# Dust Polarization of Prestellar and Protostellar Sources in OMC-3

yuhua liu Department of Earth and Planetary Sciences, Faculty of Sciences,
Kyushu University, Fukuoka 819-0395, Japan<EMAIL_ADDRESS>satoko takahashi National Astronomical Observatory of Japan, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan  Department of Astronomical Science, School of
Physical Sciences, The Graduate University for Advanced Studies, SOKENDAI,
2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  masahiro machida Department of
Earth and Planetary Sciences, Faculty of Sciences, Kyushu University, Fukuoka
819-0395, Japan<EMAIL_ADDRESS>kohji tomisaka National
Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
Department of Astronomical Science, School of Physical Sciences, The Graduate
University for Advanced Studies, SOKENDAI, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan  Josep Miquel Girart Institut de Ciències de l’Espai (ICE,
CSIC), Can Magrans s/n, E-08193 Cerdanyola del Vallès, Catalonia, Spain
Institut d’Estudis Espacials de Catalunya (IEEC), E-08034 Barcelona,
Catalonia, Spain  paul t. p. ho Institute of Astronomy and Astrophysics,
Academia Sinica, 11F of Astronomy-Mathematics Building, AS/NTU No. 1, Sec. 4,
Roosevelt Rd, Taipei 10617, Taiwan, R.O.C.  East Asian Observatory, 660 N.
A’ohoku Place, Hilo, HI 96720, USA  kouichiro nakanishi National Astronomical
Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Department
of Astronomical Science, School of Physical Sciences, The Graduate University
for Advanced Studies, SOKENDAI, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
asako sato Department of Earth and Planetary Sciences, Faculty of Sciences,
Kyushu University, Fukuoka 819-0395, Japan<EMAIL_ADDRESS>
###### Abstract

We present the Atacama Large Millimeter/submillimeter Array (ALMA)
observations of linearly polarized 1.1 mm continuum emission at ${\sim}0\hbox
to0.0pt{.\hss}\arcsec 14$ (55 au) resolution and CO ($J$=2$-$1) emission at
${\sim}1\hbox to0.0pt{.\hss}\arcsec 5$ (590 au) resolution towards one
prestellar (MMS 4), four Class 0 (MMS 1, MMS 3, MMS 5, and MMS 6), one Class I
(MMS 7), and one flat-spectrum (MMS 2) sources in the Orion Molecular Cloud 3
region. The dust disk-like structures and clear CO outflows are detected
towards all sources except for MMS 4. The diameters of these disk-like
structures, ranging from 16 au to 97 au, are estimated based on the
deconvolved full width half maximum (FWHM) values obtained from the multi-
Gaussian fitting. Polarized emissions are detected towards MMS 2, MMS 5, MMS
6, and MMS 7, while no polarized emission is detected towards MMS 1, MMS 3,
and MMS 4. MMS 2, MMS 5, and MMS 7 show organized polarization vectors aligned
with the minor axes of the disk-like structures, with mean polarization
fractions ranging from 0.6% to 1.2%. The strongest millimeter source, MMS 6,
exhibits complex polarization orientations and a remarkably high polarization
fraction of $\sim$10$\%$ around the Stokes $I$ peak, and 15$-$20$\%$ on the
arm-like structure, as reported by Takahashi et al. (2019). The origins of the
polarized emission, such as self-scattering and dust alignment due to the
magnetic field or radiative torque, are discussed for individual sources. Some
disk-like sources exhibit a polarized intensity peak shift towards the
nearside of the disk, which supports that the polarized emission originates
from self-scattering.

Polarimetry (1278) — Dust continuum emission (412) — Young stellar objects
(1834) — Magnetic fields (994)

##  1 Introduction

Magnetic fields are considered to play an important role in regulating star
formation (e.g., Shu et al., 1987; Hildebrand, 1988; Crutcher, 2012). The role
of magnetic fields could change over the size scale and the timescale in the
star formation process (Koch et al., 2022). However, the specific role of
magnetic fields during star formation has not been fully understood.

Polarized emission from dust grains at (sub)millimeter wavelengths is commonly
used as a tracer to probe the configuration of magnetic fields, as the non-
spherical dust grains tend to align their long axes perpendicular to the
magnetic fields (e.g., Hildebrand, 1988; Lazarian, 2007; Andersson et al.,
2015; Hull & Zhang, 2019). The configuration of magnetic fields has been
studied through the dust polarization observations on large-scale, spanning
from the molecular cloud-scale (a few pc) down to the protostellar envelope-
scale (a few 1000 au) by using the Submillimeter Common-User Bolometer Array
(SCUBA) camera on the James Clerk Maxwell Telescope (JCMT; e.g., Matthews &
Wilson, 2000), the Berkeley-Illinois-Maryland Association (BIMA) millimeter
array (e.g., Rao et al., 1998; Lai et al., 2001; Matthews et al., 2005), the
Submillimeter Array (SMA; e.g., Girart et al., 2006; Rao et al., 2009, 2014;
Girart et al., 2013; Tang et al., 2013; Zhang et al., 2014), the Combined
Array for Research in Millimeter-wave Astronomy (CARMA; e.g., Stephens et al.,
2013; Hull et al., 2014), and the Atacama Large Millimeter/submillimeter Array
(ALMA; e.g., Koch et al., 2018, 2022; Liu et al., 2020).

Recently, ALMA has achieved higher sensitivity and resolution enabling the
dust polarization observations on the disk-scale ($\sim$10$-$100 au; e.g.,
Kataoka et al., 2016b, 2017; Stephens et al., 2017; Ohashi et al., 2018; Lee
et al., 2018; Cox et al., 2018; Sadavoy et al., 2019). These observations
suggest that there are alternative mechanisms producing polarized emission
such as: (i) Self-scattering of thermal dust emission (e.g., Kataoka et al.,
2015, 2017; Yang et al., 2017). When the dust grains have the comparable sizes
with the observing wavelengths
($a_{\textnormal{\scriptsize{max}}}\sim\lambda/2\pi$, where
$a_{\textnormal{\scriptsize{max}}}$ is the maximum size of the dust grain and
$\lambda$ is the observing wavelength) and the radiation field has an
anisotropic distribution (Kataoka et al., 2015), the thermal emission can be
scattered by the dust grains, which induces polarized emission; (ii) Dust
grains alignment due to radiative torque with a different direction from the
magnetic field (e.g., Lazarian & Hoang, 2007a; Tazaki et al., 2017; Kataoka et
al., 2017). The dust grains are expected to align their minor axes to the
radiative flux and the polarization orientation traces the direction of
radiation anisotropy or the radiative flux; in addition to the observational
studies, theoretical studies also proposed (iii) Grain alignment by gas flow
due to the mechanical alignment introduced by Gold (1952), Lazarian & Hoang
(2007b), Hoang et al. (2018), and Kataoka et al. (2019). With this mechanism,
dust grains are aligned with the gas flow. The orientations of the
polarization vectors will depend on the Stokes numbers of the dust grains
(Kataoka et al., 2019); (iv) Reversal of self-scattering proposed by Yang & Li
(2020). Yang & Li (2020) found the reversed polarization orientation, where
the polarization vectors are parallel to the major axis of the disk. This
pattern has not been directly detected by ALMA due to the low polarization
fraction of $\sim$0.1%; (v) Magnetic field alignment in the Mie regime ($2\pi
a/\lambda\gtrsim 1$, where $a$ is the radius of the particle and $\lambda$ is
the wavelength of the light) proposed by Guillet et al. (2020). In contrast to
the magnetic field alignment mentioned earlier (e.g., Rayleigh regime: $2\pi
a/\lambda\ll 1$), the polarization vectors of the dust thermal emission in the
Mie regime tend to be parallel to the magnetic field, also known as negative
polarized emission.

The observations towards HD 142527, HL Tau, and some sources from Ophiuchus
(e.g., GSS 30 IRS 1, Oph-emb-6, and IRS 63) suggest that polarization on disk-
scale is more consistent with that expected from self-scattering and radiative
torque than other mechanisms (e.g., Kataoka et al., 2016b, 2017; Stephens et
al., 2017; Cox et al., 2018; Sadavoy et al., 2019). Moreover, polarization due
to self-scattering could be detected at the inner region of the disk (Kataoka
et al., 2016b). Therefore, dust polarization due to self-scattering is also an
effective tool to constrain the dust grain size in the optically thick region
(Kataoka et al., 2016b). Grain size constraints are important because they are
related to the dust growth, which is the first step towards understanding the
composition and properties of protoplanetary disks and the formation of
planets within them.

Our targets are the seven prestellar and protostellar sources, MMS 1$-$7, in
the Orion Molecular Cloud 3 (OMC-3) region, which are located in the northern
part of the nearest giant molecular cloud, the Orion-A Giant Molecular Cloud,
at a distance of 393 pc (Tobin et al., 2020). Previous studies have identified
several millimeter sources associated with prestellar and early stage of Class
0 sources embedded within the OMC-3 region (Chini et al., 1997). The Spatial
structure of the dust continuum sources was revealed in the size scale down to
$\sim$1800 au by the SMA observations (Takahashi et al., 2013). The
evolutionary stages of the sources are primarily identified based on the
spectral energy distribution (SED) study by Furlan et al. (2016).

The dust polarization of the OMC-3 region on large-scale was firstly observed
by Matthews & Wilson (2000) and Matthews et al. (2001) in 850 $\mu$m continuum
with the SCUBA on the JCMT, and then by Houde et al. (2004) in 350 $\mu$m
continuum with the Hertz polarimeter and Submillimeter High Angular Resolution
Camera II (SHARC II), showing that the polarization orientations are highly
ordered and aligned along the filament. Since the dust grains are considered
to align with the magnetic field on this scale, this indicates that the
interstellar magnetic field is perpendicular to the long axis of OMC-3.

Although polarization studies on the cloud- and envelope-scale have been done
extensively in the OMC-3 region, the polarization on disk-scale towards the
individual sources in this region has not been widely studied except for MMS 6
(Takahashi et al., 2019; Le Gouellec et al., 2020; Liu, 2021). In this paper,
we present polarization observations on disk-scale along with the molecular
line observations towards seven millimeter sources, MMS 1$-$7, located within
the OMC-3 region. The source names including those from other catalogues,
coordinates, Herschel Orion Protostar Survey (HOPS) IDs, bolometric luminosity
($L_{\textnormal{bol}}$), outflow associations, and evolutionary stages
(classifications) for all observed sources are summarized in Table 1. Our aim
is to examine the origins of the polarization for these young embedded sources
and also to study how the polarization characteristics change at different
evolutionary stages.

We describe the ALMA observations, data reduction, and imaging in Section 2.
The results are summarized in Section 3. We discuss the polarization origins
and examine how the source size, the outflow size, and the polarization
fraction change at different evolutionary stages in Section 4. Finally, we
present the conclusions in Section 5.

Table 1: Source Summary

Source Name |  R.A.a |  Decl.a | Other Source Name |  HOPS IDd |  $L_{\textnormal{bol}}$d | Outflow |  Classificatione | Comments
---|---|---|---|---|---|---|---|---
| (J2000) | (J2000) |  (350 $\mu$mb, 850 $\mu$mc) |  |  ($L_{\odot}$) | [Yes/No] |  |
MMS 1 | 5 35 18.03 |  $-$5 00 17.77 | CSO 5, SMM 2 |  | $<55^{f}$ | Yes |  Class 0g |
MMS 2 | 5 35 18.30 |  $-$5 00 33.01 | CSO 6, SMM 3 | HOPS-92 | 17.6 | Yes | Flat-spectrum |  Triple source (Tobin et al., 2020)
MMS 3 | 5 35 18.98 |  $-$5 00 51.63 | CSO 7, SMM 4 | HOPS-91 | 3.6 | Yes | Class 0 |
MMS 4 | 5 35 20.88 |  $-$5 00 56.25 | CSO 8, SMM 5 |  | $<56^{f}$ | No | Prestellar |
MMS 5 | 5 35 22.47 |  $-$5 01 14.43 | CSO 9, SMM 6 | HOPS-88 | 13.8 | Yes | Class 0 |
MMS 6 | 5 35 23.42 |  $-$5 01 30.35 | CSO 10, SMM 8 | HOPS-87 | 31.6 | Yes | Class 0 |
MMS 7 | 5 35 26.56 |  $-$5 03 55.04 | CSO 12, SMM 11 | HOPS-84 | 42.9 | Yes | Class I |  Binary source (Tobin et al., 2020)

aThe location of the source denotes the phase center of the ALMA observations.
bCSO sources reported by Lis et al. (1998).
cSMM sources reported by Takahashi et al. (2013).
dHOPS ID and $L_{\textnormal{bol}}$ are obtained from Furlan et al. (2016),
which can also be obtained from HOPS catalog
(https://planetstarformation.iaa.es/OMC-3).
eThe classification of the protostellar sources are defined based on the SED
study presented by Furlan et al. (2016).
fNo counterpart source is identified in the HOPS catalog, hence
$L_{\textnormal{bol}}$ is referenced from Chini et al. (1997).
gWe classify MMS 1 as a Class 0 source since the detection of a very compact
and collimated jet by Takahashi et al. (2023a, submitted to ApJ).

##  2 Observations and Data Reduction

###  2.1 1.1 mm continuum

The ALMA 1.1 mm continuum observations at Band 6 were conducted during Cycle 4
(2015.1.00341.S; P.I. S. Takahashi) on 2016 September 18, October 9, and
October 11. The ALMA observations were obtained in full polarization,
targeting MMS 1$-$7 in the OMC-3 region. The observational parameters are
listed between the second and fourth columns of Table 2. The total on-source
time of all sources was between 33 and 35 minutes. Between 38 and 44 of the
12-m antennas were operated during the observations. The full width at half
maximum (FWHM) of the ALMA primary beam is $\sim$22$\arcsec$. The projected
baselines range from 15 to 3144 m, and the minimum baseline yields in a
maximum recoverable size of $\sim$9$\hbox to0.0pt{.\hss}\arcsec 1$. Four
spectral windows (spws) in Time Division Mode (TDM), each with a total
bandwidth of 2 GHz, were allocated for the continuum observations. The number
of the effective channels varies for each spw because we flagged out some of
the channels where molecular line emissions were detected. After subtraction
of those, the obtained total effective bandwidth for the continuum observation
is $\sim$6.4$-$6.9 GHz. All the raw data were reduced with the Common
Astronomy Software Applications (CASA; CASA Team et al., 2022) package version
4.7.0.

In order to enhance the image quality and suppress the noise, we performed
self-calibration on phase by using CASA tasks “gaincal” and “applycal” for MMS
2, MMS 5, MMS 6, and MMS 7, respectively. The solution interval for self-
calibration was optimized for each source based on its structure and signal-
to-noise ratio (S/N). We did not apply self-calibration for MMS 1, MMS 3, and
MMS 4 because there were no significant improvements in both image quality and
S/N after applying self-calibration. The CLEANed images of Stokes I, Q, and U
for all sources were made using a CASA task “tclean” with Briggs weighting
(robust = 0.5). The image of Stokes $I$ shows the total intensity. The
linearly polarized intensity (PI) was derived from Stokes $Q$ and $U$ as
$\textnormal{PI}=\sqrt{Q^{2}+U^{2}-\sigma_{\textnormal{PI}}}$, where
$\sigma_{\textnormal{PI}}=\sqrt{\sigma_{\textnormal{Q}}^{2}+\sigma_{\textnormal{U}}^{2}}\sim
26\mu$Jy beam-1. Each of the PI images obtained using the pixels with an
emission detection is $\geq 3\sigma_{\textnormal{PI}}$. The polarization
fraction ${P_{\textnormal{frac}}}=\textnormal{PI}/I$ was obtained using the
debiased PI and Stokes $I$. The position angle of polarization vector (also
polarization angle) $\chi=\frac{1}{2}\textnormal{arctan}(U/Q)$ was derived
where the detection is $\geq 3\sigma_{\textnormal{PI}}$. The PI,
${P_{\textnormal{frac}}}$, and $\chi$ were computed using a CASA task
“immath”. The 1$\sigma$ error of the polarization fraction is calculated as

$\sigma_{P_{\textnormal{frac}}}=\sqrt{\frac{Q^{2}\Delta
Q^{2}}{(Q^{2}+U^{2})}+\frac{U^{2}\Delta
U^{2}}{(Q^{2}+U^{2})}+\frac{(Q^{2}+U^{2})\Delta I^{4}}{I^{4}}}.$ (1)

The average value of $\sigma_{P_{\textnormal{frac}}}$ among our sources is
estimated to be 0.01${}^{+0.01}_{-0.01}$%, where $I$, $Q$, and $U$ are the
flux values, and $\Delta I$, $\Delta Q$, and $\Delta U$ are their respective
errors. The 1$\sigma$ error of the polarization angle is calculated as

${\sigma_{\chi}}=\frac{180}{\pi}\times\sqrt{\left(\frac{\Delta
U^{2}}{4Q^{2}}+\frac{U^{2}\Delta
Q^{2}}{4Q^{4}}\right)\times\left(1+\left(\frac{U}{Q}\right)^{2}\right)^{-2}}.$
(2)

The average value of $\sigma_{\chi}$ among our sources is estimated to be
3$\hbox to0.0pt{.\hss}^{\circ}$3${}^{+3.6}_{-2.1}$. According to ALMA Cycle 4
Technical Handbook111https://almascience.org/documents-and-tools/cycle4/alma-
technical-handbook, the 1$\sigma$ systematic error of the linear polarization
fraction is 0.03% for the compact sources and 0.1% for the extended sources
within one-third of the primary beam (which corresponds to a minimum
detectable polarization fraction of 0.1% for the compact sources and 0.3% for
the extended sources). Table 3 summarizes the rms noise level measured in each
Stokes component, synthesized beam, position angle (P.A.), and solution
interval for self-calibration. The achieved rms noise level in Stokes $Q$ and
$U$ is close to the theoretical noise level of $\sim$21 $\mu$Jy beam-1, and
Stokes $I$ images are dynamic range limited.

