 shows similar results as the left panel but for
model wIA. Generally, the polarization angle dispersion function increases
with increasing intensity and shows higher values for grains containing higher
levels of iron inclusions. However, $S$ obtained from model wIA is slightly
higher than results from model rIA. This additional distortion in the
polarization pattern could be explained by the superposition of polarization
signal from grains with the right and wrong IA along the LOS. In addition, $S$
obtained inside the disk for SPM grains with $N_{\rm cl}=10^{4}$ is even
higher than model PA, i.e., the maximum $S\sim 33$ degrees of SPM grains and
$S\sim 20$ degrees for model PA, that the exceed $S$ arises from the area
where polarization vectors $\boldsymbol{P}$ changes suddenly from
$\boldsymbol{P}\perp\boldsymbol{B}$ (emission by grains with the right IA) to
$\boldsymbol{P}\|\boldsymbol{B}$ (emission by grains with the wrong IA)
(Figure 12, third and fourth rows).

Figure 18: Variation of the grain alignment efficiency determined by the
multiple between the polarization fraction $p$ with angle dispersion function
$S$ $p\times S$ with normalized intensity $I/I_{\rm max}$ obtained from model
PA (black line), model rIA (color lines, left panel) and wIA (right panel) for
different grain magnetic properties, assuming $a_{\rm max}=10\,{\mu\rm{m}}$,
and $\Theta=45^{\circ}$. In contrast to the rise of $p\times S$ with
increasing intensity toward the central region as model PA, model rIA and wIA
clearly show the reduction of $p\times S$ with intensity toward the protostar
as the evidence of the reduced grain alignment efficiency by increasing gas
randomization. The value of $p\times S$ is smaller and the reduction of
$p\times S$ with $I/I_{\rm max}$ is stronger for grains containing lower
levels of iron inclusions.

Figure 19: Effect of maximum grain size on the variation of $p\times S$ with
normalized intensity obtained from model PA (upper left panel) and model rIA
for SPM grains with high $N_{\rm cl}=10^{4}$ (left panel), SPM grains with
$N_{\rm cl}=100$ (lower left panel), and for PM grains (lower right panel).
For SPM grains, $p\times S$ obtained in the envelope increases with increasing
$a_{\rm max}$ as found in model PA due to the extension of amount of grains
with efficient magnetic alignment. However, $p\times S$ obtained in the inner
region decreases with increasing $a_{\rm max}$ because of the enhanced amount
of VLGs which are inefficiently aligned with magnetic fields. The reduction of
the grain alignment degree with grain growth becomes more significant for
grains having lower levels of iron inclusions.

Figure 17 shows the comparison between the effect of grain growth on $S-I$
relation for model PA (upper left panel) and model rIA. The upper right panel
is for SPM grains with high $N_{\rm cl}=10^{4}$, the lower left panel is for
SPM grains with low $N_{\rm cl}=100$, and the lower right panel is for PM
grains. In model PA, $S$ obtained from models with higher $a_{\rm max}$ is
higher because the $\boldsymbol{B}$-fields tangling by turbulence is reflected
better with the increasing amount of good tracer from perfectly aligned dust
grains. For model rIA, increasing $a_{\rm max}$ also induces higher $S$ as
model PA for SPM grains. However, this feature only happens in the envelope
scale beyond $\sim 200$ au where large grains can have efficient magnetic
alignment by MRAT mechanism. Moving toward the inner 200 au region, $S$
decreases with increasing $a_{\rm max}$ due to the reduced area where dust
grains can trace $\boldsymbol{B}$-fields (similar to the case of PM grains,
Figure 17). The reduced information of turbulence contained inside dust
polarization becomes stronger for grains containing lower levels of iron
inclusions (lower panels).

###  7.2 $p(\%)\times S-I/I_{\rm max}$ relation

Figure 19 shows the effect of iron inclusions on the variation of $p\times S$
with $I/I_{\rm max}$ at $\lambda=2$ mm for model PA (black line) and model rIA
(left panel) and model wIA (right panel), assuming $a_{\rm
max}=10\,{\mu\rm{m}}$. $p$ here is the polarization fraction, and $S$ is in
unit of degrees. In model PA, $p\times S$ slightly decreases with $I/I_{\rm
max}$ beyond $\sim 1000$ au then increases continuously toward the center
owing to the significant rise of $S$ with intensity (Figure 17). The reduction
of $p\times S$ in the envelope implies the subdominant effect of turbulence
over the projection effect of magnetic fields on decreasing the observed
polarization degree (Figure 14). And the rise of $p\times S$ in the inner
region could be interpreted as the sign of increasing alignment degree around
the protostar. However, taking the realistic model of grain alignment into
account, one can see that within $\sim 200$ au, $p\times S$ clearly reduces
toward the center for both PM and SPM grains, implying the significant
reduction of the grain alignment efficiency around the protostar. The
reduction of $p\times S$ versus $I/I_{\rm max}$ is more prominent for grains
with lower magnetic susceptibility. The variation of $p\times S$ shares
similar tendencies between model rIA (left panel) and wIA (right panel), but
model wIA reveals a slightly steeper slope of $p\times S$ with $I/I_{\rm max}$
because of lower $p$ induced by the co-existence of grains with the right and
wrong IA (Figures 14, right panel).

