Rapid formation of Gas Giant Planets via Collisional Coagulation from Dust Grains to Planetary Cores

Gas-giant planets, such as Jupiter, Saturn and massive exoplanets, were formed via the gas accretion onto the solid cores each with a mass of roughly ten Earth masses. However, rapid radial migration due to disk-planet interaction prevents the formation of such massive cores via planetesimal accretion. Comparably rapid core growth via pebble accretion requires very massive protoplanetary disks because most pebbles fall into the central star. Although planetesimal formation, planetary migration, and gas-giant core formation have been studied with much effort, the full evolution path from dust to planets are still uncertain. Here we report the result of full simulations for collisional evolution from dust to planets in a whole disk. Dust growth with realistic porosity allows the formation of icy planetesimals in the inner disk (>10 au), while pebbles formed in the outer disk drift to the inner disk and there grow to planetesimals. The growth of those pebbles to planetesimals suppresses their radial drift and supplies small planetesimals sustainably in the vicinity of cores. This enables rapid formation of sufficiently massive planetary cores within 0.2-0.4 million years, prior to the planetary migration. Our models shows first gas giants form at 2-7 au in rather common protoplanetary disks, in agreement with the exoplanet and solar systems.


INTRODUCTION
Gas giant planets are formed via the rapid gas accretion of solid cores each with about 10 M ⊕ in protoplanetary disks (Ikoma et al. 2000), where M ⊕ is the Earth mass. The formation of cores via the accretion of 10 km sized planetesimals is in the Jupiter-Saturn forming region estimated to be ∼ 10 7 years (Kobayashi et al. 2011), which is longer than the disk lifetime (several million years, Haisch et al. 2001). In addition, the cores undergo the fast migration caused by the tidal interaction with the disk (called "Type I" migration, Ward 1997). They are lost prior to the gas accretion if the core formation timescale is longer than the migration timescale (∼ 10 5 years, Tanaka et al. 2002). Recently, the rapid accretion of submeter-sized bodies (called "pebbles" in the context of planet formation) is argued (Ormel & Klahr 2010). Pebbles form via collisional coagulation in the outer disk and then drift to the core-growing inner disk. The accretion of such bodies may lead to the formation of massive cores in a timescale (∼ 10 5 years) comparable to the migration timescale (Lambrechts & Johansen hkobayas@nagoya-u.jp 2014). However, this process requires a massive disk because the pebble accretion is lossy. The capture rate of pebbles by a single planetary core is evaluated to be below 10% (Ormel & Klahr 2010;Lin et al. 2018;Okamura & Kobayashi 2021). Hence the total pebble mass of a few hundred Earth masses is required for the formation of a core with 10M ⊕ (see more detailed estimate in §3.2), while protoplanetary disks with such a large solid mass are very rare (Mulders et al. 2021). As a process achieving a high conversion rate from dust or pebbles to kilometer-sized or larger bodies, planetesimal formation via collisional growth of icy pebbles is one of the most probable candidates. Recent models for collisional evolution of dust grains showed that pebbles grow to planetesimals in inner disks ( 10 au) in the realistic bulk density evolution model (Okuzumi et al. 2012). This process enhances the solid surface density in the inner disk, while 10 km sized or larger planetesimals slowly accrete onto cores. If the accretion of (sub-)kilometer sized planetesimals effectively occurs prior to planetesimal growth, cores are expected to grow in a short timescale (∼ 10 4 years). In order to confirm rapid core formation, the treatment fully from dust to cores in a whole disk is required.
In this paper, we investigate the formation of solid cores of giant planets from dust grains in protoplanetary disks. In § 2, we introduce the disk model that we apply. In § 3, we analytically estimate the growth timescales of solid cores via planetesimal and pebble accretion, respectively. In addition, we also estimate the minimum disk masses required for the formation of single gas-giant cores via pebble accretion. In § 4, we model a simulation for the collisional evolution of bodies from dust to planet ("Dust-to-planet" simulation; Hereafter, DTPS), taking into account the bulk density evolution of dust aggregates. This model consistently includes planetesimal and pebble accretion. In § 5, we show the result of a DTPS, where the rapid formation of solid cores via the accretion of planetesimals formed via drifting pebbles. In § 6, we discuss the locations of giant planets in the solar system or for exoplanets, based on the results of DTPSs. In § 7, we summarize our findings.

