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OCR for page 98
Formation of the Terrestrial Planets from Planetesimals
GEORGE W. WETHERILL
Carnegie Institution of Washington
ABSTRACT
Previous work on the formation of the terrestrial planets (e.g. Safronov
1969; Nakagawa et al. 1983; Wetherill 1980) involved a stage in which on
a time scale of ~ 106 years, about 1000 embryos of approximately uniform
size (~ 1025 g) formed, and then merged on a 107 - 108 year time scale to
form the final planets. Numerical simulations of this final merger showed
that this stage of accumulation was marked by giant impacts (1027 - 1028
g) that could be responsible for providing the angular momentum of the
Earth-Moon system, removal of Mercury's silicate mantle, and the removal
of primordial planetary atmospheres (Hartmann and Davis 1975; Cameron
and Ward 1976; Wetherill 1985~. Requirements of conservation of angular
momentum, energy, and mass required that these embryos be confined to
a narrow zone between about 0.7 and 1.0 AU. Failure of embryos to form
at 1.5 - 2~0 AU could be attributed to the longer (~ 107 years) time scale
for their initial stage of growth and the opportunity of effects associated
with the growth of the giant planets to forestall that growth.
More recent work (Wetherill and Stewart 19~) indicates that the first
stage of growth of embryos at 1 AU occurs by a rapid runaway on a much
shorter ~ 3 x 104 year time scale, as a consequence of dynamical friction,
whereby equipartition of energy lowers the random velocities and thus
increases the gravitational cross-section of the larger bodies. Formation of
embryos at ~ 2 AU would occur in < 106 years, and it is more difficult to
understand how their growth could be truncated by events in the outer solar
system alone. Those physical processes included in this earlier work are not
capable of removing the necessary mass, energy, and angular momentum
9$
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99
from the region between the Earth and the asteroid belt, at least on such
a short time scale.
An investigation has been made of augmentation of outer solar system
effects by spiral density waves produced by terrestrial planet embryos in
the presence of nebular gas, as discussed by Ward (1986~. This can cause
removal of angular momentum and mass from the inner solar system.
The theoretical numerical coefficients associated with the radial migration
and eccentricity damping caused by this effect are at present uncertain.
It is found that large values of these coefficients, compression of the
planetesimal swarm by density wave drag, followed by resonance effects
following the formation of Jupiter and Saturn, "clears" the region between
Earth and the asteroid belt, and also leads to the formation of Earth and
Venus with approximately their observed sizes and heliocentric distances.
For smaller, and probably more plausible values of the coefficients, this
mechanism will not solve the angular-momentum-energy problem. The
final growth of the Earth on a ~ 108 year time scale is punctuated by giant
impacts, up to twice the mass of Mars. Smaller bodies similar to Mercury
and the Moon are vulnerable to collisional fragmentation. Other possibly
important physical phenomena, such as gravitational resonances between
the terrestrial planet embryos have not yet been considered.
INTRODUCTION
This article will describe recent and current development of theories in
which the terrestrial planets formed by the accumulation of much smaller
(one- to 10-kilometer diameter) planetesimals. The alternative of forming
these planets from massive gaseous instabilities in the solar nebula has not
received much attention during the past decade, has been discussed by
Cameron et al. (1982), and will not be reviewed here.
In its qualitative form, the planetesimal, or "meteoric" theory of planet
formation dates back at least to Chladni (1794) and was supported by
numerous subsequent workers, among the most prominent of which were
Chamberlain and Moulton (Chamberlain 1904~. Its modern development
into a quantitative theory began with the work of O.Yu. Schmidt and
his followers, most notably V.S. Safronov. The publication in 1969 of
his book "Evolutionary of the ProtoplanetaIy Swarm" (Safronov 1969)
and its publication in English translation in 1972 were milestones in the
development of this subject, and most work since that time has consisted
of extension of problems posed in that work.
The formation of the terrestrial planets from planetesimals can be
conveniently divided into three stages:
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PLANETARY SCIENCES
(1) The formation of the planetesimals themselves from the dust of
the solar nebula. The current status of this difficult question has been
reviewed by Weidenschilling et al. (~1988~.
(2) The local accumulation of these one- to 10-kilometer planetesi-
mals into ~ 1025 - 1026g "planetary embryos" revolving about the sun in
orbits of low eccentricity and inclination. Recent work on this problem has
been summarized by Wetherill (1989a), and will be briefly reviewed in this
article.
(3) The final merger of these embryos into the planets observed
today. Fairly recent discussions of this stage of accumulation have been
given by Wetherill (1986, 1988~. This work needs to be updated in order
to be consistent with progress in our understanding of stage (2~. Particular
attention will be given to that need in the present article.