Figure 19 shows the difference in the effect of grain growth on the variation
of $p\times S$ with $I/I_{\rm max}$ for model PA (upper left panel) and model
rIA (remaining panels). In model PA, the increased maximum grain size
increases values of $p\times S$ in the entire $\sim 1000$ au due to the rise
of $p$ and $S$ with $a_{\rm max}$ (Figures 14 and 17, upper left panel). At
$<400$ au, one can see that for $a_{\rm max}=1\,{\mu\rm{m}}$, $p\times S$
decreases from $\sim 400$ au to $\sim 100$ au as a result of the narrow
alignment range by high gas density, i.e., larger $a_{\rm align}$, then it
increases again toward the protostar position because small grains now can be
aligned with $\boldsymbol{B}$ by efficient RATs, i.e., smaller $a_{\rm align}$
(Figure 5). As $a_{\rm max}$ increases, this feature disappears due to the
broadening of the alignment range to larger sizes. And starting with $a_{\rm
max}\geq 10\,{\mu\rm{m}}$, $p\times S$ even rises with intensity within the
400 au region, which could be interpreted as the sign of the effective
magnetic alignment of dust grains around the protostar (Figure 19, black
line).

However, in model rIA, $p\times S$ obtained from all models of grain magnetic
properties and maximum grain size, generally, always declines toward the
center due to the reduction of the grain alignment efficiency toward the
protostar. Beyond $\sim 400$ au, $p\times S$ obtained from the model of SPM
grains (upper right and lower left panels) increases with increasing $a_{\rm
max}$ as model PA because they have efficient alignment with $\boldsymbol{B}$
by MRAT mechanism. But for PM grains, $p\times S$ declines with $a_{\rm max}$
because of the inefficient alignment of VLGs by RATs in the envelope scale.
Moving toward the central 400 au, grain growth induces the steeper reduction
of $p\times S$ with intensity, reflecting well the weak alignment degree of
almost aligned dust grains there. The suppression of grain growth on the
quantity $p\times S$ is stronger for grains containing the lower amount of
iron inclusions.

Figure 20: Effect of grain size and grain magnetic properties on the inferred
magnetic field map obtained in the inner 500 au region at $250\,{\mu\rm{m}}$.
The color code shows the polarization degree $p(\%)$, and white segments show
the $\boldsymbol{B}$ vectors obtained by rotating the polarization vector by
$90^{\circ}$. The upper row shows results for model PA, the second to fourth
rows are for model rIA with different grain magnetic properties, the left
column is for $a_{\rm max}=1\,{\mu\rm{m}}$, and the right column is for
$a_{\rm max}=100\,{\mu\rm{m}}$. In model PA, the inferred magnetic field
direction in the inner 200 au changes from parallel to perpendicular to the
disk minor axis when the maximum grain size exceeds $a_{\rm max}\geq
50\,{\mu\rm{m}}$. The change in $\boldsymbol{B}$ vectors results from the
change in the polarization mechanism from dichroic emission (at optically thin
wavelengths at 2mm) to dichroic extinction (at optically thick disk at
$250\,{\mu\rm{m}}$). In model rIA, only SPM grains with high $N_{\rm
cl}=10^{4}$ can reveal the change in the polarization pattern caused by
dichroic extinction at submillimeter wavelengths. PM and SPM grains with
$N_{\rm cl}=100$ do not reveal this feature because VLGs above
$10\,{\mu\rm{m}}$ are not able to be aligned with $\boldsymbol{B}$inside the
disk. Figure 21: Similar results as Figure 20 but for model wIA. Obviously,
only SPM grains with high $N_{\rm cl}$ can reveal in detail the change of
magnetic fields from the inner part of the envelope to the disk scale because
they can have the magnetic alignment around the protostar. However, the
inferred magnetic fields at $250\,{\mu\rm{m}}$ inside the disk are similar to
results from 2mm (Figure 12), with $\boldsymbol{B}$vectors still perpendicular
to the disk minor axis for all values of $a_{\rm max}$ from $1\,{\mu\rm{m}}$
to $100\,{\mu\rm{m}}$. This polarization signal originates from the emission
of grains with the wrong IA, which is not strongly suppressed by the
extinction of VLGs with the wrong IA as in the case of model rIA.  Figure 22:
Comparison about the variation of $p(250\,{\mu\rm{m}})$ with normalized column
density $N_{\rm H}/N_{\rm H,0}$ between model PA and model rIA for $a_{\rm
max}=1\,{\mu\rm{m}}$, $a_{\rm max}=10\,{\mu\rm{m}}$, $a_{\rm
max}=50\,{\mu\rm{m}}$, and $a_{\rm max}=100\,{\mu\rm{m}}$, from left to right,
from top to bottom, respectively. In model PA, for $a_{\rm max}\sim
5\,{\mu\rm{m}}$, $p(250\,{\mu\rm{m}})$ decreases with increasing $N_{\rm H}$
due to the extinction of large grains. As $a_{\rm max}$ increases to
$50-100\,{\mu\rm{m}}$, $p(250\,{\mu\rm{m}})$ decreases then increases again to
the center due to the change of polarization mechanism from dichroic emission
to dichroic extinction (Figure 20, upper row). However, taking into account
the realistic model of grain alignment, $p(250\,{\mu\rm{m}})$ just simply
decreases toward the center, and the reduction of $p(250\,{\mu\rm{m}})$ with
$N_{\rm H}$ is stronger than model PA due to the significant reduction of the
grain alignment in the innermost region of the protostar. The reduction of $p$
with $N_{\rm H}$ in model rIA is stronger for grains containing lower levels
of iron inclusions.

