Ring Structures by Coagulation of Dust Aggregates in Protostellar Disks observedby ngVLA
Satoshi Ohashi, 1 Hiroshi Kobayashi, 2 Nami Sakai 1
1

RIKEN Cluster for Pioneering Research, 2-1, Hirosawa, Wako-shi,
Saitama 351-0198
satoshi.ohashi@riken.jp
2
Department of Physics, Graduate School of Science, Nagoya University, Furo-cho,
Chikusa-ku, Nagoya 464-8602

Abstract
We study coagulation of dust aggregates in a Keplerian rotating disk to investigate the grain growth, leading
to the planet formation. By calculating the evolution of dust surface density and dust size, we find that the dust
grains grow in an inside-out manner because the growth timescale is roughly proportional to the orbital period.
The boundary of the dust evolution can be regarded as a growth front because large dust grains are formed inside
the boundary, while the grain growth does not occur yet outside of the boundary. Radiative transfer calculations
show that the growth front can be identified as a ring structure, and the spectral index will be one of the reliable
observing ways to identify the growth front. We discuss that the ngVLA together with ALMA will allow us to reveal
substructures in protostellar disks and testify the growth front model.
Key words: Circumstellar dust1 — Protostellar disks2 – Protoplanetary disks3 – Planet formation4
1. Introduction
The understanding of the planet formation is one of the major studies in astronomy because planets are ubiquitous in theuniverse. More than four thousands exoplanets have been discovered since the first discovery in 1995 (Mayor & Queloz
1995). Planets are known to be formed in circumstellar disksof gas and dust around young stars (Williams & Cieza 2011).
Therefore, dust grains are thought to be building blocks forplanets.
The planet formation can be regarded as size evolution of thedust grains from the typical size of 0.1 − 1 µm in Inter Stellar
Medium (ISM) to km-size of planetesimals. It is suggested
that the dust grains are grown by sticking together and form
the planetesimals in protoplanetary disks. However, the grain
growth process is poorly understood because there are several
issues for the planet formation. For example, the dust grains
move radially inward due to the drag of the gas and cannot
survive for the planet formation, which is called “radial drift
barrier” (Adachi et al. 1976; Weidenschilling 1977). One of theother issues is “mass budget problem”, which is that the dustin Class II protoplanetary disks may not be enough to explain
the averaged solid mass in exoplanets (Andrews & Williams
2005; Andrews & Williams 2007).
Early stage of the disk evolution around Class 0/I protostars
is one of the key targets to investigate the grain size evolution
in the circumstellar disks. A survey of protostellar disks in theOrion molecular clouds showed that the protostellar dust disk
masses are systematically larger than those of Class II disks bya factor of > 4 (Tobin et al. 2020). Because the decrease in dust
disk masses is expected from disk evolution by accretion, thegrain growth may need to begin during the protostellar phase.
Recent high spatial resolution observations of millimeterand sub-millimeter wavelengths with interferometers such as

the Atacama Large Millimeter/ submillimeter Array (ALMA)
and the Very Large Array (VLA) reveal a variety of picturesof disk structures from the early stage of the disk formation(Takakuwa et al. 2017; Sheehan & Eisner 2017; Sheehan &Eisner 2018; Sai et al. 2020; Tobin et al. 2020; Garufi et al.
2020) to the late stage of protoplanetary disks (van der Marelet al. 2013; Casassus et al. 2013; ALMA Partnership et al.
2015; Isella et al. 2016; Andrews et al. 2018; Fedele et al.
2018; Tsukagoshi et al. 2019). In particular, dust ring structures have been observed in many protoplanetary disks. For example, the first ALMA Long Baseline Campaign observations
show the prominent ring and gap structures of the HL Tau disk
at 30 mas resolution (ALMA Partnership et al. 2015). The Disk
Substructures at High Angular Resolution Project (DSHARP)
project in the ALMA Cycle 4 Large program (Andrews et al.
2018) has also shown that disks have gap/ring structures in thesample of large and bright disks (Huang et al. 2018).
Interestingly, the dust ring structures are observed not only
in class II protoplanetary disks, but also even in the earlier
stages of disk formation around class 0 and class I objects(Sheehan & Eisner 2017; Sheehan & Eisner 2018; Sheehan etal. 2020; Nakatani et al. 2020; Segura-Cox et al. 2020). The
formation of the ring structures in growing young disks arestudied by Ohashi et al. (2020). They suggest that the growth
front by coagulation of the dust grains can be observed as thering structure because the disk has the radial distribution of
the grain size and dust surface density. Ohashi et al. (2020)
propose that the grain growth already begin in the protostellar
disks.
The next-generation Very Large Array (ngVLA) offers highangular resolution (∼10 mas) at long wavelengths (17 − 93
GHz or 3 − 18 mm), which is sensitive to thermal emission
from dust grains with a size of millimeter to centimeter. Thanks
to its high sensitivity and high spatial resolution with the long2[Vol. ,

