# Discovery of Asymmetric Spike-like Structures of the 10 au Disk around the
Very Low-luminosity Protostar Embedded in the Taurus Dense Core MC 27/L1521F
with ALMA

Kazuki Tokuda Department of Earth and Planetary Sciences, Faculty of Science,
Kyushu University, Nishi-ku, Fukuoka 819-0395, Japan  National Astronomical
Observatory of Japan, National Institutes of Natural Sciences, 2-21-1 Osawa,
Mitaka, Tokyo 181-8588, Japan  Naoto Harada Department of Earth and Planetary
Sciences, Faculty of Science, Kyushu University, Nishi-ku, Fukuoka 819-0395,
Japan  Mitsuki Omura Department of Earth and Planetary Sciences, Faculty of
Science, Kyushu University, Nishi-ku, Fukuoka 819-0395, Japan  Tomoaki
Matsumoto Faculty of Sustainability Studies, Hosei University, Fujimi,
Chiyoda-ku, Tokyo 102-8160, Japan  Toshikazu Onishi Department of Physics,
Graduate School of Science, Osaka Metropolitan University, 1-1 Gakuen-cho,
Naka-ku, Sakai, Osaka 599-8531, Japan  Kazuya Saigo Graduate School of Science
and Engineering, Kagoshima University, 1-21-40 Korimoto Kagoshima-city
Kagoshima, 890-0065, Japan  Ayumu Shoshi Department of Earth and Planetary
Sciences, Faculty of Science, Kyushu University, Nishi-ku, Fukuoka 819-0395,
Japan  Shingo Nozaki Department of Earth and Planetary Sciences, Faculty of
Science, Kyushu University, Nishi-ku, Fukuoka 819-0395, Japan  Kengo Tachihara
Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya
464-8601, Japan  Naofumi Fukaya Department of Physics, Nagoya University,
Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan  Yasuo Fukui Department of
Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8601, Japan  Shu-
ichiro Inutsuka Department of Physics, Nagoya University, Furo-cho, Chikusa-
ku, Nagoya 464-8601, Japan  Masahiro N. Machida Department of Earth and
Planetary Sciences, Faculty of Science, Kyushu University, Nishi-ku, Fukuoka
819-0395, Japan

###### Abstract

Recent Atacama Large Millimeter/submillimeter Array (ALMA) observations have
revealed an increasing number of compact protostellar disks with radii of less
than a few tens of au and that young Class 0/I objects have an intrinsic size
diversity. To deepen our understanding of the origin of such tiny disks, we
have performed the highest-resolution configuration observations with ALMA at
a beam size of $\sim$0$\farcs$03 (4 au) on the very low-luminosity Class 0
protostar embedded in the Taurus dense core MC 27/L1521F. The 1.3 mm continuum
measurement successfully resolved a tiny, faint ($\sim$1 mJy) disk with a
major axis length of $\sim$10 au, one of the smallest examples in the ALMA
protostellar studies. In addition, we detected spike-like components in the
northeastern direction at the disk edge. Gravitational instability or other
fragmentation mechanisms cannot explain the structures, given the central
stellar mass of $\sim$0.2 $M_{\odot}$ and the disk mass of $\gtrsim$10-4
$M_{\odot}$. Instead, we propose that these small spike structures were formed
by a recent dynamic magnetic flux transport event due to interchange
instability that would be favorable to occur if the parental core has a strong
magnetic field. The presence of complex arc-like structures on a larger
($\sim$2000 au) scale in the same direction as the spike structures suggests
that the event was not single. Such episodic, dynamical events may play an
important role in maintaining the compact nature of the protostellar disk in
the complex gas envelope during the main accretion phase.

Star formation (1569); Protostars (1302); Molecular clouds (1072);
Interstellar medium (847); Circumstellar envelopes (237); Magnetic fields
(994)

††software: astropy (Astropy Collaboration et al., 2018), CASA (CASA Team et
al., 2022)

##  1 Introduction

Understanding the star formation process is a fundamental topic in astronomy
and astrophysics. The collapse of molecular clouds leading to protostar
formation (e.g., Shu et al., 1987; Andre et al., 2000) and the subsequent
interaction between stellar objects and surrounding envelopes are crucial to
deciphering the mechanism to determine stellar mass (e.g., Arce & Sargent,
2006; Machida & Hosokawa, 2013). The product of the star formation process
also provides initial conditions for planet formation, i.e., protoplanetary
disks. Multiple physical processes, such as turbulence (e.g., Padoan &
Nordlund, 2002; Offner et al., 2008), magnetic fields (e.g., Crutcher, 2012),
and self-gravity, are intricately coupled throughout the dynamical collapse
and mass accretion phases (e.g., Matsumoto & Hanawa, 2011). To observationally
elucidate such processes, recent millimeter and submillimeter facilities,
including interferometers offering high spatial and density dynamic range
observations, have been playing effective roles.