##  8 Effects of Dichroic Extinction Polarization

Lastly, we move to study the properties of dust polarization at optically
thick wavelengths where dichroic extinction becomes important. Indeed, the
disk becomes optically thick at $450\,{\mu\rm{m}}$ (see the optical depth in
the inner 500 au in Appendix C.1), but here we show results obtained at
$250\,{\mu\rm{m}}$ because this wavelength shows clearer imprint of dichroic
extinction on dust polarization. The results for $\lambda=450\,{\mu\rm{m}}$
are in Appendix C.3. We first study the effect of iron inclusions and maximum
grain size on the inferred magnetic fields from dust polarization within the
inner 500 au region in Section 8.1, and then on the variation of $p$ obtained
from Section 8.1 with gas column density $N_{\rm H}$ in section 8.2. We choose
$N_{\rm H}$ to be the parameter here instead of the intensity of dust emission
as in Section 6 because the polarization signal originating from dichroic
extinction is more sensitive with gas density than intensity. Thus, it is
easier to understand how the density structure of the disk affects the
observed polarization fraction.

###  8.1 Polarization map

####  8.1.1 Model rIA

Figure 20 shows the effect of the dust model and maximum grain size on the
magnetic field map inferred from dust polarization at $250\,{\mu\rm{m}}$. The
color code shows the polarization degree, and white segments show the inferred
magnetic field direction by rotating the polarization vector $\boldsymbol{P}$
by $90^{\circ}$, other setups are similar as in Figure 11.

In model PA (first row), from $a_{\rm max}=1\,{\mu\rm{m}}$ to $a_{\rm
max}=10\,{\mu\rm{m}}$, dust polarization observed at $250\,{\mu\rm{m}}$
reveals the similar inferred magnetic field map as detected at optically thin
2 mm (Figure 11, first row), with $\boldsymbol{B}$ vectors follow the spiral
pattern for $a_{\rm max}=1\,{\mu\rm{m}}$, and the change from the spiral field
to the pinched field along the disk minor axis for $a_{\rm max}\sim
5-10\,{\mu\rm{m}}$. When grains grow to above $50\,{\mu\rm{m}}$, one gets the
similar inferred spiral $\boldsymbol{B}$-fields within 200-500 au as cases of
$a_{\rm max}=1-10\,{\mu\rm{m}}$, but $\boldsymbol{B}$ vectors change
$90^{\circ}$ from parallel to be perpendicular to the disk minor axis in the
inner 200 au region. The $90^{\circ}$ flipping of $\boldsymbol{B}$ vectors in
the disk is caused by the change in the polarization mechanism from dichroic
emission to dichroic extinction, which is activated by the presence of aligned
VLGs above $50\,{\mu\rm{m}}$ inside the optically thick disk. In the term of
the polarization degree, $p$ slightly declines from the outer edge of the disk
at 200 au to the trough at about 100 au from the protostar and increases again
toward the center, which corresponds to the transition of the polarization
mechanism from dichroic emission (beyond 100 au) to dichroic extinction
(within 100 au).

For model rIA, PM and SPM grains with low $N_{\rm cl}=100$ (second and third
rows) generally cannot reveal in detail the morphology of magnetic fields
inside the disk as obtained at 2mm due to the alignment loss inside the disk.
But for SPM grains with high $N_{\rm cl}=10^{4}$ (fourth row), one can clearly
obtain the change of $\boldsymbol{B}$-fields within 500 au driven by gas
dynamic and the $90^{\circ}$ flipping of $\boldsymbol{B}$-fields caused by
dichroic extinction as grains grow from $a_{\rm max}=1-10\,{\mu\rm{m}}$ to
$a_{\rm max}=50-100\,{\mu\rm{m}}$ as found in model PA. However, one does not
clearly see the decrease and increase of $p$ with the disk structure as model
PA because of the weak coupling of VLGs with $\boldsymbol{B}$ within 100 au
around the protostar.

