# Cloudlet Capture Model for the Accretion Streamer onto the disk of DG Tau

Tomoyuki Hanawa,1 Antonio Garufi,2 Linda Podio,2 Claudio Codella,2 and
Dominique Segura-Cox3
1Centre for Frontier Science, Chiba University, 1-33 Yayoi-cho, Inage-ku,
Chiba 263-8522, Japan
2INAF, Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125,
Firenze, Italy
3Department of Astronomy, The University of Texas at Austin, 2515 Speedway,
Stop C1400, Austin, Texas 78712-1205, U.S.A E-mail<EMAIL_ADDRESS>(TH)

(Accepted 2024 January 30. Received 2024 January 26; in original form 2023
December 2)

###### Abstract

DG Tau is a nearby T Tauri star associated with a collimated jet, a
circumstellar disk and a streamer a few hundred au long. The streamer connects
to the disk at $\sim$50 au from DG Tau. At this location SO emission is
observed, likely due to the release of sulphur from dust grains caused by the
shock of the impact of the accretion streamer onto the disk. We investigate
the possibility that the DG Tau streamer was produced via cloudlet capture on
the basis of hydrodynamic simulations, considering a cloudlet initiating
infall at 600 au from DG Tau with low angular momentum so that the centrifugal
force is smaller than the gravitational force, even at 50 au. The elongation
of the cloudlet into a streamer is caused by the tidal force when its initial
velocity is much less than the free-fall velocity. The elongated cloudlet
reaches the disk and forms a high density gas clump. Our hydrodynamic model
reproduces the morphology and line-of-sight velocity of CS ($5-4$) emission
from the Northern streamer observed with ALMA. We discuss the conditions for
forming a streamer based on the simulations. We also show that the streamer
should perturb the disk after impact for several thousands of years.

###### keywords:

hydrodynamics — methods: numerical — stars: individual (DG Tau) — stars: pre-
main-sequence

††pubyear: 2023††pagerange: Cloudlet Capture Model for the Accretion Streamer
onto the disk of DG Tau–Cloudlet Capture Model for the Accretion Streamer onto
the disk of DG Tau

##  1 Introduction

Young stellar objects are associated with disks of gas and dust. Disks are
formed since the early stages of the star formation process due to the
conservation of angular momentum of the slowly rotating collapsing core (e.g.
Shu et al., 1993). To understand the evolution and the accretion and ejection
phenomena onto/from the disk is of key importance as disks are the birthplace
of planets. This accretion disk theory assumes that the system is essentially
symmetric around the rotation axis and that evolution is driven by angular
momentum transfer. Though this theory has succeeded in describing star
formation in a general sense, it cannot explain asymmetric features discovered
recently with high-resolution observations by the Atacama Large
Millimeter/submillimeter Array (ALMA; e.g., Yen et al., 2014; Sakai et al.,
2016; Artur de la Villarmois et al., 2019; Pineda et al., 2020, for L1489,
TMC-1A, Per-emb-2, and GSS30-IRS5, respectively).

Gas accretion onto the disk is an important phenomenon as it may affect the
mass, density, and chemical evolution of the disk. Asymmetric non-Keplerian
gaseous structures, named accretion streamers, have routinely been detected by
ALMA and high-contrast imagers (see, e.g., the review by Pineda et al., 2023).
Kinetic modeling is required to confirm that asymmetric structures are indeed
infalling streamers (e.g. Thieme et al., 2022; Valdivia-Mena et al., 2022).
Some of the most notable examples of streamers connected to the disk are found
around DO Tau (Huang et al., 2022), HL Tau (Yen et al., 2019; Garufi et al.,
2022), SU Aur (Ginski et al., 2021), DG Tau (Garufi et al., 2022), and CB 68
(Kido et al., 2023). Streamers may also feed material to evolved disks (less-
embedded in their surrounding environments) in the form of late-stage infall
events (Gupta et al., 2023; Huang et al., 2023), where the infalling material
is more-often comprised of gas initially unbound to the initial dense core
(Kuffmeier et al., 2023). These findings are pivotal to understand how the
disk accretion through streamers and the disk/environment interaction proceed
throughout the disk evolution.

In this paper, we study a particular streamer known to accrete the disk of DG
Tau, a T Tauri star located in the Taurus star forming region ($d$=125 pc,
Gaia data release 3, Gaia Collaboration et al., 2016, 2023). DG Tau is
associated with a blueshifted jet (e.g., Eislöffel & Mundt, 1998), a residual
envelope observed at large scales (e.g., Kitamura et al., 1996), and a compact
disk (e.g., Testi et al., 2002; Isella et al., 2010). The streamer, recently
observed in CS ($5-4$) and CO ($2-1$) emission in the context of the ALMA-DOT
program (Garufi et al., 2022), extends for a few hundred au to the north of
the disk as shown in Figure 1. The northern streamer consists of red and blue
components. The region where the streamer connects to the disk is associated
with SO (white contours) and SO2 emission, suggesting that an accretion shock
occurs where the streamer impacts onto the disk, causing the release of
sulphur from dust grain mantles due to dust grain sputtering and shattering
(Neufeld & Dalgarno, 1989a, b). The enhancement of the emission from S-bearing
molecules in shocks is commonly observed along jets and outflows (e.g.,
Bachiller & Pérez Gutiérrez, 1997; Lee et al., 2010; Codella et al., 2014;
Podio et al., 2021), and at the interface between the infalling envelopes and
disks (accretion shocks, e.g., Sakai et al., 2014; Oya et al., 2016).

Figure 1: The color shows the moment 1 of CS $5-4$ emission line. The red and
white contours denote the continuum at 1.3 mm and SO $4_{5}-3_{4}$ integrated
intensity, respectively. Adopted from Fig. 2d) of Garufi et al. (2022).

The flat spectrum of DG Tau indicates the existence of hollow cavities outside
the disk (see, e.g. Whitney et al., 2003). The morphology of the system
suggests that the northern streamer resides in a narrow region in the northern
cavity. We surmise an invisible tenuous gas confines the streamer thin by
pressure. The tenuous gas component prevents denser gas from accreting and
keeps the cavity transparent. It should be warm since it has significant
enough gas pressure to confine the streamer, despite the low density of the
tenuous gas. We assume that it is static for simplicity, since the streamer
resides near the disk and far from the outflow axis. The streamer is likely to
come from a place distant from DG Tau and changes its shape when approaching
DG Tau. We assume the streamer was a dense gas clump in the surrounding
molecular cloud. It changed its trajectory several thousand years ago and
began to approach DG Tau. We name it “cloudlet”in this paper. The cloudlet
consists of cold molecular gas and inherits the chemical composition of the
natal molecular cloud. Our model considers the Keplerian disk rotating around
DG Tau, the cloudlet, and the tenuous gas around the disk. Since we do not see
any interaction of the streamer with the bipolar jets and infalling envelope,
we do not include them in our model for simplicity.

