Download Making the Terrestrial Planets: N-Body Integrations of Planetary

ICARUS
136, 304–327 (1998)
IS986007
ARTICLE NO.
Making the Terrestrial Planets: N-Body Integrations of Planetary
Embryos in Three Dimensions
J. E. Chambers
Armagh Observatory, College Hill, Armagh BT61 9DG, United Kingdom
E-mail: [email protected]
and
G. W. Wetherill
Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad Branch Road NW, Washington DC 20015
Received January 2, 1998; revised May 26, 1998
We simulate the late stages of terrestrial-planet formation
using N-body integrations, in three dimensions, of disks of up
to 56 initially isolated, nearly coplanar planetary embryos, plus
Jupiter and Saturn. Gravitational perturbations between embryos increase their eccentricities, e, until their orbits become
crossing, allowing collisions to occur. Further interactions produce large-amplitude oscillations in e and the inclination, i,
with periods of p105 years. These oscillations are caused by
secular resonances between embryos and prevent objects from
becoming re-isolated during the simulations. The largest objects
tend to maintain smaller e and i than low-mass bodies, suggesting some equipartition of random orbital energy, but accretion proceeds by orderly growth. The simulations typically produce two large planets interior to 2 AU, whose time-averaged
e and i are significantly larger than Earth and Venus. The
accretion rate falls off rapidly with heliocentric distance, and
embryos in the ‘‘Mars zone’’ (1.2 , a , 2 AU) are usually
scattered inward and accreted by ‘‘Earth’’ or ‘‘Venus,’’ or scattered outward and removed by resonances, before they can
accrete one another. The asteroid belt (a . 2 AU) is efficiently
cleared as objects scatter one another into resonances, where
they are lost via encounters with Jupiter or collisions with
the Sun, leaving, at most, one surviving object. Accretional
evolution is complete after 3 3 108 years in all simulations that
include Jupiter and Saturn. The number and spacing of the
final planets, in our simulations, is determined by the embryos’
eccentricities, and the amplitude of secular oscillations in e,
prior to the last few collision events.  1998 Academic Press
Key Words: planetary formation; terrestrial planets; planetary dynamics; extra-solar planetary systems.
1. INTRODUCTION
The planetesimal theory of terrestrial-planet formation
is commonly viewed as a play in three acts. In Act One,
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Copyright  1998 by Academic Press
All rights of reproduction in any form reserved.
grains of dust near the midplane of the protoplanetary
nebula accrete one another via low-velocity collisions,
eventually forming 1- to 10-km sized ‘‘planetesimals’’
(Weidenschilling 1997). These objects are large enough to
possess nonnegligible gravitational fields that increase their
collision cross sections, aiding further growth to form
p3000-km diameter ‘‘planetary embryos.’’ The second act
is characterized by ‘‘runaway growth,’’ in which equipartition of random orbital energy between planetesimals ensures that the largest objects have orbits with low eccentricities and inclinations—orbits that are most efficient at
scooping up more material (e.g., Wetherill and Stewart
1989, Kokubo and Ida 1996). Runaway growth of the biggest objects is enhanced by gas drag acting on small collision fragments, giving them circular, co-planar orbits too
(Wetherill and Stewart 1993). In Act Three, planetary embryos strongly perturb the orbits of their neighbors until
they become crossing. Runaway growth now slows or shuts
down completely, and the embryos accrete each other in
giant impacts, leading to a handful of terrestrial planets
on widely separated orbits.
Act Three has been modeled extensively using the
Öpik–Arnold scheme to follow the dynamical and collisional evolution of disks of planetary embryos in three
dimensions (e.g., Wetherill 1992, 1994, 1996). This technique treats individual close encounters and collisions effectively and uses a simple parameterization of the important effects of the major Jupiter and Saturn resonances in
the asteroid belt. However, it does not include distant
perturbations between embryos or sequences of encounters due to node-crossing events, so the effects of secular
perturbations and resonances between embryos are beyond its ability.
The final stage of planetary accretion has also been mod-
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TERRESTRIAL-PLANET FORMATION
eled using N-body integrations in two dimensions, by Lecar
and Aarseth (1986), and Beaugé and Aarseth (1990). In
addition, Cox and Lewis (1980) carried out 2D calculations
that neglected long-range perturbations between embryos.
Numerical integrations automatically include the effects
of secular and resonant interactions between embryos.
However, calculations limited to two dimensions artificially
decrease the collisional timescale with respect to the timescale for orbital evolution.
These approximations were chosen because they require
substantially less computer time than more-realistic
N-body integrations in three dimensions. Both types of
simulation yielded plausible planetary systems, although
these were not always similar to our own. They also provided insight into the chaotic nature of planet formation
that results from the central role of close encounters—a
level of understanding that goes beyond that achievable
from analytic models.
Recent improvements in the performance of computer
workstations, and the development of a new N-body algorithm, now make it possible to carry out N-body integrations, in three dimensions, of several tens of gravitationally
interacting bodies for the p108 orbits necessary to form
the cores of the inner planets. This led us to pose the
following question. Is it possible to create a recognizable
system of terrestrial planets by integrating the orbits of a
disk of planetary embryos for p100 million years, subject
only to mutual gravitational interactions, inelastic collisions, and external perturbations from the giant planets?
If it is possible, such simulations should indicate whether
terrestrial planets such as our own are inevitable, given
the size and location of the giant planets, or whether their
formation depends critically on the nature of the disk of
embryos formed by runaway growth. (Alternatively, it may
all be a matter of luck, with the final outcome depending
on a few key events that occur essentially at random.) It
should also become possible to predict the characteristics
of terrestrial planets in extra-solar systems long before we
can determine them observationally. Conversely, if N-body
simulations involving a few dozen embryos cannot produce
something akin to the terrestrial planets, they may at least
indicate what extra physics is required to do so.
With this in mind we have carried out 27 integrations
of disks of planetary embryos, starting with objects on
isolated, nearly coplanar orbits, and following their evolution for at least 108 years. In two thirds of the simulations
we have also included the effects of the giant planets Jupiter and Saturn, assuming they formed before the accretion
of the terrestrial planets was complete. All the integrations
were performed on dedicated DEC alpha workstations,
requiring p3 years of CPU time.
The next section describes the integration method and
the initial conditions used in the simulations. Section 3
looks at the evolution of the disks of embryos, whilst Sec-
tion 4 examines the end products. In Section 5 we discuss
the results in comparison to the observed solar system.
Finally, the last section contains a summary.
2. N-BODY SIMULATIONS
We performed three sets of nine N-body integrations,
each set using a different model for the formation of the
terrestrial planets.
Model A. These integrations simulate the evolution of
a disk of planetary embryos that initially spans most of
the region currently occupied by the terrestrial planets. In
this model it is assumed that the giant planets do not
significantly influence the formation of the terrestrial planets, and hence they are not included in the integrations.
Model B. As Model A, but the effects of the giant planets are modeled by adding Jupiter and Saturn to the simulations after 107 years. The giant planets are assumed to have
their current masses and orbital elements.