DISK MODEL
Planet formation occurs in a protoplanetary disk. We consider a power-law disk model for gas and solid surface densities, Σ g and Σ s , as where Σ g,1 = 480 g/cm 2 and Σ s,1 = 8.5 g/cm 2 are the gas and solid surface densities at 1 au, respectively, and r is the distance from the host star. The given solid/gas radio is the same as that in the minimum mass solar nebula (MMSN) model beyond the snowline (Hayashi 1981). However, we apply the shallower power law index than the the MMSN model according to the observation of protoplanetary disks (Andrews & Williams 2007). The surface densities Σ g and Σ s at r = 12.5 au correspond to those in the MMSN model, while Σ g and Σ s are smaller than those in the MMSN in the inner disk. The typical sizes of observed disks are ≈ 100 au (Andrews et al. 2010). We set disk radii ≈ 108 au: Disk masses correspond to 0.037M (≈ 220M ⊕ in solid). We set the temperature at the disk midplane as The radial dependence of temperature is the same as the MMSN. However, we apply a low temperature according to Brauer et al. (2008) because of optically thick disks.

Core-Growth and Migration Timescales
We here estimate the growth timescale of a solid core growing via collisions with planetesimals. Taking into account the gravitational focusing, the growth rate is given by dM p /dt ∼ 2πGM p R p Σ s Ω/v 2 rel (e.g., Goldreich et al. 2004), where M p and R p are the mass and radius of a planetary embryo, respectively, Ω is the Keplerian frequency, v rel is the relative velocity between the core and planetesimals, and t is the time. For a solid core with M p ∼ 10M ⊕ , planetary atmosphere enhances the collisional radius of the planet. The growth rate is estimated using R e instead of R p , where R e is the enhancement radius via atmosphere (Inaba & Ikoma 2003). Assuming the relative velocity is determined by the equilibrium between gas drag and the stirring by the core, we estimate the growth timescale t grow = M p /Ṁ p (e.g., Kobayashi et al. 2010Kobayashi et al. , 2011 where m and ρ b are the mass and bulk density of planetesimals, respectively, and R e = 3R p is used from the previous estimate (see Figure 1 in Kobayashi et al. 2011) and the values related to the disk are chosen from those of the given disk at r = 7 au. Therefore, the growth timescale is comparable to or longer than the lifetimes of protoplanetary disks ∼ 10 6 years. The gravitational interaction between a solid core and the protoplanetary disk induces radial migration of the core. The orbital decay timescale for the type I migration is estimated to be (e.g., Tanaka et al. 2002) where Γ is the dimensionless migration coefficient, h g is the scale height of the disk, the values of Σ g and h g are chosen from the given disk at 7 au. In the isothermal disk, Γ ≈ 4 (Tanaka et al. 2002). The formation of planets prior to the orbital decay requires t grow t mig . Once planetary embryos reaches ∼ 10M ⊕ , the rapid gas accretion of planetary embryos occurs (e.g., Mizuno 1980). Gas giant planets formed by the gas accretion open up the gap around their orbits and the migration timescale is then much longer than the estimate in Eq. (5) because of the onset of type II migration. Therefore, the formation timescale of a massive core with 10M ⊕ is required to be comparable to or shorter than the type I migration timescale.
The collisional growth of dust grains forms pebbles, which drift inward.
The growth rate of a planetary core via pebble accretion is given by where dM F /dt is the mass flux of pebbles across the orbit of the core and ε is the accretion efficiency of drifting pebbles. From dM F /dt given by Eq. (14) of Lambrechts & Johansen (2014), the core growth timescale via the accretion of drifting pebbles is estimated to be (7) where the value of ε is used for typical pebble-sized bodies (Ormel & Klahr 2010;Okamura & Kobayashi 2021). Although the collisional cross sections for the pebble accretion are large thanks to strong gas drag for pebbles, the solid surface density of drifting pebbles is much lower because to their rapid drift. The growth timescale t grow,pe is therefore comparable to the migration timescale.

