# Clumpy Accretion in Pre-main-sequence Stars as a Source of Perturbations in
Circumstellar Disks

Tatiana V. Demidova Crimean Astrophysical Observatory,p. Nauchny,
Bakhchisaray, Crimea, 298409  Vladimir P. Grinin Pulkovo Observatory of the
Russian Academy of Sciences, Pulkovskoje Avenue 65, St. Petersburg 196140,
Russia  V.V. Sobolev Astronomical Institute, St. Petersburg University,
Petrodvorets, St. Petersburg, Russia

###### Abstract

The development of perturbations in the circumstellar disks of pre-main-
sequence stars caused by clumpy accretion was investigated. Here we perform 3D
hydrodynamical SPH simulations of disks perturbed by a recent clump accretion
event. These simulations are further explored by radiative transfer
calculations to quantify observational appearance of such disks. It was shown
that the density waves in the disks were formed at the fall of the clump.
After several revolutions they can transform into spirals and ring structures.
Their images in mm-wavelengths are very similar to those observed with ALMA in
some protoplanetary disks. We assume that the clumpy accretion may be the
source of such structures.

methods: numerical– hydrodynamics – protoplanetary disks

††software: Gadget-2 (Springel et al., 2001; Springel, 2005), RADMC-3D
Dullemond et al. (2012)

##  1 Introduction

After formation of a young star and its circumstellar disk due to the
gravitation collapse of the protostellar cloud the process of accretion of
matter from the nearest surrounding the star can continue and have a form of
the clumpy accretion. Graham (1992) was probably the first who used this term.
He tried to explain by such a way the observations of the strong extinction
events observed in some young variables. Later this type of accretion has been
considered by Hartmann & Kenyon (1996) for explanation of the FUOR’s
phenomenon. This idea is quite popular up to now (Zhu et al., 2010; Bae et
al., 2013; Hartmann & Bae, 2018). The mechanism of formation of chondrule as a
result of the clumpy accretion was discussed Tanaka et al. (1998). Obviously,
the fall of the clump on the disk should cause disturbances at the place of
the fall. It is interesting to trace how this disturbance will develop and
what structures on the disk can be caused by it.

The detection of various structures on images of protoplanetary disks is one
of the most interesting results obtained with the ALMA interferometer (see,
e.g. Cazzoletti et al., 2018; Huang et al., 2018a, b; Pérez et al., 2018).
Ring and spiral structures are most often observed. Less commonly, structures
resembling highly elongated vortices are observed. A number of papers are
devoted to theoretical studies of the formation of such structures. Their
formation is associated with perturbations in the disks caused by the motion
of the forming planets (e.g., Ruge et al., 2013; van der Marel et al., 2015;
Dong et al., 2015; Demidova & Shevchenko, 2016; Jin et al., 2016; Dong et al.,
2018), the development of various kinds of instabilities in the disks (e.g.,
Banzatti et al., 2015; Zhang et al., 2015; Birnstiel et al., 2015; Okuzumi et
al., 2016; Johansen et al., 2009; Bai & Stone, 2014; Takahashi & Inutsuka,
2014; Dullemond & Penzlin, 2018), a large scale vertical magnetic field
(Suriano et al., 2018) or with the destruction of large bodies in collisions
(Demidova & Grinin, 2019; Nayakshin et al., 2020). In all these papers, the
source of disturbance is in the disk itself.

In our paper we discuss an alternative scenario for the formation of the
observed structure. We investigate in the first time the dynamical response of
circumstellar disk on the perturbation associated with the clumpy accretion
events. Using the hydrodynamic simulations we calculate the disk images at $1$
mm and discuss the results in the context with interferometric observations of
the protoplanetary disks with ALMA.