When considering gas accretion from the envelope to the central star, we can
employ analytic models (e.g., Ulrich, 1976; Cassen & Moosman, 1981) or the
ballistic approximation (see, e.g., Sakai et al., 2014). Such models typically
consider stationary inflows of material that are symmetric about the rotation
axis, but can be leveraged to model asymmetric infall. Effects of infall that
vary with time are not considered in these analytic models. These models
assume that gas follows a free-fall orbit until it reaches the disk midplane.
In other words, the models take into account hydrodynamic effects only after
the accreting gas reaches the midplane. This approximation looks valid since
the sound speed of the gas is much lower than the infall velocity near the
star. However, the gas pressure works to disperse an accreting gas cloudlet
into a diffuse cloud if the cloudlet localized in a small region. When
discussing the formation of a streamer, we should examine the hydrodynamic
effects since streamers should be confined against the gas pressure.

We aim to make a numerical simulation reproducing the morphology and
kinematics of the DG Tau streamer revealed in CS ($5-4$) emission. Though
Garufi et al. (2022) reported two possible streamers, we concentrate on the
northern one, which consists of red-shifted and blue-shifted component as
shown in Figure 1. Note that the red-shifted arc-like 12CO emission to the
south discovered by Güdel et al. (2018) does not coincide with the CS ($5-4$)
streamer on the sky, nor was it found to be kinematically-consistent with
infall (Garufi et al., 2022). The southern arc 12CO emission is likely to be a
part of the outflow and not included in our modeling. Our model is similar to
the cloudlet capture model for TMC-1A (Hanawa et al., 2022), although these
two objects show different morphologies. TMC-1A shows highly asymmetric
features of a few 100 au scale in the CS ($5-4$) emission line. The blue-
shifted component is much stronger than the red-shifted one.

In our modeling, we focus on the formation mechanism of an elongated
structure. Although the tidal force is the main driver, that alone is not
enough. Kuffmeier et al. (2020) considered capture of a cloudlet by a young
star as a seed of second-generation disk. The cloudlet evolves not into a
streamer falling onto the star, but an arc surrounding it. Their cloudlet
expanded appreciably during the infall since they considered only one
component of gas. We assume that a warm neutral gas surrounding DG Tau
confines an infalling cloudlet to form a streamer. We aim to follow the infall
of cloudlet until its head reaches the disk.

The disk of DG Tau has been recently imaged with ALMA in continuum emission
and in several molecular tracers (Güdel et al., 2018; Podio et al., 2019,
2020). These studies constrained the disk inclination ($i=35^{\circ}$) and
position angle (PA=135∘). The disk rotation (with the NW side redshifted and
the SE blushifted) and the PA of the blueshifted jet (PA=225∘, Eislöffel &
Mundt, 1998) indicates that the disk rotates clockwise and the near-side is to
the NE. Garufi et al. (2022) obtained a best fit elliptic orbit for the
streamer under the ballistic approximation, i.e., by taking into account only
the gravity of DG Tau. The orbit is 10∘ inclined with respect to the disk
plane and above the disk plane in the northwest. They assumed that the
streamer originated from 450 au away from DG Tau with small radial (0.4 km
s-1) and rotation (0.13 km s-1) velocities. We consider a cloudlet taking a
similar orbit and examine how it evolves.

This paper is organized as follows. We describe our model and methods in §2.
We show the results of 3 selected models in §3. We show the observed channel
maps of CS ($5-4$) emission of the streamer and compare them with the output
of the hydrodynamic model in §4. We discuss the condition for the formation of
a streamer in §5. We summarize our results in §6.

##  2 Model

This section describes our modeling and methods of computation. We introduce
the basic equations and numerical methods for solving them in §§2.1. We show
the initial model of a cloudlet infalling onto a disk rotating around DG Tau
in §§2.2. We outline the methods of the mock observation in §§2.3.

###  2.1 Basic Equations

We use the hydrodynamical equations,

$\displaystyle\frac{\partial\rho}{\partial
t}+\mathbf{\nabla}\cdot\left(\rho\mathbf{v}\right)$ $\displaystyle=$
$\displaystyle 0,$ (1) $\displaystyle\frac{\partial}{\partial
t}\left(\rho\mathbf{v}\right)+\mathbf{\nabla}\cdot\left(\rho\mathbf{vv}+P\mathbf{I}\right)$
$\displaystyle=$ $\displaystyle-\rho\mathbf{\nabla}\Phi,$ (2)
$\displaystyle\frac{\partial}{\partial t}(\rho
E)+\mathbf{\nabla}\cdot\left[\left(\rho E+P\right)\mathbf{v}\right]$
$\displaystyle=$ $\displaystyle-\rho\mathbf{v}\cdot\mathbf{\nabla}\Phi,$ (3)
$\displaystyle E=\frac{|\mathbf{v}|^{2}}{2}+\frac{P}{\rho(\gamma-1)},$ (4)

to describe gas accretion onto a protostar associated with a gas disk. The
symbols, $\rho$, $P$, and $I$ denote the density, pressure, the unit tensor,
respectively, while $\mathbf{v}$ and $\Phi$ denote the velocity and
gravitational potential, respectively. The gas pressure is expressed as

$\displaystyle P$ $\displaystyle=$ $\displaystyle\frac{k}{\bar{m}}\rho T,$ (5)

where $T$, $k$, and $\bar{m}$ denote the temperature, Boltzmann constant, and
mean molecular weight, respectively. Equation (3) considers neither heating
nor cooling explicitly. However, the specific heat ratio is taken to be
$\gamma=1.05$ so that the temperature remains nearly constant. This
approximation mimics the situation where both the molecular and atomic gases
remain nearly isothermal.

We consider three gas components: cloudlet, disk, and warm neutral medium. The
cloudlet and disk consist of molecular gas while the warm neutral medium
consists of of atomic gas. The mean molecular weight is assumed to be
$\bar{m}_{\rm c}=\bar{m}_{\rm d}=2.3~{}m_{\rm H}$ for the cloudlet and disk,
and $\bar{m}_{\rm w}=1.17~{}m_{\rm H}$ for the warm neutral medium, where
$m_{\rm H}$ denotes the mass of a hydrogen atom.

We introduce a variable,

$c=\left\\{\begin{array}[]{ll}1&\mbox{(cloudlet)}\\\ 0&\mbox{(warm neutral
gas)}\\\ -1&\mbox{(disk)}\\\ \end{array}\right.,$ (6)

to trace the three components. For this purpose, we solve

$\displaystyle\frac{\partial}{\partial
t}\left(c\rho\right)+\mathbf{\nabla}\cdot\left(c\rho\mathbf{v}\right)=0,$ (7)

simultaneously. Equation (7) means that $c$ remains constant along the gas
element, i.e., the Lagrangian derivative, $Dc/dt=0$, vanishes.