wavelengths, ngVLA observations allow us to investigate thegrain sizes in the protostellar disks with optically thin regime.
In this paper, we show the idea of the growth front model andhow the growth front is observed with (sub) millimeter wavelengths. We also show the current observational problem with
ALMA and discuss the future solutions with ngVLA.

and ∼ 24 au, at t = 6.4 × 103 yr and t = 2.6 × 104 yr, respectively. Importantly, we point out that the position of the growth
front (the pebble production line) is independent of the dust
structures, disk mass, temperature, or the turbulence strength
as explained in the following discussion.

2. Ring Structure by Growth Front
2.1. ModelIn this subsection, we describe a brief summary of dust coagulation model. Detail calculations are shown by Ohashi etal. (2020).
The dust evolution considers two conditions: (1) the dust
grains grow via coagulation and (2) the dust grains have radial
drift due to decouple from the gas motion.
The mass distribution of dust particles is assumed to have asingle peak in mass mp (r) at each radial distance r. Then, assuming that the dust surface density Σd at each orbit r is dominated by particles with mass mp , we follow how the peaks ofdust surface density Σd and mass mp change with coagulation
and radial drift. According to Sato et al. (2016), the equationsof the evolution of Σd and mp are given by∂Σd 1 ∂+
(rvr Σd ) = 0,
∂t
r ∂r
√
2 πa2 ∆vpp
∂mp
∂mp
+ vr
=
Σd ,
∂t
∂r
hd

(1)
(2)

where a = (3mp /4πρint )1/3 is the particle radius, vr is the radial drift velocity of the particles, ∆vpp is the relative velocityof the particles, ρint is the internal density of dust grains, andhd is the dust scale height. The internal density of dust grains
is fixed to be a typical value of ρint = 1.4 g cm−3 for simplicity.
The initial gas surface density (Σg ) is set to be( r )−3/2
Σg = 1.7 × 103
g cm−2 ,(3)
1 AU
based on the minimum mass solar nebula (MMSN) model ofHayashi (1981). The jump of the surface density across thesnow line is ignored in this calculation because the snow line
position of ∼ 3 au is much smaller than the area to be investigated (∼ 10 − 100 au). The dust surface density (Σd ) is initiallly set to be 1% of the gas surface density.
The dust temperature is determined by a thermal equilibrium
as follows( r )−1/2 ( L )1/4
T = 280
K.
(4)
1 AU
L⊙
The initial size of the dust particle is uniformly set to 0.1µm.
2.2. Dust Evolution
The global evolution of the dust surface density Σd and particle size a of the coagulation model is shown in Figure 1 . In
this figure, the critical radius where the surface density remains
the initial state on the outside, can be identified. Furthermore,this position moves outwards over time. This position can beconsidered as a growth front, where the dust evolution proceeds. For example, the growth front corresponds to ∼ 10 au

Fig. 1. Time evolution of the surface density Σd (top panel) and radiusa (bottom panel) of dust particles as a function of orbital radius r formodels with a = 10−3 . The blue lines show the initial conditions,while the black dotted, dashed, and solid lines are the snapshots at times
t = 6.4 × 103 , 2.6 × 104 , and 1.0 × 105 yr, respectively.

