Possibility of giant planet formation by pebble accretion in Class 0/I phases
Yuki A. Tanaka 1
1Astronomical Institute, Tohoku University, 6-3 Aramaki, Aoba-ku, Sendai, Miyagi, 980-8578, Japan, 980-8578
yuki.tanaka@astr.tohoku.ac.jp

Abstract
Several observations have revealed that axisymmetric sharp gaps in very young circumstellar disks, which may
be indicative of the existence of giant planets at distant orbits. In addition, recent theoretical works suggest thatthe process called pebble accretion is important for planet formation in protoplanetary disks, because the processaccelerates the growth of planetary cores. Here we investigate the possibility of giant planet formation via pebble
accretion in much earlier phases, the gravitationally unstable disks of class 0/I young stellar objects. We find that
under the conditions of the class 0/I disks, the pebble accretion timescales can be shorter compared to the typical
protoplanetary disks due to larger gas and dust accretion rate, but also find that the accretion timescale is not always
a decreasing function of the gas accretion rate. Then we calculate a required initial mass to form cores of gas giants
within the lifetime of class 0/I phases by using estimated accretion timescales under several parameters, such asradial distances from the host star, gas accretion rates, and dust-to-gas mass ratio. In the most optimistic case, forexample the dust-to-gas mass ratio is f = 3fsolar , ∼ 10−4 M⊕ objects at 10 au can grow to 10M⊕ cores during
the typical lifetime of the Class 0/I phases, 0.5 Myr. If this process works in the Class 0/I disks, it is expected that
substructures of the disks caused by the planets can be observed by future radio observations, such as next generation
very large array (ngVLA).
Key words: Planetary systems – planets and satellites: formation – planets and satellites: physical evolution –planet-disc interactions

1. Introduction
In the traditional planet formation models, it is assumed thatkilometer-sized planetesimals are formed from dust particles
in a protoplanetary disk, after which they grow up to become
planetary embryos and/or protoplanets (e.g., ??). This process is the so-called core accretion scenario. The formation
processes of planetary systems are being actively studied boththeoretically and observationally, but there still remain several
issues to be resolved. For example, giant planet formation isthought to be difficult in the distant orbit in the gas-free condition, because the timescales to form planetary cores are toolong (e.g., ?).
Recently, an alternative model so-called pebble accretion has
been proposed; small dust particles directly form large clumps
via streaming instability (e.g., ?), and the clumps grow fast by
the accretion of migrating centimeter-sized pebbles that is enhanced by aerodynamic effects (e.g., ?). In this process, thetimescale of the growth of the cores of giant planets is shortened even in the distant region of the disks (?). Planet formation via pebble accretion in protoplanetary disks have beenstudied intensively, (e.g., ??), but the possibility of pebble accretion in Class 0/I YSOs, the earlier phases of star and planet
formation, has not been considered so far. It is shown that planetesimal formation from porous dust aggregate (??) is more
preferable in the Class 0/I phase (?). This is one of our motivation to investigate the possibility of pebble accretion in Class
0/I phases.
Another important motivation of this work is the recent
discoveries of disks with multiple rings and gaps, mainly
achieved by the Atacama Large Millimeter/submillimeter

Array (ALMA) (e.g., ?). Many mechanisms have been proposed to reproduce the gap and ring structures, and one of possible gap-carving mechanism is embedded planets in the protoplanetary disks (e.g., ?). Recent numerical works suggest that
if gaseous planets exist in the disks, clear gaps and rings similar to those in several protoplanetary disks can be created due
to disk-planet interaction (e.g., ??).
However, the radii of the observed gaps are typically several
tens of astronomical units (au), and whether planet formation
at such large distances within the early phase of evolution of
the protoplanetary disks is possible or not is still unclear. If
earlier planet formation in the Class 0/I phases is possible, theobserved gap structures around the Class I YSOs can be explained by the giant planet. Therefore, we investigate formation of solid cores of gaseous planets by pebble accretion inClass 0/I disks (?).
This article for the ngVLA memo series is based on our paper ?, so if more detailed description or discussion are needed,
please see the paper.
2. Models
2.1. Gas disk model
Here we assume one dimensional steady state structures forthe gravitationally marginally unstable disk around the Class
0/I YSOs. Several theoretical works suggest that the gravitationally unstable disk frequently forms in Class 0/I YSOs (e.g.,?). For the temperature profile of the disk, we assume that disk
temperature T is determined by the stellar irradiation. In thiscase, radial temperature profile is given as follows based on ?
and ?;