Advances in Atacama Large Millimeter/submillimeter Array (ALMA) observations
of later stages of star formation, particularly the potential planet-forming
phase, have been remarkable over the past decade. The ubiquity of
substructures such as rings and gaps within protoplanetary disks is
increasingly evident in larger disks, typically those exceeding $\sim$50 au in
size, as highlighted by numerous high-angular-resolution observations (e.g.,
Andrews et al., 2018). However, it is becoming clear that not all
protoplanetary disks are necessarily large and massive. Survey observations
across star-forming regions have shown that many disks are compact, with radii
of just a few tens of au (e.g., Cieza et al., 2019; Long et al., 2019), and
often exhibit smooth structures within the limits of their observational
resolution. The diversity of mass and size of the disk is evident even among
much younger Class 0/I objects (e.g., Ohashi et al., 2023). Some theoretical
models tend to produce massive disks (e.g., Tomida et al., 2017), and
reconciling this discrepancy between observational fact and theoretical
prediction is an urgent challenge (see Section 4.4 of Misugi et al. (2023) for
a recent theoretical explanation of the diversity in the disk formation).

We focus on the dense core system MC 27 in the Taurus region ($D\sim$140 pc;
Galli et al. 2018), where early ALMA observations with an angular resolution
of $\sim$0$\farcs$1 revealed a compact protostellar disk with a size of
$\sim$10 au (Tokuda et al., 2017). This system was considered to be a
prestellar core before Spitzer identified a very low-luminosity object (Bourke
et al. 2006; Terebey et al. 2009), whose evolutionary stage is the Class 0
phase. The dense core itself was discovered and characterized by strategically
utilizing several millimeter facilities, such as the Nagoya 4 m and the
Nobeyama 45 m telescopes (Mizuno et al., 1994; Onishi et al., 1996, 1999,
2002). The NH3 observation by Codella et al. (1997) determined the core
temperature of $\sim$10 K (L1521F in their paper). Among the dense cores in
the Taurus region, it is one of the densest protostellar cores, exhibiting
striking similarities to the well-known prestellar core L1544 (Crapsi et al.,
2004) and the recently reported first core candidate, MC35-mm (Fujishiro et
al., 2020; Tokuda et al., 2020) in their density profiles and deuterium
fractions (see also Tobin et al. 2013). In ALMA Cycle 0, the MC 27 system
emerged as the first instance in which complex arc-like gas structures with
lengths of a few thousand au were identified within a protostellar system
(Tokuda et al., 2014). Similar-sized arc structures observed at young
protostellar systems have since been argued to be accretion streams (e.g.,
Pineda et al., 2020; Lee et al., 2023), while it remains uncertain whether the
MC 27’s arc and the potential streamers are physically identical or similar.
Although several theories have been proposed regarding the origin of the arc
structures in MC 27 (see Section 4.2), their role in the star formation
process and the relationship between the peculiar structures and the compact
central disk remain to be thoroughly elucidated.

From a theoretical perspective, Machida & Basu (2020) showed that starting
with the magnetically subcritical state (Machida et al., 2018),
gravitationally collapsing cores lead to the formation of tiny protostellar
disks and weak outflows due to efficient angular momentum transport by
magnetic braking. In such environments with a relatively strong magnetic
field, magnetic interchange instability (see also Section 4) is likely to
occur near the edge of circumstellar disks to transport the magnetic flux
outward, leading to the formation of complex gas envelopes such as rings and
arcs (e.g., Zhao et al., 2011). This process provides a more dynamic direct
magnetic flux removal than commonly considered steady-state magnetic field
reduction, only with ohmic dissipation and ambipolar diffusion (e.g., Dapp et
al., 2012; Tomida et al., 2015). It may offer an alternative solution (see
Tokuda et al. 2023 and references therein) to the long-standing issue of the
magnetic flux problem in star formation (e.g., Nakano, 1984). Therefore, the
observational characterization of tiny disk systems with complex envelopes
should be an important subject to further understanding protostellar evolution
and its diversity.

This paper presents an ALMA highest-resolution ($\sim$0$\farcs$03) 1.3 mm
continuum study of the protostellar disk in MC 27. The obtained images reveal
a relatively simple structure that resolves the 10 au disk with additional
spike-like features. Incorporating a recent observational discovery, which may
be relevant to a magnetic flux transport phenomenon by interchange instability
(Tokuda et al., 2023), we propose a hypothesis for forming and maintaining
compact disks in a young phase of star formation. In Section 2, we describe
the observations and data reduction, then present 1.3 mm continuum imaging
results in Section 3. Section 4 integrates previously reported structures
around the compact disk, discussing the star formation scenario in MC 27 and
its broader implications.

##  2 Observations and Data Reduction

We carried out ALMA Cycle 5 observations of MC 27 (P.I.: K. Tokuda,
#2017.1.00480.S) with two 12 m array configurations, C43-9 and C43-10, using
the Band 6 receivers, between 2017 October 3rd and 24. The total telescope
time was $\sim$8 hours, excluding the semi-massed execution blocks. The two
basebands are allocated to obtain the continuum emission at the central
frequencies of 231.49 and 218.49 GHz. The bandwidths for each setting were 2
GHz with a channel number of 128. With the other two basebands, we targeted
the molecular lines CO ($J$ = 2–1), 13CO ($J$ = 2–1), C18O ($J$ = 2–1), and SO
($N_{J}$ = 56–45), but the challenging observations with the current ALMA
capability could not produce high-signal-to-noise-ratio maps at a comparable
resolution with the continuum data. We do not use the molecular line data
hereafter, except for proving there are no hidden companion sources (Section
4.1) because our immediate focus in this paper is the continuum emission of
the protostar vicinity within a few au.