The dust evolution can be estimated from growth timescale(Takeuchi & Lin 2005; Garaud 2007; Brauer et al. 2008;
Birnstiel et al. 2010; Birnstiel et al. 2012; Okuzumi et al. 2012,
e.g.,). The growth rate of the aggregate mass m at the midplaneis given by Tanaka et al. (2005) asdm
Σd σcol ∆v= ρd σcol ∆v = √
,
(5)
dt
2πhd
√
where ρd = Σd /( 2πhd ) is the spatial dust density at the midplane, σcol is the collisional cross section of two dust particles. Then, Equation (5) can be rewritten in terms of the growthtimescale as√( m ) m√2πh4 2π hd ρint adtgrow =
=
(6)
=
,
ṁ
Σd σcol ∆v
3 ∆v Σdwhere m = (4π/3)ρint a3 and σcol = πa2 .
Here, we focus on the millimeter sized dust grains because these grains are sensitive to millimeter-wave
emission.
√
For millimeter-sized
dust
grains,
h
∼
α
/Sth
and
∆v ∼
d
D
g
√
∆vt ∼ αD Stcs (Brauer et al. 2008). Then, we obtain

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Fig. 2. The intensity maps of the dust coagulation model at t = 6.4 × 103 yr, 1.3 × 104 yr, and 2.6 × 104 yr obtained by the radiative transfer calculations
with RADMC-3D. The observing wavelengths are λ = 870 µm. The distance is 100 pc.

(
tgrow ∼

Σg
Σd

)

1
.
ΩK

(7)

Equation (7) indicates that the dust evolution commencesfrom inside out because the growth timescale is roughly proportional to the orbital period. Importantly, the growth timescale is independent on the other parameters such as the internal density of dust grains (ρint ) and turbulence (αD ) excepting
for the dust to gas mass ratio (Σg /Σd ). Therefore, the dust
growth time is not affected by the fluffiness of dust, the disk
mass, temperature, or the strength of the disk turbulence, while
it becomes faster by increasing the dust mass ratio. Outsideof the growth front, the dust particles still remain the initialstate since the dust evolution is not proceeded yet, while thedust grains are grown inside of the growth front. As a result,the dust surface density is maximized in the growth front (seeFigure 1). Therefore, a ring structure is expected to be observedat the growth front.

Fig. 3. The ring positions of our model against time. The red line indicates the fitting to equation (8) to derive the transformation coefficientA.

2.3. Growth Front Location
Based on the dust evolution, the radiative transfer calculations are performed by using RADMC-3D (Dullemond et al.
2012). The observing wavelength is set to 0.87 mm, whichcorresponds to ALMA band 7 observations.
Figure 2 shows images of the radiative transfer calculationsof the model at t = 6.4 × 103 yr, t = 1.3 × 104 yr, and t =2.6 × 104 yr. The ring structure can be observed at the growth
front and the position moves outward over time.
The ring location can be formulated since the growth frontis roughly proportional to the orbital period. Equation (7) indicates the timescale of the dust evolution which is a function of
the Keplerian frequency ΩK . In other words, if the timescale(tgrow ) is set to the disk age (tage ), we can derive the critical radius (Rc ) where the growth front reaches because ΩK ∝ r−3/2 .
The critical radius can be estimated by transforming the equation (7) into a function of r. Thus, Equation (7) yields()1/3 ()2/3 (
)2/3
M⋆
ζd
tdisk
au
Rc = A
(8)
M⊙
0.01
1 yr
where A is the transformation coefficient and ζd = Σd /Σg . The
coefficient A can be derived by fitting the various ring positions
in time to Equation (8). Therefore, we perform the radiativetransfer calculations for the model at t = 6.4×103 yr, 1.3×104
yr, 2.6 × 104 yr, 5.2 × 104 yr, and 1.0 × 105 yr by assuming