Total Mass Required for Pebble Accretion
The required mass for the formation of a core with M p via pebble accretion is given by M F obtained from the integration of Eq. (6). Integrating Eq. (6) with the relation ε ∝ M where we assume the initial core mass is much smaller than M p . The pebble mass required for core formation, M F , is inversely proportional to ε(M p ) (see Eq. 8). Although ε 0.1 for St 0.1, we consider ε = 0.1 for St ≈ 0.1. Then the pebble mass required for core formation, M F , is estimated to be about 300(ε/0.1) −1 M ⊕ from Eq. (8). The pebble mass M F is limited by the total solid mass in a disk and thus the minimum disk mass required for a single core formation is estimated as where Σ s,1 /Σ g,1 is the metallicity in the disk. However, most protoplanetary disks are less massive than 0.1M (Andrews et al. 2010) and disks with the solid mass of 300M ⊕ are very rare even in Class 0 objects (∼ 1%, Mulders et al. 2021). Note that the required solid mass of 300 M ⊕ for pebble accretion is the minimum value. If smaller pebbles with St 0.1 are considered, the required solid mass is much more than 300 M ⊕ . Therefore, it seems difficult to explain giant exoplanets existing rather commonly (∼ 10%, Mayor et al. 2011) with pebble accretion. To reconcile this issue, we needs to increase ε due to collisional growth of drifting pebbles (see ε for St 1 in Figure 1 and the discussion by Okamura & Kobayashi 2021).
The planetesimal formation from icy pebbles would be a possible process, which achieve a high conversion rate from pebbles to kilometer-sized or larger bodies. To consider the collisional growth of pebbles into planetesimals, we need to review collisional fragmentation. The collisional simulations of icy dust aggregates shows the fragmentation velocity of aggregates, v f , depends on the interaction of monomers determined by the surface energy of ice γ ice , given by v f = 80 (γ ice /0.1 J m −2 ) 5/6 m s −1 for aggregates composed of sub-micron sized monomers . The collisional velocities for pebblesized bodies are mainly smaller than 50 m s −1 , so that collisional fragmentation is negligible if γ ice ∼ 0.1 J m −2 . The surface energy of ice was estimated to be much lower than 0.1 J m −2 from the measurement of the rolling friction force between 1.1 millimeter sized particles in laboratory experiments (Musiolik & Wurm 2019). However, the distinction between rolling and slide forces is difficult for such large particles so that Kimura et al. (2020a) explained the measurements including the temperature dependence as the slide forces given by the tribology theory with quasi-liquid layers without low γ ice . In addition, the measurement in laboratory experiments showed the tensile strength of aggregates for ice is comparable to that for silicates, implying that γ ice is as small as the silicate surface energy, γ sil (Gundlach et al. 2018). Kimura et al. (2020b) explained the measured tensile strengths for ice and silicate by the Griffith theory using γ ice ∼ γ sil ∼ 0.1 J m −2 . From a physical point of view, the surface energy should be grater than the surface tension, which is ≈ 0.08 J m −2 even in the room temperature. Therefore, collisional fragmentation is negligible for pebble growth.
We additionally discuss the effect of collisions with large m 1 /m 2 , where m 1 and m 2 are the masses of colliding bodies (m 1 > m 2 ). Erosive collisions, large m 1 /m 2 collisions with velocities higher than v f , reduce the masses of the larger colliding bodies, which inhibit the growth via collisions with large m 1 /m 2 . Krijt et al. (2015) claimed the growth of pebbles were stalled by erosive collisions under the assumption that v f for m 1 /m 2 > 100 is much smaller than that for m 1 ∼ m 2 . However, this assumption is inconsistent with the impact simulations of dust aggregates. Recent impact simulations with m 1 /m 2 100 show v f for larger m 1 /m 2 is higher than that for m 1 ∼ m 2 (Hasegawa et al. 2021). Therefore, erosive collisions are insignificant for pebble growth.

MODELS FOR DTPS
We develop a simulation for the collisional evolution of bodies from dust to planets ("Dust-to-planet" simulation; Hereafter, DTPS). We here introduce the model for the DTPS.

