Accretion among preplanetary bodies: The many faces of runaway growth

When preplanetary bodies reach proportions of ∼1 km or larger in size, their accretion rate is enhanced due to gravitational focusing (GF). We have developed a new numerical model to calculate the collisional evolution of the gravitationally-enhanced growth stage. The numerical model is novel as it attempts to preserve the individual particle nature of the bodies (like N-body codes); yet it is statistical in nature since it must incorporate the very large number of planetesimals. We validate our approach against existing N-body and statistical codes. Using the numerical model, we explore the characteristics of the runaway growth and the oligarchic growth accretion phases starting from an initial population of single planetesimal radius R₀. In models where the initial random velocity dispersion (as derived from their eccentricity) starts out below the escape speed of the planetesimal bodies, the system experiences runaway growth. We associate the initial runaway growth phase with increasing GF-factors for the largest body. We find that during the runaway growth phase the size distribution remains continuous but evolves into a power-law at the high-mass end, consistent with previous studies. Furthermore, we find that the largest body accretes from all mass bins; a simple two-component approximation is inapplicable during this stage. However, with growth the runaway body stirs up the random motions of the planetesimal population from which it is accreting. Ultimately, this feedback stops the fast growth and the system passes into oligarchy, where competitor bodies from neighboring zones catch up in terms of mass. We identify the peak of GF with the transition between the runaway growth and oligarchy accretion stages. Compared to previous estimates, we find that the system leaves the runaway growth phase at a somewhat larger radius, especially at the outer disk. Furthermore, we assess the relevance of small, single-size fragments on the growth process. In classical models, where the initial velocity dispersion of bodies is small, these do not play a critical role during the runaway growth; however, in models that are characterized by large initial relative velocities due to external stirring of their random motions, a situation can emerge where fragments dominate the accretion, which could lead to a very fast growth.     

Relative velocities among accreting planetesimals in binary systems: The circumprimary case

We investigate classical planetesimal accretion in a binary star system of separation ab?50 AU by numerical simulations, with particular focus on the region at a distance of 1 AU from the primary. The planetesimals orbit the primary, are perturbed by the companion and are in addition subjected to a gas drag force. We concentrate on the problem of relative velocities Δv among planetesimals of different sizes. For various stellar mass ratios and binary orbital parameters we determine regions where Δv exceed planetesimal escape velocities vesc (thus preventing runaway accretion) or even the threshold velocity vero for which erosion dominates accretion. Gaseous friction has two crucial effects on the velocity distribution: it damps secular perturbations by forcing periastron alignment of orbits, but at the same time the size-dependence of this orbital alignment induces a significant Δv increase between bodies of different sizes. This differential phasing effect proves very efficient and almost always increases Δv to values preventing runaway accretion, except in a narrow eb?0 domain. The erosion threshold Δv>vero is reached in a wide (ab,eb) space for small <10-km planetesimals, but in a much more limited region for bigger ?50-km objects. In the intermediate vesc<Δv<vero domain, a possible growth mode would be the type II runaway growth identified by Kortenkamp et al. [Kortenkamp, S., Wetherill, G., Inaba, S., 2001. Science 293, 1127-1129].     

Relative velocities of planetesimals and the early accumulation of planets

Safronov's statement that relative velocities of planetesimals are on the order of the escape velocity of the largest body of the population is shown to be correct only when a major part of the total mass resides in several large bodies. In the first stage of accumulation, runaway accretion produces large bodies separated by mass form the remaining population. At this stage, relative velocities of planetesimals are much smaller than those adopted earlier. This requires a modification of Schmidt's scheme of accumulation of the Earth and other terrestrial planets from material in their feeding zones. This also leads to removal of the author's arguments (Levin 1972c) in favor of a protoplanetary nebula with an extended, massive periphery.Paper presented at the Conference on Protostars and Planets, held at the Planetary Science Institute, University of Arizona, Tucson, Arizona, between January 3 and 7, 1978.     