###  2.2 Molecular line

The ALMA molecular line observations towards MMS 1$-$7 at Band 6 were
performed on 2016 January 29 during Cycle 4 (2015.1.00341.S; P.I. S.
Takahashi). Four spectral lines include CO ($J$ = 2$-$1; 230.538 GHz), N2D+
($J$ = 3$-$2; 231.322 GHz), SiO ($J$ = 5$-$4; 217.105 GHz), and C18O ($J$ =
2$-$1; 219.560 GHz), respectively. In this paper, we present CO ($J$ = 2$-$1)
data to show the molecular outflow and the extremely high-velocity (EHV) jet
associated with each observed source222Some of the molecular line data were
already presented in our previous studies (Takahashi et al., 2019; Matsushita
et al., 2019; Morii et al., 2021).. Here, “outflows” refer to the CO emissions
detected in the low-velocity range of
$v_{\textnormal{LRS}}-v_{\textnormal{sys}}\lesssim\pm$50 km s-1, while “EHV
jets” refer to the CO emissions detected in the high-velocity range of
$v_{\textnormal{LRS}}-v_{\textnormal{sys}}\gtrsim\pm$50 km s-1. The
observational parameters are listed in the fifth column of Table 2. The raw
data were reduced with the CASA (CASA Team et al., 2022) package version
4.6.0.

The CLEANed images of CO ($J$ = 2$-$1) emissions were made using a CASA task
“tclean” with Briggs weighting (robust = 0.5). The velocity interval of 2 km
s-1 was used for imaging the outflows associated with MMS 2 and MMS 7. The
velocity interval of 5 km s-1 was used for imaging both the outflows and the
EHV jets associated with MMS 5 and MMS 6. The velocity interval of 1 km s-1
was used for imaging the CO emission associated with MMS 4. The CLEANed images
of CO emissions for MMS 3 and MMS 1 were obtained from Morii et al. (2021) and
Takahashi et al. (2023a, submitted to ApJ), respectively. The integrated
intensity maps were made using a CASA task “immoment” with the 3$\sigma$
threshold. Table 4 summarizes the rms noise level, synthesized beam, position
angle (P.A.), and velocity interval for the CO images.

Table 2: ALMA Observations

Parameters | Full Polarization Continuum |  CO ($J$ = 2$-$1)
---|---|---
Observed sources | MMS 1, MMS 2, MMS 5, MMS 6, and MMS 7 | MMS 3 and MMS 4 |  MMS 1$-$7
Observing date (YYYY-MM-DD) | 2016-10-09 | 2016-10-11 | 2016-09-18 | 2016-01-29
Number of antennas | 44 | 42 | 38 | 40
FWHM of Primary beam (arcsec) | 22 | 22 | 22 | 26
PWV (mm) |  0.44$-$0.90 |  0.55$-$0.57 |  0.84$-$0.91 |  2.46$-$2.60
Phase stability rms (degree)a |  16$-$56 |  10$-$15 | 20 |  9$-$11
Polarization calibrator | J0522-3627 | J0522-3627 | J0522-3627 | -
Bandpass calibrators | J0510+1800 | J0510+1800 | J0510+1800 | J0522-3627
Flux calibrator | J0423-0120 | J0423-0120 | J0423-0120 | J0522-3627
Phase calibrators (separation from the target)b |  J0532-0307(2$\hbox to0.0pt{.\hss}^{\circ}$1) |  J0541-0541(1$\hbox to0.0pt{.\hss}^{\circ}$7) |  J0541-0211(3$\hbox to0.0pt{.\hss}^{\circ}$2) |  J0541-0541(1$\hbox to0.0pt{.\hss}^{\circ}$7)
Central frequency USB/LSB (GHz) | 257/273 | 257/273 | 257/273 | 220/231
Total continuum bandwidth; USB+LSB (GHz) | 6.875 | 6.875 | 6.438 | -
Projected baseline ranges (m) |  19$-$3144 |  19$-$3144 |  15$-$2483 |  15$-$331
Maximum recoverable size (arcsec)c | 7.3 | 7.3 | 9.1 | 9.1
On-source time (minutes) | 20 | 13 | 35 | 4

aAntenna-based phase differences measured on the bandpass calibrator.
bThe phase calibrator was observed every 8 minutes for the observations.
cOur observations were insensitive to emission more extended than this size
scale structure at the 10% of the total flux density (ALMA Cycle 4 Technical
Handbook).

Table 3: Summary of the 1.1 mm Continuum Image

Source Name | rms Noise Level |  Synthesized Beamb | P.A. |  Is Self-calibrationc | Solution Interval for
---|---|---|---|---|---
|  Stokes I a, Q, U |  |  | Applied? | Self-calibration
|  ($\mu$Jy beam-1) |  (arcsec $\times$ arcsec) | (degree) | (Yes/No) | (min)
MMS 1 | 35, 19, 18 |  0.14$\times$0.13 |  $-$53 | No | -
MMS 2 | 38, 18, 18 |  0.14$\times$0.13 |  $-$54 | Yes | 15
MMS 3 | 25, 19, 19 |  0.17$\times$0.14 | 15 | No | -
MMS 4 | 24, 19, 19 |  0.17$\times$0.14 | 15 | No | -
MMS 5 | 60, 19, 19 |  0.14$\times$0.13 |  $-$52 | Yes |  60, 30d
MMS 6 | 76, 19, 19 |  0.14$\times$0.13 |  $-$55 | Yes | 16
MMS 7 | 49, 18, 19 |  0.14$\times$0.13 |  $-$56 | Yes | 17

aStokes $I$ images are dynamic range limited.
bBriggs weighting (robust = 0.5) was used for the imaging.
cSelf-calibration was applied for the phase.
dWe applied two rounds self-calibration on MMS 5 with long solution interval
of 60 minutes and 30 minutes, respectively, because the short solution
interval gives large phase fluctuation.

Table 4: Summary of the CO ($J$ = 2$-$1) Image

Image Name | rms Noise Level |  Synthesized Beama | P.A. | Velocity Interval | Figure
---|---|---|---|---|---
|  (Jy beam-1) |  (arcsec $\times$ arcsec) | (degree) |  (km s-1) |
MMS 2 Integrated intensity | 0.32 |  1.5$\times$0.9 |  $-$78 | - |  2 (a)
MMS 5 Integrated intensity |  0.39, 0.087b |  1.5$\times$0.9 |  $-$77 | - |  2 (b)
MMS 6 Integrated intensity | 0.0925 |  1.5$\times$0.9 |  $-$78 | - |  2 (c)
MMS 7 Integrated intensity | 0.11 |  1.5$\times$0.9 |  $-$77 | - |  2 (d)
MMS 2 Channel map | 0.0058 |  1.5$\times$0.9 |  $-$78 | 2 | B2
MMS 5 Channel map | 0.0039 |  1.5$\times$0.9 |  $-$77 | 5 |  B5 (Matsushita et al., 2019)
MMS 6 Channel map | 0.0039 |  1.5$\times$0.9 |  $-$78 | 5 | B6
MMS 7 Channel map | 0.0058 |  1.5$\times$0.9 |  $-$77 | 2 | B7
MMS 1 Integrated intensity | 0.12 |  1.5$\times$0.9 |  $-$78 | - |  B1 (Takahashi et al. 2023a, submitted to ApJ)
MMS 3 Integrated intensity | 0.145 |  1.7$\times$1.0 |  $-$74 | - |  B3 (Morii et al., 2021)
MMS 4 Integrated intensity | 0.12 |  1.6$\times$0.9 |  $-$78 | - | B4

aBriggs weighting (robust = 0.5) was used for the imaging.
bDifferent rms noise levels correspond to the CO integrated intensities
obtained by integrating over different velocity ranges in the CO channel maps,
which are associated with the outflow and the EHV jet, respectively.

##  3 Results

This section is divided into three subsections. In Section 3.1, we summarize
the observations of the 1.1 mm continuum (Stokes $I$, $Q$, and $U$) and CO
($J$ = 2$-$1) emissions for individual sources. In Section 3.2, we briefly
describe the relation between the cloud- and disk-scale polarization
observations. In Section 3.3, we present the estimated physical properties of
each millimeter source, such as the dust mass, number density, and column
density.

###  3.1 The 1.1 mm continuum emission and CO (J = 2$-$1) emission

Figure 1 shows the Stokes $I$, $Q$, and $U$ for each source. We have detected
all the seven sources in Stokes $I$. Out of the seven sources, MMS 2 (Flat-
spectrum), MMS 5 (Class 0), MMS 6 (Class 0), and MMS 7 (Class I) are detected
in Stokes $Q$ and $U$, while MMS 1 (Class 0), MMS 3 (Class 0), and MMS 4
(Prestellar) are not detected in Stokes $Q$ and $U$. These three sources that
are not detected in Stokes $Q$ and $U$ have the weakest Stokes $I$ fluxes
among the observed seven sources. In addition, we estimate the 3$\sigma$ upper
limits of the polarization fractions for these three sources by assuming that
PI is detected at the Stokes $I$ peak and is comparable to the 3$\sigma$ level
of the Stokes $Q$ and $U$. The estimated 3$\sigma$ upper limits of the
polarization fractions for MMS 1, MMS 3, and MMS 4 are derived to be
$\sim$0.5$\%$, $\sim$2.1$\%$, and $\sim$3.7$\%$, respectively. These 3$\sigma$
upper limits are higher than the minimum detectable linear polarization
fraction of 0.1% for the compact sources and 0.3% for the extended sources
(ALMA Cycle 4 Technical Handbook). This suggests that our observations are
likely limited in the sensitivity.

Table 5 describes the fitting results of the 1.1 mm continuum sources. In
order to characterize the structures of the sources detected in Stokes $I$, we
performed the two-dimensional (2D) Gaussian fitting. The total flux density,
peak flux, and deconvolved size of each source were obtained by using a CASA
task “imfit”. The residual after subtracting the 2D Gaussian fitted model is
$\lesssim 2\%$ with respect to the Stokes $I$ peak flux for each source. Our
2D Gaussian fitting analysis indicates that some compact sources have
deconvolved sizes smaller than the synthesized beam with considerably small
errors, we consider these sources are spatially resolved. Fitting reliability
of the compact sources are discussed in Appendix A. The geometric mean
diameters of the fitted compact components range from $\sim$0.04$\arcsec$ to
$\sim$0.93$\arcsec$, which correspond to the linear sizes of $\sim$16 to
$\sim$364 au, respectively. The geometric mean diameters of the fitted
extended components range from $\sim$0.45$\arcsec$ to $\sim$4.02$\arcsec$,
which correspond to the linear sizes of $\sim$177 to $\sim$1581 au. For
simplicity, we will mainly refer to the linear size of each fitted component
in the subsequent sections. We estimate the inclination angle using the axis
ratio of each fitted component, assuming that they are geometrically thin
disk-like structures. The inclination angles ($i$) are listed in the last
column of Table 5. Here, the face-on and edge-on configurations are defined
with an inclination angle of $i=0^{\circ}$ and $i=90^{\circ}$, respectively.

The left panels of Figure 2 show the CO ($J$ = 2$-$1) emission overlaid with
PI. In these panels, we present the images of the integrated CO emissions
along with the PI for MMS 2, MMS 5, MMS 6, and MMS 7. The outflows associated
with MMS 2, MMS 5, MMS 6, and MMS 7, which are traced by the CO emissions
detected in the low-velocity range of
$v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ $\lesssim\pm 50$ km s-1, show a
cavity-like structure with a wide opening angle. Furthermore, we have detected
CO emissions in the EHV jets associated with MMS 5 and MMS 6 in the high-
velocity range of $v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ $\gtrsim\pm$50
km s-1 (Matsushita et al., 2019; Gómez-Ruiz et al., 2019; Takahashi et al.,
2019). The channel maps of the CO emissions for MMS 2, MMS 5, MMS 6, and MMS
7, as well as the integrated intensity images of the CO emissions for the non-
polarized detected sources MMS 1, MMS 3, and MMS 4 are presented in Appendix
B.

The right panels of Figure 2 show the PI overlaid with the Stokes $I$ and the
polarization vectors (hereafter $E$-vectors). The outflow axes are also shown
in the right panels of Figure 2, which are determined by eye from the
integrated intensity CO images in the left panels. PI of each source shows a
compact structure concentrated around the Stokes $I$ peak. In addition to the
compact PI structures, we have also detected an extended arm-like structure in
MMS 6 that extends up to $\sim$1400 au (Takahashi et al., 2019). PI peak
positional shifts towards the red-shifted CO emissions are observed in MMS 2,
MMS 5, MMS 6, and MMS 7. The positional accuracy is given by
$\sim$$(1/2)\times(\theta/S/N)$, where $\theta$ is the synthesized beam size,
and S/N is the signal-to-noise ratio of the PI image. Note that this
positional accuracy is mainly based on the S/N. We are aware that phase noise
effects may also induce the systematic positional uncertainties, possibly at
different levels in the X and Y linearly orthogonal polarization components of
the received signal. These X and Y orthogonal polarization components are
correlated to produce the cross-correlation visibilities (ALMA Cycle 4
Technical Handbook). Since the Stokes parameters are derived from different
combinations of these cross-correlation visibilities, we expect that the
systematic positional uncertainties induced by the phase noise effects may
vary slightly for the different Stokes parameters. Therefore, the PI peak
positional shift might not only be influenced by the positional accuracy based
on S/N but also by the systematic positional uncertainties induced by the
phase noise effects. For simplicity, we only discuss the PI peak positional
shift with the positional accuracy in this paper. The PI peak shifts with
respect to the Stokes $I$ peaks among MMS 2, MMS 5, MMS 6, and MMS 7 are
larger than their respective positional accuracies by a factor of
$\sim$4$-$110\. $E$-vector orientations differ among the individual sources,
such as the direction parallel to the minor axis of the disk and the azimuthal
direction. The position angle distributions of the $E$-vectors are presented
in Figure 3. To plot these histograms, hyper-Nyquist sampling (e.g., Hull et
al., 2020) was performed to regrid the images of Stokes $I$, $Q$, and $U$ to 4
pixels per synthesized beam. The regridded images of Stokes $Q$ and $U$ were
used to derive the $E$-vector position angles for each source. All histograms
include the pixels whose $\sqrt{Q^{2}+U^{2}}$ is $\geq 3\sigma$. The
polarization characteristics will be explained separately by each source in
the following subsections.

Figure 4 shows polarization fraction overlaid with Stokes $I$ and the inferred
magnetic field vectors (hereafter $B$-vectors). Except for MMS 6, low
polarization fractions of $\sim$0.4$-$2.4% are detected around the Stokes $I$
peaks of MMS 2, MMS 5, and MMS 7. High polarization fractions of
$\sim$0.4$-$10.4% and $\sim$2.7$-$47.4%, are detected around the Stokes $I$
peak and on the arm-like structure of MMS 6, respectively (Takahashi et al.,
2019). Figure 5 presents the histograms plotted from the regridded images of
the polarization fractions. We performed hyper-Nyquist sampling using a
similar approach as explained for Figure 3 to create the regridded images of
the polarization fractions. We also derived the mean polarization fractions
from these histograms. We calculated the weighted mean value of the
polarization fractions from each histogram. We then derived the standard
deviation of the mean value, and calculated the statistical errors, in which
the statistical errors were obtained by using the standard deviation and
dividing it by the square root of the total count in each histogram. The final
errors shown in the figure include both statistical and systematic errors,
where the systematic error is 0.03% for the compact sources and 0.1% for the
extended sources (ALMA Cycle 4 Technical Handbook). The mean polarization
fractions for all sources range between $\sim$0.6% and $\sim$15.7%.