Model C. As Model B, but the initial disk of embryos
is extended to encompass the region that now contains the
asteroid belt. Jupiter and Saturn are added at 107 years,
as in Model B.
The nine simulations using each model are divided into
batches of three, each batch using different values for the
surface density of solid material at 1 AU, s, and the spacing
between embryos, D. One batch each uses (s, D) 5 (6, 7),
(10, 7), and (6, 10), where s is measured in units of gcm22
and D in mutual Hill radii, RHM , where
RHM 5
S
D S
m1 1 m2
3MA
1/3
D
a1 1 a2
2
(1)
for embryos with masses m1 and m2 , and semi-major axes
a1 and a2 .
2.1. Initial Conditions
The initial conditions were chosen bearing in mind the
form of the present planetary system and the results of
simulations of the runaway-growth phase of terrestrialplanet formation (e.g., Wetherill and Stewart (1993).
Disk density. In 18/27 simulations we adopt a surface
density of solid material, s 5 6 gcm22 at 1 AU. This corresponds to the minimum mass needed to form the current
terrestrial planets. We choose a density profile that varies
as 1/a—a smaller gradient than used by some authors—in
view of the large amount of solid material (s p 10 gcm22)
required beyond the ice condensation point to form Jupiter’s core before loss of the nebula gas (Pollack et al. 1996).
As a variant on our standard initial conditions we set s 5 10
gcm22 at 1 AU in three of the simulations for each model.
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CHAMBERS AND WETHERILL
Radial extent. The 18 simulations using Models A and
B begin with a disk of embryos having semi-major axes
0.55 , a , 1.8 AU, covering most of the terrestrial-planet
region. The lower bound is a compromise between making
the simulation realistic and avoiding a short integration
timestep (and hence a large CPU overhead), which is necessary when some objects have small a. In the simulations
using Model C, we extend the outer edge of the disk to
4.0 AU to include embryos that may have formed in the
asteroid belt.
Models A and B begin with 24–40 embryos, depending
on the values of s and D, while Model C begins with 34–56
embryos. The initial disk mass varies between 1.8 and 3.2
M% for Models A and B, and between 5.0 and 8.6 M% for
Model C (which includes material in the asteroid belt).
Orbital elements. The initial eccentricities, e, and inclinations, i, are 0 and 0.18, respectively, for all embryos.
These values are somewhat arbitrary, but they quickly
become randomized once close encounters occur, and most
objects soon attain much larger values of e and i. However,
for the embryos to have formed via runaway growth there
must have been an epoch when e and i were small, and
we assume that this is still the case at the start of our
simulations. The remaining elements for each embryo—
the mean longitude and nodal longitude—are chosen randomly.
Embryo separations. We assume that the planetary
embryos that formed via runaway growth have become
isolated from one another prior to the start of our simulations due to mutual accretion events and eccentricity damping via dynamical friction. Once damping forces have been
overcome, a system containing more than two embryos
becomes unstable with respect to close encounters on a
timescale, tc , that depends exponentially on the initial orbital spacing, D, measured in mutual Hill radii (Chambers
et al. 1996).
In most of our simulations we use D 5 7, which corresponds to an embryo crossing time, tc p 5 3 104 years for
the case in which all objects have the same mass. Three
integrations per model begin with D 5 10, implying that
tc p 107 years in the equal mass case. These two values of
tc bracket the probable time required to form embryos
in the terrestrial region (Wetherill and Stewart 1993). In
practice, tc is somewhat smaller in our integrations because
the embryos begin with a range of masses (see Section
3.1). We also note that tc would be reduced if the embryos
began with non-zero eccentricities (Yoshinaga et al. 1998).
The values of D chosen for our integrations are broadly
consistent with the results of N-body integrations of embryo formation by Kokubo and Ida (1998). These authors
found that embryos formed from a population of p104
small bodies tend to have orbits spaced by about 10 mutual
Hill radii.
Embryo masses. Having chosen s and D, the mass of
each embryo is uniquely determined. These range from
0.02 M% at the inner edge of the disk to 0.1 M% at the
outer edge in Models A and B, with s 5 6 gcm23. Larger
objects are present initially in the simulations that have a
higher disk density or contain embryos in the asteroid belt.
Embryo density. The embryos’ radii are calculated assuming a density of 3 gcm23. This value is also used to
determine the new radius of an object following the accretion of a smaller body.
2.2. Integration Method
The N-body integrator used in the simulations is based
upon a second-order mixed-variable symplectic integrator
written by Levison and Duncan (1994), which uses an algorithm described by Wisdom and Holman (1991). One drawback of this fast package is that close encounters between
massive bodies are not calculated accurately, so we modified the code to integrate all objects using a Bulirsch–Stoer
method (Stoer and Bulirsch 1980), also supplied by Levison
and Duncan, whenever the separation between a pair of
objects falls below 2 Hill radii. The Bulirsch–Stoer algorithm uses a variable timestep to accurately follow the
orbital evolution during an encounter, whilst the symplectic integrator uses a fixed timestep of 10 days. The symplectic algorithm uses Jacobi coordinates, which makes it necessary to periodically re-index the objects in order of
increasing semi-major axis. This procedure minimizes numerical error introduced by having different physical and
Jacobi ordering of the objects.
Combining the symplectic and non-symplectic methods
leads to a secular growth in the energy error, typically one
part in 102 –103 over 108 years. While less than ideal, we
believe this is acceptable given that neglected effects, such
as dynamical friction due to residual planetesimals, probably have a comparable or larger effect.
Collisions between embryos are assumed to be completely inelastic, forming a single new body whose orbit
is determined by conservation of linear momentum. Our
model assumes that mass loss due to fragmentation during
a collision is negligible. This assumption is necessary in
order to avoid a rapid increase in the number of bodies
present in the integration, which would make the computational expense prohibitive.
3. EVOLUTION
Figures 1–3 show a series of snapshots, in semi-major
axis/mass space, taken from three of the simulations described above—one for each model. Each symbol in the
figures represents a surviving embryo, with the horizontal
bars depicting the perihelion and aphelion distances of
its orbit. In these and most of the other integrations the
evolution passes through four stages, described below.
TERRESTRIAL-PLANET FORMATION
307
FIG. 1. Masses and semi-major axes of the surviving objects at six epochs during a simulation using Model A, with D 5 7 and s 5 6 gcm22.
Note how embryo isolation is overcome and collisions occur (b), the disk becomes dynamically excited (c), protoplanets form—see object at 0.9 AU
(d), the small objects are swept up (e), leaving the largest surviving objects isolated from one another (f ).
3.1. Embryo Isolation Is Overcome (Figs. 1b, 2b, 3b)
The initial orbital spacing of the embryos is large enough
that no pair of objects is able to undergo close encounters
in the absence of external perturbations, due to conservation of energy, E, and angular momentum, h (Gladman
1993). When more than two embryos are present, E and
h are no longer conserved within each pair of objects, and
their isolation can be overcome.