Accretion of the asteroids: Implications for their thermal evolution

Thermal models of asteroids generally assume that they accreted either instantaneously or over an extended interval with a prescribed growth rate. It is conventionally assumed that the onset of accretion of chondrite parent bodies was delayed until a substantial fraction of the initial ²⁶Al had decayed. However, this interval is not consistent with the early melting, and differentiation of parent bodies of iron meteorites. Formation time scales are tested by dynamical simulations of accretion from small primary planetesimals. Gravitational accretion yields rapid runaway growth of large planetary embryos until most smaller bodies are depleted. In a given simulation, all asteroid‐sized bodies have comparable growth times, regardless of size. For plausible parameters, growth times are shorter than the lifetime of ²⁶Al, consistent with thermal models that assume instantaneous accretion. Rapid growth after planetesimal formation is consistent with differentiation of parent bodies of iron meteorites, but not with the assumed delay in formation of chondritic bodies. After the initial growth stage, there is an interval of slower evolution until the belt is stirred and the embryos are dynamically removed. During this interval, a fraction of asteroid‐sized bodies experience large accretional impacts, allowing bodies of the same final size to have very different histories of radius versus time. Accretion from small primary planetesimals leaves some fraction of material in bodies small enough to preserve CAIs while avoiding heating by ²⁶Al. Unheated material can be a significant fraction of the mass that remains after large embryos are removed from the Main Belt.     

Oligarchic growth of giant planets

Runaway growth ends when the largest protoplanets dominate the dynamics of the planetesimal disk; the subsequent self-limiting accretion mode is referred to as “oligarchic growth.” Here, we begin by expanding on the existing analytic model of the oligarchic growth regime. From this, we derive global estimates of the planet formation rate throughout a protoplanetary disk. We find that a relatively high-mass protoplanetary disk (∼10 × minimum-mass) is required to produce giant planet core-sized bodies (∼10 M_⊕) within the lifetime of the nebular gas (?10 million years). However, an implausibly massive disk is needed to produce even an Earth mass at the orbit of Uranus by 10 Myrs. Subsequent accretion without the dissipational effect of gas is even slower and less efficient. In the limit of noninteracting planetesimals, a reasonable-mass disk is unable to produce bodies the size of the Solar System’s two outer giant planets at their current locations on any timescale; if collisional damping of planetesimal random velocities is sufficiently effective, though, it may be possible for a Uranus/Neptune to form in situ in less than the age of the Solar System. We perform numerical simulations of oligarchic growth with gas and find that protoplanet growth rates agree reasonably well with the analytic model as long as protoplanet masses are well below their estimated final masses. However, accretion stalls earlier than predicted, so that the largest final protoplanet masses are smaller than those given by the model. Thus the oligarchic growth model, in the form developed here, appears to provide an upper limit for the efficiency of giant planet formation.     

Planetesimal dissolution in the envelopes of the forming,giant planets

We have assessed the ability of planetesimals to penetrate through the envelopes of growing giant planets that form by a “core-instability” mechanism. According to this mechanism, a core grows by the accretion of solid bodies in the solar nebula and the growing core becomes progressively more effective in gravitationally concentrating gas from the surrounding solar nebula in an envelope until a “runaway” accretion of gas occurs. In performing this assessment, we have considered the ability of gas drag to slow down a planetesimal; the effectiveness of gas dynamical pressure in fracturing and ultimately finely fragmenting it; the ability of its strength and self-gravity to resist such fracturing; and the degree to which it is evaporated due to heating by the surrounding envelope, including shock heating that develops during the supersonic portion of its trajectory. We also consider what happens if the planetesimal is able to reach the core at free-fall velocity and the ability of the envelope to convectively mix dissolved materials to different radial distances. These calculations were performed for various epochs in the growth of a giant planet with the model envelopes derived by Bodenheimer and Pollack (1986,67, 391–408). As might have been anticipated, our results vary significantly with the size of the planetesimal, its composition, and the stage of growth of the giant planet and hence the mass of its envelope. Over much of the growth phase of the core, prior to its reaching its critical mass for runaway gas accretion, icy planetesimals less than about 1 m in size dissolve in the outer region of the envelope, ones larger than about 1 m and smaller than about 1 km dissolve in the middle region of the envelope, ones larger than 1 km either reach the core interface or dissolve in the deeper regions of the envelope. Similarly rocky planetesimals smaller than about a kilometer dissolve in the middle portion of the envelope, while larger ones can penetrate more deeply. Furthermore, the convection zones of the envelopes during this stage are confined to localized regions and hence dissolved materials experience little radial mixing then. Thus, if much of the accreted mass is contained in planetesimals larger than about a kilometer, the critical core mass for runaway accretion is not expected to change significantly when planetesimal dissolution is taken into account. After accretion is terminated and the planet contracts toward its present size, the convection zone grows until it encompasses the entire envelope. Therefore, dissolved material should eventually become well mixed through the envelope. We proposed that the envelopes of the giant planets should contain significant enhancements above solar proportions in the abundances of virtually all elements relative to that of hydrogen, with the magnitude of the enhancement increasing approximately linearly with the ratio of the high Z mass to the (H, He) mass for the bulk of the planet. This prediction is in accord both qualitatively and quantitatively with the systematic increase in the atmospheric C/H ratio from Jupiter to Saturn to Uranus and Neptune and semiquantitatively with the results of recent interior models of the giant planets. It is not clear whether it is consistent with the abundances of H₂O and NH₃ in the atmospheres of some of the outer planets. Finally, the complete reduction of some dissolved materials, especially C containing compounds, is expected to consume some of the H₂ in the envelopes. Consequently, the He/H₂ ratios in the atmospheres of Uranus and Neptune may be slightly enhanced over the solar ratio. We estimate that the He/H₂ ratios for Uranus' and Neptune's atmospheres should be about 6 and 15% larger, respectively, than the solar ratio.     