##  2 Initial condition

A model consists of a young star of solar mass ($M_{\ast}=M_{\odot}$) embedded
in a gas disk with total mass is $M_{disk}=0.01M_{\odot}$. At the beginning of
simulation the disk matter was distributed azimuthally symmetrical within the
radii $r_{in}=0.5$ and $r_{out}=50$ AU. The initial density distribution of
the disk is

$\rho(r,z,0)=\frac{\Sigma_{0}}{\sqrt{2\pi}H(r)}\frac{r_{in}}{r}e^{-\frac{z^{2}}{2H^{2}(r)}},$
(1)

where $\Sigma_{0}$ is arbitrary scale parameter, which is determined by disk
mass. Hydrostatic scale height is $H(r)=\sqrt{\frac{\kappa
T_{mid}(r)r^{3}}{GM_{\ast}\mu m_{H}}}$, where $\kappa$, $G$ and $m_{H}$ are
the Boltzmann constant, the gravitational constant and the mass of a hydrogen
atom and $\mu=2.35$ is the mean molecular weight (Dutrey et al., 1994).
Following Chiang & Goldreich (1997) we determine the law of midplane
temperature distribution
$T_{mid}(r)=\sqrt[4]{\frac{\gamma}{4}}\sqrt{\frac{R_{\ast}}{r}}T_{\ast}$,
where $\gamma=0.05$ (Dullemond & Dominik, 2004). The calculations were
performed in the local thermodynamic equilibrium approximation
$P(r,z,t)=c^{2}(r)\rho(r,z,t)$, where $P$ is local pressure at the moment $t$
and $c$ is a sound speed. It was assumed that in the vertical direction along
$z$ the disk is isothermal. The temperature of the star was assumed to be
$T_{\ast}=5780$ K and star radius is $R_{\ast}=R_{\odot}$. The disk relaxed
during $600$ years, and then the remnant of the fallen gas clump was added
into it.

###  2.1 Impulse approximation to the clump-disk collision

When a clump falls onto a disk, part of its kinetic energy is converted into
thermal energy. An infrared spot should appear on the image of the disk where
the clump fell. The thermal relaxation time in protoplanetary disks at the
distance $\geq 20$ AU from the star is much less than the local orbital period
(Malygin et al., 2017). Therefore, the remnant of the fallen clump quickly
comes to thermodynamic equilibrium with the matter of the disk. It
participates in the Keplerian motion of the matter, while maintaining the
residual velocity component orthogonal to the plane of the disk.

At the initial moment of time, we assume that the remnant of the clump has
already reached thermodynamic equilibrium with the matter of the disk. The
remnant was generated as a density perturbation on the disk in the form of a
disk segment bounded by radii $R_{0}$ and step $dR$ and distributed over the
azimuthal angle $\phi$ with the axis of symmetry along negative part of x-axis
($\phi=30^{\circ}$ for all models). The density of matter in the disturbance
exceeded the local density of the disk by a factor of $\displaystyle
K=\frac{\Sigma_{cl}}{\Sigma_{d}}$, where $\Sigma_{cl}$ and $\Sigma_{d}$ are
local surface density of remnant and disk respectively. The remnant moves
prograde. The particle velocity of the remnant is equal to $\displaystyle
V(R)=L\cdot V_{k}(R)$, where $V_{k}(R)$ is Keplerian velocity at a given
distance from the star and $L$ is parameter of the problem. The velocity
vector had a residual inclination to the disk plane $\displaystyle
sin(I)=\frac{V_{z}(R)}{V(R)}$ (Fig. 1). The residual angle of inclination of
the remnant depends on the initial angle of the fall and the amount of kinetic
energy that is spent on heating the disk in the region of fall. We considered
a number of the possible options for the value and inclination of the velocity
vector. The parameters of all calculated models are given in the Table 1.

Figure 1: Particle distribution at the initial moment of the remnant of the
clump motion. Top is the projection onto the plane of the disk, bottom is a
section along the axis $y$.  Figure 2: The average value of the $z$ coordinate
of the particles in the cells of $R$, $\phi$. The model parameters are $K=3$,
$I=30^{\circ}$ and $L=0.8$. The time in years is in the upper right corner of
the pictures.  Table 1: The models parameters