Figure 4 shows the evolution of the perihelion and aphelion distances (q and Q, respectively) of the innermost
12 embryos for the first 20,000 years of a simulation using
Model A. Initially the embryos’ semi-major axes remain
almost constant, while their eccentricities, e, exhibit erratic
oscillations whose amplitude increases over time. After
about 12,000 years the eccentricities have increased sufficiently for neighboring embryos to undergo close encounters, and the initial isolation is broken. The time required
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CHAMBERS AND WETHERILL
FIG. 2. As Fig. 1 for a simulation using Model B, with the same values of D and s. In this case the first protoplanet forms at 1.1 AU (d), and
all but two objects are eventually swept up (f ).
to do this varies from one simulation to another, depending
mainly on the initial embryo separation, D. However, in
every case the embryos achieve crossing orbits eventually.
At this point the first collisions occur, reducing the total
number of objects and increasing their mean separation
in mutual Hill radii (see Eq. (1)). It is conceivable that
the surviving embryos could become isolated once more,
returning to a state of affairs similar to, but more stable
than, the start of the simulation. In practice this never
happens in our simulations because the accretion timescale
is always much longer than the timescale for increases in
e. Hence, once the first close encounters take place, the
embryos remain on crossing orbits for as long as a significant number of objects survive.
Re-isolation can occur in simulations that substantially
alter the ratio of the accretion timescale to the dynamical
timescale by constraining embryos to move in two dimensions or by artificially increasing their physical radii. In
TERRESTRIAL-PLANET FORMATION
309
FIG. 3. As Fig. 1 for a simulation using Model C, with the same values of D and s. Here protoplanets first form at 0.7 and 2.2 AU (d), most
embryos with a . 2 AU are removed by resonances rather than collisions (e), leaving just two surviving planets (f ).
trial integrations with radii enhanced by a factor of 10 we
find that re-isolation does occur, producing a final system
containing five to eight low-mass planets with very low
orbital eccentricities.
The time of the first close encounter, tc , depends on the
masses and spacing of the embryos. In the simulations with
initial spacing D 5 7, the first close encounters typically
occur after tc p 104 years, whilst integrations with D 5 10
exhibit encounters after p4 3 105 years. These times are
significantly shorter than those found by Chambers et al.
(1996) for systems of equally spaced, equal-mass planets
(tc p 5 3 104 and 107 years, respectively). However, these
authors noted that a spectrum of planetary masses can
substantially reduce the orbit-crossing time, and we suggest
that this is the cause of the rapid onset of close encounters
seen in our integrations.
Planetary embryos in different parts of the disk experience their first close encounter at different epochs, usually
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CHAMBERS AND WETHERILL
FIG. 4. Perihelion (q) and aphelion (Q) distances for the 12 innermost embryos during the first 20,000 years of a simulation using Model
A. Note the rapid orbital evolution following the first close encounter
at p12,000 years.
beginning with objects close to the inner edge of the disk.
Figure 5 shows the time of first encounter for the embryos
in four of the simulations. The times are measured in units
of the orbital period, and, given that the embryos are uniformly spaced in mutual Hill radii, one would expect the
crossing times to be roughly independent of a. This is true
for the outer part of the disk in each case, but in the inner
region the time of first encounter generally decreases with
increasing a.
This suggests that once embryos near to the inner edge
of the disk achieve crossing orbits, they significantly influence the orbital evolution of embryos further out, hastening the onset of encounters. However, a particular embryo invariably undergoes its first close approach with an
object that was initially in the same part of the disk (rather
than an interloper from elsewhere) so the effect is an indirect one—a close approach between one pair of embryos
destabilizes neighboring objects, leading to a ‘‘wave’’ of
close encounters that sweeps through the inner part of the
disk. This effect is generally limited to the region P , 2
years and is particularly marked for the simulations with
D 5 10.
Chambers and Wetherill (1996) found that the alreadyshort crossing times seen here are reduced still further
when a population of smaller objects (mass p 0.01 embryo
masses) is present in addition to the embryos. During the
earlier runaway growth stage, when planetesimals are accreting each other to form embryos, the eccentricities of
the largest objects are damped by dynamical friction and
collisions with smaller bodies (Wetherill and Stewart 1993),
and presumably the embryos remain isolated from one
another. However, as the fraction of solid disk material
contained in the embryos increases, their mutual interactions will become stronger. At the same time, the corre-
FIG. 5. Times of first close encounter for each embryo in four simulations. The times are divided by the orbital period P, and should be
independent of P if embryos are perturbed only by others in the same part of the disk. Here D is the initial embryo separation in mutual Hill radii.
TERRESTRIAL-PLANET FORMATION
sponding decrease in the total mass of the planetesimals
reduces their ability to damp the embryo’s eccentricities.
This combination is likely to produce a rapid transition to
a situation in which the embryos achieve crossing orbits
and the disk becomes dynamically excited.
3.2. The Disk Becomes Dynamically Excited
(Figs. 1c, 2c, 3c)
Once neighboring embryos have acquired crossing orbits
with significant eccentricities, strong orbital interaction can
occur due to close encounters and resonances between
embryos. This in turn leads to rapid changes in e and i,
and the disk becomes dynamically excited. This can be
seen in Figs. 1c, 2c, and 3c, where many of the embryos
have horizontal bars that overlap, indicating crossing orbits
with nonnegligible eccentricities.
Figure 6 shows the mass-weighted values of Ï(e2 1 i 2),
e, and i, for objects in two simulations using Model A. Each
of these quantities increases logarithmically with time; in
other words, very quickly at first, and then more slowly at
later times. The rate of increase is largest for simulations
with high surface densities and thus more massive embryos.
The upshot is that gravitational focusing during close encounters will diminish over time, since this is most efficient
at small relative velocities. Hence the collision probability
for a given pair of embryos will also decrease with time.
This is reflected in a short-lived burst of collisions near the
start of each simulation, followed by a monotonic decline in
the collision rate (see Section 3.3).
The embryos’ eccentricities and inclinations undergo
large variations throughout an integration. These changes
are primarily caused by secular perturbations rather than
close encounters. In the early evolution, each episode typically lasts for 5–20 thousand years (Fig. 7), and changes
in e are often correlated with the behavior of the argument
of perihelion, g, due to the Kozai resonance (Kozai 1962).
For example, the strong increase in the eccentricity of
Planet 11 at t p 50,000 years in Fig. 7 is associated with a
libration of g about 908. Kozai oscillations, often seen in
the orbits of comets, are due to the long-term interaction
of an orbit with a planar distribution of matter, which is
clearly the case here.
Secular resonances between pairs of embryos are common in each of the simulations. These produce correlated
oscillations in the eccentricities or inclinations of the two
objects, and librations of the resonant critical argument,
with periods p105 years. A common example involves a
situation in which the longitudes of perihelion of two embryos precess at the same rate, with the two orbits aligned
or anti-aligned with one another. Occasionally three, or
even four, orbits are temporarily locked together in this
way. The resulting high-frequency oscillations in e are further modulated by secular interaction with other nearby
311
embryos, or with Jupiter and Saturn in simulations that
include the giant planets.