Using the position vector $\mathbf{r}$, we approximate the gravity to be

$\displaystyle-\mathbf{\nabla}\Phi$ $\displaystyle=$
$\displaystyle-\frac{GM\mathbf{r}}{(\max{|\mathbf{r}|,a})^{3}},$ (8)

where $G$, $M$ and $a$ denote the gravitational constant, the mass of the
central star, and the unit length, respectively. Equation (8) means that we
consider only the central star, with the mass taken to be
$M=0.7~{}\mbox{M}_{\odot}$, as a source of the gravity. The gravity is reduced
artificially in the region of $|\mathbf{r}|\leq a=30~{}\mbox{au}$ so that we
can avoid numerical difficulties. The corresponding gravitational potential is
expressed as

$\displaystyle\Phi$ $\displaystyle=$
$\displaystyle\left\\{\begin{array}[]{ll}-\displaystyle\frac{GM}{|\mathbf{r}|}&(|\mathbf{r}|>a)\\\
-\displaystyle\frac{GM}{2a^{3}}\left(3a^{2}-|\mathbf{r}|^{2}\right)&(|\mathbf{r}|\leq
a)\\\ \end{array}\right.$ (11)

We take into account neither magnetic fields nor turbulent viscosity, for
simplicity.

The mass of DG Tau was assumed to be $0.3~{}\mbox{M}_{\odot}$ in the modeling
of streamer in Garufi et al. (2022). However, the mass of DG Tau is still
uncertain. Testi et al. (2002) derived $M=0.67~{}\mbox{M}_{\odot}$ from 13CO
emission from the disk by assuming Keplerian rotation and an inclination of
38∘. Podio et al. (2013) confirmed that water vapor emission line profile is
consistent with $0.7~{}\mbox{M}_{\odot}$. Hence, we adopt 0.7 M⊙ in this paper
to model both disk and streamer. Since the CS emission from the streamer is
weak, we can constrain only weakly the mass from its line-of-sight velocity.

We use cylindrical coordinates, $(r,\varphi,z)$, in our numerical simulations.
The central star of DG Tau resides at the origin while the disk is in the
plane of $z=0$. We cover the cylindrical volume of $r\leq R_{\rm out}$ and
$|z|\leq Z_{\rm out}$. The spatial resolution is constant at
$r\Delta\varphi\simeq dr=dz=\Delta r_{0}$ around the origin. Outside the
central cylindrical region, the spatial resolution is $\Delta r/r=\Delta
z/|z|=\Delta\varphi=360^{\circ}/N_{\varphi}$, where $N_{\varphi}=384$ denotes
the maximum number of numerical cells in the azimuthal direction. The
parameters are set to be $\Delta r_{0}=1.5~{}\mbox{au}$ in model B and 0.75 au
in the rest of the models. See Hanawa & Matsumoto (2021) for further details
of the numerical code.

###  2.2 Initial Model

We assume the warm gas in the cavity surrounding the disk is isothermal at
$T_{\rm w}$ and in the hydro-static equilibrium for simplicity, although we
note that realistically the environment surrounding DG Tau has a temperature
gradient and non-zero initial motions. Then, the pressure is expressed as

$\displaystyle P_{\rm w}(\mathbf{r})$ $\displaystyle=$ $\displaystyle
P_{0}\exp\left[-\frac{\bar{m}_{\rm w}\Phi(\mathbf{r})}{kT_{\rm w}}\right],$
(12)

where $P_{0}$ denotes the pressure at a large distance from DG Tau. We set the
temperature to be $GM\bar{m}_{\rm m}/akT_{\rm w}=0.45$ so that the warm gas is
weakly bound to DG Tau at the distance of 30 au. It is $T_{\rm w}=1.42\times
10^{3}~{}\mbox{K}$ in the physical unit for $\bar{m}_{\rm w}=1.17~{}m_{\rm H}$
and $M=0.7~{}M_{\odot}$. Our estimate of $T_{\rm w}$ is based on the
theoretical consideration of the thermal equilibrium of the interstellar
medium (see, e.g., Inoue et al., 2006). When the main heating source is cosmic
rays and the main coolant is ionized carbon, the temperature of the warm
component is $1000-2000$ K at a high pressure while that of the cold component
is below 100 K. Accordingly the number density is expressed as

$\displaystyle n_{\rm w}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\frac{P_{0}}{kT_{\rm w}}\exp\left[-\frac{\bar{m}_{\rm
w}\Phi(\mathbf{r})}{kT_{\rm w}}\right],$ (13)

and proportional to $P_{0}$. Since Equations (1) through (3) are proportional
to the density, our model has no specific density. We will define the
parameter $P_{0}$ later, but the choice of $P_{0}$ does not alter the
velocity.

The cloudlet, the precursor of the streamer, is a gas sphere having the radius
$R_{\rm c}$ and centered at $\mathbf{r}_{0}$ at the initial stage. We specify
the initial cloudlet center by

$\displaystyle\mathbf{r}_{0}$ $\displaystyle=$ $\displaystyle
r_{0}\cos\theta_{0}\cos\varphi_{0}\mathbf{e}_{x}+r_{0}\sin\varphi_{0}\mathbf{e}_{y}$
(14) $\displaystyle-r_{0}\sin\theta_{0}\cos\varphi_{0}\mathbf{e}_{z},$

where $\mathbf{e}_{x}$, $\mathbf{e}_{y}$, and $\mathbf{e}_{z}$ denote the
$x$\- , $y$\- and $z$-components of the unit vectors in Cartesian coordinates.

We consider two cases of orbits, elliptic and parabolic. In both the cases,
the initial velocity distribution is expressed as

$\displaystyle\mathbf{v}_{0}$ $\displaystyle=$ $\displaystyle
v_{0,r}\frac{\mathbf{r}_{0}}{\left|\mathbf{r}_{0}\right|}+v_{\varphi,0}\mathbf{e}_{z}^{\prime}\times\frac{\mathbf{r}_{0}}{\left|\mathbf{r}_{0}\right|},$
(15) $\displaystyle\mathbf{e}_{z}^{\prime}$ $\displaystyle=$
$\displaystyle\sin\theta_{0}\mathbf{e}_{x}+\cos\theta_{0}\mathbf{e}_{z}$ (16)

such that the parameter $\theta_{0}$ denotes the inclination of the orbital
plane to the $z$-axis. We specify the rotation velocity by

$\displaystyle v_{\varphi,0}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{GMr_{\rm cen}}}{\left|\mathbf{r}_{0}\right|},$ (17)

where $r_{\rm cen}$ denotes the centrifugal radius, i.e., the radius at which
the centrifugal force balances with the gravitational force. We arrange
$\varphi_{0}$ so that the periastron of the orbit is located in the plane of
$x=0$ and $y<0$.