L | I | K | R | dR | $\phi$ | Remnant mass |  | Lifetime
---|---|---|---|---|---|---|---|---
Float | Degrees | Number | AU | AU | Degrees | Jupiter mass | Sructures | yrs
0.8 | 5 | 3 | 20 | 5 | 30 | 0.11 | Arc, One-hand spiral, Horseshoe | $>600$
0.8 | 10 | 3 | 20 | 5 | 30 | 0.11 | Arc, One-hand spiral, Horseshoe | $>600$
0.8 | 20 | 3 | 20 | 5 | 30 | 0.11 | Arc, Faint two-arm spiral, Horseshoe | $>600$
0.8 | 30 | 3 | 20 | 5 | 30 | 0.11 | Arc, Faint two-arm spiral, Horseshoe | $>600$
0.8 | 10 | 5 | 20 | 5 | 30 | 0.19 | Arc, One-hand spiral, Horseshoe | $>600$
1 | 5 | 3 | 20 | 5 | 30 | 0.11 | Arc, One-arm spiral, Multi Rings, Ring | $\sim 1000$
1 | 10 | 3 | 20 | 5 | 30 | 0.11 | Arc, One-arm spiral, Ring | $>600$
1 | 20 | 3 | 20 | 5 | 30 | 0.11 | Arc, One-arm spiral, Faint two-arm spiral | $>600$
1 | 30 | 3 | 20 | 5 | 30 | 0.11 | Arc, One-arm spiral, Faint two-arm spiral | $>600$
1 | 5 | 5 | 20 | 5 | 30 | 0.19 | Arc, One-arm spiral, Multi Rings, Ring | $>600$
1 | 5 | 10 | 20 | 5 | 30 | 0.38 | Arc, One-arm spiral, Multi Rings, Ring | $>600$
1.2 | 5 | 3 | 20 | 5 | 30 | 0.11 | Arc, One-arm spiral, Multi Rings, Ring | $>600$
1.2 | 10 | 3 | 20 | 5 | 30 | 0.11 | Arc, Bright two-arm spiral | $>600$
1.2 | 20 | 3 | 20 | 5 | 30 | 0.11 | Arc, Bright two-arm spiral | $>600$
1.2 | 30 | 3 | 20 | 5 | 30 | 0.11 | Arc, Bright two-arm spiral, Asymmetric ring | $\sim 2000$
1.2 | 30 | 1 | 20 | 5 | 30 | 0.04 | Arc, Bright two-arm spiral | $>600$
0.8 | 30 | 3 | 10 | 2 | 30 | 0.04 | Arc | $\sim 100$
1 | 30 | 3 | 10 | 2 | 30 | 0.04 | Arc | $\sim 100$
1.2 | 30 | 3 | 10 | 2 | 30 | 0.04 | Arc, Faint two-arm spiral, Multi Rings | $\sim 450$

Note. — The column of “Structures” lists the types of observed asymmetries in
the order of their appearance in the disk images. The column of “Lifetimes” is
a long-lived structures lifetime.

Figure 3: The surface density multiplied by the distance from the center of
mass ($\Sigma R$). The model parameters are $K=3$, $I=10^{\circ}$ and $L=0.8$.
The time in years is in the upper right corner of the pictures.  Figure 4: The
average value of the $z$ coordinate of the particles in the cells of $R$,
$\phi$ along $x$ (left) and $y$ (right) axes after $600$ years. The model
parameters are $K=3$, $L=0.8$ and the angles are in the upper right corner of
the pictures.  Figure 5: The images at wavelength 1 mm. The color shows the
flux of radiation ($F_{\nu}$) multiplied by $R^{2}$ in conventional units. The
model parameters are $K=3$, $I=10^{\circ}$ and $L=0.8$. The time in years is
in the upper right corner of the pictures.

##  3 Methods

The evolution of the remnant in gas disk was simulated by the SPH method
(smooth particle hydrodynamics). The calculations were performed using the
code Gadget-2 (Springel et al., 2001; Springel, 2005) modified by us
(Demidova, 2016). In total, from $5\cdot 10^{5}$ to $2.5\cdot 10^{6}$
particles of the gas disk and from $5\cdot 10^{3}$ to $2\cdot 10^{5}$
particles of the perturbation were involved in the simulations. The
calculations took into account the self-gravity of the disk.