For example, Fig. 8 shows the evolution of e for four
embryos from two simulations using Model C. The figure
also shows the critical argument of a secular resonance
involving the longitude of perihelion, f, of each object and
that of another nearby embryo. Libration of the critical
argument about 0 indicates that the perihelion directions
of the two embryos are aligned, whilst oscillation about
1808 implies that the perihelion directions are antialigned.
In each case the changes in e are associated with librations
in the critical angle. At some epochs additional oscillations
are apparent, caused by secular interaction with other embryos. An object usually undergoes strong interactions with
two or three other embryos simultaneously and typically
moves back and forth between several secular resonances
during an integration.
Secular resonances with the giant planets also occur.
Figure 9 shows two cases in which the orbits of embryos
lie within the n5 resonance, where the longitude of perihelion of the protoplanet precesses at a similar rate to that
of Jupiter. This causes smooth, long-period changes in the
object’s eccentricity. This resonance influences the orbits
of objects with a , 2 AU in several of our simulations,
especially the region 0.5 , a , 0.7 AU, and occasionally
causes embryos to fall into the Sun. The analogous n6
resonance with Saturn plays an important role for objects
with 2 , a , 2.3 AU. We note that the locations of these
two secular resonances will depend to some extent on how
much nebula gas is present during the accretion of the
terrestrial planets (Lecar and Franklin 1997) and also the
degree to which the giant planets’ orbits change during
the formation of the planetary system.
The remaining diagrams in Fig. 9 show examples of
another common situation in which an embryo’s longitude
of perihelion, f, becomes almost stationary. Like the Kozai
resonance, this resonance does not directly involve the
orbits of other bodies. Whilst f is almost stationary, the
embryo’s nodal longitude and argument of perihelion usually circulate rapidly in opposite directions. This behavior
is generally accompanied by a monotonic increase in e,
which can also cause embryos to fall into the Sun.
In contrast to secular perturbations, close encounters,
which produce an abrupt jump in the orbital elements,
appear to play a minor role in determining changes in e
and i—only a handful of examples can be seen in Figs.
7–9. Close encounters mainly affect e and i by determining
the point in a secular cycle at which an object is scattered
away from one secular resonance and into another. This
in turn establishes the initial values of e and i and the
critical argument for the new resonance.
3.3. Protoplanets Are Formed (Figs. 1d, 2d, 3d)
After 3–6 million years, one or two objects of 0.3–0.5
Earth masses have formed interior to 1.5 AU in each of
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CHAMBERS AND WETHERILL
FIG. 6. Mass-weighted mean of (e2 1 i 2)1/2 (upper) versus time for two of the simulations using Model A, with s 5 6 and 10 gcm22, respectively.
Also shown are the mass-weighted means of e (middle) and i (lower). Here i is measured in radians. Note that all these quantities increase
approximately linearly in log time, with the high-s case yielding larger values.
the simulations. Such objects are apparent at 0.9, 1.1, and
0.7 AU in Figs. 1d, 2d, and 3d, respectively. In simulations
using Model C, where the disk of embryos extends into
the asteroid belt, additional large objects are present beyond 1.5 AU. However, each of these is the result of only
one or two collisional events, and their large mass is due
to the high initial masses of the embryos in this region.
Do these ‘‘protoplanets’’ form by a continuation of runaway growth or via more orderly growth? The low eccen-
tricities and inclinations present at the start of the integrations are conducive to gravitational focusing between
embryos—a necessary prerequisite for runaway growth.
However, as e and i increase this situation will change. We
can see how the effects of gravitational focusing vary during
a simulation by looking at the distribution of close-encounter distances. In the absence of focusing, the number of
encounters, N, with minimum separation, D, is given by
N Y D, while in the limit of strong focusing N Y D 1/2.
TERRESTRIAL-PLANET FORMATION
313
FIG. 7. Eccentricity and argument of perihelion, g, versus time for four embryos for the first 105 years of a simulation using Model A. Note
the general increase in e, the scarcity of sudden jumps associated with close encounters, and the correlation between the behavior of e and g,
indicating orbital evolution controlled by the Kozai resonance.
Figure 10 shows the close-encounter distribution at three
epochs of a simulation using Model A, where we have
combined the close-approach data for all the embryos present. A x 2 test indicates that we can reject the hypothesis
that focusing is absent at the first epoch (0–105 years) at
the 99.5% confidence level. Conversely, the data for the
third epoch (9–10 million years) are consistent with a lack
of gravitational focusing, while those for second epoch
(3–3.5 million years) are ambiguous—the probability that
the distribution is uniform is roughly 10%.
Given that gravitational focusing is probably significant
during at least a part of the simulations, does this
lead to runaway growth? Figure 11 shows the mass
distributions at four epochs, combining the data for all
the embryos with a , 1.5 AU in integrations using
Model C. As the embryos accrete one another the mass
distribution flattens, except for the largest objects, which
march steadily toward higher masses—i.e., orderly growth
occurs. Apparently, gravitational focusing during the
early stages of the simulations is not enough to allow
runaway growth to occur. In addition, the small number
of objects present probably makes it difficult for runaway
growth to get going before the supply of embryos is exhausted.
There is some indication that equipartition of random
orbital energy (‘‘dynamical friction’’) takes place. For example, Fig. 12 shows the eccentricities of all surviving objects in Model C after 10 million years as a function of
their mass. The largest objects tend to have lower values of
e than the smaller bodies. Conversely, the smallest embryos
generally do not have eccentricities close to zero.
A quantitative measure of the importance of dynamical
friction comes from examining the values of e and i for
each pair of objects just before they collide. Consider first
the larger of the two objects in each collision. In Model
C, 69% of these objects have inclinations below the mean
value for that epoch, while for e the figure is 56%. The
corresponding numbers for the smaller colliding body are
59 and 45%, respectively. In other words, weak equipartition of random energy takes place for objects undergoing
collisions—a necessary prerequisite for runaway growth.
Incidentally, the preference for low inclinations over low
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CHAMBERS AND WETHERILL
FIG. 8. Eccentricity versus time for four embryos during simulations using Model C. Also shown are the critical arguments for a secular
resonance between the longitudes of perihelion, f, of each object and another nearby embryo. Librations about 0/1808 indicate that the perihelion
directions of the two objects are aligned/anti-aligned.
eccentricities can be understood following the discussion
below (see Eq. (2)).
Figure 13 shows how the number of collisions varies
depending on the semi-major axis of the larger of the two
colliding bodies just prior to impact (for all the simulations
using Model C). The collision rate decreases rapidly with
increasing heliocentric distance, after a peak at about
0.7 AU, so that very few collisional events occur beyond
2 AU. Note that this is not simply a reflection of the decrease in the number density of objects with increasing a,
since this falls off more slowly.