When considering an elliptic orbit, we set the cloudlet at the apoastron
($\varphi=180^{\circ}$) at the initial stage. Accordingly, the initial radial
velocity vanishes ($v_{r,0}=0$). The eccentricity of the orbit is evaluated to
be

$\displaystyle\varepsilon$ $\displaystyle=$ $\displaystyle 1-\frac{r_{\rm
cen}}{r_{0}}.$ (18)

The orbit intersects with the disk midplane ($z=0$) at $(r,\varphi)=(r_{\rm
cen},~{}0^{\circ})$ and has the periastron distance,

$\displaystyle r_{\rm min}$ $\displaystyle=$
$\displaystyle\frac{r_{0}(1-\varepsilon)}{1+\varepsilon}.$ (19)

Then, the semi-major axis of the orbit and the orbital period are evaluated to
be

$\displaystyle a_{\rm maj}$ $\displaystyle=$
$\displaystyle\frac{r_{0}}{1+\varepsilon}=\frac{r_{0}{}^{2}}{2r_{0}-r_{\rm
cen}},$ (20) $\displaystyle P_{\rm orb}$ $\displaystyle=$ $\displaystyle
6.12\times
10^{3}\left(\frac{r_{0}}{600~{}\mbox{au}}\right)^{3/2}\left(\frac{1+\varepsilon}{2}\right)^{-3/2}~{}\mbox{yr},$
(21)

respectively. Figure 2 shows the geometry of the cloudlet and disk, the latter
of which lies with an inclination of 35∘ and the near side in the northeast.
The orbital plane of cloudlet is closer to us in the northeast than the disk
plane. Thus, the cloudlet recedes from us when approaching DG Tau. We assume
that the cloudlet is captured by turbulence at a distance of
$>500~{}\mbox{au}$ and turns into a highly elliptic or parabolic orbit.

Figure 2: This diagram denotes the geometry of the elliptic and parabolic
orbits of the cloudlet by the solid and dashed curves, respectively. The black
ellipse and square denote the disk and disk plane, respectively. The blue
solid and red dashed lines denote the blue-shifted and red-shifted jets,
respectively. The disk rotates clockwise on the sky plane as shown by the blue
and red arrows. The black arrows denote the $x$-and $y$-axis in Cartesian
coordinates in the numerical computation. The orbit of the cloudlet, shown by
the orange arc, is inclined by 10∘ from the disk plane.

When the cloudlet has a parabolic orbit, the initial radial velocity is
expressed as

$\displaystyle v_{r,0}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\left\\{-\sqrt{\displaystyle\frac{2GM}{\left|\mathbf{r}\right|}-v_{\varphi,0}^{2}}\right..$
(22)

The head of the cloudlet has a higher infall velocity than the tail. In other
words, the cloudlet head moves faster than the tail. Since the initial radial
distance is 10 times larger than the centrifugal radius ($r_{0}=10r_{\rm
cen}$), the eccentric anomaly is 154∘ at the initial stage in model C. The
periastron is located at $x$ = 0 and close to those of the elliptic orbits.
Thus the orbits are close each other near DG Tau.

We assume that the cloudlet has the same pressure as the surrounding warm
neutral medium so that it does not expand or shrink immediately. The number
density inside the cloudlet is given by

$\displaystyle n_{\rm c}(\mathbf{r})$ $\displaystyle=$
$\displaystyle\frac{P_{0}}{kT_{\rm c}}\exp\left[-\frac{\bar{m}_{\rm
w}\Phi(\mathbf{r})}{kT_{\rm w}}\right],$ (23)

for $|\mathbf{r}-\mathbf{r}_{0}|\leq R_{\rm c}$, where $T_{\rm c}$ and $R_{\rm
c}$ denote the initial temperature and radius of the cloudlet, respectively.
In the following, we assume $GM\bar{m}_{\rm c}/r_{0}kT_{\rm c}=5.0\times
10^{-3}$ so that the thermal energy is comparable with the gravitational
energy at the distance of 6000 au ($=10~{}r_{0}$) from DG Tau. The temperature
corresponds to $T_{\rm c}=28.6~{}\mbox{K}$ for $\bar{m}_{\rm c}=2.3~{}m_{\rm
H}$ and $M=0.7~{}M_{\odot}$. We assume assume $P_{0}/(kT_{\rm
c})=10^{4}~{}\mbox{cm}^{-3}$ though the density is scale free as mentioned
earlier.

We consider an isothermal disk rotating around the $z$-axis for simplicity,
though the disk should have radial and vertical temperature gradients.
Considering the gradients is beyond the scope of this paper because our
spatial resolution is limited. The velocity is expressed as

$\displaystyle\mathbf{v}$ $\displaystyle=$ $\displaystyle
v_{\varphi}(r)\mathbf{e}_{\varphi}.$ (24)

We assume that the disk is bounded in the region of $|z|\leq z_{s}(r)$,

$\displaystyle z_{\rm s}(r)$ $\displaystyle=$
$\displaystyle\beta\sqrt{(a^{2}+r^{2})\tanh\left[\frac{2\left(R_{\rm
d}-r\right)}{a}\right]},$ (25)

where $R_{\rm d}$ denotes the disk radius and $\beta$ specifies the disk
thickness. Figure 3 shows the hight of the disk upper boundary ($z_{s}$) as a
function of $r$ for $R_{\rm disk}=120~{}\mbox{au}$. The pressure distribution
is given by

$\displaystyle P_{\rm d}(r,z)$ $\displaystyle=$ $\displaystyle P_{\rm
w}[r,z_{\rm s}(r)]$ (26) $\displaystyle\times\exp\left\\{\frac{\bar{m}_{\rm
d}}{kT_{\rm d}}[\Phi(r,z_{s})-\Phi(r,z)]\right\\},$

where $T_{\rm d}$ denotes the temperature of the disk gas. We assume
$GM\bar{m}_{\rm d}/(akT_{\rm d})=0.01$, i.e., $T_{\rm d}=57.3~{}\mbox{K}$.
Equation (26) describes hydrostatic equilibrium in the vertical ($z$)
direction. The rotation velocity is set to be

$\displaystyle v_{\varphi}(r)$ $\displaystyle=$
$\displaystyle-\sqrt{\left(\displaystyle 1-\frac{\bar{m}_{\rm w}T_{\rm
d}}{\bar{m}_{\rm d}T_{\rm w}}\right)\left(\frac{\partial\Phi}{\partial
r}+\frac{\partial\Phi}{\partial z}\frac{\partial z_{\rm s}}{\partial
r}\right)},$ (27)

where we evaluate $\partial\Phi/\partial r$ and $\partial\Phi/\partial z$ at
the disk surface, $z=z_{s}(r)$, so that the sum of the centrifugal and
pressure forces balance gravity.

Figure 3: The solid curve denotes the half thickness of the disk as a function
of the radial distance from the rotation axis for $R_{\rm d}=120~{}\mbox{au}$
and $\beta=0.09$. The dashed line denotes the plane inclined 80∘ with respect
to the disk axis ($r=0$).

