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IOP PUBLISHING PHYSICA SCRIPTA Phys. Scr. T130 (2008) 014021 (10pp) doi:10.1088/0031-8949/2008/T130/014021 Accretion of planetary embryos in the inner and outer solar system S J Weidenschilling Planetary Science Institute, 1700 East Fort Lowell Road, Suite 106, Tucson, AZ 85719, USA E-mail: [email protected] Received 11 March 2008 Accepted for publication 14 March 2008 Published 16 July 2008 Online at stacks.iop.org/PhysScr/T130/014021 Abstract. Numerical simulations of accretion of planetary embryos from small planetesimals are described. In the terrestrial region, runaway growth proceeds as a wave propagating outward, producing an ‘oligarchy’ of embryos. The efficiency of accretion, i.e. the mass loss due to fragmentation, depends on the initial size of the planetesimals. In the outer region of the disk, the growth of embryos is not a localized process. At larger heliocentric distances, gravitational scattering and long-range perturbations become more significant, and tend to inhibit runaway growth. PACS number: 96.10.+i 1. Introduction 2. Runaway growth + isolation = oligarchy The formation of planetary embryos involves the accumulation of kilometer-scale planetesimals into bodies thousands of times larger in diameter. This process involves much more than simple coagulation. Collisions may produce fragmentation, with net loss of mass or complete destruction. Bodies of various sizes are subject to radial migration due to interactions with the solar nebula and with each other. Gravitational perturbations by planetary embryos may also have effects that extend well beyond the immediate vicinity of their orbits. Thus, accretion is not determined only by local conditions, but must be placed in the context of an extended swarm of bodies. Analytic modeling of accretion has yielded valuable insights into relevant processes of coagulation and gravitational interactions (Safronov 1969), but cannot account for the interplay of phenomena too complex to be reduced to tractable equations. Numerical simulations are less limited, but greater complexity does not necessarily mean more realism. Including more phenomena in a model does not ensure that they are more accurate, unless observational or experimental constraints are available. For example, accretion may be affected by the density and impact strength of planetesimals, and the structure of the solar nebula. These parameters can be guessed within plausible ranges, but not known with precision. This paper presents results of numerical simulations of formation of planetary embryos (not the final stages of planetary growth) in the inner and outer solar system. A generally accepted paradigm for the formation of the terrestrial planets involves rapid growth of a modest number (∼102 ) of embryos of sub-planetary mass, followed by a longer period of collisions between these bodies, until a few planets are left in stable orbits (Chambers and Wetherill 1998, Chambers 2001, Lissauer and Stewart 1993). The initial growth is determined by the rate at which an embryo gains mass by accreting smaller planetesimals: 0031-8949/08/014021+10$30.00 2 dM/dt = (σ/H )Vrel πr 2 (1 + Ve2 /Vrel ), (1) where σ is the surface density of accretable planetesimals, H the thickness of the swarm of bodies (σ/H = space density), √ Ve = (2G M/r ) the escape velocity of the embryo and Vrel the relative velocity with which planetesimals approach the embryo. The last term is the collisional cross-section, the geometrical area augmented by focusing due to the embryo’s gravity. If two embryos of mass M1 > M2 compete for the same population of planetesimals with the same Vrel , each grows at a rate given by (1). The larger body, with greater r and Ve , always has the larger value of dM/dt; the smaller one never overtakes it. However, the ratio of their masses, M1 /M2 , can evolve in two ways. If the gravitational term is 2 small (Ve2 /Vrel ) 1, then M1 /M2 decreases with time, and 2 approaches unity (Ohtsuki and Ida 1990). If (Ve2 /Vrel ) 1, then M1 /M2 increases without limit as long as the background population is available to supply mass. 1 © 2008 The Royal Swedish Academy of Sciences Printed in the UK Phys. Scr. T130 (2008) 014021 S J Weidenschilling These two evolutionary paths, called ‘orderly’ and ‘runaway’ growths, depend on the velocities (eccentricities and inclinations) of the planetesimals. These in turn depend on the stirring by gravitational perturbations, and damping by collisions and gas drag. Also, a collective phenomenon, dynamical