In this paper, we do not present the line tracers that kinematically define
the disk and envelope. Hence the definition of the disk and envelope is based
on the morphological structure measured from the continuum emission. We define
the disk as a structure with the linear size of $\sim$10$-$200 au, which
corresponds to the most compact component fitted by 2D Gaussian, and we define
the envelope as a structure with the linear size of $\sim$300$-$3000 au, which
corresponds to the fitted extended component. In this paper, we consider a
disk to be nearly face-on when the inclination angle is $i\leq 30^{\circ}$, a
disk to be nearly edge-on when the inclination angle is $i\geq 60^{\circ}$,
and a disk to be inclined when the inclination angle is $31^{\circ}\leq i\leq
59^{\circ}$.

Figure 1 (Continued):  Figure 1 (Continued):  Figure 1: Left panels: The
colour and white contours show the Stokes I obtained from the ALMA 1.1 mm dust
continuum data. The white contours are (a) [3, 9, 27, 81, 243, 729,
2187]$\times\sigma$ (1$\sigma$ = 35.2 $\mu$Jy beam-1), (d) [3, 9, 27, 60, 260,
460, 660, 860, 1060]$\times\sigma$ (1$\sigma$ = 36.7 $\mu$Jy beam-1), (g) [3,
9, 27, 81]$\times\sigma$ (1$\sigma$ = 25.0 $\mu$Jy beam-1), (j) [3, 9,
27]$\times\sigma$ (1$\sigma$ = 23.6 $\mu$Jy beam-1), (m) [3, 9, 27, 81, 250,
500]$\times\sigma$ (1$\sigma$ = 60.0 $\mu$Jy beam-1), (p) [3, 9, 27, 50, 70,
250, 500, 750, 1000, 1250, 1500, 1750]$\times\sigma$ (1$\sigma$ = 75.6 $\mu$Jy
beam-1), (s) [3, 9, 27, 81, 200, 400, 600, 800, 1000, 1200, 1400,
1600]$\times\sigma$ (1$\sigma$ = 49.3 $\mu$Jy beam-1). Middle panels: The
colour and black contours show the Stokes Q obtained from the ALMA 1.1 mm dust
continuum data. The black contours are (b) [-3, 3]$\times\sigma$ (1$\sigma$ =
18.8 $\mu$Jy beam-1), (e) [-21, -18, -15, -12, -9, -6, -3, 3, 6, 9, 12, 15,
18]$\times\sigma$ (1$\sigma$ = 18.4 $\mu$Jy beam-1), (h) [-3, 3]$\times\sigma$
(1$\sigma$ = 19.2 $\mu$Jy beam-1), (k) [-3, 3]$\times\sigma$ (1$\sigma$ = 18.8
$\mu$Jy beam-1), (n) [-15, -12, -9, -6, -3, 3, 6]$\times\sigma$ (1$\sigma$ =
18.5 $\mu$Jy beam-1), (q) [-216, -189, -162, -135, -108, -81, -54, -27, -18,
-15, -12, -9, -6, -3, 3, 6, 9, 12, 15, 18, 27, 54, 81, 108, 135,
162]$\times\sigma$ (1$\sigma$ = 19.0 $\mu$Jy beam-1), (t) [-45, -36, -27, -18,
-15, -12, -9, -6, -3, 3, 6, 9, 12, 15, 18, 27, 54, 81, 108, 135,
162]$\times\sigma$ (1$\sigma$ = 18.1 $\mu$Jy beam-1). Right panels: The colour
and black contours show the Stokes U obtained from the ALMA 1.1 mm dust
continuum data. The black contours are (c) [-3, 3]$\times\sigma$ (1$\sigma$ =
18.4 $\mu$Jy beam-1), (f) [-21, -18, -15, -12, -9, -6, -3, 3, 6, 9, 12, 15,
18]$\times\sigma$ (1$\sigma$ = 18.2 $\mu$Jy beam-1), (i) [-3, 3]$\times\sigma$
(1$\sigma$ = 19.0$\mu$Jy beam-1), (l) [-3, 3]$\times\sigma$ (1$\sigma$ = 18.8
$\mu$Jy beam-1), (o) [-15, -12, -9, -6, -3, 3, 6, 9, 12, 15]$\times\sigma$
(1$\sigma$ = 18.8 $\mu$Jy beam-1), (r) [-216, -189, -162, -135, -108, -81,
-54, -27, -18, -15, -12, -9, -6, -3, 3, 6, 9, 12, 15, 18, 27, 54, 81, 108,
135, 162]$\times\sigma$ (1$\sigma$ = 19.4 $\mu$Jy beam-1), (u) [-45, -36, -27,
-18, -15, -12, -9, -6, -3, 3, 6, 9, 12, 15, 18, 27, 54, 81, 108, 135,
162]$\times\sigma$ (1$\sigma$ = 18.7 $\mu$Jy beam-1). The red crosses in panel
(b) and (c) indicate the peak position of Stokes $I$ for MMS 1 as shown in
panel (a). The synthesized beam size is denoted by a filled ellipse in the
bottom left corner of all panels.

####  3.1.1 MMS 1

MMS 1 was first identified by Chini et al. (1997) in the 1300 $\mu$m survey
and classified as Class 0 source because it has a bolometric luminosity-to-
submillimeter luminosity ratio ($L_{\textnormal{bol}}/L_{\textnormal{smm}}$)
of $<$ 200 (Andre et al., 1993). This source was also identified as CSO 5 in
the 350 $\mu$m observations (Lis et al., 1998) and as SMM 2 in the 850 $\mu$m
SMA observations (Takahashi et al., 2013) as listed in Table 1 fourth column.
Despite the centrally peaked structure being as small as $\sim$1000 au, the
number density is as high as 1.5$\times$108 cm-3, as reported by Takahashi et
al. (2013). No HOPS source associated with MMS 1 was identified (Furlan et
al., 2016). Furthermore, there was no evidence to show the existence of the
molecular outflow in previous studies (Yu et al., 1997; Stanke et al., 2002;
Aso et al., 2000; Williams et al., 2003; Takahashi et al., 2008; Feddersen et
al., 2020). With our recent ALMA study by Takahashi et al. (2023a, submitted
to ApJ), we have discovered a very compact bipolar CO outflow and EHV jet
associated with this source as shown in Figure B1 (in Appendix B). One-sided
length of the outflow is $\sim$2400 au and the line-of-sight velocity of the
EHV jet is up to $\sim$65 km s-1. These characteristics suggest that MMS 1
could be one of the youngest protostars among the observed sources in this
region.

Figure 1 (a) shows that we have detected both centrally concentrated and
extended components associated with MMS 1 in Stokes $I$. This source is fitted
with one compact component and one extended component as listed in Table 5.
The compact component has a linear size of $\sim$19 au $\times$ 13 au with a
position angle (P.A.) $=154^{\circ}$, presumably tracing a very compact dust
disk around the protostar. One of the extended components has a linear size of
$\sim$684 au $\times$ 614 au (P.A. $=$ 14∘), which likely traces the inner
part of an envelope. In addition, an extended tail-like structure is detected
at the 9$\sigma$ level in Stokes $I$ pointing towards the southwest direction.
This tail-like structure is spatially correlated with the molecular outflow,
presumably tracing a swept-up gas by the outflow (Takahashi et al. 2023a,
submitted to ApJ).

MMS 1 is not detected in Stokes $Q$ and $U$ as shown in Figure 1 (b) and (c).
The 3$\sigma$ upper limit of the polarization fraction is $\sim$0.5$\%$. The
3$\sigma$ upper limit of the polarization fraction for MMS 1 is larger than
the minimum detectable polarization fraction for a compact source (0.1$\%$;
ALMA Cycle 4 Technical Handbook). Therefore, the sensitivity may be limited in
detecting the polarized emission towards the compact dust disk. On the other
hand, we notice that the 3$\sigma$ upper limit of the polarization fraction
for MMS 1 is close to the minimum detectable polarization fraction for an
extended component (0.3$\%$; ALMA Cycle 4 Technical Handbook). This indicates
that the observing sensitivity allows for the detection of the extended
polarized emission associated with the envelope and core size scale as
presented in Appendix C. Figure C1 shows PI with the magnetic field (rotate
the $E$-vectors by 90∘) along the outflow, but more in-depth observations with
shorter baselines are necessary to better characterize the magnetic field
properties.

Table 5: The 1.1 mm Continuum Fitting Results

| R.A.a | Decl.a | Peak Flux | Total Flux Density | Deconvolved Sizeb | P.A. | Inclination Angle ($i$)c
---|---|---|---|---|---|---|---
| (J2000) | (J2000) | (mJy beam-1) | (mJy) | (milliarcsec $\times$ milliarcsec) | (degree) | (degree)
MMS 1 compact component | 05 35 18.05 |  $-$5 00 17.98 | 12.885$\pm$0.057 | 14.08$\pm$0.1 | 47.5$\pm$3.0$\times$33.8$\pm$3.7 | 154$\pm$11 | 44
MMS 1 extended component 1 | 05 35 18.05 |  $-$5 00 17.96 | 1.395$\pm$0.039 | 17.34$\pm$0.53 | 616$\pm$19$\times$329$\pm$10 | 148.2$\pm$1.8 | 58
MMS 1 extended component 2 | 05 35 18.06 |  $-$5 00 18.07 | 0.4$\pm$0.013 | 65.6$\pm$1.9 | 1740$\pm$52$\times$1563$\pm$47 | 14$\pm$11 | 26
MMS 2-North-A compact component | 05 35 18.34 |  $-$5 00 32.95 | 43.0$\pm$0.085 | 64.30$\pm$0.2 | 100.0$\pm$0.9$\times$90.4$\pm$0.9 | 149.0$\pm$3.9 | 25
MMS 2-North-A extended component | 05 35 18.34 |  $-$5 00 32.97 | 1.15$\pm$0.05 | 21.5$\pm$1.0 | 622$\pm$28$\times$520$\pm$24 | 13$\pm$10 | 33
MMS 2-North-B compact componentd | 05 35 18.33 |  $-$5 00 33.21 | 1.92$\pm$0.08 | 2.01$\pm$0.14 | 41$\pm$25$\times$7$\pm$32 | 101$\pm$39 | -
MMS 2-South compact component | 05 35 18.27 |  $-$5 00 33.94 | 37.95$\pm$0.086 | 70.9$\pm$0.23 | 148.8$\pm$0.8$\times$102.3$\pm$0.8 | 173.0$\pm$1.0 | 47
MMS 2-South extended component | 05 35 18.27 |  $-$5 00 33.94 | 7.91$\pm$0.08 | 27.5$\pm$0.4 | 243.4$\pm$3.7$\times$182.7$\pm$3.2 | 5.5$\pm$2.4 | 41
MMS 3 compact component | 05 35 18.93 |  $-$5 00 51.12 | 2.849$\pm$0.037 | 4.912$\pm$0.094 | 162.8$\pm$5.6$\times$95.8$\pm$6.0 | 54.0$\pm$3.9 | 54
MMS 3 extended component 1 | 05 35 18.93 |  $-$5 00 51.18 | 0.965$\pm$0.026 | 10.09$\pm$0.3 | 542$\pm$17$\times$409$\pm$12 | 9.7$\pm$4.4 | 41
MMS 3 extended component 2 | 05 35 18.95 |  $-$5 00 51.16 | 0.166$\pm$0.01 | 14.5$\pm$0.91 | 1530$\pm$97$\times$1324$\pm$84 | 87$\pm$18 | 30
MMS 4 relatively compact component | 05 35 21.05 |  $-$5 00 58.44 | 0.4$\pm$0.016 | 14.1$\pm$0.63 | 1301$\pm$59$\times$658$\pm$30 | 131$\pm$2.5 | 60
MMS 4 extended component | 05 35 20.96 |  $-$5 00 57.21 | 0.327$\pm$0.004 | 226.5$\pm$2.8 | 11662$\pm$142$\times$1388$\pm$17 | 120$\pm$0.12 | 83
MMS 5 compact component | 05 35 22.47 |  $-$5 01 14.33 | 49.24$\pm$0.15 | 67.32$\pm$0.32 | 89.8$\pm$1.4$\times$72.1$\pm$1.7 | 1.6$\pm$3.6 | 37
MMS 5 extended component | 05 35 22.47 |  $-$5 01 14.37 | 2.30$\pm$0.07 | 72.0$\pm$2.2 | 864$\pm$26$\times$637$\pm$19 | 131.5$\pm$3.9 | 43
MMS 6 compact component | 05 35 23.42 |  $-$5 01 30.53 | 115.67$\pm$0.47 | 499.3$\pm$2.4 | 250.5$\pm$2.2$\times$243.3$\pm$2.2 | 7.4$\pm$13.8 | 14
MMS 6 extended component 1 | 05 35 23.42 |  $-$5 01 30.74 | 22.59$\pm$0.28 | 414.6$\pm$5.5 | 670.5$\pm$9.0$\times$475.7$\pm$6.4 | 111.0$\pm$1.5 | 45
MMS 6 extended component 2 | 05 35 23.42 |  $-$5 01 30.64 | 5.58$\pm$0.11 | 747$\pm$15 | 1945$\pm$40$\times$1256$\pm$26 | 168.3$\pm$1.8 | 50
MMS 7-North compact component 1 | 05 35 26.56 |  $-$5 03 55.11 | 82.31$\pm$0.14 | 213.31$\pm$0.47 | 241.5$\pm$0.6$\times$107.2$\pm$0.5 | 166.5$\pm$0.1 | 64
MMS 7-North compact component 2 | 05 35 26.56 |  $-$5 03 55.14 | 9.68$\pm$0.13 | 30.65$\pm$0.54 | 298.7$\pm$5.8$\times$116.1$\pm$3.5 | 166.0$\pm$0.8 | 67
MMS 7-South compact component | 05 35 26.54 |  $-$5 03 55.73 | 12.5$\pm$0.14 | 15.96$\pm$0.28 | 92.9$\pm$4.5$\times$45.9$\pm$7.0 | 165.8$\pm$5.1 | 60

aThe position of each component refers to the peak position in Stokes $I$,
which is determined by the 2D Gaussian fitting.
bThe errors of the deconvolved size are the 2D Gaussian fitting errors.
c$i$ is estimated using the axis ratio of each fitted component, assuming that
they are geometrically thin disk-like structure.
dThis component has a fitting error for the major axis that is approximately
half the size of the fitted major axis, and a fitting error for the minor axis
three times larger than the fitted minor axis, and thus, we consider the disk
associated with this source to be unresolved (see also Appendix A).