At the initial stage, we do not include the gas disk in early stages of
simulation ($0\leq t\leq t_{\rm d}$). We refresh the density, velocity,
pressure and the color field in the region of $r<R_{\rm d}^{\prime}$ and
$|z|\leq z_{\rm d}^{\prime}$ to insert the gas disk at $t=t_{\rm d}$.

Table 1 summarizes the parameters for the three models shown in this paper. We
set the radius of the cloudlet, $R_{\rm c}$ to reproduce the width and length
of the streamer.

Table 1: Model Parameters (see §2)

Model | $r_{0}$ | $r_{\rm cen}$ | $e$ | $|\mathbf{v}_{0}|$ | $R_{\rm d}^{\prime}$ | $Z_{\rm d}^{\prime}$ | $t_{\rm d}$ | $R_{\rm out}$ | $Z_{\rm out}$ | $\Delta r_{0}$
---|---|---|---|---|---|---|---|---|---|---
name | (au) | (au) |  | (km s-1) | (au) | (au) | (yr) | (au) | (au) | (au)
A |  600 | 60 | 0.90 | 0.32 | 181 | 18.4 | 2337 |  871 | 213 | 0.75
B | 1200 | 60 | 0.95 | 0.16 | 295 | 52.2 | 8181 | 1399 | 338 | 1.50
C |  600 | 60 | 1.00 | 1.44 | 181 | 18.4 |  904 |  871 | 213 | 0.75

The temperature is set to be $1.42\times 10^{3}$ K for the warm neutrral gas,
28.6 K for the cloudlet and 57.3 K for the disk in all the models.

###  2.3 Mock Observation

In this section we briefly illustrate the methodology adopted to produce
synthetic channel maps for comparison with observations. We use the Cartesian
coordinates,

$\displaystyle\mathbf{r}$ $\displaystyle=$ $\displaystyle
X\mathbf{n}_{1}+Y\mathbf{n}_{2}+s\mathbf{n}_{3},$ (28)

to evaluate the emission expected from our model simulation. Our line of sight
is parallel to the unit vector, $\mathbf{n}_{3}$, while $\mathbf{n}_{1}$ and
$\mathbf{n}_{2}$ denote the unit vectors in the directions of decreasing right
ascension and increasing declination, respectively. The line of sight is
related to the cylindrical coordinates by

$\displaystyle\mathbf{n}_{3}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}-\sin i\sin\varphi_{\rm obs}\\\ \sin
i\cos\varphi_{\rm obs}\\\ \cos i\end{array}\right),$ (32)

where $i$ and $\varphi_{\rm obs}$ denote the disk inclination angle and the
observer’s azimuth in the cylindrical coordinates. The other unit vectors are
expressed as

$\displaystyle\mathbf{n}_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\cos\chi_{\rm obs}\cos\varphi_{\rm
obs}+\cos i\sin\chi_{\rm obs}\sin\varphi_{\rm obs}\\\ \cos\chi_{\rm
obs}\sin\varphi_{\rm obs}-\cos i\sin\chi_{\rm obs}\cos\varphi_{\rm obs}\\\
\sin i\sin\chi_{\rm obs}\end{array}\right),$ (36)
$\displaystyle\mathbf{n}_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\sin\chi_{\rm obs}\cos\varphi_{\rm
obs}-\cos i\cos\chi_{\rm obs}\sin\varphi_{\rm obs}\\\ \sin\chi_{\rm
obs}\sin\varphi_{\rm obs}+\cos i\cos\chi_{\rm obs}\cos\varphi_{\rm obs}\\\
-\sin i\cos\chi_{\rm obs}\end{array}\right),$ (41)

where $\chi_{\rm obs}$ denotes the orientation of the major axis of the disk
on the sky. We measure the major axis counter-clockwise from $\mathbf{n}_{1}$
(West). In the following we assume $i=35^{\circ}$ and $\chi=45^{\circ}$ to
reproduce the Doppler shift due to the disk rotation.

For simplicity, we assume that the intensity at a given radial velocity is
proportional to the gas density integrated over the line of sight,

$\displaystyle\Sigma(X,Y;v)$ $\displaystyle=$
$\displaystyle\int\frac{\rho(\mathbf{r})}{\sqrt{2\pi}\sigma}\exp\left\\{-\frac{1}{2}\left[\frac{v_{\rm
los}-v}{\sigma}\right]^{2}\right\\}ds,$ $\displaystyle\mathbf{r}$
$\displaystyle=$ $\displaystyle
X\mathbf{n}_{1}+Y\mathbf{n}_{2}+s\mathbf{n}_{3},$ (44) $\displaystyle v_{\rm
los}$ $\displaystyle=$ $\displaystyle-\mathbf{v}\cdot\mathbf{n}_{3},$ (45)

where $\sigma$ denotes the local line width in units of velocity. We assume
$\sigma=0.1~{}\mbox{km~{}s}^{-1}$ to obtain a smooth channel map. For
comparison with observation, we sum up the column density along the line of
sight,

$\displaystyle\bar{\Sigma}(X,Y;v)$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{i=1}^{N}\Sigma\left(X,Y;v_{i}\right),$ (46)
$\displaystyle N$ $\displaystyle=$ $\displaystyle\frac{\Delta v}{\sigma},$
(47) $\displaystyle v_{i}$ $\displaystyle=$ $\displaystyle
v+\left(i-\frac{N+1}{2}\right)\sigma,$ (48)

over the velocity range of $v-\Delta v/2\sim v+\Delta v/2$. Since the velocity
resolution is $0.6~{}\mbox{km~{}s}^{-1}$ for the CS ($J=5-4$) emission from DG
Tau, we show the average of 6 column density differentials along the line of
sight for the direct comparison with observations. We also show mock channel
maps with higher velocity resolution for our analysis.

We note that our mock observation does not take account of self-absorption,
non-uniform excitation temperature, and chemical in-homogeneity. These factors
should affect the expected intensity. However, it is beyond the scope of this
work for us to include these factors since they are highly uncertain. The mock
channel maps enable us to compare the model and observation kinematics
directly since they provide morphology and line of sight velocity
simultaneously.

##  3 Results from the modeling

###  3.1 Model A

First, we introduce model A as a reference. The cloudlet has a radius of 90 au
and is located at 600 au away from the star at the initial stage. The density
of the cloudlet is nearly uniform and in the range of $9.96\times
10^{5}~{}\mbox{cm}^{-3}<n<1.02\times 10^{6}~{}\mbox{cm}^{-3}$ and the mass
density is 90 times higher than that of the surrounding warm gas. The density
is slightly higher on the side of the cloudlet closer to the star. The mass of
the cloudlet is $1.97\times 10^{-5}~{}\mbox{M}_{\odot}$. The initial velocity
is $0.21~{}\mbox{km~{}s}^{-1}$, and accordingly the motion is subsonic for the
assumed temperature of $T_{\rm c}=28.6~{}\mbox{K}$.