To see how this effect might arise, consider a planet that
is the largest object in its part of the disk, moving on a
low-e, low-i orbit, with semi-major axis ap . The rate at which
the planet accretes the smaller embryos in its vicinity is
dN
Q
dt
E
ap /(12e)
ap /(11e)
n(a)
3 Pnode(ap , a, e, i) 3 Pconj(ap , a, e, i) da,
P(a)
where a, e, i, and P are the orbital elements and period of
an embryo, and n(a) is the number of embryos smaller
than the planet per unit semimajor axis. In addition, Pnode
is the probability that one of the nodes of an embryo’s
orbit is close enough to the planet’s orbit to permit collisions, given by
Pnode Q 2 3
2Rp fgrav
,
2ae sin i
where Rp(ap) is the radius of the planet, and fgrav(e, i) is the
gravitational focusing factor. Finally, Pconj is the probability
that the planet is in the correct part of its orbit at conjunction for a collision to take place, given by
Pconj Q
2Rp fgrav
.
2fap
If we assume that the embryo masses, m, and their mean
spacing are comparable to their initial values, then n(a) p
1/a 3/2, and m(a) p a 3/2, which also implies that the planetary radius is given by Rp p a p1/2. Provided that embryos
315
TERRESTRIAL-PLANET FORMATION
FIG. 9. Eccentricity versus time for four embryos during simulations using Model C. Also shown in the upper diagrams are the critical arguments
for the n 5 resonance between the longitudes of perhelion, f and fJ , of the embryo and Jupiter. In the lower diagrams, the change in e is associated
with epochs when f itself is almost stationary.
in each part of the disk are subject to accretion by roughly
the same number of planets, the overall accretion rate
will be
f 2grav
dN
Y
dt
e sin i
E
f 2grav(e, i)
1
.
da
Y
ap /(11e) a 4
a 3p sin i
ap /(12e)
(2)
Note the steep dependence of the collision rate on ap
and the lack of dependence on e outside the gravitationalfocusing regime. Despite the crude assumptions used to
derive Eq. (2), the fit with the collision rate observed in
our simulations is quite good, as the 1/a 3p curve in Fig. 13
indicates. The discrepancy between theory and simulation
for a , 0.7 AU presumably reflects the effects of the disk
truncation at 0.55 AU.
The next obvious question is how the collision rate varies
with time? Figure 14 shows the fraction of the initial objects
that remain, versus time t, averaged over all the simulations
that began with an embryo separation D 5 7. The solid
lines show the actual fraction remaining, whilst the dashed
lines indicate the fraction expected according to a decay
law in which the collision probability, for a given embryo,
depends on the number of surviving objects, N, so that the
total population varies as
dN
Y 2N(N 2 1).
dt
(3)
Clearly the slopes of the curves in Fig. 14 are shallower
than the simple theory predicts, and for simulations using
Model C the number of collisions is approximately proportional to log t. This disagreement is not too surprising since
Eq. (3) ignores changes in the orbital element distributions
of the embryos—in particular the collision rate will decrease as the mean inclination of the embryos rises (see
Eq. (2)).
3.4. The Planets Become Isolated (Figs. 1f, 2f, 3f )
In the simulations using Models B and C, the giant planets Jupiter and Saturn are added at 107 years, with their
current masses and orbital elements. The immediate effect
316
CHAMBERS AND WETHERILL
FIG. 12. Eccentricity versus mass for all objects remaining at 10
million years in the simulations using Model C.
FIG. 10. Number of close encounters, N, versus distance of closest
approach, D, at three epochs of a simulation using Model A. Note that
the close-encounter data for all embryos has been combined.
is to introduce a number of strong mean-motion and secular resonances into the region a . 2 AU. Embryos in
the vicinity of these resonances undergo large increases in
eccentricity until they are removed by an impact on the
Sun, by a collision with another embryo inside 2 AU, or
by hyperbolic ejection following a close approach to Jupiter. This mechanism for clearing material from the asteroid
belt still operates today, but it is more efficient in our
simulations since the embryos are large enough that a close
encounter between a pair of embryos can often scatter one
object into a resonance, where it is quickly lost before
another encounter can scatter it out of the resonance again.
The net result is that p20 million years after the giant
FIG. 11. Cumulative mass distribution for all the simulations using
Model C, at four epochs during the integrations. Note that only objects
with a , 1.5 AU are included in the distributions.
FIG. 13. Number of collisions, as a function of semi-major axis, a,
of the larger impactor, for all integrations using Model C. The curve
shows the predicted distribution following a 1/a3 law.
TERRESTRIAL-PLANET FORMATION
FIG. 14. The fraction of surviving objects versus time for all simulations using Models A and C. The dashed lines show the expected fraction
due to a decay law of the form dN/dt Y 2N(N 2 1).
planets are added, most of the embryos with a . 2 AU
have been removed (Fig. 3e). Thus, 30 million years into
a simulation, a combination of rapid accretion in the inner
part of the disk and partial clearing of the asteroid belt by
resonances leaves three to six large ‘‘protoplanets’’ containing most of the remaining mass, together with a comparable number of smaller unaccreted embryos.
The transition from this state of affairs to a system of
isolated, ‘‘final’’ planets depends principally on the amplitude of the secular oscillations in the eccentricities of the
surviving protoplanets. These oscillations have two
sources. First, secular resonances between neighboring
planets on crossing or nearly crossing orbits, which produce
short-period (p105 year) cycles. Second, secular perturbations and/or resonances with Jupiter and Saturn, having
periods of 106 –107 years. In general the former are dominant except for a p 2.1 AU, where the n6 resonance causes
large changes in e and occasionally the region a p 0.6 AU,
where the n5 resonance can play an important role (these
values of a assume that there is no longer a significant
amount of nebula gas present and that the giant planets
have their modern orbits).
The secular oscillations in e cause the perihelion and
aphelion distances (q and Q, respectively) to change on
timescales that are short compared to the collision timescale. In order to avoid collisions with one another, protoplanets must be spaced so that the maximum value of
Q for the innermost object is less than the minimum value
of q for the second object, and so on. Further evolution
takes place until this situation is achieved, with surplus
protoplanets being thrown back and forth until they merge
with another or are scattered beyond 2 AU and removed
by a resonance.
317
For example, Fig. 15 shows the time evolution of q and
Q for each surviving object in four of the simulations,
starting from the point at which the giant planets are added.
Each pair of lines of a particular color represents q and
Q for a single object. Also shown, in blue, are q and Q
for one of the largest objects that does not survive. In each
case the protoplanets destined to remain at the end of the
simulation rarely approach one another, if at all. However,
the eccentricities of their orbits are such that there is no
room left for the ‘‘blue’’ protoplanet. Consequently, this
excess object is either accreted or ejected.