Figure 2: Left panels: The blue and red contours are the integrated intensity
of the CO ($J$ = 2$-$1) emission tracing the outflow. The cyan and magenta
contours are the integrated intensity of the CO emission tracing the EHV jet.
The CO emission in (a) is integrated from
$v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = $-$1 to $-$17 km s-1 (blue) and
$v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = 1 to 33 km s-1 (red), in (c) is
integrated from $v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = $-$5 to $-$50 km
s-1 (blue), $v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = 5 to 50 km s-1
(red), $v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = $-$50 to $-$105 km s-1
(cyan), and $v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = 50 to 70 km s-1
(magenta), respectively. The contour levels in (a) are [5, 10, 15, 20,
25]$\times\sigma$ (1$\sigma$ = 0.32 Jy beam-1), in (c) are [5, 10, 15, 20, 25,
30, 35, 40, 45, 50, 55]$\times\sigma$ (1$\sigma$ = 0.39 Jy beam-1 (red and
blue)) and [10, 20, 30, 40, 50, 60, 70, 80]$\times\sigma$ (1$\sigma$ = 0.087
Jy beam-1 (magenta and cyan)). The green crosses in (a) indicate the peak
positions of Stokes $I$ associated with MMS 2-North-A and MMS 2-South,
respectively. The synthesized beam size for continuum is denoted by a filled
dot, and for CO, it is denoted by an empty ellipse in the bottom left corner.
Illustration in the bottom left of (a) shows the proposed schematic diagram of
MMS 2 CO outflow. The red and blue lobes represent the red- and blue-shifted
CO components associated with MMS 2-North-A. The magenta and cyan lobes
represent the red- and blue-shifted CO components associated with MMS 2-South.
Right panels: Zoomed-in images of left panel images overlaid with $E$-vectors
(black line segments) and contours of Stokes $I$ (grey). The colour shows the
PI. The blue and red arrows show the outflow axes of the blue- and red-shifted
components. The contour levels in (b) are [3, 9, 27, 60, 260, 460, 660, 860,
1060]$\times\sigma$ (1$\sigma$ = 37.6 $\mu$Jy beam-1) and in (d) are [3, 9,
27, 81, 250, 500, 750]$\times\sigma$ (1$\sigma$ = 60.0 $\mu$Jy beam-1). The
$E$-vectors in all panels are set to have a uniform length. The synthesized
beam size for continuum is denoted by a filled ellipse in the bottom left
corner. The nearside of the disk is on the red-shifted CO emission. The orange
arrow in the upper right corner indicates the large-scale $E$-vector
orientation presented by Matthews & Wilson (2000). Figure 2 (Continued): Left
panels: The blue and red contours are the integrated intensity of the CO ($J$
= 2$-$1) emission tracing the outflow. The CO emission in (e) is integrated
from $v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = $-$5 to $-$40 km s-1 (blue)
and $v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = 5 to 50 km s-1 (red), in (g)
is integrated from $v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = $-$1 to $-$13
km s-1 (blue) and $v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = 1 to 7 km s-1
(red), respectively. The grey contours in (e) are the Stokes $I$. The contour
levels in (e) are [5, 10, 15, 20, 25, 30, 40, 50, 60, 70]$\times\sigma$
(1$\sigma$ = 0.0925 Jy beam-1 (red and blue)) and [5, 25, 125, 250, 500, 750,
1000, 1250, 1500, 1750]$\times\sigma$ (1$\sigma$ = 75.6 $\mu$Jy beam-1
(grey)). The contour levels in (g) are [5, 10, 15, 20, 25, 30, 40,
50]$\times\sigma$ (1$\sigma$ = 0.11 Jy beam-1 (red and blue)). The colour
shows the PI. The synthesized beam size for continuum is denoted by a filled
dot, and for CO, it is denoted by an empty ellipse in the bottom left corner.
Right panels: Zoomed-in images of left panel images overlaid with $E$-vectors
(black line segments) and contours of Stokes $I$ (grey). The blue and red
arrows show the outflow axes of the blue- and red-shifted CO components. The
contour levels in (f) are [5, 25, 125, 250, 500, 750, 1000, 1250, 1500,
1750]$\times\sigma$ (1$\sigma$ = 75.6 $\mu$Jy beam-1) and in (h) are [3, 9,
27, 81, 250, 500, 750, 1000, 1250, 1500, 1750]$\times\sigma$ (1$\sigma$ = 49.3
$\mu$Jy beam-1). The $E$-vectors in all panels are set to have a uniform
length. The synthesized beam size for continuum is denoted by a filled ellipse
in the bottom left corner. The nearside of the disk is on the red-shifted CO
emission. The orange arrow in the upper right corner indicates the large-scale
$E$-vector orientation presented by Matthews & Wilson (2000).

####  3.1.2 MMS 2

MMS 2 was first identified by Chini et al. (1997) in the 1300 $\mu$m survey,
also known as CSO 6 (Lis et al., 1998) and SMM 3 (Takahashi et al., 2013) as
listed in Table 1 fourth column. The 10 $\mu$m observations found that this
object is a binary system consisting of two infrared sources showing the Class
I-type spectral index333Spectral index ($\alpha$) describes an exponential
factor relating the flux density of a radio source to its frequency, which is
used to classify young stellar objects (YSOs) into the evolutionary stages in
the near- and mid-infrared regions, given by
$\alpha=\frac{d\textnormal{log}(\lambda
F_{\lambda})}{d(\textnormal{log($\lambda$)})}$, where $\lambda$ is the
wavelength and $F_{\lambda}$ is the flux density (e.g., Lada, 1987; Andre &
Montmerle, 1994; Greene et al., 1994). (Nielbock et al., 2003). The separation
between these two sources is $1\hbox to0.0pt{.\hss}\arcsec 3$ (Nielbock et
al., 2003). MMS 2 was also identified as HOPS-92 by Furlan et al. (2016)
through the multi-wavelength infrared observations. MMS 2 was subsequently
classified as a flat-spectrum source based on the SED study, in which its mid-
infrared (4.5$–$24 $\mu$m) spectral index is between $-\,0.3$ and 0.3 (Furlan
et al., 2016). The bolometric luminosity of this source is estimated to be
$\sim$17.6 $L_{\odot}$ (Furlan et al., 2016). In the recent 0.87 mm ALMA
observations, HOPS-92 was further resolved into three sources, HOPS-92-A-A,
HOPS-92-A-B, and HOPS-92-B (Tobin et al., 2020). A bipolar CO outflow with
one-sided length of $\sim$0.2 pc (P.A. $\sim$ 90∘) elongated along the east-
west direction, was observed in previous studies (Aso et al., 2000; Williams
et al., 2003; Takahashi et al., 2008; Tanabe et al., 2019). Recent CARMA-
Nobeyama Radio Observatory Orion (CARMA-NRO Orion) survey reported a more
elongated one-sided length of $\sim$0.48 pc associated with this outflow
(Feddersen et al., 2020). Furthermore, the Infrared Imager (IRIM) by Yu et al.
(1997) detected an east-west oriented outflow consisting of a chain of H2
knots associated with MMS 2.

As shown in Figure 1 (d), we have detected centrally condensed structures
associated with all three HOPS sources in Stokes $I$, in which our
observations are consistent with the 0.87 mm ALMA observations (Tobin et al.,
2020). In our observations, MMS 2-North-A, MMS 2-North-B, and MMS 2-South are
associated with HOPS-92-A-A, HOPS-92-A-B, and HOPS-92-B, respectively. MMS
2-North-B is detected 0$\hbox to0.0pt{.\hss}\arcsec$3 southeast with respect
to MMS 2-North-A. This source is fitted with five components as listed in
Table 5. Both MMS 2-North-A and MMS 2-South are fitted with a single compact
component and a single extended component, while MMS 2-North-B is fitted with
only a single compact component. MMS 2-North-A compact component has a linear
size of $\sim$39 au $\times$ 36 au (P.A. $=$ 149∘), and is likely tracing a
nearly face-on disk with an inclination angle of $i\sim 26^{\circ}$. The
fitted compact component of MMS 2-South has a linear size of $\sim$59 au
$\times$ 40 au (P.A. = 173∘), which likely traces a disk with an inclination
angle of $i\sim 46^{\circ}$. The extended emission of MMS 2-North-A has a
fitted linear size of $\sim$244 au $\times$ 204 au (P.A. $=$ 13∘), while the
extended emission of MMS 2-South has a linear size of $\sim$96 au $\times$ 72
au (P.A. $=$ 5.5∘). Both the northern and southern extended emissions show a
spiral arm-like structure, which is detected at the 3$\sigma$ level in Stokes
$I$. These spiral arm-like structures seem to bridge the two compact
millimeter sources, which becomes more clear when we produce the image with
the natural weighting (see Figure D1 in Appendix D). MMS 2-North-B is one the
weakest source and has a very compact structure with a linear size of $\sim$16
au $\times$ 2.8 au (P.A. = 101∘). We consider this component is unresolved
because the fitting error is larger than its fitted deconvolved size (see also
Appendix A).

Figure 2 (a) shows the outflows associated with MMS 2-North-A and MMS 2-South,
which are traced by the CO emissions detected at $\gtrsim 5\sigma$ level in
the velocity range of $v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = $-$17 to
33 km s-1 (see Figure B2 in Appendix B). We detected the outflows elongated
mainly along the east-west direction, in which the blue- and red-shifted
components are overlapping. These results are consistent with previous single
dish observations (Aso et al., 2000; Williams et al., 2003; Takahashi et al.,
2008; Tanabe et al., 2019; Feddersen et al., 2020). In addition, we resolved
the red-shifted components, which are possibly associated with MMS 2-North-A
and MMS 2-South, respectively. The red-shifted component on the eastern side
likely traces the outflow cavity and appears to be associated with MMS
2-North-A, while the red-shifted component in the southwest seems to be more
associated with MMS 2-South. We could not spatially resolved the blue-shifted
components associated with these two sources because the blue-shifted
components seem to merge with each other and appears as one along the east-
west direction. We propose a scenario for MMS 2 CO outflow (see the
illustration in the bottom left of Figure 2 (a)). For MMS 2-North-A, the red-
shifted (red) and blue-shifted (dark blue) components are detected from both
eastern and western sides (see Figures B2 (d)$-$(g)). The red- and blue-
shifted components can be connected by the parabolic lines, respectively.
Moreover, MMS 2-North-A has a nearly face-on disk and the red- and blue-
shifted components are overlapping. We think that this could be a nearly pole-
on outflow elongated along the north-south direction. MMS 2-South has a blue-
shifted (cyan) component with P.A. $\sim$ 90∘, and a red-shifted (magenta)
component in the southwest direction (see the illustration in the bottom left
of Figure 2 (a)).

As shown in Figure 2 (b), PI is detected for both MMS 2-North-A and MMS
2-South, while no PI is detected towards MMS 2-North-B. PI of MMS 2-North-A
shows a ring-like structure and the peak of the PI is located within the ring
(P.A. = 13∘). The PI peak associated with MMS 2-South is offset to the
southwest by $\sim$0$\hbox to0.0pt{.\hss}\arcsec$02 with respect to the Stokes
$I$ peak. The PI peak shifts towards the red-shifted outflow, which
corresponds to the nearside of the disk (consider that the outflow is ejected
in the perpendicular direction of the disk). The PI peak shift is larger than
the positional accuracy by a factor of $\sim$5\. The orientations of the
$E$-vectors are completely different among these two sources (see Figure 2
(b)). The $E$-vectors of MMS 2-North-A trace a ring-like structure in an
azimuthal direction, whereas the $E$-vectors of MMS 2-South are almost
parallel to the minor axis of its disk in the central disk region. The P.A. of
the $E$-vectors for MMS 2-South peaked at 66∘$-$87∘ (see Figure 3 (b)).
Additionally, the $E$-vectors of MMS 2-South are distributed azimuthally in
the outer disk region.

Figure 4 (a) shows the polarization fraction detected towards MMS 2. The
polarization fractions are detected in the ranges of 0.4$-$1.6% and 0.8$-$2.4%
for MMS 2-North-A and MMS 2-South, respectively (see Figure 5 (a) and (b)).
The differences in the polarization fractions among these two sources could be
induced by the inclination effect, which will be discussed in Section 4.2.

####  3.1.3 MMS 3

MMS 3 was first identified by Chini et al. (1997) in the 1300 $\mu$m survey,
also known as CSO 7 (Lis et al., 1998) and SMM 4 (Takahashi et al., 2013) as
listed in Table 1 fourth column. This millimeter source was classified as
Class 0 source by Chini et al. (1997) with a
$L_{\textnormal{bol}}$/$L_{\textnormal{smm}}$ of $<$ 200 (Andre et al., 1993).
Multi-wavelength infrared observations by Furlan et al. (2016) found that
HOPS-91 is associated with MMS 3, and this source is classified as Class 0
source based on the SED study. The bolometric luminosity of MMS 3 is estimated
to be $\sim$3.6 $L_{\odot}$ (Furlan et al., 2016). The classification is in
agreement with Chini et al. (1997). There was no sign of outflows towards this
source, as had been reported by previous observations (Aso et al., 2000;
Williams et al., 2003; Takahashi et al., 2008; Tanabe et al., 2019; Feddersen
et al., 2020). Recent ALMA observations by Morii et al. (2021) detected a
centrally concentrated compact continuum source and also a compact bipolar
outflow with one-sided length of 5800 au (P.A. $\sim$ 54∘) in CO ($J$ = 2$-$1)
emission for the first time (see Figure B3 in Appendix B). They could confirm
that MMS 3 is a protostellar source not a prestellar source. Morii et al.
(2021) also suggested that MMS 3 is a Class 0 source with a lower accretion
rate compared to the other protostellar sources in this region. In addition,
MMS 3 also has the lowest Stokes $I$ peak flux among all other protostellar
sources observed in this region.

Figure 1 (g) shows that we have detected both centrally concentrated and
extended components associated with MMS 3 in Stokes $I$. This source is fitted
with one compact component and two extended components as listed in Table 5.
The fitted compact component of MMS 3 has a linear size of $\sim$64 au
$\times$ 38 au (P.A. = 54∘), possibly traces a dust disk. This compact
component was also detected in the 1.3 mm continuum observations with an
angular resolution of $\sim 0\hbox to0.0pt{.\hss}\arcsec 2$ (78 au), extending
along the north-south direction and having a linear size of $\sim$176 au
$\times$ 125 au (Morii et al., 2021). The size of this compact structure
obtained from Morii et al. (2021) is about twice as large as ours. The
differences in sizes could be attributed to the differences in angular
resolutions as well as the fitting methods. First, the angular resolution of
the observations by Morii et al. (2021) is slightly lower than ours. Second,
Morii et al. (2021) fitted this source with a single component, which could
result in a larger size than fitting with multiple components, this will be
discussed in Section 4.2. Furthermore, we detected two extended components.
These two extended components have the linear sizes of $\sim$213 au $\times$
161 au (P.A. = 9.7∘) and $\sim$601 au $\times$ 520 au (P.A. = 87∘),
respectively. The larger extended component (P.A. = 87∘) presumably traces an
envelope elongated along the east-west direction.

MMS 3 is not detected in Stokes $Q$ and $U$ as shown in Figure 1 (h) and (i).
The 3$\sigma$ upper limit of the polarization fraction is $\sim$2.1$\%$. This
value is much higher than the minimum detectable linear polarization for both
compact and extended sources, which means that our observations may be limited
in sensitivity in order to detect the polarization fraction at a minimum of
3$\sigma$ level towards this source.

Figure 3: Histograms of the $E$-vector position angles for MMS 2-North-A, MMS
2-South, MMS 5, MMS 6-Center, MMS 6-Arm-like structure, MMS 7-North, and MMS
7-South. By taking into account of the 180∘ ambiguity in linear polarization
position angles, we added 180∘ to any $E$-vector position angle that is
$<0^{\circ}$ to better present the distributions of the $E$-vector position
angles. The $E$-vector position angles are measured from north towards east.

####  3.1.4 MMS 4

MMS 4 was first identified by Chini et al. (1997) in the 1300 $\mu$m survey
and classified as Class 0 source based on the
$L_{\textnormal{bol}}$/$L_{\textnormal{smm}}$ of $<$ 200 (Andre et al., 1993).
This source was also identified as CSO 8 (Lis et al., 1998) and SMM 5
(Takahashi et al., 2013) as listed in Table 1 fourth column. There was no
evidence of molecular outflow towards this source and no infrared counter part
had been detected for this source (Yu et al., 1997; Stanke et al., 2002; Aso
et al., 2000; Williams et al., 2003; Takahashi et al., 2008; Feddersen et al.,
2020). Takahashi et al. (2013) revealed that MMS 4 has a triangular-shaped
structure with a linear size of 7870 au $\times$ 2150 au. The gas number
density of this source is a factor of 10 smaller than most of the sources
observed in this region (Takahashi et al., 2013), which is comparable to the
other prestellar cores (e.g., $\sim$1$\times 10^{6}$ cm-3; Caselli et al.,
2019). Therefore, MMS 4 was considered as a prestellar source and the youngest
source among the sources presented in this paper. In addition, Hirano et al.
(2023) proposed that the nucleus of MMS 4 is either on the brink of forming
the first hydrostatic core or is a potential candidate for the first
hydrostatic core.

Figure 1 (j) shows an elongated structure with axis ratio of $\sim$0.1
detected towards MMS 4 in Stokes $I$. This source is fitted with a relatively
compact component and a largely extended component as listed in Table 5. We
refer to the compact component detected in MMS 4 as a “$relatively$ compact
component” because it is larger than the compact components of the other
sources by a factor of $\sim$4$-$50\. The relatively compact component has a
linear size of $\sim$511 au $\times$ 259 au (P.A. = 131∘), which is extended
to the northwest with respect to the Stokes $I$ peak. In addition, there is a
narrow and extended emission with a linear size of $\sim$4583 au $\times$ 546
au (P.A. = 120∘), which is elongated along the southeast-northwest direction.
This elongated structure is consistent with the structure of SMM 5 detected by
Takahashi et al. (2013), which likely traces either the outer part of the
envelope or the inner part of a core.