Collisional Evolution
Collisions between bodies lead to planet formation. The surface number density n s of bodies with mass m at the distance r from the host star with mass M * evolves via collisions and radial drift. The governing equation is given by where K(m 1 , m 2 ) is the collisional kernel between bodies with masses m 1 and m 2 and v r is the radial drift velocity. We where v drag and v mig are the radial drift velocity due to gas drag and the type I migration. We model as (see Appendix A for v drag and Tanaka et al. 2002, for where Γ = 4 is the dimensionless migration coefficient (Tanaka et al. 2002), e and i are the orbital eccentricity and inclination, St is the dimensionless stopping time due to gas drag, and Here, c s is the isothermal sound velocity. The dimensionless stopping time, St, called the Stokes parameter, is given by (Adachi et al. 1976) where ρ g is the mid-plane gas density, h g = c s /Ω is the gas scale height, and λ mfp is the mean free path. As discussed above, collisional fragmentation is negligible for St 1. For further collisional growth, fragmentation is unimportant until planetary embryo formation (Kobayashi et al. 2016;Kobayashi & Tanaka 2018). We ignore collisional fragmentation even after embryo formation because of the uncertainty of collisional outcome models. This crude assumption is good to compare with the studies for pebble accretion, in which collisional fragmentation is also ignored except for consideration of pebble sizes (Bitsch et al. 2018;Lambrechts et al. 2019;Johansen et al. 2019). In addition, collisional fragmentation for pebble formation works negatively for pebble accretion because of low ε for St 10 −2 ( Figure 1). Therefore, we consider only the collisional merging.
Collision-less interactions among bodies induce the evolution of their eccentricities and inclinations, which is sensitive to the mass spectrum of bodies. We calculate the e and i evolution together with the mass evolution, taking into account the mutual interaction between bodies such as viscous stirring and dynamical friction, gas drag, and the perturbation from the turbulent density fluctuation (Kobayashi & Tanaka 2018). The detailed treatment of e and i evolution is described in Appendix D. We developed a simulation for planetesimal accretion (St 1), which perfectly reproduces the result obtained from the direct N body simulation (Kobayashi et al. 2010).
For St 1, we calculate P additionally using the scale height and the relative velocity. For St 1, the scale height for bodies with m 1 and St 1 is given by (Youdin & Lithwick 2007) where α D is the dimensionless turbulent parameter. We introduce the relative scale height between m 1 and m 2 as h s,1,2 = [π(h 2 s,1 + h 2 s,1 )/2] 1/2 .
The relative velocity is given by v 2 rel,gas = ∆v 2 B + ∆v 2 r + ∆v 2 where ∆v B , ∆v r , ∆v θ , ∆v z , and ∆v t are the relative velocities induced by the Brownian motion, radial and azimuthal drifts, vertical settling, and turbulence, respectively (detailed description in Appendix E). For St 1 1 and St 2 1, K is expressed using h s,1,2 and v rel,gas as (Okuzumi et al. 2012), We therefore expand Eq. (16) to apply the case for St 1 using h s,1,2 and v rel,gas . The collisional probability P is the function of (ẽ 2 1,2 + i 2 1,2 ) 1/2 andĩ 1,2 , which represent the relative velocity and the relative scale height, respectively. Therefore, we use the greater values of (ẽ 2 1,2 +ĩ 2 1,2 ) 1/2 or v rel,gas /r H,1,2 Ω andĩ 1,2 or h s,1,2 /r H,1,2 for the function of P. We then calculate the collisional Kernel K for any St. Using this method, we calculate K for St 1 , St 2 1, which corresponds to Eq. (20). Therefore, we apply this method for bodies from dust grains to planets.

Bulk Density
The collisional growth of dust grains produces the fractal dust aggregates, whose ρ b = (3m/4πs 3 ) is lower than the original material. The stopping time St depends on ρ b . The evolution of ρ b is significantly important for collisional growth for St 1. We model ρ b as where ρ mat = 1.4 g/cm 3 is the material density, corresponding to the density of compact bodies or monomer grains in dust aggregates, s mon = 0.1 µm is the monomer radius, m mon = 4πρ mat s 3 mon /3 is the monomer mass, E roll = 4.74 × 10 −9 erg is the rolling energy between monomer grains, and G is the gravitational constant.