####  8.1.2 Model wIA

Figure 21 shows similar results as Figure 20 but for model wIA. Similar to
model rIA, only SPM grains with high $N_{\rm cl}=10^{4}$ (fourth row) can
reveal the change of $\boldsymbol{B}$-fields driven by gas dynamic and grain
growth within 500 au. However, when grains grow to $a_{\rm
max}=50-100\,{\mu\rm{m}}$, surprisingly, the inferred magnetic field inside
the disk does not change from parallel to perpendicular to the disk minor axis
as model PA even though VLGs here are large enough to activate the effect of
dichroic extinction at submillimeter wavelengths. The polarization signal in
this case still comes from the emission of VLGs with the wrong IA (Figure 12,
lower right panel), which could be explained by the weak extinction of VLGs
induced by the co-existence of absorber with right and wrong IA along the LOS.

###  8.2 Polarization degree and gas column density

The upper left panel of Figure 22 shows the comparison of the relation
$p(250\,{\mu\rm{m}})-N_{\rm H}$ obtained in the inner 500 au region between
model PA (black line) and model rIA (colors lines) with different grain
magnetic properties, assuming $a_{\rm max}=1\,{\mu\rm{m}}$. In model PA, the
polarization degree declines continuously from $p\sim 20\%$ at 500 au to
$p\sim 1\%$ in the densest region inside the disk due to the narrower of the
alignment range (similar to results from 2mm, Figure 14, case $a_{\rm
max}=1\,{\mu\rm{m}}$). In model rIA, the polarization degree produced from SPM
grains with high $N_{\rm cl}=10^{4}$ share similar behavior with $N_{\rm H}$
and values as model PA. SPM grains with lower $N_{\rm cl}=100$ and PM grains
also show the reduction of $p$ with increasing $N_{\rm H}$, but with smaller
polarization degree due to the inefficient alignment of sub-micron grains with
$\boldsymbol{B}$ around the protostar.

The upper right panel of Figure 22 shows similar results as the upper left
panel but for $a_{\rm max}=10\,{\mu\rm{m}}$. In the PA model, $p$ obtained in
the inner 500 au region generally be higher than the case of $a_{\rm
max}=1\,{\mu\rm{m}}$ due to the extension of the alignment range. However,
they still show the decline of $p$ with $N_{\rm H}$ owing to the dichroic
extinction of aligned micron-sized inside the disk. In model rIA, one also
obtains the reduction of $p$ with gas column density, but with a much steeper
slope than model PA due to the additional contribution from the low alignment
degree of micron-sized grains with $\boldsymbol{B}$ here. Grains with lower
embedded iron inclusions produce smaller $p(\%)$ and stronger declination of
$p$ with $N_{\rm H}$ due to their weaker alignment degree with magnetic
fields.

The lower left panel of Figure 22 shows similar results as the upper right
panel but for $a_{\rm max}=50\,{\mu\rm{m}}$. In model PA, the polarization
fraction at $250\,{\mu\rm{m}}$ decreases from $p\sim 10\%$ at 500 au to $p\sim
5\%$ when $N_{\rm H}\sim 0.05/N_{\rm H,max}$ ($\sim 150$ au) then increases
again to $p\sim 10\%$ when $N_{\rm H}\sim 0.5N_{\rm H,max}$ ($\sim 100$ au).
The decrease and increase of $p$ with gas column density from 500 to $\sim
100$ au are driven by the increased extinction efficiency of aligned VLGs,
which changes the polarization mechanism from dichroic emission in the outer
edge of the disk to dichroic extinction inside the disk (Figure 20, first
row). Within $\leq 100$ au, $p(250\,{\mu\rm{m}})$ slightly decreases again
from $p\sim 10\%$ to $p\sim 7\%$ with increasing $N_{\rm H}/N_{\rm H,max}$ due
to the trap of polarization signal by high gas-radiation interactions. In
model rIA for SPM grains with $N_{\rm cl}=10^{4}$, one also sees the decrease
and increase of $p(250\,{\mu\rm{m}})$ with increasing $N_{\rm H}$ as model PA.
However, the rise of $p$ with $N_{\rm H}$ within $\sim 150$ au is not
prominent as model PA because, in reality, VLGs are not well coupled with
$\boldsymbol{B}$ enough to clearly activate the effect of dichroic extinction
inside the disk. For grains with lower values of $N_{\rm cl}$,
$p(250\,{\mu\rm{m}})$ just simply decreases continuously with increasing gas
column density due to the misalignment of VLGs within $\sim 200$ au around the
protostar. The tendency of $p-N_{\rm H}$ and the values of $p$ for all
realistic models of grain magnetic properties are the same as grains grow from
$a_{\rm max}=50\,{\mu\rm{m}}$ to $a_{\rm max}=100\,{\mu\rm{m}}$ (lower right
panel).

##  9 Discussion

In this section, we will discuss more details about our findings and how to
connect dust physics with observations via synthetic modeling of polarized
dust emission.