MMS 4 is not detected in Stokes $Q$ and $U$ as shown in Figure 1 (k) and (l).
This source is the youngest and the emission is also the weakest among all
other sources. The 3$\sigma$ upper limit of the polarization fraction is
$\sim$3.7$\%$, which is much higher than the minimum detectable linear
polarization. As with MMS 3, our observations may be limited in sensitivity.

Although MMS 4 is identified as a prestellar source, it is possible that a
compact core has not yet developed. Furthermore, the number density of $\sim$2
$\times 10^{9}$ cm-3 is quite high, of which the value is comparable to that
of the compact component of the Class 0 source MMS 3 (number densities can be
found in the last column of Table 7). This value is also approaching the
theoretical value expected for the first hydrostatics core ($\gtrsim 10^{10}$
cm-3; e.g., Larson, 1969; Masunaga & Inutsuka, 2000; Machida et al., 2008;
Joos et al., 2012). In addition, we have detected a low-velocity blue-shifted
CO emission associated with the relatively compact Stokes $I$ emission of MMS
4 (see Figure B4 in Appendix B). This blue-shifted CO emission is elongated
towards the western direction with respect the Stokes $I$ peak, with a
velocity ranging from $v_{\textnormal{LSR}}=6$ to $7$ km s-1. The high number
density and marginal detection of CO emission indicate that MMS 4 might be in
the first hydrostatics core stage, which is in agreement with Hirano et al.
(2023).

Figure 4: Polarization fraction (greyscale) overlaid with the contours of
Stokes $I$ (blue) and the inferred $B$-vectors (green line segments) for (a)
MMS 2, (b) MMS 5, (c) MMS 6, and (d) MMS 7. The blue contours correspond to
[3, 9, 27, 60, 260, 460, 660, 860, 1060]$\times\sigma$ (1$\sigma$ = 36.7
$\mu$Jy beam-1) in panel (a), [3, 9, 27, 81, 250, 500, 750]$\times\sigma$
(1$\sigma$ = 60.0 $\mu$Jy beam-1) in panel (b), [5, 25, 125, 250, 500, 750,
1000, 1250, 1500, 1750]$\times\sigma$ (1$\sigma$ = 75.6 $\mu$Jy beam-1) in
panel (c), and [3, 9, 27, 81, 200, 400, 600, 800, 1000, 1200, 1400,
1600]$\times\sigma$ (1$\sigma$ = 49.3 $\mu$Jy beam-1) in panel (d). The
inferred B-vectors are obtained by rotating the $E$-vectors (in Figure 2) by
90∘, assuming that magnetic field is the polarization origin. The $B$-vectors
in all panels are set to have a uniform length. The $B$-vectors and the
polarization fraction are shown where the PI is $\geq$ 3$\sigma$ level. The
red and blue arrows show the outflow axes for the red- and blue-shifted CO
components as presented in Figure 2.

####  3.1.5 MMS 5

MMS 5 was identified by Chini et al. (1997) in the 1300 $\mu$m survey. This
source is also known as CSO 9 (Lis et al., 1998) and SMM 6 (Takahashi et al.,
2013) as listed in Table 1 fourth column. Multi-wavelength infrared
observations by Furlan et al. (2016) found that HOPS-88 is associated with MMS
5. The bolometric luminosity of this source is estimated to be $\sim$13.8
$L_{\odot}$ (Furlan et al., 2016). MMS 5 is classified as a Class 0 source
(Chini et al., 1997; Furlan et al., 2016). A compact bipolar CO outflow with
one-sided length of $\sim$0.1 pc (P.A. $\sim$ $-$90∘), which is elongated
along the east-west direction, as reported in the CO outflow surveys by Aso et
al. (2000), Williams et al. (2003), Takahashi et al. (2008), Tanabe et al.
(2019), and Feddersen et al. (2020). Moreover, Yu et al. (1997) and Stanke et
al. (2002) reported a chain of H2 emission knots along the east-west
direction, among which three knots associated with the blue-shifted lobe of
the CO outflow were also reported by Takahashi et al. (2008). Recent ALMA CO
(2$-$1) observations by Matsushita et al. (2019) spatially resolved the CO
outflow showing the cavity-like structure, which extends up to $\sim$14000 au
(P.A. $\sim$ 79∘). Additionally, Matsushita et al. (2019) also detected an EHV
jet, which extends up to $\sim 10000$ au (P.A. $\sim$ 96∘) with a line-of-
sight velocity of 100 km s-1. The EHV jet was also detected in CO (6$-$5)
observations by Gómez-Ruiz et al. (2019). Furthermore, Matsushita et al.
(2019) imaged an envelope on a $\sim$2000 au scale in the C18O (2$-$1)
emission, revealing a rotational motion of the envelope. The rotation could be
fitted by either Keplerian rotation or rotation due to angular momentum
conservation Matsushita et al. (2019).

As shown in Figure 1 (m), we have detected the Stokes $I$ emission associated
with MMS 5. The detected emission shows a centrally concentrated compact
component with the V-shaped extended emission opening to the northwest
direction. This source is fitted with a single compact component and a single
extended component as listed in Table 5. The compact component has a linear
size of $\sim$35 au $\times$ 28 au (P.A. = 1.6∘), which traces a dust disk
around the protostar. The western arm of the extended V-shape structure seems
to be partially correlated with the blue-shifted component of the EHV jet,
while the northern arm of the extended V-shape structure is more or less
correlated with the wide-opening blue-shifted component of the outflow. This
implies that the extended emission traces the warm dust caused by the
interaction between the outflow and the surrounding dense gas.

The molecular line observations for MMS 5 were reported by Matsushita et al.
(2019) as described earlier. Figure 2 (c) shows the outflow and EHV jet
associated with MMS 5, which are traced by the CO emissions detected at
$\gtrsim 5\sigma$ in the velocity ranges of
$v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = $\pm$5 to $\pm$50 km s-1 and
$v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = $-$105 to 70 km s-1,
respectively (see Figure B5 in Appendix B). Both blue- and red-shifted
components of the outflow show a cavity-like structure as seen in Figure 2
(c). The red-shifted component of the outflow has a projected length of
$\sim$7000 au with P.A. $\sim$ 69∘ extends along the northeast-southwest
direction, while the blue-shifted component of the outflow has a projected
length of $\sim$7900 au with P.A. $\sim$ $-95^{\circ}$ extends along the east-
west direction. The red-shifted component of the EHV jet has a length of
$\sim$4300 au with P.A. $\sim$ 82∘, while the blue-shifted component of the
EHV jet has a length of $\sim$6200 au with P.A. $\sim$ $-$98∘.

As shown in Figure 2 (d), PI is mainly detected around the Stokes $I$ peak in
the central $\sim$0$\hbox to0.0pt{.\hss}\arcsec$25 (100 au) region, whereas a
small portion of the outlying component is detected at the $\sim$0$\hbox
to0.0pt{.\hss}\arcsec$3 western side with respect to the Stokes $I$ peak. We
found that the PI exhibits a peak offset of $\sim$0$\hbox
to0.0pt{.\hss}\arcsec$03 towards the nearside of the disk with respect to the
Stokes $I$ peak (see Figure 2 (d)). The peak shift is larger than the
positional accuracy by a factor of $\sim$4\. The $E$-vectors seem to peak at
P.A. = 100∘$-$116∘ (see Figure 3 (c)), which are almost parallel to the minor
axis of the centrally compact component traced by the Stokes $I$ emission.

Figure 4 (b) shows the polarization fraction detected towards MMS 5. The high
polarization fractions of $\gtrsim$ 4$-$6% are mainly originated from the
outlying PI component rather than from the central region of the disk. We plot
the histogram with a focus only on the central 100 au region of MMS 5 to avoid
the interference of noise from those outlying regions. The centrally compact
component has a polarization fraction of 0.4$-$1.3% (see Figure 5 (c)).

####  3.1.6 MMS 6

MMS 6 was first identified in the 1300 $\mu$m survey by Chini et al. (1997).
This millimeter source is also known as CSO 12 (Lis et al., 1998) and SMM 8
(Takahashi et al., 2013) as listed in Table 1 fourth column. Furthermore, this
source was identified as HOPS-87 in the multi-wavelength infrared observations
(Furlan et al., 2016), and was subsequently classified as a Class 0 source due
to its positive mid-infrared (4.5–24 $\mu$m) spectral index and a bolometric
temperature ($T_{\textnormal{bol}}$) of $<$ 70 K (Furlan et al., 2016). The H2
knots possibly associated with MMS 6 were reported by Yu et al. (1997), Stanke
et al. (2002), and Williams et al. (2003). The CO outflow was not clearly
observed towards this source (Aso et al., 2000; Williams et al., 2003;
Matthews et al., 2005; Takahashi et al., 2008). MMS 6 was resolved as MMS
6-main and MMS 6-NE in SMA 0.9 mm continuum observations by Takahashi et al.
(2009), for which MMS 6-main was identified as a bright and the most compact
source in the OMC-3 region. Takahashi & Ho (2012) detected a very compact
bipolar outflow with one-sided length of $\sim$1000 au in CO (3–2) and HCN
(4–3) line observations towards MMS 6-main. All above evidences suggest that
MMS 6 could be a protostellar core at the earliest stage (Takahashi & Ho,
2012; Takahashi et al., 2012).

As shown in Figure 1 (p), we have detected a centrally condensed Stokes $I$
emission associated with MMS 6. In the northern region, we have also detected
an extended emissions showing an arm-like structure, which is mostly elongated
along the northeast-southwest direction. Here, we refer to the centrally
condensed emission as MMS 6-Center (within central $\sim$500 au region), and
the extended emission as arm-like structure (see Figure 1 (p)). MMS 6 is one
of the youngest sources, while it has the highest Stokes $I$ peak flux among
all other sources.

As shown in Figure 2 (e), MMS 6 is associated with the most compact CO outflow
among MMS 2, MMS 5, and MMS 7. The outflow traced by the CO emission is
detected at $\gtrsim 5\sigma$ in the velocity range of
$v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = $-$40 to 50 km s-1 (see Figure
B6 in Appendix B). At $v_{\textnormal{LSR}}-v_{\textnormal{sys}}$ = 50 km s-1,
we have detected a red-shifted component that may trace an EHV jet (see Figure
B6 (j)). The projected lengths of the red- and blue-shifted CO components are
comparable with each other, which the one-sided length is $\sim$1200 au with
P.A. $\sim$ 6∘ (see Figure 2 (e)). The high angular resolution images of the
outflow and EHV jet traced by CO (2$-$1) and SiO (5$-$4) emissions obtained
through the same project, will be presented in Takahashi et al. (2023b, in
preparation).

In Figure 2 (f), the PI associated with MMS 6-Center exhibits a peak that is
offset by $\sim$0$\hbox to0.0pt{.\hss}\arcsec$04 towards the nearside of the
disk with respect to the Stokes $I$ peak. The PI peak shift is larger than the
positional accuracy by a factor of $\sim$110\. Such a high value results from
its extremely large S/N ($\sim$175). $E$-vectors are peaked at P.A. =
130∘$-$150∘ (see Figure 3 (d)). We have also detected the arm-like structure
in the PI from the northern extended emission. The PI associated with this
arm-like structure extends up to $\sim$1400 au along the northeast-southwest
direction. The $E$-vectors are mainly parallel to the minor axis of this
extended arm-like structure (Takahashi et al., 2019).

As shown in Figure 4 (c), MMS 6-Center exhibits a high polarization fraction
of 0.4$-$10.4%, and the arm-like structure displays a significantly high
polarization fraction of 2.7$-$47.4% (See Figure 5 (d) and (e)). MMS 6
polarization analysis was thoroughly presented by Takahashi et al. (2019).

Figure 5: Histograms of the polarization fractions for MMS 2-North-A, MMS
2-South, MMS 5, MMS 6-Center, MMS 6-Arm-like structure, MMS 7-North, and MMS
7-South. The vertical dashed line and $\mu$ indicate the mean polarization
fraction. The errors of the mean, which includes both statistical and
systematic errors, is shown in each histogram and is denoted by the shaded
region. The method for calculating the mean and its errors is explained in the
beginning of Section 3.

####  3.1.7 MMS 7

MMS 7 was first identified by Chini et al. (1997) in the 1300 $\mu$m survey,
and was classified as Class 0 source based on the
$L_{\textnormal{bol}}$/$L_{\textnormal{smm}}$ of $<$ 200 (Andre et al., 1993).
This source is also known as CSO 12 (Lis et al., 1998) and SMM 11 (Takahashi
et al., 2013) as listed in Table 1 fourth column. The multi-wavelength
infrared observations by Furlan et al. (2016) found that HOPS-84 is associated
with MMS 7. This source was subsequently classified as Class I source due to
its Class I-type SED (Furlan et al., 2016). The bolometric luminosity of MMS 7
is estimated to be $\sim$42.9 $L_{\odot}$ (Furlan et al., 2016). This source
was detected as a single source in previous observations (Chini et al., 1997;
Lis et al., 1998; Takahashi et al., 2006, 2008, 2013; Furlan et al., 2016).
Recent ALMA observations by Tobin et al. (2020) resolved HOPS-84 into two
sources, HOPS-84-A and HOPS-84-B in the 0.87 mm continuum image. In this
paper, MMS 7-North and MMS 7-South are associated with HOPS-84-A and
HOPS-84-B, respectively. A giant bipolar molecular outflow elongated along the
east-west direction was observed in the CO observations (Aso et al., 2000;
Williams et al., 2003; Takahashi et al., 2006), and the red-shifted lobe of
the CO outflow is significantly more elongated than the blue-shifted lobe
(Takahashi et al., 2008; Tanabe et al., 2019; Feddersen et al., 2020). The
size of the blue-shifted lobe is $\sim$0.12 pc and the red-shifted lobe
extends up to $\sim$0.83 pc (Feddersen et al., 2020). Furthermore, MMS 7 was
also identified as VLA 4 in the 3.6 cm VLA detection. The observed flux
associated with VLA 4 might be due to free-free emission from the radio jets
(Reipurth et al., 1999, 2004). The faint knots along the CO outflow were also
detected by Takahashi et al. (2008). Takahashi et al. (2006) detected a disk-
like envelope with a fan-shaped structure around MMS 7 by Nobeyama Millimeter
Array (NMA) in H13CO+(1$-$0) emission. Moreover, Ren et al. (2021) identified
the filamentary infall motion towards MMS 7 region using the ALMA and SMA.

As shown in Figure 1 (s), we have detected centrally condensed Stokes $I$
emissions associated with both MMS 7-North and MMS 7-South. MMS 7-South is
detected $\sim$0$\hbox to0.0pt{.\hss}\arcsec$7 (275 au) southwest with respect
to MMS 7-North. MMS 7 is fitted with two components, in which both MMS 7-North
and MMS 7-South are fitted with a single compact component as listed in Table
5. The compact component of MMS 7-North has a linear size of $\sim$95 au
$\times$ 42 au (P.A. = 166.5∘), which possibly traces a compact dust disk. MMS
7-South has a linear size of $\sim$37 au $\times$ 17 au (P.A. = 165.8∘), which
is about twice more compact than MMS 7-North and also shows a disk-like
structure. The disk of MMS 7-North is elongated almost perpendicular to its
associated outflow. The disks of both MMS 7-North and MMS 7-South are nearly
edge-on with an inclination angle of $i\sim 64^{\circ}$ and $i\sim
60^{\circ}$, respectively.

Figure 2 (g) shows the outflow associated with MMS 7, which is traced by the
CO emission detected at $\gtrsim 5\sigma$ in the velocity range of
$v_{\textnormal{LSR}}-v_{\textnormal{sys}}=-13$ to 7 km s-1 (see Figure B7 in
Appendix B). We detected a blue-shifted component of the CO emission showing a
cavity-like structure extended to $\sim$10000 au along the northeast-southwest
direction with P.A. $\sim$ 80∘. The red-shifted component of the CO emission
with P.A. $\sim$ 106∘ is detached into two parts along the northeast-southwest
direction.

In Figure 2 (h), PI is detected for both MMS 7-North and MMS 7-South. A strong
PI of MMS 7-North is detected, and its peak exhibits an offset of
$\sim$0$\hbox to0.0pt{.\hss}\arcsec$02 towards the nearside of the disk with
respect to Stokes $I$ peak. The peak shift is larger than the positional
accuracy by a factor of $\sim$9\. On the other hand, only a small and weak
patch of PI is detected within the central 30 au region of MMS 7-South. The
$E$-vectors of both MMS 7-North and MMS 7-South are mainly parallel to the
minor axes of their disks with P.A. = 64∘$-$84∘ and P.A. =
75${}^{\circ}-$$80^{\circ}$, respectively (see Figure 3 (f) and (g)). Near the
edge of the disk of MMS 7-North, the orientation of the $E$-vectors is more or
less in an azimuthal direction (see Figure 2 (h)).