We used the Common Astronomy Software Application package (CASA Team et al.,
2022), v.6.5.5, in the data reduction and imaging. We employed the tclean
algorithm with the multi-scale deconvolver in the analysis. The clean mask
regions were manually defined. We experimented with several visibility
weightings and found that a Briggs weighting with a robust parameter of $-$1.0
provided the best balance between resolution and sensitivity, effectively
capturing the disk and its associated extra components (see Section 3). The
resultant beam size and rms noise level are 0$\farcs$024 $\times$ 0$\farcs$017
with a position angle (P.A.) of 15.1°and 20 $\mu$Jy beam-1, respectively. Note
that due to the faint nature of the object, the intensity and structure
somewhat depend on the imaging parameters. In Appendix A, we present
additional images with different weighting parameters (Figure
LABEL:fig:contmap_change).

##  3 Results

Figure LABEL:fig:contmap (left) displays images of the 1.3 mm continuum
obtained (see Section 2). The new data represent the highest-resolution ALMA
images of an extremely faint protostellar disk, with a 1.3 mm flux of $\sim$1
mJy as the primary target. It should be noted that there are instances in
cluster-forming regions where additional faint sources are reported within the
field of view, separate from the main target (e.g., Sharma et al., 2023).

The structure extending in a north-south direction at the center is identified
as a disk component associated with the protostar. In addition to the main
feature, the extra components are visible in the analyses with different
imaging parameters (see Figure LABEL:fig:contmap_change). In the case of the
images with a Briggs weighting with robust parameters of $-$1 and $-$2, there
are three components extending from the main source, each with emissions of
$\gtrsim$3$\sigma$ significance. In the following, we refer to the central
source as the $``$disk$"$ and the protruding structures that extend to the
northeast as the $``$spikes.$"$

We characterize the disk and spike properties. Applying a 2D Gaussian fit to
the central bright source in Figure LABEL:fig:contmap (left) yields the
observed size in FWHM of 0$\farcs$058 $\times$ 0$\farcs$025 (P.A. =
168.5$\arcdeg$), with a beam-deconvolved size of 0$\farcs$054 $\times$
0$\farcs$018 (P.A. = 165.9$\arcdeg$). The major axis is roughly perpendicular
to the rotation axis of the disk measured in a molecular line (Tokuda et al.,
2017), suggesting that the north-south elongation is more likely to be
physical, rather than a result of the beam-smearing effect. The disk flux is
measured as $\sim$1.0 mJy, which corresponds to 75–80% of the total 1.3 mm
flux, 1.4 mJy, of the detected emission over the region displayed in Figure
LABEL:fig:contmap (left). The total H2 gas mass ($M_{\rm gas}$) is derived
from the following equation:

$M_{\rm gas}=\frac{F_{\rm 1.3mm}d^{2}}{\kappa_{\rm 1.3mm}B(T_{\rm d})},$ (1)

where $F_{\rm 1.3mm}$ is the total flux, $d$ is the distance to the source
(140 pc), $\kappa_{\rm 1.3mm}$ is the dust opacity per interstellar matter,
$B$ is the Planck function, and $T_{\rm d}$ is the dust temperature. Assuming
that the entire disk is optically thin, with $T$ = 20 K and $\kappa_{\rm
1.3\,mm}$ = 0.01 cm2 g-1 (e.g., Ossenkopf & Henning, 1994), the $M_{\rm gas}$
is estimated to be $\sim$4 $\times$10-4 $M_{\odot}$. The derived mass
represents one of the lightest categories within the current ALMA survey
across low-mass star-forming regions at the distance of 100–200 pc (see Long
et al. 2019; Ohashi et al. 2023). The H2 column density ($N_{\rm H_{2}}$) is
estimated using

$N_{\rm H_{2}}=\frac{F_{\rm 1.3mm}^{\rm beam}}{\Omega_{\rm A}\mu_{\rm
H_{2}}m_{\rm H}\kappa_{\rm 1.3mm}B(T_{\rm d})},$ (2)

where $F_{\rm 1.3mm}^{\rm beam}$ is the 1.3 mm intensity at a position,
$\Omega_{\rm A}$ is the beam solid angle, $\mu_{\rm H_{2}}$ is molecular
weight per hydrogen molecule ($\mu_{\rm H_{2}}$ = 2.8), and $m_{\rm H}$ is the
H atom mass. $F_{\rm 1.3mm}^{\rm beam}$ at the disk edge and peak are
$\sim$0.06 mJy beam-1 and $\sim$0.28 mJy beam-1, respectively. Applying the
same assumptions to derive $M_{\rm total}$, $N_{\rm H_{2}}$ are $\sim$5
$\times$1024 cm-2 at the disk’s edge and $\sim$2 $\times$1025 cm-2 at the disk
peak.

Next, we characterize the spike-like feature. Figure LABEL:fig:contmap (right)
shows the residual 1.3 mm image subtracting the Gaussian component described
above. This figure justifies that the observed 1.3 mm emission having excess
features in addition to the Gaussian-like component. The total flux of the
excess (spikes) is $\sim$0.36 mJy. Under the same assumptions as for the disk
calculations, the measured flux corresponds to a mass of $\sim$1 $\times$10-4
$M_{\odot}$ and a column density of $\sim$(3–5) $\times$1024 cm-2. Table 1
summarizes the disk and spike properties derived in this section. Note that
the spike structures are faint; thus, the measured properties somewhat depend
on the imaging parameters and analysis methods. We have conducted the same
analysis with lower-resolution data (Figure LABEL:fig:contmap_change (left))
to outline the possible range of quantities in Table 1. Estimating additional
systematic errors is challenging, but the possible uncertainties for each
quantity could be a factor of $\sim$2\. In any case, it is quite evident that
there are entities with masses amounting to several tens of percent of the
protostellar disk. What we present as our results may be extremely simple, but
such protruding structures have been identified for the first time in the
disks of young protostellar systems.