###  9.1 Grain alignment in protostellar envelopes and disks

The alignment of grains with magnetic fields is the key to using polarized
dust emission to trace magnetic fields and study dust physics. However, in
contrast to the well-determined behavior of aligned dust grains in ISM and MCs
(Reissl et al. 2020b), the strong gas randomization in dense protostellar
environments would reduce the magnetic alignment of grains, which hinders the
usage of dust polarization within thousands au around the protostar (Hoang,
2022).

The detailed picture of grain alignment efficiency in protostellar cores and
disks is revealed numerically in studies by Hoang (2022), Hoang et al. (2022)
and in following synthetic modeling of polarized dust emission using the
analytical model of the Bok globule by Giang et al. (2023). The magnetic
alignment in dense protostellar environments is found to be size- and spatial-
dependence, with VLGs of size $\geq 10\,{\mu\rm{m}}$ requiring embedded iron
inclusions for having efficient internal and external alignment around the
protostar. Results obtained in the realistic MHD simulation of the self-
collapsing low-mass core in Section 4.2 are consistent with previous findings.
Particularly, the maximum alignment size of PM grains is limited to $\sim
10\,{\mu\rm{m}}$ in protostellar envelopes and $\sim 1\,{\mu\rm{m}}$ inside
the disk owing to the limitation of slow Larmor precession (Figure 6, left
panel). For grains below $a_{\rm max,JB}^{\rm Lar}$, RAT is the primary
external alignment mechanism of PM grains with typical $f_{\rm high-J}\sim
0.25$ (Figures 9). PM grains at high-J can have efficient IA due to their
suprathermal rotation (Figure 8), but grains at low-J always have slow
internal relaxation regardless of their position within 1000 au around the
protostar (Figures 8).In contrast, considering $30\%$ of iron locked inside
dust grains under the cluster form, all SPM grains with $N_{\rm cl}=10^{4}$
from $a_{\rm align}$ (Figure 5) up to $100\,{\mu\rm{m}}$ will have perfect
magnetic alignment beyond $\sim 200$ au owing to efficient MRAT mechanism
(Figures 6, 8, and 9, right panel). However, within $\sim 100$ au around the
protostar, SPM grains of size $\geq 40\,{\mu\rm{m}}$ cannot be aligned with
$\boldsymbol{B}$, and grains above $\geq 20\,{\mu\rm{m}}$ will be aligned with
$\boldsymbol{B}$ by only RATs due to strong gas randomization effect. The
internal alignment of SPM grains inside the disk is also inefficient, given
the maximum size for fast internal relaxation at high- and low-J attractors of
$\sim 20\,{\mu\rm{m}}$ and $\sim 1\,{\mu\rm{m}}$, respectively (Hoang et al.
2022, Giang et al. 2023). Indeed, Giang et al. (2023) found that within 100 au
around the protostar, VLGs at high-J can have fast internal relaxation owing
to their rapid rotation rate driven by efficient RATs. However, with the very
high gas density up to $n_{\rm H}\sim 10^{9}-10^{11}\,{\rm{cm}}^{-3}$ (Figure
1), gas randomization inside the disk is strong enough to eliminate the
enhanced Barnett relaxation by high rotational rate, forcing grains to have
slow internal relaxation at high-J in our simulated protostellar disk.

The internal alignment efficiency of large grains inside the protostellar
envelope and disk will be improved (Hoang et al. 2022) if considering both
nuclear relaxation (Lazarian & Draine 1999, Lazarian & Hoang 2008) and
inelastic relaxation (Lazarian & Efroimsky 1999, Efroimsky 2000). As shown in
Hoang et al. (2022), the timescale of inelastic relaxation is $\tau_{\rm
iER}\sim a^{11/2}St^{-3}$ (Efroimsky 2000) and Barnett relaxation is
$\tau_{\rm BR}\sim a^{7}St^{-2}$, given $St=J/J_{\rm th}$ the suprathermal
rotation number (Hoang et al. 2022) which characterizes the ratio of the grain
angular momentum over the grain thermal angular momentum. As $\tau_{\rm iER}$
increases with grain sizes slower, decreases with increasing the grain
rotational rate faster, and does not depend on the grain magnetic properties
as Barnett relaxation, inelastic relaxation can help both PM and SPM grains
above $1\,{\mu\rm{m}}$ in size to achieve fast internal relaxation at both
high and low-J attractors. Especially, inelastic relaxation can lead SPM
grains with $N_{\rm cl}\geq 10^{2}$ of size $10-100\,{\mu\rm{m}}$ inside the
disk to have efficient IA at high-J owing to their high rotation rate around
the protostar. Similarly, nuclear relaxation can dissipate rotational energy
faster than Barnett relaxation by a factor of $\sim 10^{6}$ due to its
stronger Barnett-equivalent magnetic fields (Lazarian & Hoang 2008). However,
due to the saturation of the magnetic response inside suprathermal grains,
nuclear relaxation may be only important in driving the internal alignment of
large micron-sized grains at low-J attractors. The rotation rate suitable for
operating nuclear relaxation will be extended with the presence of iron
inclusions (Lazarian & Hoang 2008, Lazarian & Hoang 2019), which consequently
strengthens the IA for more large SPM grains at low-J in the inner envelope
and inside the disk. Besides the two mechanisms mentioned above, irregular
grains (not oblate grains as in our model) can also undergo internal alignment
by the Amplitude Barnett Relaxation mechanism (ANR, Lazarian & Hoang 2019).
ANR dissipates rotational energy via the change in the amplitude of
$\boldsymbol{J}$ during the IA period of wobbling grains. Therefore, it can
play some role in improving the IA of large grains facing slow internal
relaxation by Barnett relaxation. We expect that it can work for both grains
at high and low-J because the grain wobbling always appears (Lazarian 1994) as
a consequence of the rotational dissipation-induced fluctuation inside them.
Given the complex dependence of inelastic relaxation, nuclear relaxation, and
ANR on grain rotational rate, grain magnetic susceptibility, and also
inelasticity (for inelastic relaxation scenario), these new features must be
incorporated into POLARIS to precisely quantify their realistic strength in
driving the internal alignment of VLGs in protostellar environments.