Figure 4 (d) shows the polarization fraction detected towards MMS 7. MMS
7-North exhibits a polarization fraction of 0.4$-$1.7%, and MMS 7-South
exhibits a polarization fraction of 0.9$-$1.1% (see Figure 5 (f) and (g)).

###  3.2 Multi-scale E-vector orientations and outflow direction

####  3.2.1 Comparison of E-vectors at 100$-$1000 au and 10000 au

In the right panels of Figure 2, we compare the $E$-vector orientations on the
disk-scale ($\sim$55 au) for MMS 2, MMS 5, MMS 6, and MMS 7 with the
$E$-vector orientation associated with the filamentary cloud-scale ($\sim$7500
au), as reported by the 850 $\mu$m SCUMBA/JCMT observations towards OMC-3
region (Matthews & Wilson, 2000). We found that there is no clear correlation
in the $E$-vectors between the cloud-scale and the disk-scale for MMS 2, MMS
5, MMS 7, and MMS 6-Center. Note that MMS 6 image also shows the $E$-vector
orientation of the extended arm-like structure coincides with the $E$-vector
orientation observed in the filamentary cloud-scale (see Figure 2 (f)). The
non-correlation in the $E$-vector orientations between the filamentary cloud-
scale and the disk-scale could potentially suggest different polarization
origins between the filament and the individual sources. Alternatively,
assuming that polarization on disk-scale is originated from the magnetic
field, there might be no $B$-vector correlation between the cloud-scale and
the disk-scale.

####  3.2.2 Comparison of E-vector orientations at 100 au and outflow
directions

The right panels of Figure 2 also present the relation between the $E$-vector
orientation and the outflow axis. The $E$-vector orientations seem to
correlate with the outflow axes for MMS 2-South, MMS 5, MMS 6, and MMS 7-North
within a deviation of $\sim$$10^{\circ}$$-$$40^{\circ}$. If we assume the
polarization origin is magnetic field444The dust polarization towards these
sources can be explained by multiple mechanisms and the details of the
polarization origins will be discussed in 4.1. Here, we assume the magnetic
field is the polarization origin to show the inferred magnetic field
orientation., the orientation of the inferred $B$-vectors for each source is
obtained by rotating the $E$-vectors by 90∘ as shown in Figure 4. The outflow
axes are perpendicular to the inferred $B$-vectors for most of sources, which
can be explained by the toroidal magnetic field configuration, as discussed in
Takahashi et al. (2019) for MMS 6.

###  3.3 Physical properties

The dust mass $M_{\textnormal{dust}}$ of the each component is estimated as

$M_{\textnormal{dust}}=\frac{F_{\lambda}d^{2}}{\kappa_{\lambda}B_{\lambda}(T_{\textnormal{dust}})},$
(3)

where $F_{\lambda}$ is the total flux density, $d$ = 393 pc is the distance to
the source (Tobin et al., 2020), $B_{\lambda}(T_{\textnormal{dust}})$ is the
Planck function as a function of dust temperature $T_{\textnormal{dust}}$,
$\kappa_{\lambda}$ is the dust opacity and we adopted
$\kappa_{\lambda}=0.00144$ cm2g-1 at $\lambda=1.1$ mm, which is interpolated
from the coagulation model of dust grains coated with thin ice mantles at a
gas density of 108 cm-3 (Ossenkopf & Henning, 1994). We assume a gas-to-dust
mass ratio of 100. We estimate the dust temperature from previous studies with
two different scenarios. One is from Tobin et al. (2020), where the dust is
heated by stellar radiation. The dust temperature $T_{\textnormal{dust}}$
adopted from Tobin et al. (2020) is

$T_{\textnormal{dust}}=T_{0}\left(\frac{L_{\textnormal{bol}}}{L_{\odot}}\right)^{0.25}\left(\frac{r}{50\textnormal{au}}\right)^{-0.46},$
(4)

where $T_{0}$ = 43 K, $L_{\textnormal{bol}}$ is the bolometric luminosity, and
$r$ is the radius (we use the geometric mean of the semi-major and semi-minor
axes of each deconvolved component). $L_{\textnormal{bol}}$ is obtained from
the HOPS catalog. MMS 1 is not detected in HOPS sources and given that the
lowest $L_{\textnormal{bol}}$ among the detected HOPS sources in OMC-3 region
is $<0.05$ $L_{\odot}$, we assume that MMS 1 has a bolometric luminosity of
$L_{\textnormal{bol}}\simeq 0.05$ $L_{\odot}$. The other scenario is that the
dust is heated by disk accretion heating (Xu & Kunz, 2021). The dust
temperatures adopted from disk accretion heating model are interpolated from
Figure 16 (leftmost panel: the case of 0.1 $M_{\odot}$) in Xu & Kunz (2021).
We notice that for most of the compact components, the temperatures derived
from these two models are comparable. However, for disk accretion heating
model, temperature is mostly affected by the inner region ($r$ $\lesssim$ 30
au) of the disk. Especially for MMS 1 compact component, the temperature
derived by disk accretion heating is about 10 times higher than that by
stellar radiation. The extended components were not significantly affected by
disk accretion heating. The derived dust temperatures for both models are
listed in Table 6. Since many components have the radii larger than 30 au, we
use the temperature derived from stellar radiation (Tobin et al., 2020) to
estimate the dust mass. For the prestellar source MMS 4, we consider that the
dust is being heated by the external sources (Zucconi et al., 2001).

Table 6: Dust Temperature ($T_{\textnormal{dust}}$)

|  Stellar Radiation Heatingc | Disk Accretion Heating
---|---|---
| $T_{\textnormal{dust}}$ | $T_{\textnormal{dust}}$
| [K] | [K]
MMS 1 compact component |  47.5$\pm$1.4 | 400
MMS 1 extended component 1 |  15.6$\pm$0.2 | 19
MMS 1 extended component 2 |  8.6$\pm$0.1 | 14
MMS 2-North-A compact component |  138.5$\pm$0.4 | 140
MMS 2-North-A extended component |  60.8$\pm$0.9 | 18
MMS 2-North-B compact componenta |  $\leq$305.0 | 550
MMS 2-South compact component |  122.9$\pm$0.3 | 95
MMS 2-South extended component |  96.0$\pm$0.5 | 25
MMS 3 compact component |  82.2$\pm$1.4 | 90
MMS 3 extended component 1 |  44.6$\pm$0.4 | 19
MMS 3 extended component 2 |  26.8$\pm$0.6 | 14
MMS 4 relatively compact componentb | - | -
MMS 4 extended componentb | - | -
MMS 5 compact component |  140.8$\pm$1.0 | 150
MMS 5 extended component |  50.7$\pm$0.5 | 17
MMS 6 compact component |  103.4$\pm$0.3 | 23
MMS 6 extended component 1 |  70.6$\pm$0.3 | 18
MMS 6 extended component 2 |  44.2$\pm$0.3 | 14
MMS 7-North compact component 1 |  135.9$\pm$0.2 | 39
MMS 7-North compact component 2 |  127.1$\pm$1.1 | 27
MMS 7-South compact component |  205.8$\pm$7.8 | 220

aMMS 2-North-B is unresolved, with only the upper limit on
$T_{\textnormal{dust}}$ provided.
bMMS 4 is a prestellar core and $T_{\textnormal{dust}}$ for MMS 4 is
extrapolated from Zucconi et al. (2001), in which the
temperature for compact and extended components are 5 and 7 K, respectively.
The dust is considered to be heated by the external sources such as cosmic
rays, therefore the outer part has higher temperature than the inner part.
cThe errors for $T_{\textnormal{dust}}$ are obtained by propagating only the
errors for deconvolved size.

Table 7: Physical Properties

|  Radius ($r$)a |  Dust Mass ($\textit{M}_{\textnormal{dust}}$)b |  Column Density ($N_{\textnormal{H}_{2}}$)c |  Number Density ($n_{\textnormal{H}_{2}}$)d
---|---|---|---|---
| (au) |  (10-4 $M_{\odot}$) |  (1025 cm-2) |  (1010 cm-3)
MMS 1 compact component |  7.9$\pm$0.5 |  0.812$\pm$0.027 |  7.901$\pm$1.059 |  66.976$\pm$9.974
MMS 1 extended component 1 |  88.5$\pm$1.9 |  4.098$\pm$0.077 |  0.317$\pm$0.015 |  0.239$\pm$0.012
MMS 1 extended component 2 |  324.1$\pm$6.9 |  41.703$\pm$0.929 |  0.240$\pm$0.012 |  0.050$\pm$0.003
MMS 2-North-A compact component |  18.7$\pm$0.1 |  1.160$\pm$0.004 |  2.012$\pm$0.024 |  7.201$\pm$0.094
MMS 2-North-A extended component |  111.8$\pm$3.6 |  0.938$\pm$0.015 |  0.045$\pm$0.003 |  0.027$\pm$0.002
MMS 2-North-B compact componente |  $\leq$3.4 |  $\leq$0.016 |  $\leq$0.857 |  $\leq$17.073
MMS 2-South compact component |  24.2$\pm$0.1 |  1.450$\pm$0.004 |  1.492$\pm$0.014 |  4.113$\pm$0.043
MMS 2-South extended component |  41.4$\pm$0.5 |  0.731$\pm$0.004 |  0.257$\pm$0.006 |  0.415$\pm$0.011
MMS 3 compact component |  24.5$\pm$0.9 |  0.154$\pm$0.003 |  0.155$\pm$0.012 |  0.421$\pm$0.035
MMS 3 extended component 1 |  92.5$\pm$2.0 |  0.625$\pm$0.006 |  0.044$\pm$0.002 |  0.032$\pm$0.002
MMS 3 extended component 2 |  279.7$\pm$12.5 |  1.652$\pm$0.046 |  0.013$\pm$0.001 |  0.003$\pm$0.000
MMS 4 relatively compact component |  181.8$\pm$5.9 |  31.036$\pm$1.387 |  0.568$\pm$0.045 |  0.209$\pm$0.018
MMS 4 extended component |  790.6$\pm$6.8 |  219.018$\pm$2.708 |  0.212$\pm$0.004 |  0.018$\pm$0.000
MMS 5 compact component |  15.8$\pm$0.2 |  1.193$\pm$0.009 |  2.891$\pm$0.089 |  12.227$\pm$0.420
MMS 5 extended component |  145.8$\pm$3.1 |  3.851$\pm$0.043 |  0.110$\pm$0.005 |  0.050$\pm$0.002
MMS 6 compact component |  48.5$\pm$0.3 |  12.258$\pm$0.038 |  3.153$\pm$0.043 |  4.346$\pm$0.065
MMS 6 extended component 1 |  111.0$\pm$1.0 |  15.351$\pm$0.071 |  0.754$\pm$0.015 |  0.454$\pm$0.010
MMS 6 extended component 2 |  307.1$\pm$4.5 |  46.718$\pm$0.365 |  0.300$\pm$0.009 |  0.065$\pm$0.002
MMS 7-North compact component 1 |  31.6$\pm$0.1 |  3.925$\pm$0.006 |  2.377$\pm$0.013 |  5.028$\pm$0.030
MMS 7-North compact component 2 |  36.6$\pm$0.7 |  0.606$\pm$0.006 |  0.274$\pm$0.010 |  0.501$\pm$0.021
MMS 7-South compact component |  12.8$\pm$1.0 |  0.191$\pm$0.007 |  0.703$\pm$0.118 |  3.666$\pm$0.684

a$r$ is the geometric mean of the semi-major and semi-minor axes of each
deconvolved component. The errors for $r$ are obtained by propagating the
errors for deconvolved size.
bThe errors for $\textit{M}_{\textnormal{dust}}$ are obtained by propagating
the errors for $\textit{T}_{\textnormal{dust}}$ and total flux density. For
MMS 4, the errors for $\textit{M}_{\textnormal{dust}}$ are obtained by
propagating only the errors for total flux density.
cThe errors for $N_{\textnormal{H}_{2}}$ are obtained by propagating the
errors for $\textit{T}_{\textnormal{dust}}$ and $r$.
dThe errors for $n_{\textnormal{H}_{2}}$ are obtained by propagating the
errors for $N_{\textnormal{H}_{2}}$ and $r$.
eMMS 2-North-B is unresolved, with only the upper limits on $r$,
$M_{\textnormal{dust}}$, $N_{\textnormal{H}_{2}}$, and
$n_{\textnormal{H}_{2}}$ provided.

The column density $N_{\textnormal{H}_{2}}$ is estimated as

$N_{\textnormal{H}_{2}}=\frac{M_{\textnormal{H}_{2}}}{\pi
r^{2}\mu_{\textnormal{H}_{2}}m_{\textnormal{H}}},$ (5)

where $M_{\textnormal{H}_{2}}$ is the gas mass obtained by multiplying the
dust mass $M_{\textnormal{dust}}$ by 100, $r$ is the geometric mean radius of
each deconvolved component, $\mu_{{}_{\textnormal{H}_{2}}}=2.8$ is the mean
molecular weight per hydrogen molecule (Kauffmann et al., 2008), and
$m_{\textnormal{H}}=1.67\times 10^{-24}$ g is the mass of hydrogen atom. The
number density is estimated as $n_{\textnormal{H}_{2}}\sim
N_{\textnormal{H}_{2}}/{r}$ by assuming each source is spherical. Table 7
summarizes the physical properties of all components, where all the estimates
are based on the temperature derived by stellar radiation (Tobin et al.,
2020). For the extremely compact disk such as MMS 1 compact component, the
dust mass estimated from the temperature derived by accretion heating is
$\sim$$0.085\times 10^{-4}$ $M_{\odot}$, which is about 10 times smaller than
the dust mass estimated from the temperature derived by stellar radiation.

The compact components of MMS 1, MMS 2-North-B, MMS 5 and MMS 7-South have
diameters smaller than the beam size by a factor of $\sim$2$-$8\. The compact
sizes could result in their high column densities and number densities
compared to the components of other sources.

##  4 Discussion

Figure 6: Illustration of the $E$-vector orientations. The $E$-vectors are
denoted by the blue solid line segments. The major axis of the disk is in the
vertical (north-south) direction. Top panel: $E$-vector orientation predicted
by dust alignment due to toroidal magnetic field. The black dashed line
denotes the toroidal magnetic field. Middle panel: $E$-vector orientation
predicted by self-scattering. The two intensity rings for face-on
configuration are represented by different greyscales in the left panel.
Bottom panel: $E$-vector orientation predicted by radiative torque. The black
solid lines denote the radiation field.

###  4.1 Origin of the polarization

In this section, we discuss the polarization origin of each source. As
explained in Section 1, there are six mechanisms producing polarized emission,
which will be explained in detail in the following paragraphs. Among these
mechanisms, the polarization fraction of the reversal of self-scattering (Yang
& Li, 2020) is expected to be $\sim$0.1%, which is equivalent to the current
ALMA systematic calibration uncertainty. Thus, we do not discuss this
mechanism further with our sources.

  1. 1.

Dust alignment by (toroidal) magnetic field: As the radiative torque rotates
the non-spherical dust grains (Draine & Weingartner, 1996), the spinning dust
grains are expected to align their major axes perpendicular to the local
magnetic field and emit polarized emission (e.g., Hildebrand, 1988; Lazarian,
2007; Andersson et al., 2015; Hull & Zhang, 2019). Polarization due to the
dust alignment by magnetic field has been mostly studied on large-scale (e.g.,
Rao et al., 1998; Matthews & Wilson, 2000; Girart et al., 2006; Koch et al.,
2022). Some observations showed that the polarization fraction could be as
high as $\gtrsim$ 10% (e.g., Rao et al., 1998; Matthews & Wilson, 2000), in
which the high polarization fraction may be attributed to the limited
sensitivity of both the SMA and CARMA. It is possible that a weak magnetic
field could produce a low polarization fraction. However, this has not been
confirmed by observations. Ohashi et al. (2018) and Cox et al. (2018)
suggested that dust grain with small size ($\lesssim$ 100$\mu$m; Ohashi et
al., 2018) is more easily aligned with magnetic fields. It should be noted
that it is difficult for large-sized dust grains ($>100\,\mu$m) to be aligned
due to the magnetic field (Ohashi et al., 2018; Yang, 2021). On small-scale (a
few 100 au), in the presence of the toroidal magnetic field within the disk
(Brandenburg et al., 1995), the induced $E$-vectors are predicted to point in
the radial direction for a face-on disk (Cho & Lazarian, 2007) and more or
less parallel to the disk minor axis for inclined and edge-on disks (Lazarian,
2007) as shown in the top panel of Figure 6.