Note that the final spacing of the planets is determined
by the values of e while there are still ‘‘too many’’ protoplanets present. The eccentricities are often subsequently
reduced by interactions with residual small embryos, which
are then accreted or ejected. For example, the ‘‘green’’
planet in Fig. 15c undergoes a substantial decrease in e at
p50 million years due to interactions with a much smaller
body. A series of close encounters with the smaller embryo
nudges the protoplanet into the n5 resonance, producing
a rapid drop in e (Fig. 16). The residual embryo is subsequently removed, leaving the larger body on a low-eccentricity orbit. This late-stage orbital damping occurs in several of our simulations, but does little to alter the semimajor axes of the final planets, these being determined
earlier in the evolution.
4. THE FINAL STATE OF THE SIMULATIONS
Figures 17–19 show the final states of the integrations
using Models A, B, and C, respectively. The figures indicate
the osculating orbit of each surviving planet, with the
planet itself represented by a filled symbol whose radius
is proportional to the radius of the body. The same data
are given in Tables I, II, and III, except that the values of
e and i have been averaged over the last 106 years of the
integration. The column headed ‘‘last event’’ refers to the
time at which the last accretion or ejection took place.
Where close encounters are still taking place, it is assumed
that the last event time will be greater than the length of
the integration.
The results of simulations using Model A are given in
Fig. 17. In each case 108 years have elapsed since the start
of the calculation. In most cases the evolution is incomplete
in the sense that several objects are still able to undergo
close encounters with one another. However, the collision
rate has slowed to almost zero, so it is not apparent how
much longer would be required to achieve a set of isolated
objects. In many of the simulations the innermost objects
have achieved non-crossing orbits. These planets tend to
have smaller i than objects further from the Sun, and they
usually contain most of the mass. However, it is quite likely
that the configuration of these objects will change due to
subsequent interaction with objects at larger a. The one
318
CHAMBERS AND WETHERILL
FIG. 15. Time evolution of the aphelion, Q, and perihelion, q, distances (same color for each body) for each of the objects that survives until
the end of four simulations. Jupiter and Saturn are added to the simulation at 107 years. In each case, the blue lines indicate Q and q for a large
object that is accreted or ejected before the end of the simulation.
example of a ‘‘completed’’ simulation—number 5A—
contains five approximately equal-mass planets, with a
roughly geometric orbital spacing.
The simulations using Model B were continued for 108
years or until close encounters had ceased, whichever was
longer. The final state of each simulation is shown in Fig.
18. Generally there are two objects with a , 1.7 AU, while
in several cases a third object lies further from the Sun.
In the two simulations that yielded three planets, these
have roughly geometric orbital spacings. This is also true
for the three outer planets in simulation 7B, although the
two inner planets lie closer together than a geometric law
would predict. In each case the mean spacing is somewhat
larger than the planets we observe in the inner solar system.
As with Model A, the planets closest to the Sun tend to
be the largest. In addition, there is considerable scatter in
the mean values of e and i from one integration to another.
Simulations using Model C (shown in Fig. 19) usually
produced only one or two planets interior to 2 AU. Several
of the simulations also contain an object in the asteroid
belt, although these are invariably very large compared to
a typical asteroid, due to the large embryo masses used in
our calculations. It is possible that some or all of these
objects have orbits that are unstable over the age of the
solar system. In only one of the two simulations that ended
with .2 isolated planets do the objects have geometrically
spaced orbits. In almost all the simulations the orbital
spacing is larger than in the inner part of the solar system,
TERRESTRIAL-PLANET FORMATION
FIG. 16. Time evolution of the semi-major axis, a, eccentricity, e,
and critical argument for the n 5 resonance for the ‘‘green’’ planet in Fig.
15. Note the rapid drop in e when the critical argument starts to circulate
slowly, following a close encounter at p50.8 million years.
and the planets are also typically spaced more widely than
those from simulations using Model B. Once again, the
objects closest to the Sun tend to be the largest, and these
usually have smaller eccentricities than their ‘‘asteroidal’’
cousins. Two of the simulations produced only a single,
giant terrestrial planet, roughly two and a half times as
massive as Earth.
In general, the surviving objects in our simulations have
time-averaged values of e and i that are substantially larger
than those of Earth and Venus (e p 0.03 and i p 28). Earthsized planets with eccentricities of 0.2 are not uncommon
in our simulations. The large mean values of e occur early
in each integration and lead to correspondingly large maximum values, emax , during each secular oscillation. This in
turn requires that protoplanets remain widely spaced to
avoid scattering or accreting one another. In general, emax
is too large to permit more than two, or occasionally three,
final planets to form in the simulations that contain Jupiter
and Saturn. Thus, the small number of terrestrial planets
319
seen in our calculations, and their large eccentricities, are
directly related to one another.
Approximately two thirds of the ‘‘completed’’ simulations contain a large (.0.5M%) planet lying within the Sun’s
habitable zone. This is the region in which a geologically
active planet can support liquid water. The conservative
estimate for the habitable zone adopted here is 0.95 ,
a , 1.37 AU (Kasting et al. 1993). It is quite likely that
the region is somewhat larger, in which case a greater
proportion of our simulations would contain a habitable
planet.
Looking now at the overall distribution of the surviving
objects, Fig. 20 shows the masses and semi-major axes of
all the remaining objects at the end of the integrations.
In Model A, the survivors encompass a broad range of
heliocentric distances, with the largest objects clustered
around 1 AU and an extended tail of smaller bodies out
to 3 AU. The inner edge of the distribution is quite sharp
and almost identical to the inner cutoff of the initial disk
of embryos at 0.55 AU.
In Model B the remaining objects occupy a narrower
range of heliocentric distance, with only one survivor having a . 2 AU. The few objects that entered this region
during each integration were either scattered back to
smaller values of a or removed by resonances with the
giant planets. The largest bodies lie closer to the Sun than
in Model A, in a cluster centered on 0.6 AU, with a second
group between about 1.0 and 1.4 AU. All the surviving
objects with a . 1.5 AU are of less than one third of an
Earth mass.
In Model C the picture is different again. Now most of
the large objects are confined to a region a , 1.2 AU, with
a tail of smaller objects extending outward. The difference
between the two regions is clear: most of the planets interior to 1.2 AU have masses greater than 1 M% , whilst most
of the objects with a . 1.2 AU are less massive than Earth.
Table IV shows the fates of the embryos according to
their initial location in the disk, giving the fraction that
are incorporated into surviving planets versus those that
were ejected or collided with the Sun. All the initial material remains at the end of simulations using Model A.
The addition of the giant planets, and their associated
resonances, reduces the number of surviving objects to
p85% in Model B—a fraction which is independent of
initial semi-major axis, a0 . The majority of the remainder
are lost via hyperbolic ejection.
In Model C the fraction of survivors with a0 , 1 AU is
the same as in Model B. Exterior to 1 AU the proportion
of objects that survive decreases monotonically with increasing distance from the Sun. Clearly, the presence of
material beyond 2 AU reduces the fraction of embryos
with 1 , a0 , 2 AU that survive until the end of the simulation (57% in Model C compared to 83% in Model B). The
additional embryos in Model C remove many objects with
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CHAMBERS AND WETHERILL
TABLE I
Surviving Objects for Each of the Simulations Using Model A
Simulation
code
Last event
(106 year)
a
(AU)
e
i
(deg.)