The densities ρ s , ρ m , and ρ l almost correspond to the bulk density for small, intermediate, and large bodies, respectively. For small dust, collisional growth occurs without collisional compaction. Eq. (22) is determined by the model given in the previous study (Okuzumi et al. 2012) under the assumption of the collisional evolution between same-mass bodies without collisional compaction, which is almost similar to the density evolution with the fractal dimension ∼ 2 (Okuzumi et al. 2012). For large bodies, the bulk density increases with increasing mass by self-gravity compaction until compact bodies with ρ b = ρ mat and the equilibrium density is given by Eq. (24) (Kataoka et al. 2013).
For intermediate bodies, the bulk density is most important for St ∼ 1, which determined the fate of bodies. The bulk density is determined by the compression due to ram pressure of the disk gas (Kataoka et al. 2013). We estimate ρ b at St = 1 under the assumption of the Epstein gas drag, It should be noted that ρ b for St = 1 is smaller than that given by Eq.(25) at r 10 au because the Stokes gas drag is dominant for St ∼ 1 at the inner disk. Therefore ρ b for St ∼ 1 becomes up to ∼ 10 −3 g/cm 3 so that we simply choose the value of ρ m = 10 −3 g/cm 3 according to the estimate. Figure 2 shows the radii or St of bodies in the model as a function of mass.

RESULT
We perform a DTPS for the collisional evolution of bodies drifting due to gas drag and Type I migration in a protoplanetary disk. We set a disk with the inner and outer radii of ≈ 3 au and ≈ 108 au, whose gas surface density is inversely proportional to r (see Eq. 1). The disk mass corresponds to 0.036M (total solid mass ≈ 210M ⊕ ), which is smaller than the required mass for the pebble accretion (see Eq. 9). Solid bodies initially have a mass m = 5.9 × 10 −15 g (corresponding to a radius of 0.1 µm). We set the turbulent strength to be α D = 10 −3 . Figure 3 shows the surface density of bodies whose masses are similar to m within about a factor of 2, as a function of m and the distance from the host star. Dust growth occurs around r ≈ 5 au at t ≈ 560 years ( Figure  3a). Largest bodies reach at m ∼ 10 13 g at 3 au. The drift of bodies is controlled by the gas coupling parameter of bodies St (see Eq. 15). Bodies have highest drift velocities at m ∼ 10 8 , corresponding to St = 1 (Figure 2). For low-density bodies, the collisional growth timescale is much shorter than the drift timescale so that large bodies with St 1 are formed via collision growth (Okuzumi et al. 2012). Dust collisional growth propagates from the inner to outer disk (Figure 3b, see also Ohashi et al. 2021). Dust growth front reaches 20, 50, 90 au and the outer boundary at t ≈ 1.5 × 10 4 , 5.6 × 10 4 , 1.2×10 5 , and 2.1×10 5 years, respectively (Figure 3c-f). Radial drift is more dominant than collisional growth for bodies with St ∼ 1 beyond 10 au. The drifting bodies grow to planetesimals in the disk inside 10 au.
In the early growth (Figure 3a,b), the total solid surface densities are mainly determined by largest bodies. At t ≈ 6 × 10 4 years (Figure 3c), the runaway growth of bodies with m = 10 13 -10 16 g occurs at r 6 au. The solid surface density of planetesimal-sized bodies (m ∼ 10 18 g) becomes dominant. Planetary embryos with m ∼ 10 24 g are formed at r 10 au via the runaway growth ( Figure 3d). The further growth of embryos occurs via collisions with planetesimals ( Figure 3e). The largest planetary embryos exceed 10 Earth masses even at t ≈ 2 × 10 5 years (Figure 3f).