Stochastic coalescence model for terrestrial planetary accretion

A nonequilibrium stochastic coalescence model for terrestrial planetary accretion is developed by using an approximation to the Safronov-Golovin solution for the scalar transport equation with linear kernel. According to this model, formation of comparatively massive objects occurs quite rapidly during the early stages of accretive evolution in a given terrestrial planetesimal population, while during late growth stages, an increasingly substantial fraction of total population mass becomes incorporated into progressively fewer, relatively very large bodies. The model also implies that the (conservative) growth rate of the population's largest member varies directly as its mass, and further suggests that this growth rate may not decline significantly until very nearly final planetary mass is attained.     

Planetary embryos and planetesimals residing in thin debris discs

We consider constraints on the planetesimal population residing in the discs of AU Microscopii (AU Mic), β Pictoris (β Pic) and Fomalhaut taking into account their observed thicknesses and normal disc opacities. We estimate that bodies of radius 5, 180 and 70 km are responsible for initiating the collisional cascade accounting for the dust production for AU Mic, β Pic and Fomalhaut's discs, respectively, at break radii from the star where their surface brightness profiles change slope. Larger bodies, of radius 1000 km and with surface density of the order of 0.01 g cm⁻², are required to explain the thickness of these discs assuming that they are heated by gravitational stirring. A comparison between the densities of the two sizes suggests the size distribution in the largest bodies is flatter than that observed in the Kuiper belt. AU Mic's disc requires the shallowest size distribution for bodies with radius greater than 10 km suggesting that the disc contains planetary embryos experiencing a stage of runaway growth.     

Dust to planetesimals: Settling and coagulation in the solar nebula

The behavior of solid particles in a low-mass solar nebula during settling to the central plane and the formation of planetesimals is examined. Gravitational instability in a dust layer and collisional accretion are considered as possible mechanisms of planetesimal formation. Non-Keplerian rotation of the nebula results in shear between the gas and a dust layer. This shear produces turbulence within the layer which inhibits gravitational instability, unless the mean particle size exceeds a critical value, ～1 cm at 1 AU. The size requirement is less stringent at larger heliocentric distances, suggesting a possible difference in planetesimal formation mechanisms between the inner and outer nebula. Coagulation of grains during settling is expected in the solar nebula environment. Van der Waals forces appear adequate to produce centimeter-sized aggregates. Growth is primarily due to sweepup of small particles by larger ones due to size-dependent settling velocities. A numerical model for computing simultaneous coagulation and settling is described. Relative velocities are determined by gas drag and the non-Keplerian rotation of the nebula. The settling is very nonhomologous. Most of the solid matter reaches the central plane as centimeter-sized aggregates in a few times 10³ revolutions, but some remains suspended in the form of fine dust. Drag-induced relative velocities result in collisions. The growth of bodies in the central plane is initially rapid. After sizes reach ～10³ cm, relative velocities decrease and the growth rate declines. Gas drag rapidly damps the out-of-plane motions of these intermediate-sized bodies. They settle into a thin layer which is subject to gravitational instability. Kilometer-sized planetesimals are formed by this composite process.     