Figure 4 shows the infall of cloudlet in model A in a series of snapshots.
Each panel denotes the cloudlet and disk by volume rendering. The color
denotes the number density of the molecules, $\rho|c|/\bar{m}_{\rm c}$, for
the region of $|c|\geq 0.5$. The viewing angle is specified by $\varphi_{\rm
obs}=5.6^{\circ}$ and $\chi=45^{\circ}$ so that each panel mimics the
observational image of DG Tau on the sky plane. The warm gas ($|c|<0.5$) is
less dense than the colorbar threshold and invisible in the panels, though it
fills the vast space outside of the disk and cloudlet.

Figure 4: Each panel denotes the cloudlet and disk in model A at a given epoch
with top (left) and side (right) views. The color denotes the number density
of molecules. The movie associated with this figure includes stages up tp
$t\leq 4859$ yr and shows the disk evolution after impact

.

At $t=2125~{}\mbox{yr}$, the cloudlet is prolate and its head is $\sim$300 au
away from the star. The radial motion dominates over rotation at this stage.
The acceleration in the radial direction and elongation are due to gravity and
tidal force, respectively. As mentioned in §2, we fixed the disk and its
motion at the initial values. We followed the change in the region of $r\leq
R_{\rm d}^{\prime}$ and $|z|\leq Z_{\rm d}$ from $t=2337~{}\mbox{yr}$. At
$t=2473~{}\mbox{yr}$, the head of the cloudlet is close to the disk outer
edge. The maximum density of the cloudlet is $1.25\times
10^{6}~{}\mbox{cm}^{-3}$ and only slightly higher than the initial value. The
pressure of the surrounding warm gas is nearly uniform in the region far from
DG Tau ($\sqrt{r^{2}+z^{2}}>120~{}\mbox{au}$). Thus, the cloudlet does not
shrink or expand appreciably, though the shape is highly elongated like
streamer. Finally, at $t=2499~{}\mbox{yr}$, the head of the cloudlet hits the
disk from the upper surface. The main part of the cloudlet follows the head to
form an accretion streamer.

Figure 5 shows a zoom of the stage shown in the lower right panel of Figure 4,
when the streamer impacts the disk. Figure 6 shows the same stage as a cross
section at $x=0$. The northern side of the disk is deformed by the accretion.
We see the shock-compressed gas at the intersection of the cloudlet and disk.
We expect that SO molecules are formed in the shock due to dust grains
sputtering and the release of S into the gas phase. We also see the spiral
shock waves in the disk. Since the disk is rotating around DG Tau, any
dynamical impact tends to result in trailing spirals. In model A, the streamer
does not penetrate the disk but instead deflects at the surface. We note that
the cloudlet and disk have almost the same pressure in our modeling. Both of
them are in pressure equilibrium with the warm gas. The disk is set to have
twice higher temperature, and accordingly a lower density, than the cloudlet
in model A. However, we do not claim that the disk gas is less dense than the
cloudlet. If we had taken account of the vertical temprature gradient, the
density should be higher around the midplane. We think that our model
overestimates the impact of the streamer on the disk.

Figure 5: A zoom-in view of the disk and streamer in Model A at $t=$ 2647 yr.
The SW quarter of the disk is removed to show the vertical structure.  Figure
6: The cross section view of the disk and streamer shown in Fig. 5.

Figure 7 and associated animation show the infall of cloudlet by the bird’s
view from $t$ = 0 yr to 3099 yr. The viewing angle is set so that the disk
rotation axis is 35∘ inclined with respect to the line of sight and the
$x$-axis lies on the position angle of PA = 315∘. The cloudlet evolves into a
structure elongated toward the North of DG Tau. We use the same viewing angle
for the Mock observation.

Figure 7: The bird’s eye view for the stage at $t~{}=~{}2647~{}\mbox{yr}$ in
model A. The color denotes the number density of the molecules. The associated
animation shows time evolution of model A by the bird’s eye view from
$t=0~{}\mbox{yr}$ to 4762 yr.

The accretion streamer should last for a few thousand years. We can estimate
the duration from the initial size of the cloudlet. The tail is 180 au behind
from the head and the initial velocity is $0.21~{}\mbox{km~{}s}^{-1}$. This
means that the delay of the tail is $2.7\times 10^{3}~{}\mbox{yr}$.

The impact of the streamer disrupts a part of the disk and leaves a
significant imprint. The disk shows asymmetric features such as arms and
spirals at least for several thousands years (see the animation associated
with Figure 4). The disturbance may evolve into multiple rings after several
disk rotations as demonstrated by Demidova & Grinin (2022). However, our model
does not include any physical dissipation processes, and long-time evolution
is beyond the scope of this paper.

###  3.2 Model B

We have constructed model B to examine the effects of the initial distance.
The cloudlet is 1200 au away from DG Tau at the initial stage of model B. We
have reduced the spatial resolution to cover a larger computational box and
reduce computational cost. Since the initial distance is twice as large as the
other models, the timescale for the cloudlet capture is three times longer.

Figure 8 shows the capture of the cloudlet in a series of snapshots. While the
cloudlet is spherical at the initial stage ($t=0$ yr), it changes first to
prolate and later forms into a streamer. The cloudlet is elongated toward DG
Tau at $t=7366$ yr, though it is still far from DG Tau.

Figure 8: The same as Fig. 4 but for model B. See the animation associated
with this figure for further details of the change in the cloudlet form.

We stopped following the evolution at $t=8661~{}\mbox{yr}$ in model B. We
could not follow the further evolution because of the lower spatial
resolution. At this stage, the head of the streamer is immersed within the
disk but the remaining part extends outside the disk.

The cloudlet is elongated more in model B. The elongation is due to the
initial lower velocity. If the initial velocity further lower, a smaller
cloudlet can evolve into a streamer.

###  3.3 Model C

We have constructed model C to examine the case in which the clouldlet follows
a parabolic orbit. The cloudlet is spherical and 600 au away from DG Tau at
the initial stage, $t=0$ yr. The initial velocity of the cloudlet is $\sim
1.4~{}\mbox{km~{}s}^{-1}$ though it is not uniform in the cloudlet. This
infall velocity is higher than the sound speed of the cloudlet while it is
lower than that of the warm gas.

Figure 9 shows the infall of the cloudlet in a series of snapshots. The
cloudlet is slightly streched in the radial direction but the elongation is
less prominent. Model C fails to reproduce the streamer, i.e., an elongated
structure. The orbit does not follow that of a bullet. The changes in the
shape and orbit are due to the drag force working on the cloudlet. The drag
force is roughly proportional to the square of the relative velocity between
the cloudlet and warm gas. Another factor is the larger initial velocity.
Since the drag force works only on the front side, the rear side catches up to
the head. We examine the drag force in §5. The higher initial velocity also
suppresses the elongation.