λ = 870 µm, M⋆ = 1M⊙ , and ζd = 0.01. Then, we identify
the ring position. Figure 3 shows the time evolution of the ring
position. The error of the ring position indicates the full widthhalf maximum (FWHM) derived by the direct measurements ofthe synthetic images. By fitting the ring positions to equation(8), A is derived to be 0.026.
3. Observational Study of the Growth Front with ngVLA
We discuss the growth front and observations for the dust
ring structures. As shown by Ohashi et al. (2020), the growth
front location is consistent with the observed ring positions of
the Class 0/I sources.
3.1. Spectral Index Distributions
The important evidence of the growth front would be thetransition of the dust grain size across the growth front. Inside
the growth front, dust grains as large as millimeter or centimeter size are formed because the grain growth proceeds. On theother hand, outside the growth front, small dust grains (∼ 1 µmsize) are still remained.
The spectral index will be one of the reliable ways to investigate the dust grain size distribution. Based on the dust evolution model and radiative transfer calculations, we derive the4[Vol. ,

Fig. 4. The spectral index (α) maps of the dust coagulation model at t = 6.4 × 103 yr, 1.3 × 104 yr, and 2.6 × 104 yr. The spectral index α is derived
by the intensities of the 870 µm and 7 mm images.

Fig. 5. (Left two panels) the VLA 7 mm dust continuum image and the spectral index map of L1527 (Nakatani et al. 2020; Ohashi et al. 2020). The
spectral index is derived by the intensities between 3 mm and 7 mm wavelengths. (Right two panels) the ALMA 1.3 mm dust continuum image and thespectral index map of IRS 63 (Segura-Cox et al. 2020). The ring positions are shown in the white lines. The spectral index map is derived by the ALMA
1.3 mm and 3 mm dust continuum emission.

spectral index between 0.87 mm and 7 mm observing wavelengths. As shown in Figure 4, the growth front model predicts that the spectral index will change across the growth
front. Therefore, observations of dust continuum emission withmultiple wavelengths allow us to identify the evidence of thegrowth front for the ring formation mechanism in the protostellar disks.
3.2. A Case Study of the Growth Front: L1527 and IRS 63
We study the protostellar disks around L1527 and IRS 63
by using ALMA and VLA archival data. These disks have aring (L1527) or multiple ring structures (IRS 63), and thesering positions are consistent with the growth front (Ohashi etal. 2021). By using 0.87 mm and 7 mm dust continuum emission for L1527, and using 1.3 mm and 3 mm dust continuum
emission for IRS 63, we derive the spectral index distributions.
Figure 5 shows the spectral index maps of the L1527 and
IRS 63 disks. The both disks show that the spectral index haveα < 2 nearby the protostars, and that seems to increase to α ∼2.3 in the outer regions. Even though the increasing trend of
the spectral index with outer radii is consistent with the growth
front model, the observed spectral indices of α ∼ 1.6 − 2.3 aremuch lower than the predictions of α ∼ 2.6 − 3.7 by the growth

front model.
The discrepancy of the observations and model may becaused by high optical depth of the submillimeter dust continuum emission. If the emission becomes optically thick, itfollows the black body radiation. Then, the spectral index is
α = 2 regardless of the grain size. In this case, the spectral index does not imply the grain size. By taking into account thebeam dilution of the observations for these disks, the spectral
indices can be larger than 2 but lower than that of the opticallythin case.
We also point out that these disks have anomalously low(i.g., < 2.0) spectral indices nearby the protostars. Such low
millimeter spectral indices can be achieved by optically thickemission with temperature gradients. Short wavelengths tend
to be optically thicker and are opaque at the outer region of thedisks. By taking into account the temperature gradient of thedisk, the shorter wavelengths observe the outer cold regions.
Thus, the spectral indices can be low values (i.g., < 2.0).
The self-scattering of dust grains (Kataoka et al. 2015) may
also affect the intensity of the dust thermal emission and spectral index at millimeter wavelengths (Soon et al. 2017; Ueda etal. 2020). Liu (2019) and Zhu et al. (2019) pointed out that the
spectral index decreases lower than 2 due to the scattering ef-