Table 1: Physical properties of the disk and spikes based on the 1.3 mm continuum emission Component | Size (arcsec in FWHM) | P.A. ($\arcdeg$) |  $F_{\rm 1.3mm}$ (mJy) |  $M_{\rm gas}$ ($M_{\odot}$) | H2 column density (cm-2)
---|---|---|---|---|---
Disk (Gaussian) |  $\sim$0.058 $\times$ 0.025 | 177.9 | 1.0–1.1 |  $\sim$4 $\times$10-4 |  $\sim$(5–20) $\times$1024
Spikes (Residual) | $\cdots$ | $\cdots$ | 0.3–0.36 |  $\sim$1 $\times$10-4 |  $\sim$(3–5)$\times$1024

##  4 Discussion

###  4.1 What is the origin of the spike-like structures around the disk?

We first discuss what the newly identified spike-like structure could be.
Given its extension from the disk, it is initially presumed to be a structure
that has fragmented from the disk. The enclosed mass of the protostar is
estimated to be $\sim$0.2 $M_{\odot}$ (Tokuda et al., 2017), and the mass of
the disk is extremely light at $\sim$4 $\times$10-4 $M_{\odot}$ (Section 3).
Following a simplified formula of Kratter & Lodato (2016), the Toomre Q value
is estimated to be $>>$1, suggesting a substantially stable state of the disk.
Therefore, it is unlikely that the structure results from the disk undergoing
gravitational fragmentation. Turbulent fragmentation could be a possible
mechanism for forming wide binaries (Offner et al., 2010), but the separation
between the central disk and spikes is quite small in our case. Moreover, the
available high-resolution molecular line measurements (Tokuda et al., 2017),
including the preliminary analysis of the 12CO data in the present data set,
have not revealed any signs of a velocity field that would suggest the
presence of additional protostellar sources, making it unlikely that the
structure is a circumstellar disk of hidden companions.

Given that the known fragmentation processes mentioned above cannot explain
the formation of spike-like structures, we propose magnetic interchange
instability (Parker, 1979; Kaisig et al., 1992; Lubow & Spruit, 1995; Stehle &
Spruit, 2001) as a plausible mechanism. Magnetic buoyancy acts when the
magnetic field becomes strong relative to the mass loading toward the central
gravitational source (Lubow & Spruit, 1995), and then the magnetic flux
advects outward. Figure B1 in the Appendix illustrates the magnetic field and
gas dynamics due to interchange instability based on relevant numerical
studies described below. Initially, a disk and envelope system is present.
After interchange instability occurs, magnetic flux escapes from an edge of
the disk and forms a cavity within which the leaked magnetic flux and a small
amount of gas expand outward, maintaining a nearly elliptical shape (e.g.,
Zhao et al., 2011; Matsumoto et al., 2017). The surrounding gas and magnetic
field from the envelope flow into the disk through channels along the cavity
edge (e.g., Machida & Basu, 2020).

It is possible that the spike is tracing a part of this ring structure,
because the outer part farther from the protostar likely has a lower column
density below the detection limit. The ring structure, due to interchange
instability, grows at roughly the speed of sound (see Section 4.2), and thus
it could be a recent event that occurred within the last few hundred years.

We roughly estimate the magnetic field strength conditions necessary to (1)
initiate the creation of and (2) further develop interchange-instability-
originated structures, such as spikes, rings, and arcs, at the edge of the
protostellar disk, denoted as $B_{\rm init}$ and $B_{\rm dev}$, respectively.
According to the formulae of Lubow & Spruit (1995), the condition required to
trigger the instability can be described as

$B_{\rm init}^{2}>\frac{2\pi\Sigma_{0}GM_{*}}{r^{2}},$ (3)

where $G$ is the gravitational constant, $\Sigma_{0}$ is the surface density
at the edge of the disk, and $M_{*}$ is the mass of the central star. From our
observations, $\Sigma_{0}$ is approximately $\sim$5 $\times$1024 cm-2 at the
edge of the disk, $r$ $\sim$4 au (Section 3). Given an $M_{*}$ of 0.2
$M_{\odot}$ (Tokuda et al., 2017), the required magnetic field strength
$B_{\rm init}$ is on the order of $\sim$1 G. Subsequently, for the structures
formed by interchange instability to develop further, the magnetic pressure
must overcome the ram pressure from the infalling envelope onto the disk. This
condition is given by