friction, causes bodies with different masses and velocities to exchange kinetic energy tending towards equipartition, i.e. more massive bodies tend to have lower velocities than less massive ones (Stewart and Wetherill 1988). It has been shown (Ohtsuki and Ida 1990, Wetherill and Stewart 1989) that such size dependence of velocities is necessary for runaway growth. Under most conditions (one exception will be seen below), gravitational stirring produces such a result, and runaway growth ensues. Figure 1. Impact strength Q ∗ versus size used in the simulations in this paper. In the strength dominated regime (D < 1 km), Q ∗ has a slight dependence on the projectile/ target mass ratio, as well as the impact energy. 2.1. Outcome of runaway: isolation mass Runaway growth can be quite rapid (and generally is faster than orderly growth), but its essential property is not its rate, but the resulting size distribution. A trivial initial size difference eventually yields one largest body containing much of the available mass, with the others much smaller. Its growth cannot proceed without limit, because the embryo and smaller bodies are orbiting the Sun. At some point, the remaining planetesimals are in orbits too far from the embryo to be swept up by it. In the restricted three-body problem, an embryo of mass M and semimajor axis a and a particle of negligible mass can √ never collide if their semimajor axes differ by more than 2 3× the Hill radius, RH = a(M/3M )1/3 . If the embryo is in a swarm of particles, RH increases as it accretes them, but the process halts at a limiting ‘isolation mass’ √ Miso = (8π 3σ )3/2 a 3 /(3M )1/2 , (2) into the Sun by gas drag. Thus, the fate of a significant fraction of the swarm depends on the impact strength of the planetesimals. Impact strength, Q ∗ , is defined as the energy per unit mass (erg g−1 ) that shatters a target body and removes half of its mass (Davis et al 1994). For small bodies, Q ∗ is simply the material strength, which is controlled by flaws in the material. As a larger body contains more flaws, it is easier to initiate fracture at its weakest point, and Q ∗ decreases with size. For larger bodies, additional energy is needed to propel the fragments to escape velocity, and the remaining mass may be a gravitationally bound ‘rubble pile’. When gravitational binding energy is significant, Q ∗ increases with size. Experimental laboratory-scale impacts and models of the collisional evolution of the asteroid belt (Bottke et al 2005, Davis et al 1994, Durda et al 1998), suggest that the minimum Q ∗ occurs at about kilometer size (figure 1). (Lissauer and Stewart 1993). Miso varies as a 3 and σ 3/2 . Most solar nebula models assume σ ∝ a −3/2 to a −1 , so that Miso increases with heliocentric distance. Accretion does not strictly follow the conditions of the restricted three-body problem; other processes, e.g. gas drag, collisions between planetesimals and perturbations by other embryos, can have some effect. However, Miso gives a fairly good estimate of the size attained by an embryo during runaway growth. Its collisional influence extends to a few RH , and its strongest gravitational perturbations affect planetesimals within a few times that distance. In a solar nebula of moderate mass, σ does not vary significantly over that range. In an extended swarm of planetesimals, the same process occurs at intervals of ∼5–10 RH , yielding embryos of comparable size and fairly regular orbital spacing; this outcome is known as ‘oligarchy’. This results in a bimodal size distribution of embryos and small bodies. Runaway stops before the small planetesimals are exhausted. Perturbations by the embryo and its neighbors stir the swarm until Vrel is comparable with the embryo’s Ve , and the decrease in gravitational focusing slows accretion (Ida and Makino 1993). In typical simulations (Kokubo and Ida 1996, 1998), comparable fractions of the mass of the swarm are in small bodies and embryos at the end of runaway growth. If only coagulation occurred, the remaining planetesimals would eventually be swept up by the embryos. However, if they shatter each other in collisions, small fragments are lost 3. Methods of numerical simulation 3.1. N-body simulations The most straightforward approach to planetary accretion is simply to integrate the orbits of many planetesimals simultaneously, allowing them to merge when they collide. This allows mutual gravitational perturbations, orbital evolution and collision rates to be computed precisely. Progress in computer technology and algorithms for computing