The simulated region was divided by $200\times 30\times 90$ cells in spherical
coordinates ($R,\theta,\phi$), in which the average values of the SPH particle
density were determined. We assume that dust particles with a size of 1, 10,
100 microns and 1 mm are well mixed with the gas, and are distributed
according to the law$\frac{dn(s)}{ds}\propto s^{-3.5}$, where n is the
concentration and s is the size of the dust grain (Dohnanyi, 1969). The dust
to gas mass ratio in the disk was $0.01$ as the average in interstellar
medium. The dust opacity is calculated using Mie theory for Magnesium-iron
silicates (Dorschner et al., 1995). RADMC-3D (Dullemond et al., 2012) code was
used for the 3-D radiative transfer calculations.

Figure 6: The same as in Fig. 3 for model parameters $K=3$, $I=10^{\circ}$ and
$L=1$.  Figure 7: The same as in Fig. 5 for model parameters $K=3$,
$I=10^{\circ}$ and $L=1$.  Figure 8: The same as in Fig. 5 for model
parameters $K=3$, $L=1$ after $600$ years. The angle of $I$ is in the upper
right corner of the pictures.  Figure 9: The same as in Fig. 3 for model
parameters $K=3$, $I=10^{\circ}$ and $L=1.2$.  Figure 10: The same as in Fig.
4 for model parameters $K=3$, $L=1.2$. Figure 11: The same as in Fig. 5 for
model parameters $K=3$, $L=1.2$ after $600$ years. The angle of $I$ is in the
upper right corner of the pictures.  Figure 12: The same as in Fig. 5 for
model parameters $K=1$, $I=30^{\circ}$, $L=1.2$ after $600$ years.  Figure 13:
The same as in Fig. 5 for two models with $2.5\cdot 10^{6}$ particles. The
models parameters $K=3$, $I=5^{\circ}$, $L=1$ (left) and $K=3$,
$I=30^{\circ}$, $L=1.2$ (right) at the time $1000$ years.  Figure 14: The same
as in Fig. 13 for model with parameters $K=3$, $I=30^{\circ}$, $L=1.2$ at the
time $2000$ years.  Figure 15: The same as in Fig. 4 for models with
parameters $K=3$, $I=5^{\circ}$, $L=1$ (top) and $K=3$, $I=30^{\circ}$,
$L=1.2$ (bottom). The time in years is in the upper right corner of the
pictures.  Figure 16: The same as in Fig. 5 for models with parameters $K=3$,
$I=30^{\circ}$, $R_{0}=10$ AU and $dR=2$ AU after $36.5$ years. The parameter
of $L$ is in the upper right corner of the pictures.  Figure 17: The same as
in Fig. 16 for models with parameters $K=3$, $I=30^{\circ}$, $L=1.2$ for the
time $400$ years.  Figure 18: The average value of the $z$ along $y$ axes
after $365$ (left) and $1460$ (right) years. The model parameters are $K=3$,
$I=30$. The values of $L$ are in the upper right corner of the pictures.

##  4 Results

The emergence of the remnant of the clump of matter in the disk leads to the
propagation of density waves in horizontal and vertical directions. The
strongest perturbations arise at large values of the parameters $K$, $L$, and
$I$, as expected. Due to the differential rotation of the Keplerian disk the
remnant stretches and transform during the time. A local increase in the
surface density leads to the appearance of corresponding large-scale
inhomogeneities in the disk images.

###  4.1 Perturbations at large radii

For the models discussed here, the initial position of the disturbing remnant
of the clump was set equal to $R_{0}=20$ AU with a step of $dR=5$ AU.
Calculations have shown that the parameter $L$, which characterizes the
kinetic energy of the falling clump, has the greatest influence on the type of
disturbance in the disk. Therefore, we will sequentially discuss the three
energy regimes considered in our models.