$\frac{B_{\rm dev}^{2}}{8\pi}>\rho v^{2},$ (4)

where $\rho$ is the density of the infalling envelope and $v$ is the velocity
of the gas falling toward the disk. Note that we cannot use the
observationally derived densities at the disk edge (Section 3) to apply as the
property of infalling material, because the disk is not smoothly connected to
the envelope (Tsukamoto et al., 2023b). At a few au from the protostar, we
adopt a value of 10-13 g cm-3 based on numerical work (Masunaga & Inutsuka,
2000). This value represents the critical density at which magnetic
dissipation becomes effective, as indicated by (Nakano et al., 2002; Machida
et al., 2007). It is considered that the rotationally supported disks form in
the region exceeding the critical density (Tsukamoto et al., 2023b). A
freefall velocity at $r$ = 4 au is $\sim$9 km s-1, which is determined by the
central stellar mass of 0.2 $M_{\odot}$. In this case, the required magnetic
field strength $B_{\rm dev}$ is approximately 1 G. Therefore, the magnetic
field strength necessary for the formation and subsequent development of
structures due to interchange instability, though local, is strong.
Unfortunately, determining the actual magnetic field strength through
observations remains challenging, especially in a protostellar disk scale, due
to some physical and technical reasons (e.g, Liu et al., 2023). The
abovementioned field strength is slightly stronger than $\lesssim$1 G, which
is typically seen in nonideal magnetohydrodynamics simulations that involve
magnetic diffusion (Masson et al., 2016; Tsukamoto et al., 2017, 2023a).
However, if the parental core is penetrated by a strong average magnetic
field, the gravitational contraction amplifies the magnetic field, especially
on a smaller scale, creating favorable conditions for triggering interchange
instability (Machida & Basu, 2020). In the following, we discuss the magnetic
field conditions at the envelope on the dense core scale based on the
available observations.

Submillimeter continuum polarimetric observations of the MC 27 core have
allowed for the determination of the magnetic field strength. Using the
single-dish James Clerk Maxwell Telescope (JCMT), Soam et al. (2019) and
Fukaya et al. (2023) detected polarized emission and estimated a plane-of-sky
magnetic field strength ($B_{\rm pos}$) of 70–400 $\mu$G based on the Davis-
Chandrasekhar-Fermi method (Davis, 1951; Chandrasekhar & Fermi, 1953),
providing a mass-to-flux ratio normalized by its critical value
(2$\pi$$G^{1/2}$)-1, $\mu_{\rm c}$ of $\sim$1.5–3, which is a magnetically
supercritical state as a whole. We here reevaluate the magnetic field strength
and mass-to-flux ratio within a 3000 au radius from the central protostar,
which is relevant to the ALMA-observed envelope-scale (see also Section 4.2).
Fukaya et al. (2023) reported a polarization vector dispersion of
$\sim$11$\arcdeg$ from 450 $\mu$m at the close vicinity of the protostar. The
dense gas tracer N2H+ with a single dish at a similar resolution yields an
FWHM velocity width of 0.28 km s-1 (Tatematsu et al., 2004). The two values
provide a magnetic field strength of $\sim$510 $\mu$G using the same other
assumptions (Soam et al., 2019; Fukaya et al., 2023). Based on the formula
from Crutcher (2004), $\mu_{\rm c}=7.6\times 10^{21}N({\rm H{{}_{2}}})/B_{\rm
pos}$ is calculated to be $\sim$0.5 with a column density of 3 $\times$1022
cm-2 at a radius of 3000 au (Tokuda et al., 2016). Note that it is not
necessarily a strong constraint, but the low $\mu_{\rm c}$ value cannot
exclude the central region having a strong magnetic field to realize the
magnetically subcritical state.

Note that the conditions under which interchange instability occurs are not
necessarily constrained tightly, as they involve nonlinear, three-dimensional
magnetohydrodynamic phenomena. Interchange instability can occur even in
weaker magnetic field conditions on a dense core scale with $\mu_{\rm c}$ of
$\gtrsim$2-3 (Zhao et al., 2011; Joos et al., 2012; Machida et al., 2014).
Furthermore, the characteristics of dust grains and their growth can influence
magnetic flux transport in the disk, potentially triggering interchange
instability even when $\mu_{\rm c}$ is as high as $\sim$5 (Tsukamoto et al.,
2023a). In any case, if interchange instability occurred in the protostellar
disk of MC 27, we propose that it could explain the formation of the observed
spike structures. This hypothesis can also simultaneously explain the larger
structures, as discussed in Section 4.2.

In general, the onset of interchange instability depends on the resistivity
evolution, which depends on the ionization degree evolution. In high-
molecular-density regions such as the one studied in this article, the
ionization degree is essentially determined by the dust grain-size
distribution, because the surfaces of dust grains are the main recombination
sites of charged particles. However, we are currently missing information on
the dust grain size distribution in such a high-density region. Therefore, if
our interpretation of the current observation is correct, the identification
of the interchange instability may enable us to discriminate various models of
resistivity in numerical simulations. Thus, further studies in this direction
may provide valuable information on the evolution of the grain-size
distribution, which is crucial in our understanding of planet formation.