( r )−3/7
T = Tirr ≃ 56
K.
10 au

[Vol. ,
(1)

The aspect ratio of the gas disk Ĥgas is written as follows:)−1/2 (
(
Hgas
cs
M∗
r )2/7
Ĥgas ≡
(2),=
≃ 4.8 × 10−2
r
rΩ
M⊙
10 au
where Hgas , Ω, M∗ , and M⊙ are the pressure scale height of
the gas disk, rotational frequency,
stellar mass, and solar mass,√
respectively, and cs = kB T /mg is the sound speed, where
kB is the Boltzmann constant, and mg (∼ 3.9 × 10−24 g) is themean molecular mass of the gas.
For the structures of the marginally gravitationally unstablegas disk, we use steady-state solution given by (?). We obtain()1/2 (
M∗r )−12/7
2
Σgas ≃ 6.7 × 10
g cm−2 , (3)
M⊙
10 au
(
)
( r )9/14
Ṁgas
α ≃ 3.2 × 10−2
(4)
10−6 M⊙ yr−1
10 au
for gas surface density Σgas , and α viscosity.
2.2. Dist disk model
For dust disk, we use the analytical model introduced by ?
in which radial migration of dust particle and dust growth aretaken into account. In this model, radial dependence of dust
size is written as)1/2(( r )−47/28
Ṁgas2
adust ≃ 3.8 × 10
10 au
10−6 M⊙ yr−1
(
)(
)−1
M∗
ρint
cm,(5)
M⊙
0.1 g cm−3and dust surface density is written as()1/2( r )−31/28
Ṁgas
g cm−2 ,Σdust ≃ 1.0
10 au
10−6 M⊙ yr−1(6)where Ṁgas , ρint are gas accretion rate, and internal density ofthe dust particle, respectively.
2.3. Pebble accretion timescale
For the derivation of the timescale of pebble accretion onto
planetesimals and/or planetary embryos, we use the analyticalmodel by ?. Previous theoretical works have shown that theaccretion mode of the pebbles are divided into two regimes,
the 3D and 2D regimes (e.g., ?). In addition, the accretion
mode also can be divided into another two regimes, Bondiand Hill regimes, mainly depending on the mass of planetesimals/planetary embryos (?). In the general condition of thedisk around Class 0/I YSOs, however, the accretion mode willbe 3D and Bondi regime, with Epstein drag regime for dust
particle (?). Therefore, the pebble accretion timescale can be
written as)−1()−3/4 (
M∗
Ṁdust5 −2tacc,3D ≃ 9.4 × 10 κM⊙
10−4 M⊕ yr−1

Accretion timsscale [year]2

Mdot=5×10-7Msol/yr
Mdot=1×10-6Msol/yr
Mdot=2×10-6Msol/yr
Mdot=3×10-6Msol/yr
Mdot=1×10-6Msol/yr, w/o κ

106

105

104
10

100
Radius [au]Fig. 1. Radial profiles of the accretion timescales with different gas
accretion rates. We fixed M∗ = M⊙ , Ṁdust = f Ṁgas , f = 0.01,
and M = 10−3 M⊕ in this plot. The magenta dotted, red solid, green
dashed, and blue dot-dashed lines correspond to the gas accretion rates
5 × 10−7 M⊙ yr−1 , 1 × 10−6 M⊙ yr−1 , 2 × 10−6 M⊙ yr−1 , and3 × 10−6 M⊙ yr−1 , respectively. The black dotted line shows theaccretion timescale when Ṁgas = 10−6 M⊙ yr−1 and κ = 1 are assumed, in order to show the importance of the reduction factor.

(

Ṁgas
10−6 M⊙ yr−1

)1/4( r )65/56
yr,
10 au

(7)