Figure 9: The same as Fig. 4 but for model C. See the animation associated
with this figure for further details of the change in the cloudlet form.

##  4 Comparison with observations

###  4.1 CS Channel maps

The upper half of Figure 10 shows the channel maps of CS ($J=5-4$) line
emission obtained with ALMA at a spatial resolution of
$0.16^{\prime\prime}\times 0.13^{\prime\prime}$ and a spectral resolution of
0.6 km s-1 in the context of the ALMA-DOT program. Each panel covers an area
of $6.0^{\prime\prime}\times 6.0^{\prime\prime}$ centered at the continuum
peak. The details of the observations are given in Garufi et al. (2021, 2022).
The CS line emission probes both the rotating gas in the disk and an accretion
streamer north of the disk, which impacts on the NE disk side. Due to dust
opacity effects in the inner disk region and/or due to excitation conditions,
CS only probes the outer disk from a radius of $\sim 30$ au out to $\sim 130$
au. The streamer has two components, red-shifted and blue-shifted.

Figure 10: Comparison of CS ($5-4$) emission (Podio et al., 2020) and mock
channel maps for model A at $t$ = 2647 yr. The upper half denotes the line
intensity by color and the column density per unit velocity by blue contours.
The lower half shows the latter by color.

The color bars are given in the last row.

###  4.2 Other Molecules and Dust Continuum

Podio et al. (2019) reported a ring in the H2CO emission line observed with
ALMA. From the same data set, Podio et al. (2020) detected 1.3 mm continuum
emission originating from the disk. The radius of the dust disk is $\sim 70$
au, and we find weak asymmetry in H2CO or dust emission with the limited
angular resolution data. The level of asymmetry is much lower than the
disturbance shown at a later stage in our hydrodynamical models. Ohashi et al.
(2023) have analyzed the archival data (2015.1.01268.S) to show that the dust
continuum emission at 1.3 mm is smooth and does not show any substructures at
a spatial resolution of 5 au. These observations therefore suggest that the
disk midplane is for the most part undisturbed. In turn, this may indicate
that either the entire disk has not been perturbed yet by the streamer, or
that the disk is stiff and the dust at the midplane is not significantly
affected by the streamer. We consider the former case when comparing our
hydrodynamic model with the observations. The former possibility is supported
by the localized nature of the SO emission. Unless the SO molecules are
deposited back onto the dust grains within a few hundreds of years, the SO
emission should form an arc extended along the gas stream, like in the case of
HL Tau described by Garufi et al. (2022).

The ALMA maps obtained by Bacciotti et al. (2018); Podio et al. (2019, 2020)
show that both the molecular emission (H2CO and CS ($5-4$)) and the continuum
emission at 0.8 and 1.3mm are enhanced in the NW side, i.e. in the region
where the streamer impacts the disk See Figures 1 and 3 from Podio et al.
(2020), Fig. B.1 from Podio et al. (2019), and Fig. 2 from Bacciotti et al.
(2018). The lack of SO emission along the gas stream in the disk, as observed
in the case of HL Tau (Garufi et al., 2022), could be due to the lower angular
resolution and signal-to-noise ratio of the SO observations for DG Tau (beam
size $\sim 0.35\arcsec$ and rms $\sim 0.7$ mJy beam-1).

###  4.3 Model A

We compare model A with the ALMA observation since it gives the bet among our
models. The third and fourth rows of Figure 10 are the mock channel maps based
on the stage at $t=2647$ yr in model A. Each map covers a square of 250 au
$\times$ 250 au centered at DG Tau, which is marked by the cross on each map.
The spectral resolution is set to be 0.6 km s-1 for comparison with the
observations.

The channel maps show the rotating disk and the Northern streamer. We find a
strong emission spot at 7.6 km s-1. This spot corresponds to the shock
compressed cloudlet head. Thus, it corresponds to the SO emission spot. Model
A cannot reproduce the blue-shifted streamer component seen in the
observations. See also Figure 7 for the morphology of the cloudlet. It shows
the elongated cloudlet colliding with the disk at the stage shown in the mock
observation. The associated animation shows the formation of the streamer and
evolution of the disk after the impact by bird’s eye view.

##  5 DISCUSSIONS

A cloudlet approaching DG Tau transforms into a streamer under some
circumstances. As long as the drag force is weak, the tidal force elongates
the cloudlet, and the warm surrounding gas compresses it from the side. We
examine the strength of the drag force.

Newton’s law gives the drag force (resistance) acting on a body moving at
velocity $v$,

$\displaystyle F_{\rm drag}$ $\displaystyle=$ $\displaystyle C_{\rm
D}S\rho_{\rm w}\frac{v^{2}}{2},$ (49)

where $\rho_{\rm w}$, $S$ denote the density of the ambient (warm) gas and the
cross section of the body, respectively. The symbol, $C_{\rm D}$, denotes a
non-dimensional number, and is evaluated to be $\approx 0.4$. Equation (49) is
the same as Equation (7) of Weidenschilling (1977), who evaluated the drag
force acting on a solid particle in the solar nebula. The adopted value of
$C_{\rm D}$ is appropriate when the Reynolds number is very large, i.e., the
body size is much larger than the mean free path of the gas molecule. The drag
force not only decelerates but shortens the cloudlet since it works only on
the front side.

We compare the drag force with the tidal force, since the latter works to
elongate the cloudlet. It pulls the front side toward the star and pushes the
rear side away. When the cloudlet is a gas sphere of radius $a_{c}$, the tidal
force is estimated to be

$\displaystyle F_{\rm tidal}$ $\displaystyle=$ $\displaystyle\frac{\pi
GM\rho_{\rm c}a_{c}{}^{4}}{r_{c}{}^{3}}.$ (50)

Here, the tidal force is defined as the difference between the force extending
the head forward and that extending the tail backward. The drag force works in
the direction against the cloudlet motion while the tidal force works in the
direction towards the star. Though the drag force has an azimuthal component,
the radial part is dominant except for near the periastron and apoastron since
we consider parabolic or highly eccentric orbits. Thus, we assume it has only
the radial component for simplicity. Then the ratio of these forces is
expressed as

$\displaystyle\frac{F_{\rm tidal}}{F_{\rm drag}}$ $\displaystyle=$
$\displaystyle\frac{1}{C_{\rm D}}\frac{\rho_{\rm c}}{\rho_{\rm
w}}\left(\frac{a_{c}}{r_{c}}\right)^{2}\frac{2GM}{r_{c}v^{2}}.$ (51)

The tidal force is dominant when (1) the cloudlet is dense ($\rho_{\rm
c}\gg\rho_{\rm w})$, (2) the velocity is low, and (3) the viewing angle of
cloudlet from the star is relatively large. Note that the velocity is
normalized by the gravity in Equation (51).