###  4.2 Summary of the MC 27/L1521F studies and a new star formation scenario

The previous ALMA observations investigated MC 27 at various resolutions and
molecular lines, and here we first summarize the detailed characteristics. MC
27 is a dense core with a mass of $\sim$3–4 $M_{\odot}$, which is a general
feature of similar targets in the Taurus region (Onishi et al., 2002). What
separates it from the others is its high central density, which is estimated
to be $\sim$106 cm-3 at a single-dish-resolution scale, $\sim$2000–3000 au
(Onishi et al., 1999; Crapsi et al., 2004; Kauffmann et al., 2008). At its
center lies the protostar, with a very low internal luminosity of $\sim$0.07
$L_{\odot}$, surrounded by a bipolar scattered light that spans several
thousand au, likely created by an outflow activity (Bourke et al., 2006;
Terebey et al., 2009). This bipolar nebula aligns roughly perpendicular to the
north-south elongation of the central disk (Figure LABEL:fig:contmap). On the
other hand, the bipolar outflows observed in molecular line emission with ALMA
are remarkably more compact than the scattered light nebula (Tokuda et al.,
2014), spanning only about a few hundred au, and do not connect to the central
disk (Tokuda et al., 2016, 2018). The absence of a large outflow corresponding
to the Spitzer scattered light was puzzling. Instead, arc-like structures
spanning $\sim$2000 au were observed in HCO+, 13CO, and C18O, not entirely
coinciding with the edge of the scattered light (Tokuda et al., 2014, 2018).

Using data from Tokuda et al. (2014, 2018), Figure 2 reconstructs the
characteristic spatial and velocity structures of the inner $r\sim$3000 au
region of the vicinity of the protostar. Panel (a) shows the arc-like
structures in HCO+ and millimeter sources. MMS-1 is the protostellar source,
which corresponds to the very low-luminosity protostar, and its disk size is
just $\sim$10 au based on our new spatially resolved image (Figure
LABEL:fig:contmap). MMS-2 and MMS-3 are starless condensations with a volume
H2 density of $\sim$106–107 cm-3. The particularly high-density source, MMS-2,
is located in the southwest direction of the protostar. Panel (b) shows the CO
(3–2) peak brightness temperature map. The high intensity (30–60 K) along the
filamentary cloud means that the kinematic temperature is remarkably high. The
warm CO structure appears to exist inside the arc structures. Panels (c) and
(d) illustrate a slightly larger scale of C18O with a spatial filter
supplement, using the Atacama Compact Array data. Beyond the HCO+ arc
structures and warm filamentary gas, there is an even more extended gas
outward. Looking at the velocity structure, the southern arc seems to have a
gentle velocity gradient. The systemic velocity of the dense core itself is
$\sim$6.6 km s-1 as measured by the single-dish studies (Onishi et al., 1999;
Tatematsu et al., 2004), and thus the bulk velocity of the arcs and their
surroundings is the same as the parental core. The inner side, closer to the
protostar, appears to be redshifted. The above features are also schematically
summarized in Figure 3, which is discussed in more detail later.

Figure 2:  (a) Color scale image showing the HCO+ (3–2) moment 0 map with a
velocity range of 6.7–7.1 km s-1. The white contours show the 1.3 mm continuum
image with a contour step of 0.36 mJy beam-1 (reproduced data from Tokuda et
al. 2014). The contours in the other panels are the same as those in panel
(a). Ellipses give the angular resolutions of each panel’s image in the lower
left corners. (b) Color scale image showing the peak brightness temperature
map of CO (3–2) with a velocity range of 0.2–6.2 km s-1 (reproduced data from
Tokuda et al. 2018). The white dotted lines highlight the warm filamentary gas
components. The same lines are also illustrated in the other panels. (c) Color
scale image shows the C18O (2–1) moment 0 map with a velocity range of 6.4–7.5
km s-1 (reproduced data from Tokuda et al. 2018). (d) Color scale showing the
C18O (2–1) moment 1 map.

For the star formation scenario in MC 27, particularly the origin of the
distinctive arc structures, we initially proposed gravitational torques
resulting from interactions among multiple protostellar sources, which were
believed to be hidden in the dense condensations MMS-2 and MMS-3 (Tokuda et
al., 2014; Matsumoto et al., 2015). However, subsequent observations,
including the present highest-resolution study, have clarified that no such
point sources likely exist within the dense sources. In addition, as we
discussed in Section 4.1, the dense region of the MC 27 core may have a strong
magnetic field ($\mu_{c}$ $\sim$0.5), which could effectively suppress the
fragmentation and subsequent formation of binary and multiple stellar systems
(Machida et al., 2008; Hennebelle & Fromang, 2008; Wurster & Bate, 2019). This
finding may exclude the multiple system-driven scenarios as an origins of the
arc-like structure, although another promising system has been discovered (Lee
et al., 2023). Tokuda et al. (2017, 2018) argued that the small ($\sim$10 au)
size of the protostellar disk was also once attributed to dynamical truncation
by multiple components, but this possibility can be similarly dismissed. The
presence of possible shock-heated gas, as observed in CO (3–2), located around
the disk, led us to propose turbulent stripping as truncating the disk. A
colliding gas flow onto a face-on disk can truncate the outer edge of the
protostellar disk, but it requires a high-velocity ($\sim$20 km s-1) and dense
(5 $\times$106 cm-3) stream (Wijnen et al., 2016). Such extreme conditions are
unlikely in the low-mass dense core, necessitating the reconsideration of the
overall description of the star formation scenario in this particular system.

We discuss the formation of the large-scale arc structures and the newly
detected spikes in MC 27 through magnetic interchange instability (see Section
4.1). Unlike the previously mentioned scenarios, magnetic interchange
instability does not impose physically inconvenient conditions, such as a
highly turbulent environment and/or multiple protostellar interactions.
Magnetic interchange instability allows for the creation of asymmetric
structures around the disk (Zhao et al., 2011; Matsumoto et al., 2017) and
efficient angular momentum transportation through magnetic braking. Because
the disk is the primary outflow driver due to magnetocentrifugal force (e.g,
Machida & Hosokawa, 2013; Commerçon et al., 2022), interchange instability
disrupts the accretion disk and may terminate the outflow activity as well as
the accretion, at least temporarily.