Mass
(Earth 5 1)
Component
embryos
1A
.100
2A
.100
3A
.100
4A
.100
5A
37
6A
.100
7A
.100
8A
.100
9A
.100
0.63
0.96
1.23
1.57
1.81
1.97
0.58
0.77
1.05
1.45
1.73
2.22
2.35
0.87
1.50
1.81
1.85
2.06
2.36
0.52
0.87
1.04
2.25
2.50
2.57
0.61
0.82
1.14
1.63
2.28
0.58
1.14
2.02
2.25
0.59
1.17
2.21
2.80
3.15
0.64
0.99
1.48
2.01
2.57
2.58
0.57
0.82
1.12
1.81
1.92
2.47
0.09
0.05
0.05
0.13
0.13
0.04
0.08
0.08
0.08
0.04
0.08
0.16
0.08
0.04
0.08
0.07
0.12
0.16
0.12
0.40
0.49
0.10
0.24
0.14
0.31
0.08
0.08
0.06
0.06
0.07
0.06
0.04
0.05
0.05
0.30
0.18
0.34
0.11
0.15
0.32
0.14
0.18
0.16
0.21
0.10
0.09
0.10
0.08
0.28
0.11
0.04
2
3
3
5
5
2
4
4
3
2
5
6
7
6
6
6
12
5
5
8
34
2
32
5
11
5
4
2
6
2
3
2
6
8
7
6
6
16
33
12
11
10
9
25
10
5
3
3
10
5
9
0.54
0.39
0.30
0.08
0.03
0.49
0.31
0.26
0.45
0.48
0.13
0.07
0.13
1.15
0.13
0.26
0.09
0.08
0.11
0.22
0.19
2.20
0.10
0.45
0.08
0.69
0.79
0.65
0.35
0.76
0.96
1.29
0.58
0.41
0.49
0.87
0.30
0.14
0.07
0.58
0.34
0.41
0.27
0.16
0.12
0.29
0.39
0.67
0.11
0.28
0.14
17
9
4
1
1
8
9
10
8
6
4
1
2
30
2
4
1
1
2
5
1
17
1
7
1
12
7
6
2
5
14
11
3
4
11
9
2
1
1
12
3
4
3
1
1
7
6
7
1
2
1
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TERRESTRIAL-PLANET FORMATION
TABLE II
Surviving Objects for Each of the Simulations Using Model B
Simulation
code
Last event
(106 year)
a
(AU)
e
i
(deg.)
Mass
(Earth 5 1)
Component
embryos
1B
127
2B
83
3B
185
4B
86
5B
50
6B
72
7B
88
8B
259
9B
73
0.71
1.08
1.74
0.50
1.39
0.76
1.76
0.68
1.27
0.31
1.27
0.60
1.09
0.59
0.79
1.40
2.35
0.61
1.37
0.50
0.98
1.88
0.10
0.08
0.14
0.16
0.24
0.16
0.22
0.13
0.09
0.12
0.65
0.17
0.29
0.08
0.10
0.04
0.07
0.37
0.18
0.07
0.26
0.21
5
8
14
2
5
4
13
2
6
15
17
2
6
6
7
4
6
17
20
32
3
5
0.89
0.59
0.25
1.05
0.45
1.31
0.29
1.85
1.01
1.18
0.51
1.59
0.70
0.46
0.49
0.68
0.18
0.94
0.40
0.24
1.11
0.28
24
12
3
28
8
33
4
21
9
16
5
21
4
5
11
6
1
13
7
6
12
2
a0 , 2 AU by scattering them into the asteroid belt where
they are lost via resonances with the giant planets.
Figure 21 shows the composition of the surviving objects
in terms of the initial location of the embryos incorporated
into each object. The graph combines the data from all
the simulations using Model C. Final planets with 0 , a
, 1 AU tend to be composed mainly of embryos from the
inner part of the disk—the region between 0 and 2 AU.
Planets with 1 , a , 2 AU contain only a small amount
of material from closer to the Sun, but a substantial fraction
TABLE III
Surviving Objects for Each of the Simulations Using Model C
Simulation
code
Last event
(106 year)
a
(AU)
e
i
(deg.)
Mass
(Earth 5 1)
Component
embryos
1C
153
2C
67
3C
229
4C
61
5C
266
6C
86
7C
8C
124
239
9C
68
0.90
2.83
0.68
1.51
0.70
1.31
0.60
1.06
2.49
0.45
1.61
0.53
1.13
2.54
0.81
0.98
1.56
1.08
0.20
0.36
0.17
0.03
0.13
0.16
0.29
0.19
0.11
0.14
0.02
0.24
0.15
0.41
0.34
0.15
0.14
0.18
4
35
5
23
9
11
7
5
17
6
8
15
9
39
7
9
18
8
1.67
0.05
1.33
0.49
1.52
0.22
2.04
0.87
0.81
1.23
1.09
1.44
1.53
0.07
2.74
1.37
0.39
2.65
27
1
27
3
33
1
23
4
2
15
5
13
11
1
27
14
3
19
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CHAMBERS AND WETHERILL
FIG. 17. The osculating orbits of each surviving object at the end of the nine simulations using Model A. Each body is represented by a symbol
whose radius is proportional to the radius of the object.
of their mass comes from the asteroid belt. Surviving objects in the asteroid belt are almost entirely composed of
material from this region, although we note that the small
number of such objects in our simulations makes this conclusion somewhat uncertain.
5. DISCUSSION
The final states of the 27 simulations in Figs. 17–19 vary
considerably. This despite the fact that the simulations in
the same row of each figure (e.g., 1B, 2B, and 3B) have
identical initial conditions, except for the values of the
embryo’s angular elements. The evolution in each simulation is highly stochastic, making it difficult to predict the
kind of planetary system that a particular protoplanetary
disk will produce.
One thing is clear however: each of our simulations
produces a planetary system that is qualitatively different
from our own. On the plus side, the simulations yield a
small number of terrestrial planets moving on isolated or-
TERRESTRIAL-PLANET FORMATION
323
FIG. 18. As Fig. 17 for Model B.
bits. In addition, they frequently produce two planets with
semi-major axes and masses comparable to Earth and
Venus. However, the results differ from the inner solar
system in a number of respects:
1.
2.
3.
than
4.
There are generally too few terrestrial planets.
The planets have orbits that are too widely spaced.
The orbits have values of e and i substantially larger
Earth and Venus.
The outermost object in the terrestrial-planet region
(a , 2 AU) is generally much more massive than Mars
(p0.1M%).
5. No analogues of Mercury are present.
Some of these points are easier to address than orders.
The absence of objects with orbits similar to Mercury
(a p 0.4 AU) is presumably due to our decision to truncate
the inner disk at 0.55 AU. The steep gravitational potential
gradient in this region limits the extent to which objects
can be scattered inward from the inner edge of the disk.
324
CHAMBERS AND WETHERILL
FIG. 19. As Fig. 17 for Model C.