Collisional growth successfully forms bodies with m 10 10 g (St 1) only at r 10 au (Figures 3d-f). To overcome the drift barrier at St ≈ 1, bodies with St ≈ 1 should grow via collisions much faster than their radial drift. The requirement for this condition is that bodies with St = 1 feels gas drag in the Stokes regime (Okuzumi et al. 2012). Therefore, bodies for St = 1 has s 9λ mfp /4 (see Eq. 15); For bulk densities and disk conditions given in § 4.2 and 3.2, Eq. (26) corresponds to r grow 24 au, where r grow is the radius inside which pebbles can grow to planetesimals. Therefore, collisional growth results in planetesimals with St 1 for r 10 au. The radial drift of planetesimals with St 1 is much slower than that of pebbles St 1; the pile-up results in the enhancement of solid surface densities at r 10 au (Figures 3d-f and 4b). Pebbles formed in the whole disk with the total solid mass M solid,disk finally drift inward across r grow , so that the enhanced surface density is estimated to be M solid,disk /πr 2 grow ≈ 18(r grow /10 au) −2 g cm −2 (compare with Figure 4b). Figure 4a shows the mass of largest planetary embryos in each annulus of the disk. Planetary embryos acquire 10 M ⊕ around r ≈ 6-7 au at t ≈ 2 × 10 5 years. Such rapid formation of massive embryos is achieved via the pile-up of bodies in r < 10 au ( Figure 4b). As explained above, bodies with St < 1 drift inwards from the outer disks until the bodies grow to St 1 in r < 10 au. The solid surface density increases to 20 g cm −2 at 7 au in 2×10 5 years (see Figure 4b), the formation of cores with 10 M ⊕ requires the surface density of 3 g cm −2 . Therefore, only about 15 % of bodies are needed for the core  formation in the enhanced disk. The Σ s enhancement effectively accelerates the growth of cores. However, the growth rate depends on the mass spectrum of bodies accreting onto cores. We additionally investigate which masses of bodies mostly contribute to the core growth. The cumulative accretion rate of bodies onto the largest core in the annulus at r = 6.75 au is shown in Figure 5a. The contribution of pebbles (St < 1) to the accretion rate is minor, because the solid surface density of bodies with St < 1 is tiny (Figure 5b). Collisions with 100 m-10 km sized bodies of m = 10 9 -10 19 g mainly contribute to the accretion rate, while the solid surface density is mainly determined by planetesimal-sized bodies of m ∼ 10 19 g ( Figure 5 and see also Figure 3). The atmospheric collisional enhancement promotes the accretion of sub-kilometer-sized bodies of m ∼ 10 16 g, which are in the course of growing to planetesimals. The growth of planetary cores additionally increases their Hill radii so that collisions between planetary embryos occur. The embryo accretion therefore increases the total accretion rate by a factor of about 1.5 additionally (Chambers 2006;Kobayashi et al. 2010).
We again estimate the growth timescale of cores via planetesimal accretion in this condition using Eq. (4) in § 3.1. The solid surface density of planetesimals or planetesimal precursors increases to 15 g/cm 2 ( Figure   5b). As mentioned above, the contribution of planetesimal precursors to the accretion is significant. The enhancement factor R e /R p proportional to m −1/9 is higher for small planetesimals (Kobayashi et al. 2011). The growth timescale is then estimated to be t grow ≈ 1.5 × 10 4 (Σ s /15 g cm −2 )(m/10 16 g) 11/25 years, corresponding to the accretion rate of 4.0 × 10 24 g/yr for M p = 10M ⊕ . This value is consistent with the accretion rate at t ≈ 1.9 × 10 5 years (see Figure 5a). The accretion of planetesimals with mass m = 10 15 -10 19 g induces the rapid growth of planetary embryos. Such planetesimals are produced via collisional growth of planetesimal precursors with m ∼ 10 8 − 10 15 g. The bulk density of such planetesimal precursors for m 10 13 g is given by ρ b ∝ m 2/5 (see § 4.2). Their collisional timescale among planetesimal precursors with mass m and radius r p is given by which is estimated to be t col ≈ 4.5×10 3 years at 7 au for m = 10 13 g and Σ s = 0.2 g cm −2 according to Figure 5b. On the other hand, the drift timescale of planetesimal precursors is given by where η is the dimensionless parameter depending on the pressure gradient. The drift timescale is estimated to be t drift ≈ 3.5 × 10 5 years for m = 10 13 g. Therefore, t col /t drift ≈ 1.3 × 10 −2 (Σ s /0.2 g cm −2 ) −1 1 independent of m. For such planetesimal precursors, the collisional growth timescale is much shorter than the drift timescale.