Figure 3 summarizes the large- and small-scale picture of MC 27. Our newly
proposed star formation scenario for MC 27 is as follows. Initially, just
after the protostar formation, the first episode of interchange instability
occurrs. A parcel of gas at the edge of the disk begins to leak out from the
disk into the infalling or circumstellar envelope, accompanying a large amount
of the magnetic flux, leading to the formation of ring-like structures or arcs
as parts of incomplete ring features. If the rings and arcs were formed due to
interchange instability, the interior would be a cavity with low gas density
but a strong magnetic field. The leaked gas collides with the circumstellar
material, and shocks should occur at the (inner) edge of the rings and arcs,
which is consistent with the observed characteristics of higher temperatures
inside the arcs (Figures 2 and 3) and the partial detection of shock-related
tracers such as SO and methanol (Favre et al., 2020). Meanwhile, the arcs
themselves expand outward on the order of the sound speed, $c_{\rm s}$ (Zhao
et al., 2011; Matsumoto et al., 2017). If slight density asymmetries existed
during the protostar formation, the interchange-instability-driven arc would
expand toward a specific direction (Krasnopolsky et al., 2012; Tokuda et al.,
2023). Assuming their expansion speed as $c_{s}$, 0.2 km s-1 at 10 K, the
dynamical time for these structures would be $\sim$5 $\times$105 years (=2000
au/0.2 km s-1). If we assume that the small-scale spike-like structures with a
few au, developed at a similar speed, we are likely observing a very recent
event, on the order of 100 years. If we interpret the former time, $\sim$5
$\times$105 years, as the same as the mass accretion history, growing to the
currently identified stellar mass of 0.2 $M_{\odot}$, the time-averaged
accretion rate would be $\sim$4 $\times$10-6 $M_{\odot}$ yr-1. This is
consistent with theoretical predictions just after the protostar formation
(e.g., Hunter, 1977; Shu, 1977; Whitworth & Summers, 1985; Tomisaka, 1996).
The rarefaction wave could also spread outward on a similar timescale, $c_{\rm
s}t$ (Shu, 1977), potentially explaining the currently observed two-power-law
H2 column density, $N$(H2), profile in the MC 27 system, which shows a
shallower slope than the outer ($\gtrsim$3000 au) part (Tokuda et al., 2016).
The rarefaction wave is thought to spread just after forming the first
hydrostatic core (Larson, 1969). Because interchange instability occurs at the
protostellar disk edge due to magnetic diffusion (Machida & Basu, 2020), the
relevant structure growth must begin at least after the disk formation, i.e.,
the second collapse. The presence of the arcs inside the radius where the
column density profile transits, i.e., the front of the rarefaction wave
(Figure 3a), does not contradict the theoretical expectations described above.

During the active mass accretion phase, a powerful outflow likely formed,
creating the cavity traced by Spitzer’s scattered light. If the mass accretion
and ejection were continuous, corresponding components should be observable in
the molecular line emission, but their absence suggests that the outflow has
been at least temporarily halted. Indeed, the current low luminosity indicates
an almost negligible mass accretion rate of $\lesssim$ 2 $\times$10-8
$M_{\odot}$ (Tokuda et al., 2017). The most recent interchange instability
event could have disrupted the disk’s asymmetry and temporarily quenched the
outflow (e.g., Figure 7c in Machida & Basu 2020), and we see the compact
outflowing components as the remnants in this case (Figure 3(b)). The
directions of the large-scale arc and the disk’s spike are consistent with
each other, which indicates the interchange-instability-driven structure
cannot expand toward the higher-density region around MMS-2. In summary, we
suggest that at least two times episodes of drastic magnetic field leakage due
to interchange instability realize the complex gas distribution and maintain
the disk size as quite small, $\sim$10 au. It should be noted that Matsumoto
et al. (2017) and Machida & Basu (2020) have already proposed the origin of MC
27’s large-scale arc structures as being due to interchange instability. By
combining the current observational evidence, including the ALMA highest-
resolution image, we reconstruct and consistently explain the overall star-
formation scenario for the first time.

Figure 3: A schematic summary of the characteristic gas structure at the
protostellar vicinity of the MC 27 dense core system.

###  4.3 Implications for the broader picture of star and protostellar disk
formation

While MC 27 presents a highly peculiar system, it is crucial to examine
whether the star formation scenario that we devise here in Section 4.2 is
generally applicable. Prior to this MC 27 work, a ring structure with a
diameter of $\sim$7000 au possibly driven by interchange instability was
reported in the Corona Australis (CrA) IRS 2 Class I system (Tokuda et al.,
2023). Unlike MC 27, the single ring feature suggests a solitary interchange
instability event. The CrA IRS 2 system, formed at the edge of a dense
cluster-forming complex, likely did not experience external disturbances
during the ring’s growth phase, preserving a relatively pristine shape.
However, conditions conducive to interchange instability could be frequently
realized at the disk’s edge (Machida & Basu, 2020), and, indeed, theoretical
calculations often reveal multiple ring structures (e.g, Stehle & Spruit,
2001), which are also likely to be disturbed by gas accretion and outflow.