orbits (Durda et al 1998) render this approach ever more useful and applicable to a wider variety of problems. However, it is useful for simulating the late-stage of evolution of planetary embryos after completion of runaway growth, when their number may be in the hundreds (Chambers 2001). A full-scale simulation of accretion of an Earth-sized planet from kilometer-scale planetesimals would require simultaneous integration of ∼1012 bodies for millions of orbits, a task beyond the present- or next-generation of computers. N-body simulations of early stages of runaway growth (Kokubo and Ida 1996, 1998, 2000) cover only a limited range of heliocentric distance. To make the problem tractable, they used relatively few bodies of larger mass; 2 Phys. Scr. T130 (2008) 014021 S J Weidenschilling collision cross-sections and gas drag effects were increased artificially to simulate the dynamical behavior of smaller planetesimals. Kominami et al (2005) and Daisaka et al (2006) modeled later stages of accretion, but used enhanced cross-sections with scaling relations to extrapolate their results to smaller bodies. N-body methods are unable to treat fragmentation, as the number of bodies would quickly become intractable after a few collisions. Therefore, such simulations typically assume that all collisions result in coagulation. alter eccentricities and inclinations at rates proportional to the degree of overlap. Gas drag causes orbital decay and transfer of mass between zones. A similar multi-annulus code has been developed independently by Kenyon and Bromley (2001, 2002). Bodies of size above some threshold (typically ∼1024 g, or about the size of the largest asteroids, but Miso ), are treated as individual entities. When growth of the continuum in a zone produces a statistical entity above the threshold mass, it is assigned discrete values of mass, semimajor axis, eccentricity and inclination, which evolve during the subsequent model time. Its interactions with the continuum are computed for the statistical rates of collisions and stirring. In the cases shown here, discrete orbits are not integrated directly. Angular elements are assumed to be uniformly distributed, and encounters within RH modeled as stochastic scattering events that produce impulsive changes in their orbital elements. Similar codes that integrate orbits of discrete bodies have been developed by F Marzari (unpublished) and Bromley and Kenyon (2006). Discrete bodies also affect bodies (both continuum and discrete) in distant orbits by long-range gravitational perturbations. These are modeled by stochastic impulses during synodic encounters (Weidenschilling 1989). This stirring is distinct from the viscous stirring term that accounts for non-crossing orbits of small bodies in the continuum distribution. These features of fragmentation, stochastic scattering, and distant perturbations produce some interesting effects on the evolution of the planetesimal swarm, as will be seen below. 3.2. Statistical methods Rather than tracking individual bodies, a population can be represented by the numbers of bodies in specified ranges of size, usually logarithmic intervals of mass. Interactions are modeled by analytic expressions, e.g. for collision rates as a function of the space density of bodies and relative velocity (cf equation 1). Collisions during a timestep move a certain amount of mass between size bins. Fragmentation can be treated, as that outcome distributes mass into smaller bins with an appropriate size distribution. Individual gravitational interactions are not treated, but modeled by continuum equations for the rates of change of eccentricity and inclination (Stewart and Ida 2000). Such simulations have been performed extensively by Wetherill and co-workers (Chambers and Wetherill 1998, Inaba et al 2003, Wetherill and Stewart 1989, 1993), and by Kenyon and Luu (1998, 1999). The statistical method is useful for modeling the early evolution of a swarm of small planetesimals, but becomes less so for later stages. The expressions for rates of collisions and stirring depend on quantities that are implicitly assumed to be uniform through some volume being modeled, including the space density of bodies. This assumption is valid in the earliest stage of evolution, but is soon violated during runaway growth. The presence of a dominant embryo means that the space density and velocity distribution of small planetesimals vary on a spatial scale comparable to RH . It becomes increasingly difficult to represent collision rates and gravitational interactions as the swarm evolves. Continuum stirring equations do not account for sudden large changes of orbital elements that may result from close encounters with embryos. 