Besides MRAT mechanism, the external alignment of dust grains also can be
improved with higher $f_{\rm high-J}$ if grains at low-J can be lifted to
stably aligned with $\boldsymbol{B}$ at high-J by the gas bombardment (Hoang &
Lazarian 2008). This process takes about $5-10$ time of the gas damping
timescale $\tau_{\rm gas}$, but given small $\tau_{\rm gas}$ inside
protostellar disks, gas bombardment may play an important role in enhancing
the net magnetic alignment of dust grains within few hundred au around the
protostar. The increased amount of grains at $f_{\rm high-J}$ not only
increases the observed degree of polarization but also strengthens the origin
of dust polarization from magnetically aligned dust grains (see Section 9.7).
Furthermore, they will validate the perpendicular orientation of the dust
polarization $\boldsymbol{P}$ vectors with $\boldsymbol{B}$, given grains at
high-J always tend to have the right IA regardless of fast or slow internal
relaxation. This issue is very important (see below Section 9.2.1) in
inferring $\boldsymbol{B}$ field morphology inside the disk.

So in conclusion, considering Barnett relaxation as the major internal
alignment mechanism of dust grains, it is impossible to assume the fast
internal relaxation for all aligned dust grains inside the protostellar
envelope and disk. The alignment size range is not only determined by the
minimum alignment size by RATs, $a_{\rm align}$, but also by the rate of
Larmor precession compared to the gas randomization, $a_{\rm max,JB}^{\rm
Lar}$. Besides, the external alignment mechanism, RATs or MRAT, will be size-
dependent and is controlled by the grain magnetic susceptibility, local gas
density, and magnetic field strength inside the core. Theoretically, the
alignment mechanism will determine the observed degree of dust polarization
(Section 6). The internal alignment determines the orientation of polarization
vectors with local magnetic fields, and the alignment range determines the
area where dust polarization can trace $\boldsymbol{B}$-fields (Section 5).
Therefore, detailed modeling of grain alignment with their magnetic properties
is required to maximize the usage of dust polarization in probing magnetic
fields and dust physics around the protostar.

###  9.2 The reliability of dust polarization in tracing magnetic fields in
protostellar envelopes and disks

####  9.2.1 Inferred magnetic fields from dust polarization

In Section 5.1, we found that in general, PM and SPM grains can trace the
large-scale magnetic field in the envelope of $\sim 500-1000$ au scale because
they can be aligned with $\boldsymbol{B}$ there. For SPM grains, most of them
have fast internal relaxation and tend to be aligned with $\boldsymbol{B}$ at
high-J attractors. As a result, their polarization vectors are always
perpendicular to the orientation of magnetic fields. For PM grains, although
$\sim 75\%$ of grains will be aligned with $\boldsymbol{B}$ at low-J
attractors by RATs, the net polarization signal obtained in the envelope still
be dominant by the emission of $25\%$ of grains with perfect right internal
alignment and perfect external alignment at high-J (Figure 29, upper panels).
Thus, one can confidently rotate polarization vectors $\boldsymbol{P}$ by
$90^{\circ}$ to infer the magnetic field morphology in the envelope regardless
of the magnetic properties of dust grains.

In the protostellar disk, the situation is different. In Section 5.2, we show
that PM grains cannot trace $\boldsymbol{B}$-fields inside the disk because
grains above $1\,{\mu\rm{m}}$ cannot be aligned with $\boldsymbol{B}$ in this
area. Their observed polarization signal thus only provides magnetic field
information of foreground envelope, whose polarization vectors are always
perpendicular to $\boldsymbol{B}$. On the contrary, polarized dust emission
from SPM grains can be used to trace $\boldsymbol{B}$ fields inside the disk
because micron-sized and VLGs can have magnetic alignment in this region with
the help of iron inclusions. The detailed morphology of magnetic fields around
the protostar increases with increasing amount of iron inclusions locked
inside SPM grains. For example, when $\Theta=45^{\circ}$ and $a_{\rm
max}=10-100\,{\mu\rm{m}}$, SPM grains with $N_{\rm cl}=100$ only can capture
the spiral pattern of magnetic fields beyond $\sim 200$ au from the protostar
(Figure 11, third row). While SPM grains with higher $N_{\rm cl}=10^{4}$ have
the capability to trace the transition of $\boldsymbol{B}$-fields, from the
spiral pattern beyond 200 au to the uniform field along the disk minor axis in
the inner 200 au region as expected in model PA (Figure 11, fourth row).