Orbital resonances and planetary formation sites

Orbital resonances may have played an important role in determining the locations where the planetesimal swarm eventually accreted into full-size planets. Several pairs of planets do indeed have commensurable orbital periods at present, but the case for control of planet formation by resonances is weakened by the fact that many pairs are not commensurable and that those which are do not necessarily exist at the strongest resonances. However, the mass loss and redistribution that occurred in the early solar system evolution can substantially alter the positions of planets and planetary embryos within the swarm. A cascaded resonance structure is hypothesized where planetesimal growth was accelerated at 2:1 interior and 1:2 exterior resonances with an early-formed Jupiter producing runaway growth of planetary embryos. These embryos produce their own resonances which, in turn, lead to additional embryos in a process that successively propagates inward and outward to generate a resonant configuration of embryos. In this manner, the early presence of Jupiter imposed a harmonic structure on the accumulating planetesimal swarm. For an accretion disk with surface density obeying a power law of index ?1.2 the positions of the planetary embryos can be moved into a reasonably good agreement with most of the present planetary positions that is as good as that given by the Titius-Bode law.     

Planetary growth with collisional fragmentation and gas drag

As planetary embryos grow, gravitational stirring of planetesimals by embryos strongly enhances random velocities of planetesimals and makes collisions between planetesimals destructive. The resulting fragments are ground down by successive collisions. Eventually the smallest fragments are removed by the inward drift due to gas drag. Therefore, the collisional disruption depletes the planetesimal disk and inhibits embryo growth. We provide analytical formulae for the final masses of planetary embryos, taking into account planetesimal depletion due to collisional disruption. Furthermore, we perform the statistical simulations for embryo growth (which excellently reproduce results of direct N-body simulations if disruption is neglected). These analytical formulae are consistent with the outcome of our statistical simulations. Our results indicate that the final embryo mass at several AU in the minimum-mass solar nebula can reach about ∼0.1 Earth mass within 10⁷ years. This brings another difficulty in formation of gas giant planets, which requires cores with ∼10 Earth masses for gas accretion. However, if the nebular disk is 10 times more massive than the minimum-mass solar nebula and the initial planetesimal size is larger than 100 km, as suggested by some models of planetesimal formation, the final embryo mass reaches about 10 Earth masses at 3-4 AU. The enhancement of embryos’ collisional cross sections by their atmosphere could further increase their final mass to form gas giant planets at 5-10 AU in the Solar System.     

Are Phobos and Deimos the result of a giant impact?

Despite many efforts an adequate theory describing the origin of Phobos and Deimos has not been realized. In recent years a number of separate observations suggest the possibility that the martian satellites may have been the result of giant impact. Similar to the Earth–Moon system, Mars has too much angular momentum. A planetesimal with 0.02 Mars masses must have collided with that planet early in its history in order for Mars to spin at its current rate (Dones, L., Tremaine, S. [1993]. Science 259, 350–354). Although subject to considerable error, current crater-scaling laws and an analysis of the largest known impact basins on the martian surface suggest that this planetesimal could have formed either the proposed 10,600 by 8500-km-diameter Borealis basin, the 4970-km-diameter Elysium basin, the 4500-km-diameter Daedalia basin or, alternatively, some other basin that is no longer identifiable. It is also probable that this object impacted Mars at a velocity great enough to vaporize rock (>7 km/s), which is necessary to place large amounts of material into orbit. If material vaporized from the collision with the Mars-spinning planetesimal were placed into orbit, an accretion disk would have resulted. It is possible that as material condensed and dissipated beyond the Roche limit forming small, low-mass satellites due to gravity instabilities within the disk. Once the accretion disk dissipated, tidal forces and libration would have pulled these satellites back down toward the martian surface. In this scenario, Phobos and Deimos would have been among the first two satellites to form, and Deimos the only satellite formed—and preserved—beyond synchronous rotation. The low mass of Phobos and Deimos is explained by the possibility that they are composed of loosely aggregated material from the accretion disk, which also implies that they do not contain any volatile elements. Their orbital eccentricity and inclination, which are the most difficult parameters to explain easily with the various capture scenarios, are the natural result of accretion from a circum-planetary disk.     

Nebular shock waves generated by planetesimals passing through Jovian resonances: Possible sites for chondrule formation