####  4.1.1 Sub Keplerian perturbations

In this case, when the parameter is $L=0.8$, due to the differential rotation
of the Keplerian disk the remnant stretches and transform to a piece of arc
reembling a cyclonic vortex and then turn into a spiral during one revolution
around the star ($\sim 125$ years). Since the matter of the disk is involved
in its motion the spiral dips and thickening of the density are visible in the
disk (Fig. 3). In the central part of the disk, the spiral splits into two
branches. The spiral structure quickly ($\sim 300$ years) twists into an
asymmetric ring, which, scattering in the disk, retains the asymmetry until
the end of the calculations ($600$ years). This asymmetry rotates with the
disk.

The wave also passes in the vertical direction along the disk, which extends
inside and to the edge of the disk. Perturbation twists the central plane of
the disk. The maximum distortion of the disk plane occurs near the axis $y$,
but does not coincide with its position. The inner parts of the disk inclines
relative to the periphery. Over time, the radius of the outer boundary of the
inclined area increases. It distorts 30 AU at the time of $600$ years (Fig.
2). In this case, the tilt of the disk plane relative to the initial position
is approximately equal to $0.2^{\circ}$ at $I=5^{\circ}$ and $0.9^{\circ}$ in
the case of $I=30^{\circ}$. An increase in the angle of $I$ does not affect
the speed of propagation of global perturbation to the edge of the disk (Fig.
4). Increasing the initial density (parameter K) of the clump increases the
vertical distortion of the disk.

The perturbations described above show themselves on the images of
protoplanetary disks. The form of asymmetric structures in images corresponds
to the perturbations of the density of the disk matter. However, on the
periphery of the disk is visible a shadow from the matter above the disk
plane. The asymmetric ring-shaped structure on the images has a horseshoe-
shaped form (Fig. 5). Calculations have shown that the minimum value of $K$ in
which this structure is visible on the images, equal to $3$ (about $0.1$ of
Jupiter mass). An increase in the parameter $K$ to $5$ increases the
brightness of the structure and their lifetime, but its horseshoe form is
preserved.

####  4.1.2 Keplerian perturbations

For this class of models of the phase of the disintegration of the clump
remnant during the first orbit similar to the previous one (Fig. 6). A piece
of arc is converted to the one-hand spiral. It twisted into a symmetric ring-
shaped structure during the next few revolutions ($\sim 500$ years).

An increase of the radius of distortion of the central plane of the disk
relative to the initial position is faster than in the previous case. It
reaches $40$ AU at the time $600$ years. The maximum inclination angle is
$0.2^{\circ}$ in the case $I=5^{\circ}$ and $0.8^{\circ}$ at $I=30^{\circ}$.
The direction of maximum distortion in the vertical direction is also close to
the axis $y$, but does not coincide with it.

The visible asymmetry on the images of the disk are also appropriate for
surface density (Fig. 7). In this case, the propagation of the density wave
gives multi-lane image of the protoplanetary disk at a certain point in time
(right image of Fig. 7). An increase in the angle of $I$ affects the image of
the ring-shaped structure. Two symmetric weakly pronounced spirals are visible
on the images instead of the ring, if the value of $I\geq 20$ (Fig. 8).

####  4.1.3 Super Keplerian perturbations

In this case, the clump matter motion in the protoplanetary disk causes severe
density perturbations and significantly distorts the disk in the vertical
direction. As in previous cases, during one convolution of the clump, the
vortex-like structure is stretched into a spiral, which is converted into two
spirals during the next revolution (Fig. 9). Each of the spirals is
logarithmic, they are shifted by phase relative to each other by
$180^{\circ}$. The form of the spirals weakly depends on the angle of
inclination of $I$.

Disk distortion in the vertical direction differs from the models described
above for this case. The periphery of the disk is distorted, and the inner
parts of the disk deforms weaker. The waves are still propagating along the
disk in a vertical direction at the time of 600 years as seen from Fig. 10. In
this case, the distortion of the inner region depends on the inclination angle
and has the opposite character for $I<20$ and $I\geq 20$.

It become seen noticeable asymmetry of the spirals on the periphery of the
disk in images with an increase in the angle of inclination of $I$ (Fig. 11).

For this class of models, the case $K=1$ (corresponds to the mass of the
remnant $\sim 12M_{\oplus}$) was considered. In this case, the perturbation is
weaker; however, the two-arm spiral can also be identified in the disk images.
It is more pronounced at a larger inclination angle $I$ (Fig. 12).