However, the alignment of SPM grains with $\boldsymbol{B}$ in the disk
accidentally activates the problem related to the alignment direction of
grains having slow internal relaxation with magnetic fields. As shown in
Section 4.3 (see also Figures 27 and 27), almost micron-sized grains above
$1\,{\mu\rm{m}}$ inside the disk will have slow internal relaxation at both
high and low-J attractors. Since RATs play the major role in aligning large
SPM grains within the disk with $f_{\rm high-J}=0.25$ (Figures 28 and 29),
emission from grains at low-J may dominant over grains at high-J to decide the
net polarization pattern. In contrast to grains with fast internal relaxation
whose alignment direction is well-determined to be perpendicular with
$\boldsymbol{B}$, the alignment direction of grains with slow internal
relaxation is still not well understood. As shown in Hoang & Lazarian (2009),
grains at high-J may still have right IA due to the spinning torque of RATs,
but grains at low-J may have right or wrong IA, with unclear conditions
supporting their orientation in space. We show Section 5.2 the polarization
pattern obtained in two separate cases. If all grains with slow internal
relaxation have the right IA, they will produce
$\boldsymbol{P}\perp\boldsymbol{B}$, and rotating $\boldsymbol{P}$ by
$90^{\circ}$ is reasonable, yielding the correct magnetic field map inside the
disk as obtained in model PA (Figure 11, fourth row). However, if grains with
slow internal relaxation at low-J have the wrong IA, emission from aligned
dust grains at low-J will dominant and produce the net polarization vectors
$\boldsymbol{P}\|\boldsymbol{B}$. Consequently, rotating $\boldsymbol{P}$ by
$90^{\circ}$ induces the wrong inferred pattern of magnetic fields in the disk
(Figure 12, fourth row).

Table 3: Effects of maximum grain size and dust model on the recovery rate of the entire protostellar core Model | PM | SPM, $N_{\rm cl}=100$ | SPM, $N_{\rm cl}=10^{4}$
---|---|---|---
PA, $a_{\rm max}=10\,{\mu\rm{m}}$ | $83.39\%$
rIA | $72.31\%$ | $83.51\%$ | $84.92\%$
wIA | $64.30\%$ | $78.02\%$ | $83.25\%$
PA, $a_{\rm max}=100\,{\mu\rm{m}}$ | $83.76\%$
rIA | $71.83\%$ | $82.46\%$ | $83.53\%$
wIA | $69.26\%$ | $81.25\%$ | $82.92\%$
Table 4: Recovery rate in concentric rings focusing in the envelope and disk scale for different grain magnetic properties, assuming $a_{\rm max}=10\,{\mu\rm{m}}$ Model | Grain type |  $<100$ au | 100-500 au | 500-2000 au
---|---|---|---|---
PA |  | 93.37$\%$ | 90.90$\%$ | 82.72$\%$
rIA | PM | 44.58$\%$ | 59.39$\%$ | 73.70$\%$
| SPM, $N_{\rm cl}=100$ | 54.82$\%$ | 82.70$\%$ | 82.59$\%$
| SPM, $N_{\rm cl}=10^{4}$ | 78.61$\%$ | 89.60$\%$ | 82.72$\%$
wIA | PM | 43.37$\%$ | 56.32$\%$ | 71.19$\%$
| SPM, $N_{\rm cl}=100$ | 45.78$\%$ | 75.39$\%$ | 82.26$\%$
| SPM, $N_{\rm cl}=10^{4}$ | 32.83$\%$ | 86.44$\%$ | 82.72$\%$

So in conclusion, we can freely rotate polarization vectors of dust
polarization obtained beyond $\sim 200$ au by $90^{\circ}$ to infer the
magnetic field morphology in the protostellar envelope, regardless of
examining the grain magnetic properties. But for the polarization signal
obtained within the inner $200$ au, careful examination of its origin must be
done before further investigating the magnetic field information embedded in
dust polarization. As discussed in the above paragraph, we can estimate the
region where dust polarization trace $\boldsymbol{B}$ fields (envelope or
disk) based on the grain magnetic properties (we will discuss this issue in
the following Section 9.5.2). But in terms of the orientation between
$\boldsymbol{P}$ and $\boldsymbol{B}$, we indeed do not have a clear answer
due to the undetermined alignment direction of large micron-sized grains with
slow internal relaxation with $\boldsymbol{B}$ inside the disk. We refer
readers to Section 9.5.1 where we discuss the method to constrain the grain
alignment direction inside the disk via multi-wavelength observations based on
our synthetic results obtained in Sections 5.2 and 8.1. However, as discussed
in Section 9.1, more SPM grains above $\sim 1\,{\mu\rm{m}}$ may have fast
internal relaxation at both high and low-J if we consider both nuclear
relaxation and inelastic relaxation effects. The higher amount of grains with
fast internal relaxation will reduce the uncertainty in rotating
$\boldsymbol{P}$ to infer $\boldsymbol{B}$. However, the detailed impact of
the mentioned internal relaxation mechanisms on aligned dust grains is not
quantified and should be addressed in future studies to provide a definitive
answer to this question.