where Ṁdust is dust accretion rate, and M⊕ is the mass of the
Earth. In this equation, κ is the reduction factor introduced by?. This factor can be approximated to unity when Stokes number St is relatively small and the mass of a planetesimal is large(?). However, under the condition of typical Class 0/I YSOs,
the reduction of accretion efficiency by the factor becomes important (?).
If we assume that the dust accretion rate is related to thegas accretion rate as Ṁdust = f Ṁgas , where f is the dust-togas mass ratio, and also assume Ṁgas = 10−6 M⊙ yr−1 and
f = 0.01 as a fiducial model, we obtain the accretion timescale
as follows;()−3/4 (M∗r )65/56
4 −2tacc,3D ≃ 2.8 × 10 κyr. (8)M⊙
10 au
3. Results
3.1. Accretion timescale
As expressed by equation (??), the accretion timescale depends on the stellar mass, accretion rates of the gas and dust,
radial distance, and dust-to-gas mass ratio. Here we assume
that M∗ = M⊙ , Ṁdust = f Ṁgas , and we fix the dust-to-gas
mass ratio as f = 0.01. The accretion timescale also depends
on the mass of the planetesimal M because the reduction factor
κ is the function of M ?. Here we assume that M = 10−3 M⊕ .
Fig. ?? shows the radial dependence of the accretion
timescale. We also show the accretion timescale when κ = 1 isassumed as a reference. If the reduction factor is neglected, theradial dependence of the accretion timescale is tacc ∝ r65/56 ,as described by equation (??), and therefore the timescale at∼100 au is approximately ten times longer than that at ∼10 au.
The black dotted line in Fig. ?? shows this relation. However,No. ]

1026

10-2

1025

10-3
Mdot=5×10-7 Msol/yr
Mdot=1×10-6
Msol/yr
Mdot=2×10-6 Msol/yr
Mdot=3×10-6 Msol/yr
Mdot=10-6 Msol/yr, w/o kappa

-4

10

10-5

10

Initial mass [g]

1027

10-1

1024
1023
100

Radius [au]Fig. 2. Radial dependence of the required initial mass to form a 10
M⊕ core within the disk lifetime with different gas accretion rates.
The horizontal axis is the radial distance from the central star inunits of au. The vertical axes are the required initial mass, the left-hand side axis is in units of M⊕ , and the right-hand side is inunits of gram. M∗ = M⊙ , Ṁdust = f Ṁgas , tlife = 0.5 Myr, andf = 0.01 are assumed in this plot. The magenta dotted, red solid, green
dashed, and blue dot-dashed lines correspond to the gas accretion rate,
5 × 10−7 M⊙ yr−1 , 1 × 10−6 M⊙ yr−1 , 2 × 10−6 M⊙ yr−1 , and3 × 10−6 M⊙ yr−1 , respectively, and the blue dashed horizontal line
shows M = 10−7 M⊕ . The black dotted line shows the required initial
mass when 1 × 10−6 M⊙ yr−1 and κ = 1 are assumed.

10-5

101
f=0.01100

Initial maass [MEarth]

The definition of the accretion timescale is described as follows;Mtacc ≡.
(9)Ṁ
We integrate equation (??) for the disk lifetime tlife to derive
the required initial mass to form the core of the gas giants. If
the core of the gas giant is formed via the pebble accretion in
the class 0/I disk within its lifetime, it is a promising pathway
to the early formation of the gas giants in distant orbits. In thiscalculation we set the final mass to 10 M⊕ , because runaway
gas accretion will occur when a planetary core becomes heavier
than this value (e.g., ?).
Figure ?? show the radial dependence of the required initial
mass to form the 10 M⊕ core, with the difference gas accretion
rate assuming tlife = 0.5 Myr, M∗ = M⊙ , Ṁdust = f Ṁgas , andf = 0.01, respectively. The figure indicates that when the gas
accretion rate is Ṁgas = 10−6 M⊙ yr−1 , the required initial
mass is ∼ 5.0 × 10−4 M⊕ at 10 au. Therefore, if the embryo
whose mass is ∼ 5.0 × 10−4 M⊕ forms at 10 au, it can growto the core of the gas giant within the lifetime of the class 0/I
phases. Since the accretion timescale is much longer in the
distant region, the required initial mass have to be much larger.
The timescale of pebble accretion strongly depends on thegas accretion rate and the dust-to-gas mass ratio, therefore therequired initial mass to form the core of the gas giant will alsochange significantly depending on these factors. Fig. (??)
shows the radial dependence of the required initial mass toform the 10 M⊕ core, with the difference gas accretion rate
assuming tlife = 0.5 Myr, M∗ = M⊙ , Ṁdust = f Ṁgas , andf = 0.01, respectively. Fig (??) also shows the dependence ofthe required initial mass, but the dust-to-gas mass ratio is set tolarger value, f = 3fsolar , and fsolar = 0.0149 is the value forsolar nebula (?). The blue dashed lines shows the contour fora required initial mass. From left to right, 10−4 M⊕ , 10−3 M⊕ ,10−2 M⊕ , and 10−1 M⊕ , respectively.
Because the pebble accretion timescale is proportional to

1028

100

Gas accretion rate [Msol/yr]