Abstract— The primordial asteroid belt contained at least several hundred and possibly as many as 10,000 bodies with diameters of 1000 km or larger. Following the formation of Jupiter, nebular gas drag combined with passage of such bodies through Jovian resonances produced high eccentricities (e = 0.3‐0.5), low inclinations (i < 0.5°), and, therefore, high velocities (3–10 km/s) for “resonant” bodies relative to both nebular gas and non‐resonant planetesimals. These high velocities would have produced shock waves in the nebular gas through two mechanisms. First, bow shocks would be produced by supersonic motion of resonant bodies relative to the nebula. Second, high‐velocity collisions of resonant bodies with non‐resonant bodies would have generated impact vapor plume shocks near the collision sites. Both types of shocks would be sufficient to melt chondrule precursors in the nebula, and both are consistent with isotopic evidence for a time delay of ?1‐1.5 Myr between the formation of CAIs and most chondrules. Here, initial simulations are first reported of impact shock wave generation in the nebula and of the local nebular volumes that would be processed by these shocks as a function of impactor size and relative velocity. Second, the approximate maximum chondrule mass production is estimated for both bow shocks and impact‐generated shocks assuming a simplified planetesimal population and a rate of inward migration into resonances consistent with previous simulations. Based on these initial first‐order calculations, impact‐generated shocks can explain only a small fraction of the minimum likely mass of chondrules in the primordial asteroid belt (?10²⁴‐10²⁵g). However, bow shocks are potentially a more efficient source of chondrule production and can explain up to 10–100 times the estimated minimum chondrule mass.     

N-Body simulations of growth from 1 km planetesimals at 0.4 AU

We present N-body simulations of planetary accretion beginning with 1 km radius planetesimals in orbit about a 1 M_⊙ star at 0.4 AU. The initial disk of planetesimals contains too many bodies for any current N-body code to integrate; therefore, we model a sample patch of the disk. Although this greatly reduces the number of bodies, we still track in excess of 10⁵ particles. We consider three initial velocity distributions and monitor the growth of the planetesimals. The masses of some particles increase by more than a factor of 100. Additionally, the escape speed of the largest particle grows considerably faster than the velocity dispersion of the particles, suggesting impending runaway growth, although no particle grows large enough to detach itself from the power law size-frequency distribution. These results are in general agreement with previous statistical and analytical results. We compute rotation rates by assuming conservation of angular momentum around the center of mass at impact and that merged planetesimals relax to spherical shapes. At the end of our simulations, the majority of bodies that have undergone at least one merger are rotating faster than the breakup frequency. This implies that the assumption of completely inelastic collisions (perfect accretion), which is made in most simulations of planetary growth at sizes 1 km and above, is inappropriate. Our simulations reveal that, subsequent to the number of particles in the patch having been decreased by mergers to half its initial value, the presence of larger bodies in neighboring regions of the disk may limit the validity of simulations employing the patch approximation.     

Planet formation: Mechanism of early growth

Experiments in vacuum (approx. 0.5 to 1 mbar) and in air quantify mechanics of collisions, rebound, and fragmentation at low velocities (1–50 m/sec), under the conditions usually postulated for the preplanetary environment in the primitive solar nebula. Such collisions have been little studied experimentally. Contrary to widespread assumptions, accretionary growth of the largest meteoroid- and asteroid-sized bodies in a given swarm results spontaneously from the simple mechanics of these collisions, without other ad hoc sticking mechanisms. The smaller bodies in the swarm are less likely to grow. Granular surfaces form, either by gravitational collapse of dust swarms or by rapid formation of regolith surfaces on solid planetesimals; these surfaces strongly promote further growth by retarding rebound. Growth of large bodies increases modal collision velocities, causing fragmentation of smaller bodies and eventual production of interstellar dust as a by-product planetesimal interactions.     

Evolution of Planetesimal Velocities Based on Three-Body Orbital Integrations and Growth of Protoplanets

We obtain the viscous stirring and dynamical friction rates of planetesimals with a Rayleigh distribution of eccentricities and inclinations, using three-body orbital integration and the procedure described by Ohtsuki (1999, Icarus137, 152), who evaluated these rates for ring particles. We find that these rates based on orbital integrations agree quite well with the analytic results of Stewart and Ida (2000, Icarus 143, 28) in high-velocity cases. In low-velocity cases where Kepler shear dominates the relative velocity, however, the three-body calculations show significant deviation from the formulas of Stewart and Ida, who did not investigate the rates for low velocities in detail but just presented a simple interpolation formula between their high-velocity formula and the numerical results for circular orbits. We calculate evolution of root mean square eccentricities and inclinations using the above stirring rates based on orbital integrations, and find excellent agreement with N-body simulations for both one- and two-component systems, even in the low-velocity cases. We derive semi-analytic formulas for the stirring and dynamical friction rates based on our numerical results, and confirm that they reproduce the results of N-body simulations with sufficient accuracy. Using these formulas, we calculate equilibrium velocities of planetesimals with given size distributions. At a stage before the onset of runaway growth of large bodies, the velocity distribution calculated by our new formulas are found to agree quite well with those obtained by using the formulas of Stewart and Ida or Wetherill and Stewart (1993, Icarus106, 190). However, at later stages, we find that the inclinations of small collisional fragments calculated by our new formulas can be much smaller than those calculated by the previously obtained formulas, so that they are more easily accreted by larger bodies in our case. The results essentially support the previous results such as runaway growth of protoplanets, but they could enhance their growth rate by 10-30% after early runaway growth, where those fragments with low random velocities can significantly contribute to rapid growth of runaway bodies.     