####  9.2.2 Recovery rate

To quantify how well dust polarization can capture the orientation of
protostellar magnetic fields, Valdivia et al. (2022) calculate the magnetic
field angle difference between MHD simulations and synthetic observations of
dust polarization in different cores with different initial setups of magnetic
fields and turbulence. They conclude that dust polarization from aligned dust
grains is the robust tracer of magnetic fields in $\sim 500-1000$ au scale,
with the ability to recover $\sim 90\%$ of magnetic field orientation in this
region. However, this paper considers standard RAT theory with the constant
$f_{\rm high-J}=0.25$ and assumes that all grains from $a_{\rm align}-a_{\rm
max}=20\,{\mu\rm{m}}$ have fast internal relaxation and can be aligned with
$\boldsymbol{B}$. As discussed in Section 9.1, this assumption is not always
valid, even in the protostellar envelope. Therefore, even polarized dust
emission generally can trace $\boldsymbol{B}$-fields in the envelope, how
accurately is this method must be re-examined with the new grain alignment
model shown in Section 4.2.

Following Valdivia et al. (2022), we calculate the angle difference
$\Delta\phi$ between the integrated magnetic field from the MHD simulation
shown in the right panel of Figure 2 and the inferred magnetic fields from
dust polarization $\phi_{\rm B,syn}$ derived by rotating $\boldsymbol{P}$ by
$90^{\circ}$. Here, we consider the observed direction is along the face-on
direction ($\Theta=0^{\circ}$) and polarized dust emission can trace magnetic
fields if $\Delta\phi\leq 20^{\circ}$. The recovery rate is then defined as
the ratio of the area where dust polarization can capture
$\boldsymbol{B}$-field orientation over the studied area.

Table LABEL:tab:recover_rate shows the recovery rate of dust polarization from
PM grains (first column), SPM grains with low $N_{\rm cl}=100$ (second
column), and SPM grains with high $N_{\rm cl}=10^{4}$ (third column). Three
first rows show results obtained from model PA, rIA, and wIA with $a_{\rm
max}=10\,{\mu\rm{m}}$, and the next three rows are for the model with $a_{\rm
max}=100\,{\mu\rm{m}}$. In general, the recovery rate is smaller when we
consider the realistic model of grain alignment in protostellar environments.
In particular, PM grains only can cover $\sim 70-75\%$ of magnetic fields in
both the envelope and disk due to their weak alignment degree with magnetic
fields. In contrast, SPM grains can recover $80-90\%$ of $\boldsymbol{B}$
within $\sim 2500$ au around the protostar depending on the amount of iron
inside dust grains. The recovery rate slightly declines with increasing
$a_{\rm max}$ due to the increased amount of VLGs with inefficient magnetic
alignment. This parameter is also smaller for model wIA as a result of the
superposition of polarization signal from grains with the right and wrong IA
along the LOS.

Table LABEL:tab:recover_rate_ring_a100 shows similar results as Table
LABEL:tab:recover_rate but for different concentric rings of thickness from
500-2000au (envelope scale), 100-500 au (inner envelope), and within 100 au
(the disk scale), assuming $a_{\rm max}=100\,{\mu\rm{m}}$. In general, the
accuracy of $\boldsymbol{B}$-field morphology inferred from dust polarization
decreases toward the center and decreases with decreasing amount of iron
inclusions embedded inside aligned dust grains. In particular, PM grains can
recover $\sim 70\%$ beyond $>500$ au and $\sim 45\%$ in the disk, while SPM
grains can recover higher percentages of $\sim 80-90\%$ in the envelope but
only $\sim 50-78\%$ inside the disk. The recovery rate declines with
increasing maximum grain sizes, and it is smaller if grains with slow internal
relaxation at low-J have wrong alignment direction.

However, one can see that for SPM grains with $N_{\rm cl}=10^{4}$ in model
wIA, the recovery rate within $<100$ au is very small of $\sim 30\%$ due to
the wrong interpretation of dust polarization radiating from wrong aligned
grains (Figure 12, lower right panel). This problem thus emphasizes the
importance of accurately determining the alignment direction of grains with
$\boldsymbol{B}$-fields before deciding how to get $\boldsymbol{B}$ vectors
from $\boldsymbol{P}$.

###  9.3 Origin of the polarization hole