Download Accretion of planetary embryos in the inner and outer solar system

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;
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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
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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
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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
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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
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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
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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.
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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. This work was supported by NASA
Outer Planets Research Program, grant no. NNG05GG89G.
Computations were made possible by a grant from the NASA
Center for Computational Sciences.
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Figure 7. Mass versus semimajor axis of discrete bodies in the full
disk simulation. The dotted line shows Miso with a step at 4.5 AU
due to H2 O condensation. Runaway growth and scattering of ‘seed
bodies’ produces a few large embryos beyond the snow line, but
growth is inhibited beyond 15 AU.
7. Summary
Numerical simulations using the multi-zone code support the
conventional paradigm of runaway growth and oligarchy for
accretion in the inner solar system. Type 1 migration may
cause loss of the more massive embryos into the Sun, unless
the gas is removed from the inner region of the disk within
a few million years. Runaway growth leaves a substantial
fraction of the starting mass in bodies comparable to the
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