3.2. Growth of planets by the pebble accretion

101

Initial mass [MEarth]

the effect of the reduction factor κ cannot be neglected for thecondition of Class 0/I YSOs. This effect is shown in Fig. ??;the accretion timescale described by the red solid line becomeslarger than the black dotted line that κ is ignored. As shownin the figure, the accretion timescale increases rapidly in the
distant region.
Fig. ?? shows that the accretion timescale decreases with theincrease of the gas accretion rate in the region within ∼ 20 au.
The dependence of the accretion timescale on the dust accre−1tion rate is ∝ Ṁdust(equation (??)), and the dependence of the
1/4
timescale on the gas accretion rate is ∝ Ṁgas . Hence the resultant dependence of the timescale on the accretion rate becomes−3/4∝ Ṁgas , when κ is neglected. In the distant region where>r ∼ 20 au, however, the accretion timescale slightly increases
with the gas accretion rate. This is caused by the dependenceof St on the gas accretion rate. The larger gas accretion ratecauses the larger St, and hence smaller reduction factor. Since
the critical mass is large in the distant region, the effect of the
reduction factor becomes significant.

3

10-1
10-2

10-6

10-3
10-4
10

100
Radius [au]Fig. 3. Map of the required initial mass in a radius-gas accretion rateplane. The color bar corresponds to the required initial mass to form
a 10 M⊕ core. M∗ = M⊙ , Ṁdust = f Ṁgas , tlife = 0.5 Myr, andf = 0.01 are assumed in this plot. The blue dashed lines shows thecontour for a required initial mass. From left to right, 10−4 M⊕ ,10−3 M⊕ , 10−2 M⊕ , and 10−1 M⊕ , respectively.

4[Vol. ,
of such substructures will benefit our understanding of the starand planet formation processes.
101
f=3.0 fsolar
100

References
Initial maass [MEarth]Gas accretion rate [Msol/yr]

10-5

10-1
10-2

10-6

10-3
10-4
10

100
Radius [au]Fig. 4. Maps of the required initial mass in the radius-gas accretion rateplane. The color bars and axes are the same as in Figure ??. M∗ = M⊙ ,Ṁdust = f Ṁgas , and tlife = 0.5 Myr are assumed, and the dust-to-gas
mass ratio is 3 fsolar . The blue dashed lines shows the contour for arequired initial mass same as in Fig. ??.
−1
Ṁdust, the larger amount of dust lead to faster accretion. The
larger dust-to-gas mass ratio leads to smaller required initialmass, due to the shorter accretion timescale. For dust rich condition, as shown in Fig (??), a larger dust component strongly
enhances the accretion rate of the pebbles and reduces the required initial mass. In the case of f = 3fsolar , a ∼ 10−3 M⊕embryo at >∼ 20 au can grow to a 10 M⊕ core. This results
suggest that if the planetary embryo whose mass is ∼ 10−3 M⊕
is formed, it can grow to the core of the gas giant, and thus thegiant planet can form during the Class 0/I phases.

4. Future prospects for ngVLA
We showed that the required initial mass to form the
10 M⊕ core that can grow into the gas giant at 10 au is∼ 5.0 × 10−4 M⊕ when we assume the gas accretion rateis ∼ 10−6 M⊙ yr−1 , the dust-to-gas mass ratio is 0.01, andthe disk lifetime is 0.5 Myr. In addition, for the most optimistic case, for example, the formation of the core of the
giant planet is possible at ∼ 10 au from the host star whenṀgas ∼ 3.0 × 10−7 − 10−6 M⊙ yr−1 and f = 3fsolar , and theembryo mass is comparable to Ceres’s mass, ∼ 10−4 M⊕ .
It is unclear that such large objects can form in the disk round
Class 0/I YSOs, so both theoretical and observational works
will be needed. As described by equation (??), the size of thedust grain can be a few centimeter or larger in Class 0/I phases(??). The dust grain of this size is difficult to observe by current radio telescopes such as ALMA. However, future facilities that using longer wavelengths such as ngVLA, cm-sized
dust grain in Class 0/I YSOs will be detectable. Therefore,
it is expected that dust evolution process in younger stage ofstar and planet formation will be observable, and the possibility of the planet formation and growth via pebble accretion inClass 0/I YSOs will be testable. In addition, if a gaseous planet
already formed in Class 0/I YSOs, the disk-planet interaction
will generate substructures in the disk similar to that observedin many protoplanetary disks. These substructures will also be
observable by future radio telescopes, and future observations

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