Pebbles with St ∼ 0.1 drift from the outer disk, and the collisional growth among the pebbles produces planetesimal precursors prior to their drift. Because t col t drift as estimated above, planetesimal precursors grow without significant drift in the inner disk (< 10 au) until gravitational scatterings by planetary cores, which lead to the uniform distribution of planetesimal precursors around cores. The surface density of planetesimal precursors is much smaller than that of planetesimals ( Figure 5b). Planetesimal precursors are maintained via the supply from the growth of pebbles drifting from the outer disk. This mechanism leads to the sustainable accretion of small planetesimals, resulting in the rapid growth within 0.2 Myr.
If the scattering of planetesimal precursors by a solid core is comparable to the drift, such solid cores may open gaps up in a planetesimal-precursor disk, which would reduce the accretion rate of small planetesimals (Levison et al. 2010). However, the collisional growth timescale of planetesimal precursors is much shorter than the gap opening timescale comparable to t drift , so that planetesimals are then formed via collisonal growth of precursors prior to gap opening. Therefore, the rapid growth is achieved without the gap opening in the solid disk.

DISCUSSION
We show the rapid core formation at 6-7 au in 2 × 10 5 years. We obtain the similar result for a weak turbulent level of α D = 10 −4 , with which the simulation results in the core formation at ≈ 7 au in 3 × 10 5 years. A sufficient massive core starts gas accretion and type II migration. The orbital samimajor axis of a gas giant with Jupiter mass resulting from the gas accretion and type II migration is about 0.9 times that of the original core (Tanaka et al. 2020). Therefore, the first gas giant is formed around 6 au. The giant planets in the solar system may experience migration. Outward migration of Neptune can explain the orbital eccentricities of Plutinos in the Kuiper belt. The exchange of angular momentum between Nepture and Jupiter via interactions with planetesimals requires the original orbit of Jupiter at ∼ 6 au (Minton & Malhotra 2009). Therefore, the formation of a core at 6-7 au is consistent with the origin of Jupiter.
On the other hand, the formation location of gas-giant cores depends on the solid/gas ratio, which is set to 0.017 in the simulation according to the minimum-mass solar nebula model (Hayashi 1981). We additionally carry out a simulation for the solid/gas ratio of 0.01, whose disk includes solids ≈ 120 M ⊕ . The core formation occurs at 3 × 10 5 years at 3-4 au. The population of exoplanets found via radial velocity surveys is high around 2-3 au (Fernandes et al. 2019). The occurrence location of gas giants may be explained by the typical solid/gas ratio of ∼ 0.01.
For much smaller solid/gas ratios (solid masses smaller than 100M ⊕ ), the growth timescales of cores are longer than the migration timescales, resulting in the difficulty of gas giant formation. In addition, small disk masses tend to make the planetesimal forming radii r grow small. If r grow is smaller than the snow line, the enhancement of solid surface density due to the mechanisms shown in §. 5 is less effective, depending on the outcomes of the sublimation of icy pebbles. Furthermore, disk sizes are also important. If disk sizes are much smaller than 100 AU, pebble supply is stalled prior to core formation. Such difficulties of core formation may be important to discuss the gas giant occurrence rate (∼ 10%, Mayor et al. 2011).

SUMMARY
Gas giant planets are formed via rapid gas accretion of massive solid cores prior to type I planetary migration of cores (timescale of ∼ 10 5 years). Core formation via accretion of 10 km sized or larger planetesimals requires times much longer than the migration timescale. Pebbles form via collisional coagulation in outer disks, which drift into the inner disk. The core-growth timescale via pebble accretion is much shorter than that via the accretion of 10 km sized planetesimals. However, pebble accretion mostly losses drifting pebbles, which requires more than 300 M ⊕ in solid for single-core formation ( §. 3). To reconcile the issue, we need to consider collisional growth of pebbles to planetesimals.