In model C, the drag force is dominant over the tidal force mainly because the
initial velocity is high. In the other models, the tidal force is dominant
since the initial velocity is low ($v^{2}\ll GM/r_{c}$). The drag force
compresses the head of the cloudlet and the compression propagates in the
cloudlet at the sound speed. When the cloudlet’s motion is subsonic, the
compression propagates through the cloudlet and the shape changes little.
Otherwise, the cloudlet has a highly compressed layer near the head as shown
in model C.

Once the cloudlet is elongated, the elongation continues. The elongation
reduces the drag force through the reduction in the cross section, and
enhances the tidal force. As shown in model B, an elongated cloudlet
transforms into a streamer. Equation (51) provides a constraint on the early
phase of streamer formation.

As far as we know, direct observations of the ambient warm gas have been not
yet reported, although it is expected to be present (see, e.g., Figure 1 of
Dutrey et al., 2014). The morphology of streamers could give indirect evidence
of the warm gas and a constraint on the temperature.

The impact of the cloudlet depends on the inclination of the orbital plane and
disk thickness. When the orbit is coplanar with the disk, the cloudlet
collides with the outer edge of the disk. When the orbit is inclined, the
cloudlet hits the disk surface. In our model, $\theta_{0}$ specifies the
orbital plane. When $(H/r)\tan\theta>1$, the orbit is coplanar, where $H$ and
$r$ denote the disk height and radius, respectively. Since the SO emission
appears at a radius of 50 au in DG Tau, the streamer’s orbit should be
inclined with the disk. We assumed the inclination of 10∘ in our models in
accordance with Garufi et al. (2022). The ratio of the height to radius of the
CS emitting layer is estimated to be $H/r\sim 0.09$ by Garufi et al. (2021)
from the asymmetry between the near and far sides of the disk. Model A is
consistent with these observations. The orbital plane of the cloudlet should
be located above the disk surface in the northeast.

Our model implies the possibility that we could detect the velocity gradient
along the streamer if the spectral resolution of the observations had been
higher. The velocity gradient is also shown in the ballistic modeling of
Garufi et al. (2022). Applying the ballistic approximation, we have made a
moment 1 (intensity-weighted mean velocity) map, Figure 11. The colored curve
denotes the cloudlet’s orbit projected on the sky plane while the colored
ellipse denotes the rotating disk. The orbital elements are the same as those
of model of A1. The streamer is accelerated in the region where the projected
distance is less than $r_{\rm proj}\la 100~{}\mbox{au}$. The projected orbit
is curved only near the periastron. The accretion streamer indicates that the
specific angular momentum is already low in the region of $r_{\rm
proj}>100~{}\mbox{au}$.

Figure 11: Synthetic moment 1 map for the streamer and disk of DG Tau. The
streamer is assumed to be a part of the elliptic orbit. The orbital element is
the same as that for the cloudlet center of model A. The streamer is red-
shifted.

The SO emission is a tracer of recently shocked, compressed gas as these
molecules are rapidly formed in shocks due to the release of sulphur from the
dust grain mantles (e.g., Neufeld & Dalgarno, 1989a, b). van Gelder et al.
(2021) estimated the lifetime of the SO molecule to be deposited back onto the
dust as several hundreds of years in their modeling accretion shocks at the
disk-envelope interface. We expect that the SO molecules flow out from the
impact point since the gas disk is rotating. The Keplerian rotation velocity
is 3.5 km s-1 at the distance of 50 au for the assumed mass of 0.7 M⊙. This
velocity is equivalent to 74 au per century. The localization of the SO
emission on the disk suggests two possibilities: (1) the impact of the
streamer is a recent event, (2) the lifetime of SO molecule is shorter in DG
Tau than expected in models. This would be consistent with the observations
that the disk of DG Tau has relatively minor asymmetries in the dust emission
(e.g., Podio et al., 2019). Alternatively, as noted above, the lack of SO
spiraling through the disk may be due to the lower resolution of the
observations or fainter emission in DG Tau compared to HL Tau.

We discuss the dependence of our model on the disk gas temperature since we
neglected the temperature gradient in the disk. The temperature should be
lower near the mid plane and in the outer part of the disk. When the
temperature is lower, the gas density is higher for a given surface pressure.
If the disk is more massive, the larger inertia should soften the dynamical
impact.

As summarized by Pineda et al. (2023), many young stellar objects show
asymmetric accretion features feeding disks, though the morphologies have
variety in shape and dynamics. The form may reflect the environment and
evolutionary stage. HL Tau shows an accretion streamer similar to that of DG
Tau. Both DG Tau and HL Tau classical T Tauri stars, and both have relatively
symmetric disks. More evolved stars are often associated with asymmetric
extended disks (e.g., Boccaletti et al., 2020). On the other hand, some Class
0 sources such as TMC 1A often show asymmetry but not streamers (Aso et al.,
2015; Sakai et al., 2016). DG Tau is also associated with another type of
asymmetry, the southern arc (Garufi et al., 2022). Because the southern arc
has a large velocity offset at a large projected distance, it is unbound to DG
Tau and unlikely to be an accreting streamer.

##  6 SUMMARY

We have shown that our hydrodynamic model can reproduce an accretion streamer
infalling to DG Tau by assuming the following conditions.

First, we considered a warm neutral medium of atomic gas to prevent expansion
of the streamer. We can see the warm neutral medium neither in the molecular
emission lines nor in the dust continuum. Though the density is low, it has a
gas pressure comparable to that of the molecular gas because of the relatively
high temperature.

Second, the infalling cloudlet has small radial and rotation velocities at the
initial stage. If the rotation velocity is large, the cloudlet cannot reach
the disk because of the large centrifugal force. If the radial velocity is
large, the drag force prevents the elongation of the cloudlet.

Third, the inclination of the cloudlet’s orbit should be higher than the
opening angle of the disk, $z/r$. Otherwise, the spot of SO emission at the
impact zone should appear at the disk outer edge.

Fourth, the accretion streamer should have reached the disk only recently
($<10^{3}~{}\mbox{yr}$) in DG Tau. Otherwise, we would see more dramatic
perturbations in the observations of the disk.

## Acknowledgements

We thank an anonymous reviewer for the valuable comments. This work was
supported by JSPS KAKENHI grant Nos. JP18H05222, JP19K03906, JP20H05847, and
JP20H00182. CC and LP acknowledge the PRIN-MUR 2020 MUR BEYOND-2p
(Astrochemistry beyond the second period elements, Prot. 2020AFB3FX). D. S.-C.
is supported by an NSF Astronomy and Astrophysics Postdoctoral Fellowship
under award AST-2102405.

## Data Availability

This paper makes use of the following ALMA data: ADS/JAO.ALMA#2015.1.01268.S.
ALMA is a partnership of ESO (representing its member states), NSF (USA) and
NINS (Japan), together with 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 ESO, AUI/NRAO and NAOJ.

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