4. Accretion simulations: the inner solar system This section presents results for two simulations of accretion in the region of terrestrial planets and the asteroid belt, between 0.5 and 4 AU. The nebular surface density varies as 1/a, with values of 2500 g cm−2 for gas and 8.4 g cm−2 for solids at 1 AU. The initial swarm mass is ∼7 M⊕ . Much of the mass beyond 1.5 AU would have been removed at a later time by dynamical effects of Jupiter (Chambers and Wetherill 2001). In these simulations, Jupiter is assumed to have formed later, and does not affect the accretion of embryos. The gas deviates from Keplerian rotation by 1.8 × 10−3 due to a radial pressure gradient (Weidenschilling 1977). Drag causes orbits of small bodies to decay, and damps eccentricities and inclinations (Adachi et al 1976). Large bodies interact with the nebula via tidal forces, which damp eccentricities and inclinations, and cause inward ‘Type 1’ migration (Tanaka et al 2002, Tanaka and Ward 2004). Small bodies are most affected by aerodynamic drag, and large embryos by tidal forces; intermediate sizes, tens to a few hundred kilometer in diameter, are almost unaffected by the gas. Tidal decay of embryos would cause them to be lost into the Sun in a few million years, unless the gas is removed on a comparable timescale; the gas density is assumed to decay exponentially on a timescale of 2 My. The initial population of planetesimals has a single size, either 10 or 1 km. Fragmentation is allowed, with fragments smaller than 0.125 km lost from the swarm. In principle, some of this mass could be reaccreted by larger bodies, but the lifetime of a 100 m fragment against loss by gas drag is only ∼104 years, which is short compared with the simulation time. 3.3. Hybrid methods Some of the limitations of N-body and statistical methods may be avoided by a hybrid approach that treats small bodies as a statistical continuum, and large ones as individuals. Such an approach was developed by Spaute et al (1991), and expanded by collaborators. A brief listing of its features is given here; for a more detailed description see Weidenschilling et al (1997). In this code, the swarm is divided into sub-populations in intervals of orbital semimajor axis. Within each radial zone, small bodies are represented by logarithmic mass bins, with collisional and gravitational interactions computed from continuum equations. One important difference is that the zones are not isolated; where orbits of bodies in different zones overlap due to their eccentricities, they interact both collisionally and gravitationally. Collisions may transfer mass between zones, and viscous stirring and dynamical friction 3 Phys. Scr. T130 (2008) 014021 S J Weidenschilling Figure 2. Distributions of sizes and eccentricities at 0.1, 1 and 5 My, for initial planetesimal size 10 km (mass ∼1018 g). The histograms show the number of bodies in each mass bin for each zone of semimajor axis, and the mean eccentricity of the bodies in that bin. Scales are logarithmic; semimajor axes span the range 0.5–4 AU. Bins smaller than 1018 g are populated by fragments from collisional disruption of larger bodies. remains beyond ∼2 AU even after 5 My. A few embryos grow significantly larger than Miso , aided by radial migration which allows them to encounter planetesimals from a wider range of a than the constraints of the 3-body problem, and to collide with other embryos. The largest embryos reach a substantial fraction of M⊕ , and migrate inward ∼1 AU from their starting distances by 5 My (figure 3). 4.1. Initial size 10 km Growth is fastest near the inner edge of the swarm, where σ is highest and the collision timescale shortest. Embryos accrete by runaway growth in a ‘wave’ that propagates outward (figure 2). Eccentricities initially decrease with increasing size due to dynamical friction, allowing runaway growth. At each distance, as embryos approach Miso their perturbations dominate the stirring of the smaller continuum bodies, whose eccentricities become almost independent of mass. Size bins smaller than 10 km are populated by fragments as impact velocities increase, but some signature of the original size 4.2. Initial size 1 km It is often assumed that the outcome of accretion is not sensitive to the initial planetesimal size. Miso depends only 4 Phys. Scr. T130 (2008) 014021 S J Weidenschilling of growth sweeps outward at about ten times the