  2. 2.

Self-scattering of thermal dust emission: Self-scattering exhibits a strong
dependence on the wavelength and the polarization fraction is highest when
$a_{\textnormal{\scriptsize{max}}}\sim\lambda/2\pi$ (Kataoka et al., 2015).
Theoretical studies have shown that self-scattering is more efficient with
larger dust grains ($\gtrsim$100 $\mu$m) (e.g., Kataoka et al., 2015, 2016a).
Therefore, self-scattering is expected to be a dominant mechanism on disk-
scale rather than on the cloud- or envelope-scale. Dust polarization due to
self-scattering is also considered to be an effective tool for constraining
grain sizes within the inner region of the disk (Kataoka et al., 2015). In
addition to the dependence on grain sizes and wavelengths, the polarization
patterns also depend on the scattering angle. This means that the polarization
configurations could possibly change with the disk inclination (e.g., Kataoka
et al., 2015; Yang et al., 2016; Pohl et al., 2016). As shown in the middle
panel of Figure 6, two radiative intensity rings will be produced for a face-
on disk. Within the inner ring, the radiative intensity is larger in the
azimuthal direction than in the radial direction. The scattering of the photon
moving in an azimuthal direction towards the observer, will induce a radial
orientation for the $E$-vectors. In contrast, on the outer ring, intensity
flow radially is predominant over the azimuthal one. The scattering photons
will induce an azimuthal orientation for the $E$-vectors (Kataoka et al.,
2015, 2016b). Kataoka et al. (2016b) explained that the light scattered at the
optically thick region is isotropic, resulting in the cancellation of fluxes,
which in turn produces little or no polarization, particularly at the center
of the disk. In the optically thin region, the anisotropy of the radiation
field causes the differences in the incoming fluxes, producing the
polarization as a residual. As the flux becomes more anisotropic from the
center to the edge, the polarization fraction is observed to increase outwards
(Kataoka et al., 2015, 2016b; Ohashi et al., 2018). For an inclined disk, Yang
et al. (2017) demonstrated that self-scattering can produce a near-far side
asymmetry in PI within the optically thick region. This results in a deviation
between the PI peak and Stokes $I$ peak, meaning a positional shift of the PI
peak relative to the Stokes $I$ peak. Such a phenomenon could be due to the
larger grains on the surface layer of the disk not having fully settled to the
mid-plane (Yang et al., 2017). Yang et al. (2016) and Yang et al. (2017) have
analytically shown that the PI becomes elongating along the major axis and the
$E$-vectors become aligning with the minor axis in the central disk region as
the disk is tilted away from the observer. In some cases, the $E$-vectors
remain in the azimuthal direction on the outer edge of the disk as the
polarization from anisotropic radiation field is dominated in the azimuthal
direction. In the case of an edge-on disk, similar to an inclined disk, an
near-far side asymmetry arises, causing the $E$-vectors to align more parallel
to the minor axis of the disk (see the middle panel of Figure 6).

  3. 3.

Dust alignment due to radiative torque: This mechanism has been proposed by
Lazarian & Hoang (2007a), which is a direct dust grain alignment due to
radiation field. Tazaki et al. (2017) applied this mechanism to investigate
the protoplanetary disks around a T Tauri star. Tazaki et al. (2017) performed
three-dimensional (3D) radiative transfer calculations, and demonstrated that
the grains in a disk could align their minor axes with the direction of the
radiative flux. As the radiative flux is expected to point radially in a disk,
it will then induce the $E$-vectors in the azimuthal direction for a face-on
disk, in an elliptical direction for an inclined disk (Kataoka et al., 2017),
and parallel to the major axis for an edge-on disk as illustrated in the
bottom panel of Figure 6. Tazaki et al. (2017) also suggested that the
polarization fraction mainly depends on the maximum grain size
$a_{\textnormal{max}}$ and the polarization fraction increases as
$a_{\textnormal{max}}$ decreases. The radiative alignment is not easily to be
detected by observations, except for the polarization of HL Tau in the ALMA
3.0 mm observations. The dust polarization towards this source has been
interpreted as originating from radiative torque because of the elliptical
pattern of the $E$-vectors (Stephens et al., 2017; Kataoka et al., 2017).
However, such an interpretation still remains controversial. Yang et al.
(2019) suggested that the radiative torque will produce a circular pattern
rather than an elliptical pattern for an inclined disk, which is inconsistent
with the dust polarization pattern observed in HL Tau. Further observations
might be necessary to validate this mechanism.

In addition to the well-studied mechanisms mentioned above, theoretical
studies have suggested new polarization mechanisms as shown below. However,
these mechanisms have not yet been thoroughly tested by observations. In the
following, we explain such processes:

  1. 4.

Grain alignment by gas flow due to mechanical alignment: This mechanism was
originally proposed by Gold (1952), which was widely discussed within the
framework of interstellar medium. Gold (1952) proposed that polarization
orientation is parallel to the gas flow, while Lazarian & Hoang (2007b) and
Hoang et al. (2018) proposed helical grain alignment where the polarization
orientation is perpendicular to the gas flow. Kataoka et al. (2019) opted for
helical grain alignment as the possible mechanism that works in the
protoplanetary disk, as the gas flow is supposedly subsonic. This mechanism
will produce the $E$-vectors in an azimuthal direction for a face-on disk, in
a circular direction for an inclined disk. In addition, Kataoka et al. (2019)
found that the polarization orientations could depend on the Stokes numbers of
the dust grains, where the Stokes number $\rm{St}$ refers to the ratio of dust
stopping time $t_{\rm s}$ to the Keplerian timescale $t_{\rm K}$ as
$\rm{St}=t_{\rm s}/t_{\rm K}$ 555 The stopping time is defined as
$t_{s}=a_{\rm d}\rho_{\rm s}/(v_{\rm th}\rho)$ where $a_{\rm d}$, $\rho_{\rm
s}$, $v_{\rm th}$ and $\rho$ are the dust grain size, material density,
thermal velocity of molecular gas and gas density, respectively. . When the
Stokes numbers are smaller than unity ($\rm{St}<1$), the dust grains are well
coupled with the gas in the disk. In this case, the $E$-vectors will be
oriented in the azimuthal direction. As the Stokes numbers are approaching
unity ($\rm{St}\sim 1$), the coupling between the dust grains and gas becomes
weaker and the gas drug force acting on dust grains cannot be ignored. In this
case, the $E$-vectors will take on a spiral pattern. Once the Stokes numbers
are larger than unity by a factor of 10 (i.e., $\rm{St}>10$), the $E$-vectors
will be pointed in a radial direction.

  2. 5.

Magnetic field alignment in the Mie regime: Guillet et al. (2020) found that
$E$-vectors could align parallel to the magnetic field in the Mie regime when
the grain size is $\sim$1 mm. As a result, the $E$-vectors align themselves
with toroidal magnetic field in the azimuthal direction for both face-on and
inclined disks. This mechanism is an alternative way to explain the azimuthal
$E$-vectors detected on the outer edge of the disk.

There are cases where a polarization pattern could be explained by multiple
polarization mechanisms. For example, the azimuthal pattern of $E$-vectors
could be explained by either radiative torque, grain alignment by gas flow or
magnetic field alignment in the Mie regime. Moreover, the morphology could
appear as a mixture due to multiple mechanisms. In such cases, multi-
wavelength observations are necessary to disentangle the polarization origins.
However, since our data is only limited to a single wavelength, it is
difficult to conclude the origins of the polarization for our sources. In the
following subsections, we list the possible polarization mechanisms for each
source. Some mechanisms that could explain most of the polarization features
such as $E$-vector orientation, polarization fraction, and PI peak positional
shift, whereas some mechanisms could only explain one of the features such as
the $E$-vector orientation. We also discuss which mechanism is more likely to
be the origin of the polarization for each source.

####  4.1.1 MMS 2-North-A

MMS 2-North-A is a flat-spectrum source. We detected a compact disk-like
structure around MMS 2-North-A, which is nearly face-on with an inclination
angle of $i\sim 25^{\circ}$. PI of MMS 2-North-A shows a ring-like structure,
and the $E$-vectors on the ring show an azimuthal pattern. In addition, a
nearly pole-on outflow elongated along the north-south direction is possibly
associated with MMS 2-North-A. The mean polarization fraction is estimated to
be $\sim$1.0%.

The observed dust polarization could be explained by the poloidal magnetic
field. The $B$-vectors point in the radial direction as shown in Figure 4 (a).
Considering that this disk is nearly face-on, the poloidal magnetic field may
dominate over the toroidal one. The $E$-vectors are expected to be aligned
perpendicular to the poloidal magnetic field, that is in the azimuthal
direction. Thus, the $B$-vectors are expected to point in the radial direction
as illustrated in Figure 7. This is different from the case of dust alignment
by toroidal magnetic field. In the synthetic polarization maps of protostellar
source NGC 1333 IRAS 4A, Frau et al. (2011) has shown that the magnetic field
tend to be aligned in the radial direction for face-on configurations. By
comparing the Stokes $Q$ and $U$ images of MMS 2-North-A with those of NGC
1333 IRAS 4A (see the bottom panel of Fig.4 and Fig.5 in Frau et al. (2011)),
we found that the morphologies are similar between NGC 1333 IRAS 4A and MMS
2-North-A. Therefore, it is possible that the polarization pattern in MMS
2-North-A is produced by poloidal magnetic field.

The observed dust polarization could also be attributed to self-scattering of
thermal dust emission. The $E$-vectors trace the PI dust ring in an azimuthal
direction. This configuration is consistent with that due to self-scattering
for a face-on disk as shown in Figure 6 middle panel (e.g., Kataoka et al.,
2015, 2016b; Pohl et al., 2016). The mean polarization fraction is $\sim$1.0%,
which can also be explained by self-scattering where a few percent is
expected. Depolarization seems to occur at the center of the disk. Such a
feature has also been seen in the disk of HD 142527, which can be explained by
the optical effects (e.g., Kataoka et al., 2015, 2016b).

Radiative torque could also explain the $E$-vector orientation, where the
$E$-vectors are expected to point in an azimuthal direction for a face-on
disk. The depolarization occurring at the center of the disk has also been
seen in HL Tau at 3.1 mm observations (Stephens et al., 2017). Polarization
patterns of HL Tau at 3.1 mm are consistent with radiative torque model, and
Stephens et al. (2017) explained that the depolarization could be due to beam-
averaging of the azimuthal $E$-vectors within the beam. Although the disk of
MMS 2-North-A is viewed slightly more face-on than HL Tau disk, since the
difference in inclination is about $\sim$$10^{\circ}-$$20^{\circ}$, we expect
the morphologies might be similar.

Both grain alignment by gas flow and magnetic field alignment in the Mie
regime could explain the azimuthal pattern of the $E$-vectors. However, for
grain alignment by gas flow, this explanation holds true only when the Stokes
numbers are smaller than unity. Since the studies associated with these two
grain alignment mechanisms primarily focus on the morphology of the
$E$-vectors (Kataoka et al., 2019; Guillet et al., 2020), we could not delve
into the other polarization features like polarization fraction and PI
morphology expected by these two grain alignment mechanisms.

Overall, self-scattering and radiative torque are the more likely origins of
the polarization for MMS 2-North-A, because the azimuthal $E$-vector
orientation, depolarization, and the low polarization fraction are consistent
with both mechanisms. However, the depolarization can also be due to smoothing
over the complicated structure at the central beam, as seen in HL Tau. The
unresolved polarization structures could lead to a decrease in the
polarization fraction. Therefore, other mechanisms cannot be ruled out without
verification from multi-wavelength observations at a higher angular
resolution.

Figure 7: Illustrations of MMS 2-North-A with outflow in the presence of the
poloidal magnetic field is shown in the left panel. The black thin solid lines
indicate the poloidal magnetic field. The blue line segments on the middle
panel indicate the $E$-vectors predicted by dust alignment due to poloidal
magnetic field. The purple line segments in the right panel indicate the
$B$-vectors, obtained by rotating the $E$-vectors by 90∘.

####  4.1.2 MMS 2-South and MMS 7-North

In this section, we will explain MMS 2-South and MMS 7-North together due to
their similar polarization features such as $E$-vector orientation and PI peak
positional shift. The brief summary for these two sources is as follows.

MMS 2-South and MMS 7-North are the flat-spectrum and Class I sources,
respectively. We detected a compact and inclined disk-like structure around
MMS 2-South with an inclination angle of $i\sim 45^{\circ}$, and a compact and
nearly edge-on disk-like structure around MMS 7-North with an inclination
angle of $i\sim 64^{\circ}$. For both sources, the PI is elongated along the
major axis of the disk, and the peak of PI shifts towards the nearside of the
disk with respect to the Stokes $I$ peak. The $E$-vectors in the central
region are mostly parallel to the minor axis of the disk, while on the outer
edge, the $E$-vectors are more azimuthal. The mean polarization fractions of
MMS 2-South and MMS 7-North are estimated to be $\sim$1.2% and $\sim$1.0%,
respectively.

The configurations of the $E$-vectors observed in MMS 2-South and MMS 7-North
could be explained by dust alignment due to magnetic field. The dust
polarization in the central disk regions is likely to be produced by the
toroidal magnetic field as the $E$-vectors are mostly parallel to the minor
axes of the disks. In contrast, the $E$-vectors are azimuthal near the outer
disk edges (see Figure 2 (b) and (h)). These patterns could be interpreted as
the dust grains being aligned by a toroidal magnetic field in the central disk
region and a poloidal magnetic field near the disk edge, which have been seen
in HH 111 (Lee et al., 2018). However, the near-far side asymmetry in the
central disk region could not be explained by this mechanism.

For MMS 2-South and MMS 7-North, the observed dust polarization patterns are
broadly consistent with the expected outcomes of self-scattering for inclined
and nearly edge-on disks. In these two sources, the $E$-vectors in the central
regions are predominantly aligned with the minor axes of both disks. Near the
outer edges, the $E$-vectors point in the azimuthal direction. The orientation
of $E$-vectors in the optically thick central region is mainly induced by the
inclination effect, while on the outer edge is mostly dominated by the
intrinsic polarization from the anisotropic radiation field (Yang et al.,
2016). The directional change of the $E$-vectors can also be explained by a
change in direction of the thermal radiative flux from inner disk to outer
disk (Kataoka et al., 2016b). Furthermore, the directional change of the
$E$-vectors observed in MMS 7-North mainly occurs on the nearside of its disk,
unlike for MMS 2-South, where it occurs on both the nearside and farside (see
Figure 2 (b) and (h)). We expect that this may come from the inclination
effect because MMS 7-North is viewed more edge-on than MMS 2-South. A similar
configuration has also been observed in HH 111 (Lee et al., 2018), which is a
nearly edge-on disk of a Class I protostar, exhibiting distinct $E$-vector
orientations on the nearside and farside of its disk. The $E$-vectors on the
nearside are parallel to the minor axis of the disk, whereas they are almost
perpendicular to the minor axis on the farside. Lee et al. (2018) suggested
that this configuration could possibly be produced by a disk surrounded by
rings and gaps that have marginally different inclinations compared to the
disk. As a result, the $E$-vectors become less parallel to the minor axis of
the disk. In addition, the PI peak shift towards the nearside has been
observed for both sources, and this positional shift of the PI peak can be
explained by the optical depth effect (Yang et al., 2017). These two sources
have the low polarization fractions of $\sim$1.2% and $\sim$1.0%, which are
consistent with the polarization fraction expected for the self-scattering,
where a few percent ($\sim$1$-$2%) is expected.

There is another possibility that the polarization could be caused by a
combined mechanisms involving both self-scattering and radiative torque. We
note that the some $E$-vectors in the central disk region are not completely
parallel to the minor axis of the disk, but slightly rotate away from the
minor axis. Near the edge, the $E$-vectors are distributed in the azimuthal
direction. A similar feature has been seen in HL Tau at 1.3 mm observations
(Stephens et al., 2017). The $E$-vectors that are not entirely aligned with
the minor axis of the disk, along with the $E$-vectors in the azimuthal
direction observed at the edge, could be interpreted as a mixed morphology
caused by a combination of self-scattering and radiative torque.

Grain alignment by gas flow and magnetic field alignment in the Mie regime are
unlikely to be the origins of the polarization for these two sources. The
$E$-vectors are expected to be oriented elliptically for an inclined disk due
to these two grain alignments, which are inconsistent with the $E$-vector
orientations for both sources.