This explains why only two surviving objects have a ,
0.5 AU.
The high mass of ‘‘Mars’’ in our simulations may be
due to the large masses of the embryos that were initially
present in this region (say 1.2 , a , 2.0 AU). The rapid
falloff in accretion rate with distance from the Sun (see
Section 3.3) implies that the mass of a planet that forms
in the vicinity of Mars largely depends on the masses of
the embryos which began there. To illustrate this, Fig. 22
shows the sizes of all the surviving objects, where ‘‘size’’
indicates the fraction of the embryos in the simulation that
collided to make up that planet. For example, if a planet
is the result of the accumulation of 20 embryos, and 40
were present initially, then the planet’s size is 0.5. The
difference between the inner and outer terrestrial region is
now more marked than in Fig. 20, especially for simulations
that include the giant planets. Surviving objects with a .
1.2 are often composed of only one or two embryos. In
contrast, planets closer to the Sun usually contain .10
embryos. Hence, if the embryos in the Mars region were
actually smaller than those used in our simulations, the
low mass of the red planet becomes easier to understand.
TERRESTRIAL-PLANET FORMATION
325
FIG. 21. Fraction of the mass of surviving planets that comes from
different regions of the initial disk. The three curves show the mass
fractions for final planets with 0 , a , 1, 1 , a , 2 and 2 , a , 3 AU.
FIG. 20. The masses and semi-major axes of all surviving objects at
the end of the simulations.
This raises the interesting possibility that Mars represents a leftover planetary embryo that underwent little
or no further accretion after the cessation of the earlier
runaway-growth phase of planet formation. The accretion
rate 1.5 AU from the Sun may have been so slow that
most of the embryos that began here were scattered inward
and swept up by Earth and Venus or scattered outward
TABLE IV
Fraction of the Initial Population of Embryos That Are Incorporated into Surviving Objects, as a Function of the Embryos’
Initial Semi-major Axes
Initial a (AU)
Model A
Model B
Model C
0–1
1–2
2–3
3–4
1.00
1.00
—
—
0.84
0.83
—
—
0.84
0.57
0.31
0.18
FIG. 22. The sizes and semi-major axes of all surviving objects. Here
size is defined as the fraction of the embryos initially present in a simulation that accumulated to form each planet.
326
CHAMBERS AND WETHERILL
and removed via resonances in the asteroid belt, before
they could accrete one another. In this scenario, Mars is
the sole embryo that managed to avoid these fates. Once
all its competitors were removed there was no longer a
mechanism to scatter Mars out of its orbit, and its status
changed to that of a final planet.
The problem of the large eccentricities and inclinations
arises at an early stage in our simulations, as mutual perturbations between embryos rapidly increase the mean values
of e and i (Fig. 6). Collisions between objects often diminish
these quantities during the final stages of accretion, but
the effect is insufficient to yield the essentially circular
orbits observed for Earth and Venus.
This result is intimately connected with the small number
of planets produced by our simulations and their wide
spacing. The large eccentricities of the embryos, compounded by the large secular oscillations about the mean
values, imply that neighboring protoplanets must be widely
spaced to avoid scattering or accreting one another. This
in turn limits the number of final planets, since they are
restricted to the region between the inner edge of the disk
and the point at which strong resonances with the giant
planets occur (a p 2 AU).
The large values of e and i are especially marked in
Model C, which may be due to the large masses of the
embryos in the asteroid belt. It it quite possible that objects
this large never formed here, which would go some way
toward remedying the problem. However, the values of e
and i are also too large in simulations using Model B,
where there are initially no embryos beyond 1.8 AU.
Clearly something else is wrong, and with hindsight we
suggest that it is the initial mass distribution used in our
calculations. Earlier, during the runaway growth phase of
the evolution, equipartition of random energy (dynamical
friction) between planetary embryos and planetesimals ensures that the largest objects have orbits with small e and i.
This effect is sufficient to overcome perturbations between
embryos, which will tend to excite these quantities. However, once the embryos contain a substantial fraction of
the solid disk material, dynamical friction will become too
weak to damp down inter-embryo perturbations unless
these perturbations are also weak. This can only occur if
the remaining embryos have orbits which are tens of Hill
radii apart, which in turn implies that only a handful of
embryos are present at this stage.
Hence, a more likely configuration at the start of the
final stage of terrestrial-planet formation is a handful (,10)
of widely separated protoplanets (with masses of say 0.03–
0.3 M%) and a few tens of smaller ‘‘failed embryos.’’ These
objects would presumably be accompanied by a population
of residual planetesimals containing a small fraction of the
total mass. In this scenario, many of the protoplanets are
destined to become true planets and so the final locations
of the planets are already determined, to some extent,
during the runaway growth phase. We plan to test the
scenario outlined here in subsequent simulations.
6. SUMMARY
The principal conclusions drawn from our simulations
are:
• A disk of initially isolated planetary embryos quickly
becomes dynamically excited once mutual perturbations
allow embryos to achieve crossing orbits. At this point e
and i increase rapidly.
• Once isolation is overcome, the evolution of e and i
is driven largely by secular perturbations rather than close
encounters. Secular resonances between pairs of embryos
are common, and the n 5 (a p 0.6 AU) and n 6 (a p 2.1
AU) resonances also play a role in some simulations.
• Orderly growth takes place despite the presence of
gravitational focusing early in the integrations. However,
weak dynamical friction is apparent even though only a
few tens of objects are present.
• Accretion occurs most rapidly in the inner part of
the disk, with the collision rate decreasing sharply as a
increases. Almost no collisions occur beyond 2 AU. Instead, most material with a . 2 AU is removed by meanmotion and secular resonances with the giant planets.
• Embryos with 1.2 , a , 2 AU tend to be scattered
outward and removed by resonances or scattered inward
and accreted by planets closer to the Sun, before they can
accrete one another. Thus, Mars may be an unaccreted
embryo that remained in this region.
• Accretion is complete after 108 years in 60% of simulations that include Jupiter and Saturn, and in all such simulations after 3 3 108 years. Simulations that neglect the giant
planets still have several objects with crossing orbits at
108 years.
• The most common outcome is a pair of large planets
interior to 2 AU. Occasionally a smaller object remains
further from the Sun. The surviving planets tend to have
time-averaged eccentricities and inclinations that are substantially larger than those of Earth and Venus.
• The number and spacing of the planets in our simulations are largely determined by the mean values of e, and
the amplitude of secular oscillations in e, during the final
stages of accretion.
ACKNOWLEDGMENTS
We are grateful to Alan Boss, Martin Duncan, Hal Levison, Jack
Lissauer, and Derek Richardson for fruitful discussions during the preparation of this paper. Thanks also to Conel Alexander, Geoff Coxhead,
Sandy Keiser, and Martin Murphy for computer support, and to Hal
Levison and Martin Duncan for providing the routines that formed the
basis of our computer code. This work was partly supported by NASA
Grants NAG5-3656, NAG5-4285, and NAG5-4386—thank you.
TERRESTRIAL-PLANET FORMATION
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