rate for the 10 km swarm. The behavior of eccentricities is similar, on the shorter timescale. However, the largest embryos are less massive than for the first case. This is because the final mass of the swarm is only about half its starting value. Most of this mass loss occurs within the first million year. Recall that runaway growth tends to produce a bimodal size distribution, comprising embryos and bodies near the starting size. The starting condition with 1 km bodies corresponds to the minimum value of Q ∗ , the weakest objects. These are so easily destroyed by collisions that about half of the total mass of the swarm is ground down to small fragments and lost from the system. This mass is not available to be swept up later, and the growth of the embryos effectively stops at ∼Miso ; their smaller masses then result in less migration due to nebular tidal torques. Figure 5 shows the evolution of the discrete bodies for this case. The outcome of accretion differs significantly between these cases, due to the choice of initial planetesimal size and the size dependence of impact strength. One would expect that any initial size &10 km would result in very little mass loss, while all values <1 km would lose about half the starting mass. More realistically, it is unlikely that planetesimals formed simultaneously with a single size, and the median size may have been a function of heliocentric distance. Thus, the efficiency of accretion may have varied with location, depending on the formation mechanism for planetesimals. 5. The outer solar system The accretionary behavior of small planetesimals in the outer part of the solar nebula has not been studied in as much detail as in the terrestrial region. Ida and Lin (2004) used scaling laws obtained from N-body simulations of accretion in the terrestrial zone, and applied them to model formation of cores of the outer planets. However, Levison and Stewart (2001) and Thommes et al (2003) found that an oligarchy of embryos would not accrete to form the cores of the four giant planets on plausible timescales, unless the mass of the swarm was implausibly large. Few direct simulations of accretion have been performed starting with small planetesimals beyond the ‘snow line’ for H2 O condensation (generally assumed to be ∼4–5 AU). Inaba et al (2003) used a statistical code to model growth of planetesimals of initial size ∼10 km, in six zones between 5.2 and 29 AU. Each zone evolved independently, except that small fragments, down to ∼meter size, produced in the outer zones were transported inward by gas drag and added to the inner zones. However, the timescales for growth increase with distance, so the slowly evolving outer zones had relatively little effect on the inner ones. Inaba et al found runaway growth, but it did not produce embryos comparable in mass to the cores of the outer planets, unless the nebular surface density was large. In this section, results are presented for a simulation spanning a range of heliocentric distance comparable to the full size of the solar nebula, from 0.5 to 30 AU. It will be shown that there are significant differences in the mode of accretion between the inner and outer regions. The nebular model is similar to that used in the previous simulations of the inner region, except that σ has a ‘step’ at 4.5 AU, where H2 O Figure 3. Mass versus semi-major axis for the large (>2 × 1024 g) discrete bodies, for initial planetesimal size of 10 km. Horizontal bars show perihelion and aphelion distances. The curved dotted line shows Miso as defined in equation (2). on surface density, but planetesimal size has a strong effect on the timescale of runaway growth. Figure 4 shows results obtained with identical parameters to the one discussed above, except with an initial diameter of 1 km. Runaway growth proceeds much faster in this case. The collision timescale is inversely proportional to planetesimal size. A certain number of collisions must occur before a dominant body emerges due to a series of stochastic impacts that allow it to become larger than its neighbors, initiating runaway growth. The wave 5 Phys. Scr. T130 (2008) 014021 S J Weidenschilling Figure 4. Distributions of size and eccentricity for the second case, with the same initial conditions except the planetesimal size are 1 km (M ∼ 1015 g). Evolution is similar to the case for 10 km bodies, but runaway growth of embryos is much faster. The final embryos are smaller due to loss of mass from the swarm by collisional disruption of the smaller planetesimals. is assumed to be condensed and the