Interchange instability also allows for the formation of asymmetric arc
structures in the protostellar envelope without initially imposing strong
turbulence (see Offner et al. 2008). These arc-like structures, as incomplete
rings, probably correspond to streamers in numerous protostellar targets.
While further individual studies are necessary to discuss the origins of such
structures observationally, arc-like structures or rings spanning a few
thousand astronomical units, similar to those discovered in MC 27, have been
frequently identified in Class 0/I targets (e.g., Alves et al., 2020; Murillo
et al., 2022; Mercimek et al., 2023; Sai et al., 2023; Sharma et al., 2023).
The possibility that these are parts of rings formed through interchange
instability cannot be dismissed in favor of a simple accretion interpretation.
If this is the case, the disks formed in a strongly magnetized condition that
drives the interchange instability, as in MC 27, are expected to remain small
without showing a powerful outflow. Aso et al. (2019) presented some Class 0
objects with extensive envelopes but no detectable outflows, suggesting that
the protostars may have formed from magnetically subcritical cores (see the
discussion in Machida & Basu 2020). High-resolution observations targeting
small disks to search for spike structures and exploring the relationship
between disk size and the presence or absence of a surrounding arc-like
envelope, as well as outflow activities, will be beneficial to gain
observational insights into indirect pieces of evidence for the role of the
magnetic field in star formation and subsequent protoplanetary disk formation.
This type of study can potentially offer insights into the magnetic flux
problem in the star-formation process.

##  5 Summary

We have conducted the highest-resolution ($\sim$0$\farcs$03) ALMA observations
at a wavelength of 1.3 mm of the puzzling dense core MC 27/L1521F in Taurus,
which harbors a very low-luminosity protostar. This measurement marks the
first instance of executing such a high-resolution study on a faint
protostellar source with a 1.3 mm flux of $\sim$1 mJy. Our results spatially
resolved the 10 au disk and captured spike-like features extending
northeastward from the disk. We propose that this faint feature could be the
result of a recent event involving magnetic flux advection, because other
possible mechanisms, such as gravitational instability and turbulent
fragmentation, cannot explain the new feature. The dynamic magnetic flux
transport events (interchange instability) occurring in this system are not
single; at least one previous event has played a crucial role in shaping the
compact disk of this system and making the large-scale arc-like feature, which
resembles recently reported accretion streamers. We suggest that the
interchange instability might be at work in other protostellar systems hosting
compact ($\lesssim$10 au) disks surrounded by arc- or ring-like structures. By
investigating the history of magnetic flux transport imprinted in complex
envelopes, we can deepen our understanding of the diversity of star formation,
particularly the origin of small-sized disks, and subsequent planet formation
as inferred from grain-size distributions that satisfy the conditions for the
occurrence of interchange instability.

We would like to thank the anonymous referee for useful comments that improved
the manuscript. This paper makes use of the following ALMA data:
ADS/JAO.ALMA#2011.0.00611.S, 2015.1.00340.S, and 2017.1.00480.S. ALMA is a
partnership of ESO (representing its member states), the NSF (USA), and NINS
(Japan), together with the NRC (Canada), MOST, and ASIAA (Taiwan), and KASI
(Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA
Observatory is operated by the ESO, AUI/NRAO, and NAOJ. This work was
supported by an NAOJ ALMA Scientific Research grant (Nos. 2022-22B) and
grants-in-aid for scientific research (KAKENHI) of the Japan Society for the
Promotion of Science (JSPS; grant Nos. JP18H05436, JP18H05437, JP20H01945,
JP20H05645, JP21H00046, JP21H00049, JP21K03617, JP21K13962, JP23K03464, and
JP23H00129). The main content observations for this study were conducted more
than 6 years before the writing of this paper. We have long regretted not
being able to release these valuable ALMA data (2017.1.00480.S) in a
publication format, but the serendipitous discovery of a large-scale gas ring
in the Corona Australis region (Tokuda et al., 2023) provided a breakthrough
for our astronomical interpretations in this study. We would like to express
our gratitude once again to the objects, including MC 27, that have shown us
the dynamic aspects of star formation.

##  Appendix A 1.3 mm continuum imaging with different CLEAN weighting
parameters

Figure LABEL:fig:contmap_change presents two imaging results with different
Briggs weighting parameters, robust = 0.0 and $-$2.0 (Uniform), to ensure the
fidelity of the faint features of the disk and excess, i.e., spike structures.
The resultant beam sizes of the Briggs and Uniform images are 0$\farcs$029
$\times$ 0$\farcs$020 (P.A. = 18.0°) and 0$\farcs$024 $\times$ 0$\farcs$017
(P.A. = 17.6°), respectively. The rms noise levels are $\sim$11 $\mu$Jy beam-1
(robust = 0.0) and $\sim$27 $\mu$Jy beam-1 (Uniform). Typically, Uniform
weighting is expected to enhance angular resolution, but in this instance, it
resulted in a resolution comparable to that obtained with a robust parameter
of $-$1.0 (as illustrated in Figure LABEL:fig:contmap). The use of a robust
parameter of $-$1.0 provided the advantage of higher sensitivity in our data.

##  Appendix B A Schematic view of interchange instability

Figure B1 illustrates the schematic view of interchange instability around the
disk.

Figure B1: A schematic view of the magnetic field and gas distributions due to
interchange instability around a protostellar disk.

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