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Fig. 6. Example images from ALMA 0.87 mm dust continuum observations toward selected protostellar disks having ring structures. The data is obtainedby the VANDAM survey (Tobin et al. 2020). The synthesized beams are drawn in the lower left corner; the typical ALMA synthesized beam is 0 1 arcsec.

fect because the scattering of (sub)millimeter-sized dust grains
makes the (sub)millimeter thermal emission fainter for optically thick emission. As a result, the spectral index becomesflatter or steeper than the case without scattering effect.
3.3. Future Observations by ngVLA
In the previous subsection, we show that the protostellar
disks of L1527 and IRS 63 have lower spectral indices than thegrowth front model possibly due to the optically thick emission even though the ring structures of these disks are consistent with the growth front locations. Furthermore, recent
ALMA observations suggest that sub-millimeter dust continuum emission may be optically thick in bright ring regions andinner area around protostars in protoplanetary disks (Zhu etal. 2019; Ohashi et al. 2020; Ueda et al. 2020). Therefore, it
may be difficult to measure the dust grain size in such regionswith ALMA sub-millimeter dust continuum observations. In
order to reveal the dust distributions and investigate the growth
front, multi wavelengths observations in optically thin regime
are needed.
ngVLA will allow us to observe dust grains at midplane ofdisks with optically thin dust continuum emission because theabsorption opacity is much lower in longer wavelength. The
absorption opacity at 7 mm wavelength is almost an order ofmagnitude lower than that at 0.87 mm wavelength. Thus, theoptical depth at 7 mm is roughly an order of magnitude lower
than that at 0.87 mm.
According to the ngVLA key performance, the entire disksof L1527 and IRS 63 can be imaged at 7 mm wavelength witha spatial resolution of 0.1 arcsec within an integration time ofa few minutes for each source. Therefore, ngVLA is an important telescope to reveal the disk structures and investigate thedust grain sizes at the midplane of the disks.
Recent ALMA survey observations with 0.87 mm dust continuum emission have been performed toward various star
forming regions (Tobin et al. 2020). These observations also
show that some fraction of protostellar disks have substructures such as rings and lopsided structures (Figure 6). It may
be possible that the ring structures are explained by the growth
front because the ring positions are similar to the growth front
locations. To investigate the dust distributions of these disks,observations with longer wavelengths are needed to derive thespectral indices.

Fig. 7. The intensity maps of the dust coagulation model at
t = 6.4 × 103 yr, 1.3 × 104 yr, and 2.6 × 104 yr obtained by theradiative transfer calculations with RADMC-3D. The observing wavelength is 7 mm and the spatial resolution is 0.1 arcsec. The distance is
100 pc.

However, these disks are too faint to detect emission of 3
mm or 7 mm wavelength by using current telescopes such as
ALMA and VLA. On the other hand, ngVLA will allow us
to detect the emission even at 7 mm wavelength above 3σ atring structures with an integration time of <∼ 2 hrs, thanks to
its high sensitivity capability. Even for the MMSN model, the
7 mm dust continuum emission can be imaged with a spatial
resolution of 0.1 arcsec with an integration time of ∼ 3 hrs.
The emission is sharply dropped in the growth front because
smaller dust grains emit inefficiently in longer wavelengths,
while larger dust grains emit efficiently (Figure 7). Therefore,
ngVLA provides us with a great important possibilities to investigate the grain growth not only for bright protostellar disks
but also for typical faint protostellar disks.
It should be noted that the collisional fragmentation is ignored in our calculation. By comparing the observed spectral
index with the model in detail, we will also further investigate
the collisional fragments inside of the growth front.
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