solids/gas ratio increases by a factor of four. Icy planetesimals might have lower density and/or impact strength, but this was not included in the model. The initial planetesimal size was taken to be 1 km, and fragmentation was allowed, with minimum size 0.125 km. Tidal damping and migration of the large bodies were not included in this simulation. The evolution of the size distribution is plotted in figure 6. As before, a wave of runaway growth propagates outward from the inner edge of the swarm. Another wave begins at the snow line due to the higher surface density. The initial evolution in the outer disk is qualitatively similar to that in the inner; however, some differences develop in time. At 0.5 My, a large embryo of about 5 M⊕ appears near 10 AU, while the wave of embryos has only reached ∼6 AU. A small embryo from that region, of mass ∼0.01 M⊕ , has close encounters with one or more of the larger embryos, and is scattered outward into an orbit with large eccentricity and a∼10 AU (figure 7). At that distance, the swarm still consists of small bodies a few kilometer in size, and dynamically cold (with low eccentricities). The scattered body’s orbit is circularized by dynamical friction and collisions with the local background. Its mass is negligible compared with the total population of planetesimals in that region, but is still >104 times the mass of the next largest body. This size advantage approximates the conditions of the restricted three-body problem; although the background population is stirred, approach velocities to the embryo remain low due to conservation of the Jacobi 6 Phys. Scr. T130 (2008) 014021 S J Weidenschilling a planetary core forms, it excites eccentricities in a large region of the swarm about its orbit. Any other scattered seed body arriving after this excitation will not experience runaway growth; thus, only a few cores (at the most) can form in this manner. Monarchical growth occurs only in the outer disk for two reasons. The higher surface density beyond the snow line means that to accrete the first embryos there are large (Miso > M⊕ ). Also, gravitational scattering produces larger changes of orbital elements, due to the lower orbital velocities; planetary embryos are more mobile at larger heliocentric distances. The scattering of smaller embryos by encounters with larger ones is robust (Levison and Stewart 2001), but its consequences are less certain. The multi-zone code yields rapid damping of eccentricities of scattered bodies, decoupling them from the scattering embryos and allowing them to begin runaway growth. However, H Levison (personal communication), using another orbital integrator with a different treatment of dynamical friction, does not find such rapid circularization. Among the issues that need to be addressed are the role of inclinations (velocities of scattered bodies with inclinations greater than the swarm thickness are damped less efficiently), and the validity of expressions for dynamical friction at large eccentricities. 6. Can runaway growth occur in the outer disk? In the terrestrial region, runaway growth propagates outward in a rather uniform wave, producing a regularly spaced series of embryos. This trend does not continue indefinitely in the outer disk. Recall that runaway growth requires that larger bodies have an advantage in gravitational cross-section; this depends on dynamical friction to give them low velocities relative to the background population (Kokubo and Ida 1996, 1998). A ‘monarch’ can stifle competition near its own orbit by stirring the swarm, but even in regions far from such bodies, where no seed bodies are implanted in the outer disk, the wave of growth eventually ceases. In this simulation, no embryos form beyond ∼13 AU after 1 My. At the start, eccentricities decrease with increasing mass in the outer region, but by 1 My, their distribution becomes flat, with values independent of mass. This behavior is due to long-range perturbations of the embryos inside 10 AU (the radial coordinate is logarithmic; the outer half of the disk is compressed in this view). Eccentricities excited in this manner are independent of mass. This velocity distribution effectively prevents runaway growth. Tests with simpler disk models (no fragmentation and no step in surface density due to H2 O condensation) show similar behavior. Long-range perturbations become dominant in the outer disk due to geometry. Rates of viscous stirring and dynamical friction are proportional to the volume density of planetesimals at a given location. However, the