Planetesimals to protoplanets – II. Effect of debris on terrestrial planet formation

In this paper, we extend our numerical method for simulating terrestrial planet formation to include dynamical friction from the unresolved debris component. In the previous work, we implemented a rubble pile planetesimal collision model into direct N -body simulations of terrestrial planet formation. The new collision model treated both accretion and erosion of planetesimals but did not include dynamical friction from debris particles smaller than the resolution limit for the simulation. By extending our numerical model to include dynamical friction from the unresolved debris, we can simulate the dynamical effect of debris produced during collisions and can also investigate the effect of initial debris mass on terrestrial planet formation. We find that significant initial debris mass, 10 per cent or more of the total disc mass, changes the mode of planetesimal growth. Specifically, planetesimals in this situation do not go through a runaway growth phase. Instead, they grow concurrently, similar to oligarchic growth. The dynamical friction from the unresolved debris damps the eccentricities of the planetesimals, reducing the mean impact speeds and causing all collisions to result in merging with no mass loss. As a result, there is no debris production. The mass in debris slowly decreases with time. In addition to including the dynamical friction from the unresolved debris, we have implemented particle tracking as a proxy for monitoring compositional mixing. Although there is much less mixing due to collisions and gravitational scattering when dynamical friction of the background debris is included, there is significant inward migration of the largest protoplanets in the most extreme initial conditions (for which the initial mass in unresolved debris is at least equal to the mass in resolved planetesimals).     

Growth of large,late-stage planetesimals

The late stage of terrestrial planets' growth determined many of their fundamental properties, including their thermal properties and petrology, their impact records, and possibly the existence of the Moon. A critical result of late-stage models, which bears on observable properties, is the size of the largest planetesimals that grew near, and later impacted,those that became full-size planets. There has been considerable misinterpretation of previous models regarding the relation between the size of planetesimals and their relative velocities. Furthermore, some models neglect the possible decrease in relative velocity as control is transferred from the largest to the second-largest body in an accreation zone. Evidence that Venus helped stir Earth-zone planetismals is not copelling. When models are evaluated, the results are found to depend strongly on uncertain initial conditions. The size of the second-largest planetesimal in the Earth's zone might range from ～300 to ～2500 km, with corresponding accretion times of ～7 × 10⁶ and ～10⁸ years, respectively. Both extremes are generated from plausible initial conditions and both seem consistent with observed planetary properties.     

The rotation rates of accreting planetesimals

There are obtained upper limits for the relative velocity at infinity of accreting planetesimals for a nearly constant mass of the largest accreting planetesimal and also in the case of variable mass. We conclude, that while the larger planets cannot be brought to the stage of rotational instability by stochastic collisions, the asteroids could be brought. provided that the relative velocities in the asteroid belt were larger than about 2 km s^–1.     

Formation of gas giant planets: core accretion models with fragmentation and planetary envelope

We have calculated formation of gas giant planets based on the standard core accretion model including effects of fragmentation and planetary envelope. The accretion process is found to proceed as follows. As a result of runaway growth of planetesimals with initial radii of ∼10 km, planetary embryos with a mass of ∼10²⁷ g (∼ Mars mass) are found to form in ∼10⁵ years at Jupiter's position (5.2 AU), assuming a large enough value of the surface density of solid material (25 g/cm²) in the accretion disk at that distance. Strong gravitational perturbations between the runaway planetary embryos and the remaining planetesimals cause the random velocities of the planetesimals to become large enough for collisions between small planetesimals to lead to their catastrophic disruption. This produces a large number of fragments. At the same time, the planetary embryos have envelopes, that reduce energies of fragments by gas drag and capture them. The large radius of the envelope increases the collision rate between them, resulting in rapid growth of the planetary embryos. By the combined effects of fragmentation and planetary envelope, the largest planetary embryo with 21M_⊕ forms at 5.2 AU in 3.8×10⁶ years. The planetary embryo is massive enough to start a rapid gas accretion and forms a gas giant planet.