####  4.1.4 Long-term dynamics

For the models described above, the number of SPH particles was $5\cdot
10^{5}$. These models have a lower resolution compared to the models, the
calculations of which involved $2.5\cdot 10^{6}$ particles. However, the
calculations showed that at the initial phases of clump destruction, the
images of disks obtained on the basis of models with a small number of
particles show the same structures as for more accurate models. However, to
study the long-term dynamics of the fallen clump remnant, higher resolution is
required. We have calculated two limiting cases that correspond to the
parameters that cause the minimum ($K=3,I=5,L=1$) and maximum
($K=3,I=30,L=1.2$) disturbance in the disk.

Over time, density waves scattered and the all structure settles down to the
plane of the disk. The ring of the first model stretches along the radius and
loses brightness mixing with the matter of the protoplanetary disk with time.
It is faintly noticeable at $1000$ years after the fall of the clump (Fig. 13
left). Than it is not visible against the background of the disk matter. For
the second model spiral waves are still noticeable after $1000$ years (Fig. 13
right), but after $\sim 2000$ years they completely disappear. An asymmetric
ring structure can be seen on the disk by the time of 2000 years (Fig. 14).

The charateristic time of the disk dynamic relaxation after the fall of the
clump also depends on the place of its fall. For example, with $R_{0}=30$ AU
and the same clump parameters as in the previous model, the lifetime of
spirals and ring structures generated by its fall increases to $4\times
10^{3}$ years. At $R_{0}=50$ AU, the characteristic relaxation time of
disturbances on the disk is even longer: $\sim 10^{4}$ years.

On the Fig. 15 it is seen that the plane of the disk settles to its original
position over time. However, even 2000 years after the fall of the clump, a
slight inclination of the disk plane near the axis $y$ remains. For the first
model, it is equal to $\sim 0.14^{\circ}$, and for the second, it is $\sim
0.72^{\circ}$.

###  4.2 Perturbations at small radii

In this class of models the perturbation was located near the star at the
distance $R_{0}=10$ AU with the step of $dR=2$ AU. The clump had parameters
$\phi=30^{\circ}$, $K=3$ and $I=30^{\circ}$. The parameter $L$ was varied. The
mass of the clump was about $13M_{\oplus}$.

Fig. 16 shows images of the disk for three models with the L parameter $0.8$,
$1$ and $1.2$ after one period ($36.5$ years). Bright dense structures and
areas of shadow, which are caused by matter rising above the plane of the
disk, are visible on the disk. But all structures are scattered after next
period for models with $L\leq 1$. For case $L=1.2$ waves propagate along the
disk, which in time from 250 to 500 years can be seen in the images as a
multi-lane structure (Fig. 17).

In the vertical direction, the disk is distorted in all considered cases. The
maximum distortion is achieved near the $y$ axis as for models distant from
the star. The Fig. 18 shows the average values of $z$ along the y-axis for two
points in time $365$ and $1460$ years. One can see in all cases, the
perturbation propagates outward from the inner part of the disk, tilting its
central plane. With an increase in L, the final inclination of the disk
increases, but remains within $0.5^{\circ}$.

##  5 Conclusion

Calculations have shown that at the initial stages of the remnant of the clump
disintegration, the structures that are visible in the images of the
protoplanetary disk are similar for the entire set of the models. However, the
shape of the final long-lived structure primarily depends on the kinetic
energy of the falling clump.