We thus develop a simulation for the collisional evolution of bodies from dust to planet ("Dust-to-planet" simulation; Hereafter, DTPS), which describes both planetary growth modes via planetesimal accumulation and pebble accretion ( § 4). The result of the DTPS shows as follows.
1. Collisional coagulation of dust aggregates forms planetesimals in the inner disk ( 10 au) thanks to the realistic bulk densities of dust aggregates, while pebbles drift inwards from the outer disk ( 10 au). Planetesimal formation via collisional growth of pebbles increases the solid surface density at 6-9 au by the factor of ∼ 10 ( Figure 4).
2. Planetesimals growing from pebbles are relatively small, whose accretion rate to planetary cores per unit surface density is relatively high. Both the enhanced solid surface density and the high accretion rate accelerate the growth of cores significantly. The accretion rate becomes 4 × 10 24 g at about 7 au in 2 × 10 5 years ( Figure 5), corresponding to the core-growth timescale of 2 × 10 4 years, which is much shorter than Type I migration times of cores. Solid cores are formed in 6-7 au without significant type I migration, which are likely for the formation of Jupiter in the Solar System. 1, the drift velocity induced by gas drag is given by (Adachi et al. 1976;Inaba et al. 2001) where K and E are the complete elliptic integrals of the first and second kinds, and we ignore the higher order terms of e and i because the terms are small enough (Kobayashi 2015). On the other hand, for St 1 v drag is given by (Adachi et al. 1976) We combine the both regimes as (Kobayashi et al. 2010), The collisional radius is enhanced by an atmosphere. Inaba & Ikoma (2003) derived the enhancement factor for the radius, where v rel is the relative velocity, r H = (m/3M * ) 1/3 r is the Hill radius, ρ a is the atmospheric density at ξs from the center of the body. We derive ρ a according to Inaba & Ikoma (2003). The pressure P a , temperature T a , and density ρ a of the atmosphere at the distance r a from the body have the relations as follows.
where the dimensionless pressureP a = P a /P o , temperatureT a = T a /T o , densityρ a = ρ a /ρ o , distancer a = r a /r o are scaled by the pressure, temperature, density, and distance at the outer boundary, respectively, a r is the radiation density constant, κ a is the opacity, and L a is the luminosity. The luminosity is given by whereṁ is the accretion rate of the body, σ SB is the Stefan-Boltzmann constant, and the function MAX(x, y) gives the larger of x and y.
We set the outer boundary values and opacity as follows.
T o = 170 K, P o = P a,170 K , r o = r a,170 K , κ a = 2ζ + 0.01cm 2 g −1 for 170 K < T a ≤ 1700K, T o = 1700 K, P o = P a,1700 K , r o = r a,1700 K , κ a = 0.01cm 2 g −1 for T a > 1700K, where P is the gas pressure at the disk mid-plane, c s is the isothermal sound velocity, k hc is the heat capacity ratio, the function MIN(x, y) gives the smaller of x and y, the subscripts of 170 K and 1700 K indicate the values at T a = 170 K and 1700 K, respectively, and ζ is the reduction factor of the atmospheric opacity. A massive planetary body acquire an atmosphere. Small dust grains decrease until the formation of massive bodies. We therefore apply ζ = 10 −4 .
This collisional probability corresponds to that between dust grains with St 1 , St 2 1.

D. RANDOM VELOCITY EVOLUTION
For St 1, collisional evolution depends on e and i. We consider the e and i evolution due to gravitational interaction (Ohtsuki et al. 2002), gas drag (Adachi et al. 1976), and collisional damping (Ohtsuki 1992). On the other hand, a body have a orbit determined by the Kepler law for St 1. The orbital elements of e and i do not indicate the motion of bodies. However, the collisional velocity is determined by v rel,gas instead of e and i. Therefore, we calculate the e and i evolution via the following equations.
We only consider the leading-order terms of e and i, because the higher-order terms of e and i are negligible for e and e with which we are concerned (Kobayashi 2015). The collisional damping terms, de 2 /dt| c and di 2 /dt| c are given via the random velocities of collisional outcomes according to Kobayashi et al. (2010).