Overall, self-scattering is more likely to be the origin of the dust
polarization observed from MMS 2-South and MMS 7-North than the dust alignment
by magnetic field. Both mechanisms could explain the $E$-vector orientation.
However, the PI peak positional shift and low polarization fraction are more
consistent with self-scattering. In addition, dust polarization could also
possibly be caused by the mixture of self-scattering and radiative torque,
which needs to be verified with multi-wavelength observations.

####  4.1.3 MMS 7-South

MMS 7-South is a Class I source. We detected a compact disk-like structure
around MMS 7-South, which is nearly edge-on with an inclination angle of
$i\sim 60^{\circ}$. PI is only detected in the immediate central region around
the Stokes $I$ peak. The detected PI is relatively weak and PI peak positional
shift has not been seen from this source (see Figure 2 (h)). Only a few
$E$-vectors have been observed and these $E$-vectors appear to be more or less
parallel to the minor axis of the disk. The mean polarization fraction is
estimated to be $\sim$1.0%.

There are only two features that could be used to discuss the origin of the
polarization, and these are the $E$-vector orientation and the polarization
fraction. The $E$-vector orientation could be explained by either self-
scattering or magnetic field. However, the low polarization fraction is more
consistent with self-scattering.

Overall, self-scattering is more likely to be the origin of dust polarization
observed in MMS 7-South since the $E$-vector orientation and low polarization
fraction are consistent with those expected from self-scattering. However, the
$E$-vector orientation is also consistent with dust alignment by magnetic
field. Moreover, this source is possibly unresolved due to its compact size
and weak polarization detection. Therefore, we could not rule out other
mechanisms with the limited detection from this single wavelength
observations.

####  4.1.4 MMS 5

MMS 5 is a Class 0 source. We detected an inclined disk-like structure around
MMS 5 with an inclination angle of $i\sim 37^{\circ}$. PI is detected in the
central region (within $\sim$100 au). The peak of the PI shifts towards the
nearside of the disk with respect to the Stokes $I$ peak. The fitted disk has
a P.A. $\sim$ 1.6∘, and the $E$-vectors peaked around 100∘. This suggests that
the $E$-vectors could potentially be parallel to the minor axis of the disk.
The mean polarization fraction is estimated to be $\sim$0.6%.

MMS 5 is in the younger Class 0 stage and has a small disk size of $\sim$32
au, in which the disk size is about twice smaller than the synthesized beam.
Therefore, the detection of a lower polarization fraction for MMS 5 as
compared with the other sources could be due to either the undeveloped disk or
the beam dilution. If the $E$-vectors are indeed parallel to the minor axis of
the disk as we expect, then this configuration can be explained by dust
alignment due to toroidal magnetic field or self-scattering. The positional
shift of the PI peak towards the nearside of the disk and the low polarization
fraction, are consistent with self-scattering.

The observed dust polarization in MMS 5 contradicts the alignment mechanisms
due to radiative torque, gas flow, and magnetic field in the Mie regime, as
the $E$-vector orientation is inconsistent with expectations from these
mechanisms.

Overall, the dust polarization observed in MMS 5 is more likely to be
originated from self-scattering, as the $E$-vector orientation, polarization
fraction and PI peak positional shift are consistent with those expected from
self-scattering. However, we could not conclusively rule out the other
mechanisms with the limited detection from this single wavelength
observations. Multi-wavelength observations are needed to confirm the origin
of the polarization for this source.

In summary, the dust polarization observed in our sources could be explained
by multiple mechanisms. Self-scattering is likely the primary polarization
origin. However, we could not definitely rule out other mechanisms due to
their similarities in the polarization morphologies. It is essential to
disentangle the polarization origins through multi-wavelength observations,
which can provide valuable insights into better constraining the dust grain
size. Our results have shown that dust polarization observed in Class 0/I
sources at 1.1 mm is possibly arising from self-scattering, indicating that
there is dust growth at the early stages (dust grains may have grown to as
large as $\sim$170 $\mu$m, which is estimated by
$a_{\textnormal{\scriptsize{max}}}\sim\lambda/2\pi$; Kataoka et al., 2015). As
for future work, we propose to observe these sources with multiple
wavelengths. Previous studies have shown that polarization patterns observed
at shorter wavelengths (e.g., Band 7) is highly consistent with self-
scattering, some clear observational results such as the uniform $E$-vectors
along the minor axis of the disk and PI peak positional shift (e.g., Stephens
et al., 2017; Lee et al., 2018). We also expect to have more clear detection
towards MMS 5 at shorter wavelengths as the polarization detection towards
this source at Band 6 is very limited. At longer wavelengths (e.g., Band 3),
we predict the $E$-vectors will be distributed in the azimuthal direction if
radiative torque is the dominant mechanism. If the maximum grain size has
grown to as large as 500 $\mu$m, it might still be possible to observe
$E$-vectors parallel to the minor axis of the disk due to self-scattering.
However, the polarization fraction may be slightly higher compared to that
detected at Band 6. If there is no significant change in the polarization
properties, such as the $E$-vector orientation and polarization fraction, it
suggests that the observed dust polarization has no strong dependence on
either wavelength or maximum grain size, which is more consistent with dust
alignment by magnetic field.

###  4.2 Evolutionary tendency

Figure 8: The length of the CO outflow versus the disk diameter of the sources
(excluding MMS 4 and MMS 7-South because no disk-like structure is detected
towards MMS 4 and no clear outflow is detected towards MMS 7-South). The
outflow length is estimated as the arithmetical average length of the blue-
and red-shifted CO components. The outflow lengths for MMS 1 (blue) is
obtained from Takahashi et al. (2023a, submitted to ApJ), for MMS 3 (blue) is
obtained from Morii et al. (2021), and for MMS 6 (blue) is obtained from
Takahashi et al. (2019), respectively. Among the sources, elongated outflows
were detected towards MMS 2, MMS 5, and MMS 7 from previous single-dish
observations (Takahashi et al., 2008; Tanabe et al., 2019; Feddersen et al.,
2020). Since the interferometric data might be affected by the missing flux,
we plot the outflow lengths for MMS 2-North, MMS 2-South, MMS 5, and MMS
7-North with the lengths obtained from the single-dish observations by
Takahashi et al. (2008) denoted in red, Tanabe et al. (2019) denoted in green,
and Feddersen et al. (2020) denoted in purple, respectively. The disk
diameters for all the sources are obtained from this paper. The circle,
triangle, and diamond denote the Class 0, Class I, and flat-spectrum sources,
respectively. The line in the horizontal direction represents the 2D Gaussian
fitting errors for the fitted disk diameter. The line in vertical direction
shows the minimum to maximum lengths of the blue- and red-shifted CO
components obtained from each data set. The outflow length is corrected using
the inclination angle $i$ for each source as listed in the last column of
Table 5.

In this section, we discuss the correlation between disk size, outflow size,
polarization fraction, and their associations with the evolutionary stages of
these protostellar sources. We define the disk size based on the most compact
component fitted by the 2D Gaussian as listed in the second column of Table 7.
We classify the sources primarily based on the SED study by Furlan et al.
(2016) as listed in the eighth column of Table 1. MMS 1 was not detected by
Furlan et al. (2016), and is classified as a Class 0 source since a very
compact and collimated jet is detected by Takahashi et al. (2023a, submitted
to ApJ). For the sources within the same evolutionary stage such as MMS 1, MMS
5, and MMS 6, we compare the mean length of outflow to determine the evolution
of each source, where the mean length is obtained by averaging the lengths of
the blue- and red-shifted CO components.

Figure 8 presents the mean length of outflow versus disk diameter along with
the SED classification of each protostellar source. We found a clear positive
correlation between the disk size and outflow size except for MMS 6, in which
MMS 6 has the smallest outflow size but is measured with the largest disk
size. The larger outflow size is likely to indicate a longer lifetime since
the outflow velocities are typically comparable. The longer lifetime also
allows the disks to grow in size in principle. Therefore, the outflow size and
the disk size may be correlated, but this does not necessarily imply a direct
causal relationship, such as a larger disk driving a larger outflow.

Figure 8 also clearly shows that that the more evolved sources have the larger
outflows. In addition, our results show a possible trend where the disk size
appears to increase from the Class 0 to the Class I/flat-spectrum phase, which
is consistent with the expected formation and evolution of disks in rotating
and collapsing cloud cores (e.g., Vorobyov, 2011; Tomida et al., 2017). Such a
trend is also consistent with the observational results by Yen et al. (2017),
in which they showed that the radii of Keplerian disks grow from the Class 0
to the Class I phase.

On the contrary, Tobin et al. (2020) showed that the disk size decreases from
Class 0 to flat-spectrum phase. Such an opposite trend could be attributed to
several factors. One of them could be due to the different Gaussian fitting
methods. Tobin et al. (2020) fitted each source with a single Gaussian
component, while we fitted each source with multiple Gaussian components. Our
disk sizes are estimated from the most compact components as shown in the
second column of Table 7. We found that if we fit the source with a single
component, the radius for the compact component will become larger than when
fitting with multiple components. Such an increase in size is especially
prominent in the case of Class 0 sources. That is, the average increase in
radius for our Class 0, Class I, and flat-spectrum sources is 110%, 7% and
13%, respectively. Tobin et al. (2020) also pointed out that the decrease of
dust disk radii through evolutionary phase may be partially influenced by a
systematic bias caused by measuring the radii using Gaussian fitting, in
particular for Class 0 sources. In addition, Tobin et al. (2020) expected that
the decrease in the disk radii in the more evolved sources come from radial
drift of the dust grains. In this process, the large grains in the outer disk
are decoupled from the gas and gradually drift towards the central star,
resulting in a loss of disk dust mass and a reduction in disk size (e.g.,
Weidenschilling, 1977; Birnstiel et al., 2010; Trapman et al., 2020). Note
that although Yen et al. (2017) has demonstrated a general trend in which
Class I sources typically have larger disk sizes than Class 0 sources, there
are some samples inconsistent with this trend, i.e., some Class 0 sources have
larger disks than Class I sources. Since we have a limited number of samples,
we need to expand the number of samples to make fair comparisons and confirm
the trend in the future.

dia-

Figure 9: The central 60-au polarization fraction versus the disk diameter of
the sources (excluding MMS 4 because no disk-like structure is detected). The
central 60-au polarization fraction is the mean polarization measured within
central 60 au region of each disk. The circle, triangle, and diamond denote
the Class 0, Class I, and flat-spectrum sources, respectively. Since there is
no polarized emission detected towards MMS 1 and MMS 3, 3$\sigma$ upper limits
of the polarization fractions are plotted with the black arrows for these two
sources. The horizontal red line represents the 2D Gaussian fitting errors for
the fitted disk diameter. The vertical red line represents the errors for the
central 60-au polarization fraction. We estimate the central 60-au
polarization fraction and its errors using the same method as described in
Section 3 for the mean polarization fraction presented in Figure 5.

Figure 9 shows the central 60-au polarization fraction measured over the same
size scale towards each source as a function of disk diameter. For all sources
that polarized emissions are detected, the mean polarization fraction is
estimated within the central 60 au region to focus solely on the compact dust
disk. For the non-polarized detected sources MMS 1 and MMS 3, we plot the
3$\sigma$ upper limit of the polarization fraction as the central 60-au
polarization fraction. In general, we observe a central 60-au polarization
fraction of $\lesssim$ 1% for most of the sources, with the exception of MMS
6. Such a low percentage is consistent with the expectation for the
polarization resulting from self-scattering as discussed in Section 4.1.

Furthermore, the central 60-au polarization fraction varies slightly among the
sources. We expect that the variation of the polarization fraction could also
be affected by the inclination of each source. It is important to highlight
that, despite MMS 2-North being in a more evolved evolutionary stage, it
exhibits the lowest polarization fraction, which is comparable to that of MMS
5. One explanation could be that the unresolved polarization results in a
decrease in the polarization fraction as mentioned in Section 4.1.1. Another
plausible explanation could be the inclination effect. Given that MMS 2-North
is viewed nearly face-on, it is likely that depolarization occurs at the
central disk region in this face-on configuration, particularly if the
polarization arises from self-scattering (Kataoka et al., 2016b). Therefore,
we detected less $E$-vectors and a much lower polarization fraction than the
other sources, particularly in the central disk region. Similarly, by
comparing the polarization fractions within the two binary systems, MMS 2 and
MMS 7, we found that the polarization fractions of MMS 2-North-A and MMS
7-South are lower than that of MMS 2-South and MMS 7-North, respectively. The
former two sources are viewed more face-on than the latter two sources, and
the face-on sources are likely to have a lower polarization fraction than
edge-on sources. This result is consistent with our expectation that the
polarization fraction originating from self-scattering is likely to be
influenced by the inclination of the disk.

Finally, among our protostellar samples, MMS 6 stands out as an exceptional
source due to its inconsistency with the observed trend, where the disk size
and outflow length increases with the evolutionary stage as seen in the other
sources. Although MMS 6 has the most compact outflow and is at the earliest
evolutionary stage (Class 0), it is brightest in Stokes $I$ (Takahashi et al.,
2009, 2012; Takahashi & Ho, 2012; Takahashi et al., 2013, 2019). In fact, the
measured Stokes $I$ peak flux is a factor of 2$-$34 brighter than those
measured in the other protostellar sources among our samples. MMS 6 is
physically larger than other sources and contains more stuff around. The
radiative flux is possibly affected by the amount of material and the dust
temperature. The details will be discussed in Takahashi et al. (2023b, in
preparation). Moreover, MMS 6 has shown a significantly higher polarization
fraction than other sources. We expect that the high polarization fraction is
resulting from the polarization due to dust alignment by magnetic field
(Takahashi et al., 2019; Le Gouellec et al., 2020; Liu, 2021; Le Gouellec et
al., 2023).

##  5 Conclusion

We present the results of ALMA 1.1 mm dust linear polarization observations at
${\sim}0\hbox to0.0pt{.\hss}\arcsec 14$ (55 au) and CO ($J=2-1$) observations
at ${\sim}1\hbox to0.0pt{.\hss}\arcsec 5$ (590 au) towards seven millimeter
sources, MMS 1$-$7 in OMC-3. The results are summarized as follows.

  1. 1.

We have detected MMS 1$-$7 in Stokes $I$. Except for the prestellar source MMS
4, we have detected bright and compact disk-like structures towards six other
sources. The disk size was fitted using the 2D Gaussian, ranging from 16 au to
97 au in linear size. The measured disk size is positively correlated with the
evolutionary stage. Out of the seven sources, MMS 2 (Flat-spectrum), MMS 5
(Class 0), MMS 6 (Class 0), and MMS 7 (Class I) are detected in Stokes $Q$ and
$U$. MMS 1 (Class 0), MMS 3 (Class 0), and MMS 4 (Prestellar core) are not
detected in Stokes $Q$ and $U$ with a 3$\sigma$ upper limit of 0.5%, 2.1%, and
4.7%, respectively.

  2. 2.

We detected clear CO outflows towards MMS 1, MMS 2, MMS 3, MMS 5, MMS 6, and
MMS 7. The outflow axes in MMS 2, MMS 5, MMS 6, and MMS 7 are mostly aligned
perpendicular to the major axes of the disks. The outflow size primarily has a
positive correlation with the evolutionary stage and the disk size, with MMS 6
as an exception. Regarding MMS 4, we have detected a relatively compact
millimeter source (geometric mean diameter $\sim$ 364 au) with a number
density as high as $\sim$2 $\times 10^{9}$ cm-3, while no clear bipolar
outflow is detected. However, we have a marginal detection of the compact low-
velocity ($v_{\textnormal{LSR}}$ = 6 to 7 km s-1) blue-shifted CO emission
elongated to the western direction with respect to this millimeter source,
possibly implying the presence of the first hydrostatics core (Hirano et al.,
2023).

  3. 3.

The compact linearly polarized emissions associated with the dust disks are
detected towards MMS 2, MMS 5, MMS 6, and MMS 7. $E$-vector orientations
differ among individuals, such as the direction parallel to the minor axis of
the disk and the azimuthal direction. There is no strong correlation between
the cloud-scale and disk-scale $E$-vector orientations. The mean polarization
fractions for MMS 2, MMS 5, and MMS 7 are between 0.6$\%$ and 1.2$\%$, while
MMS 6 exhibited a relatively high mean polarization fraction of 3.5$\%$ around
Stokes $I$ peak, and 15.7$\%$ on the arm-like structure.

  4. 4.

The dust polarization observed in these sources can be explained by multiple
mechanisms, while self-scattering is the more likely polarization origin, as