During the first revolution of the center of the remnant (at the initial
moment of time), an arc-like structure resembling a vortex is visible in the
disk image. Similar structures are observed, for example, in objects HD
135344B (Cazzoletti et al., 2018) and HD 143006 (Pérez et al., 2018). At the
next stage of evolution, the image shows a tightly wound spiral, as, for
example, in the case of object HD 163296 (Huang et al., 2018b). In the case
when the residual velocity of the remnant does not exceed the Keplerian
velocity, a ring is a long-lived structure, which can also be asymmetric. In
this case, at a certain moment in time, the passage of a wave over the disk
can give a multi-lane structure in the image. The ring-shaped structure is
visible in the images of a number of objects, for example, HD 169142 (Fedele
et al., 2017), HD 97048 (van der Plas et al., 2017), RU Lup, Elias 24, AS 209,
GW Lup (Huang et al., 2018b). In the case of a high kinetic energy of a clump,
a two-armed spiral appears on the disk image, which, after several thousand
years, transforms into an asymmetric ring. Two-arm spirals were obtained on
images of objects Elias 27, IM Lup, WaOph 6 (Huang et al., 2018a). The median
age of these sources is $\sim 1$ Myr (Huang et al., 2018b), and the youngest
sources have estimated ages about a few tenths of Myr (Huang et al., 2018a).

Since the velocity vector of the remnant has a residual inclination relative
to the plane of the disk, it is distorted. However, over time, the inclination
of the plane of the disk relative its original position decreases. The tilt
angle at the end of calculations does not exceed $1^{\circ}$. Probably, for a
more noticeable change in the inclination of the disk, a large mass of the
clump remnant is required. In this work, we were looking for the minimum mass
of the remnant, which develops into a structure visible in the image of the
protoplanetary disk. It turned out that in the case of a high-energy fall, the
minimum mass of the remnant of the clump is $\sim 10M_{\oplus}$. If the
residual velocity of the remnant does not exceed Kepler one, then the minimum
mass is $\sim 0.1M_{J}$.

Ring-shaped structures formed from the material of the remnant of the fallen
clump and the protoplanetary disk are long-living structures and can exist for
more than $3000$ years. Thus with a sufficiently large mass of the falling
clump, the evolution of the disturbance can lead to the formation of a planet
on an inclined orbit. It should be noted that Toomre parameter in the disks
under consideration is equal to $\sim 40$ at a distance of $20$ AU.
Consequently, in a more massive disk, the fall of a clump several times denser
than the material of the disk can trigger the process of gravitational
collapse and the formation of a planet.

The fall of the clump near the star can cause not only a FUOR flare, but also
a strong increase in circumstellar extinction, leading to a deep and prolonged
weakening of the optical brightness of the star and an increase in its
infrared radiation. Such events were observed in three young objects: V1184
Tau (Grinin et al., 2009), RW Aur (Shenavrin et al., 2015; Koutoulaki et al.,
2019) and AA Tau (Covey et al., 2021).

Let us make a rough estimate of the additional mass of circumstellar matter,
which can be added to the mass of the disk during the lifetime of a star due
to episodic falls of clumps. Suppose that the average age of stars with
protoplanetary disks $10^{6}$ years (Huang et al., 2018b) and the average
lifetime of one disturbance $3\times 10^{3}$ years. This will give the
probability of observing one event $P_{1}\sim 3\times 10^{-3}$. The
probability close to unity will be obtained when $\sim 3\times 10^{3}$ clumps
fall during the disk lifetime ($\sim 1$ Myr). If the clump mass of $0.1M_{J}$
is necessary to create a strong disturbance (see above), the disk mass will
increase during this time by $0.03M_{\odot}$ at an average accretion rate on
the disk of $3\times 10^{-8}M_{\odot}yr^{-1}$. Such an increase in mass is not
critical for a typical disk mass of $0.01-0.2M_{\odot}$ and an accretion rate
onto a young star of $\sim 10^{-7}-10^{-8}M_{\odot}yr^{-1}$.

Thus, the mechanism of the clumpy accretion in protoplanetary disks can
explain the formation of the main types of structures identified in images of
protoplanetary disks. In addition, one clump falling at an angle onto a
protoplanetary disk can produce multi-lane bright ring-shaped structures. So
far, we have demonstrated here the fundamental possibility of obtaining the
observed structures on protoplanetary disks in the model of the clumpy
accretion. Appreciate the complexity of this process, it needs a more detailed
consideration, taking into account the thermal regime in the perturbed region
of the disk.

Acknowledgments. It is a pleasure to thank the referee for valuable and useful
remarks. Authors acknowledge the support of Ministry of Science and Higher
Education of the Russian Federation under the grant 075-15-2020-780
(N13.1902.21.0039).

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