rate of stirring by synodic encounters of bodies in non-crossing orbits is proportional to the surface density of material in those bodies. As surface density falls off more slowly than volume density, distant perturbations eventually become dominant at sufficiently large heliocentric distances. If the code is modified to suppress long-range perturbations, dynamical friction remains effective for damping eccentricities of the larger continuum bodies in the outer zones. Runaway Figure 5. Mass versus semimajor axis for discrete bodies in the case with initial size 1 km. Runaway growth is faster, but final embryos are smaller due to mass loss by collisional grinding of small bodies. parameter. This condition allows runaway accretion with the maximum gravitational enhancement of its collisional cross-section (Greenzweig and Lissauer 1992). A single body, with no competition from neighboring embryos, approaches the isolation mass within a few thousand orbital periods: not oligarchy, but ‘monarchy’ (Weidenschilling 2005). Scattered ‘seed bodies’ can produce objects comparable in mass to the cores of the giant planets on short timescales, if the background population is dynamically cold. Once 7 Phys. Scr. T130 (2008) 014021 S J Weidenschilling Figure 6. Evolution of sizes and eccentricities for a simulation spanning the full disk from 0.5 to 30 AU. The initial planetesimal size is 1 km (1015 g). The ‘snow line’ for H2 O condensation is at 4.5 AU (log a = 0.65), where the surface density of solids jumps by a factor of 4. By 105 years, Earth-sized embryos form in this region. At 106 years, three large embryos have grown near 10 AU from ‘seed’ bodies scattered outward. Long-range perturbations by these bodies raise eccentricities in the outer disk and shut off runaway growth. growth then proceeds in an outward wave without interruption, but at an ever-slower pace at greater heliocentric distances. This suppression of runaway growth in the outer disk was not seen in earlier simulations of accretion in the Kuiper Belt, for a variety of reasons. Kenyon and Luu (1998, 1999) used a single-zone statistical simulation, while Kenyon (2002) applied a multi-annulus model only for the region beyond 40 AU. Kenyon and Bromley (2004) and Kenyon et al (2008) included perturbations by Neptune acting on Kuiper Belt objects, but assumed that the planet took 100 My to reach its present mass. Those simulations did not include influence of large embryos that may have accreted earlier at smaller heliocentric distances. Effects of long-range perturbations become apparent only in ‘holistic’ simulations that span the entire disk beyond the snow line. Embryos with masses comparable Neptune’s are sufficient to stifle runaway growth; formation of gas giants is not necessary. The distance at which distant perturbations by large embryos becomes significant depends on the mass of the disk and the gradient of its surface density, but is ∼10–15 AU for the disk model assumed here. The ‘Nice Model’ for the evolution of the outer planets (Tsiganis et al 2005) assumes an initial condition with the four giant planets in the range ∼5–15 AU, and a massive disk of small planetesimals beyond that distance. Such a configuration may not require precise timing for accretion of gas by Jupiter and Saturn, or dissipation of the solar nebula, but may be a natural consequence of self-termination of runaway growth by the cores of the outer planets. 8 Phys. Scr. T130 (2008) 014021 S J Weidenschilling initial size; these are subject to mutual collisions and loss by fragmentation after formation of embryos. The efficiency of planetary formation depends on the initial size(s) of the planetesimals. If <1 km, collisional disruption depletes the residual small bodies. Initial sizes >10 km make them much more resistant to disruption due to gravitational binding energy, and little mass is lost. In the outer solar system beyond the ice line, non-local effects influence accretion. Embryos are more mobile than in the inner solar system, due to larger masses and lower orbital velocities. Runaway growth may be triggered by seed bodies scattered outward into dynamically cold regions of the swarm. At larger heliocentric distances, long-range perturbations become more effective relative to local gravitational stirring. This effect reduces the effectiveness of dynamical friction and can inhibit runaway growth in the outer part of the disk. Acknowledgments I thank the Nobel Foundation for supporting my participation in this Symposium. 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