FORMATION OF THE ORIGINAL PI~ETESIMALS
The original solid material in the solar nebula was most likely con-
centrated in the micron size range, either as relic interstellar dust grains,
as condensates from a cooling solar nebula, or a mixture of these types of
material. The fundamental problem with the growth of larger bodies from
such dust grains is their fragility with regard to collisional fragmentation,
not only at the approximate kilometers per second sound speed velocities
of a turbulent gaseous nebula, but even at the more modest ~ 60 m/see
differential velocities associated with the difference between the gas veloc-
ity and the Keplerian velocity of a non-turbulent nebula (Whipple 1973;
Adachi et al. 1976; Weidenschilling 1977~. Agglomeration under these con-
ditions requires processes such as physical "stickiness," the imbedding of
high-velocity projectiles into porous targets, or physical coherence of splash
products following impact. Despite serious efforts to experimentally or
theoretically treat this stage of planetary growth, our poor understanding
of physical conditions in the solar nebula and other physical properties of
these primordial aggregates make it very difficult.
Because of these difficulties, many workers have been attracted to the
possibility that growth of bodies to one- to 10-kilometer diameters could be
accomplished by gravitational instabilities in a central dust layer of the solar
nebula (Edgeworth 1949; Safronov 1960, Goldreich and Ward 1973~. Once
bodies reach that size, it is plausible that their subsequent growth would
be dominated by their gravitational interactions. Weidenschilling (1984)
however has pointed out serious difficulties that are likely to preclude the
development of the necessary high concentration and low relative velocity
(approximately 10 centimeters per second) in a central dust layer. Therefore
the question of how the earliest stage of planetesimal growth took place
remains an open one that requires close attention.
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GROWTH OF PLANETESIMALS INTO PLANETARY EMBRYOS
101
If somehow the primordial dust grains can agglomerate into one- to
10-kilometer diameter planetesimals, it is then necessary to understand the
processes that govern their accumulation into larger bodies.
The present mass of the terrestrial planets is ~ 1028g, therefore about
10~° 10km (~ 10~8g) bodies are required for their formation. It is com-
pletely out of the question to consider the gravitationally controlled orbital
evolution of such a large swarm of bodies by either the conventional meth-
ods of celestial mechanics, or by Monte Carlo approximations to these
methods. Therefore all workers have in one way or another treated this
second stage of planetary growth by methods based on gas dynamics, partic-
ularly by the molecular theory of gases, in which the planetesimals assume
the role of the molecules in gas dynamics theory. This approach is similar
to that taken by Chandrasekhar (1942) in stellar dynamics. Nevertheless,
the fact that the planetesimals are moving in Keplerian orbits rather than
in free space requires some modification of Chandrasekhar's theory.
The most simple approach to such a "gas dynamics" theory of plan-
etesimals is to simply assume that a planetesimal grows in mass ~I) by
sweep up of smaller bodies in accordance with a simple growth equation:
dM 2
d;t. = Herr p5VFg,
(1)
where R is the physical radius of the growing planetesimal, pi iS the
surface mass density of the material being swept up, V is their relative
velocity, and Fg represents the enhancement of the physical cross-section
by "gravitational focussing," given in the two-body approximation by
Fg = (1 ~ 28),
(2)
where ~ is the Safronov number, ~ = An, and Ve is the escape velocity of
the growing body.
Although it is possible to gain considerable insight into planetesimal
growth by simple use of equation (1), its dependence on velocity limits its
usefulness unless a way is found to calculate the relative velocity. Safronov
(1962) made a major contribution to this problem by recognition that this
relative velocity is not a free parameter, but is determined by the mass
distribution of bodies. The mass distribution is in turn determined by the
growth of the bodies, which in turn is dependent on the relative velocities
by equation (1~. Thus the mass and velocity evolution are coupled.
Safronov made use of Chandrasekhar's relaxation time theory to de-
velop expressions for the coupled growth of mass and velocity. He showed
that a steady-state velocity distribution in the swarm was established as
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PLANETARY SCIENCES
a result of the balance between "gravitational stirring" that on the aver-
age increased the relative velocity, and collisional damping, that decreased
their relative velocity. The result was that the velocity and mass evolution
were coupled in such a way that the relative velocity of the bodies was
self-regulated to remain in the proper range, i.e. neither too high to pre-
vent growth by fragmentation, nor too low to cause premature isolation
of the growing bodies as a result of the eccentricity becoming too low.
In Safronov's work the effect of gas drag on the bodies was not included.
Hayashi and his coworkers (Nakagawa et al. 1983) complemented the work
of Safronov and his colleagues by including the effects of gas drag, but
did not include collisional damping. Despite these differences, their results
are similar. The growth of the planetesimals to bodies of ~ 1025 - 1026g
begins with a steep initial distribution of bodies of nearly equal mass. With
the passage of time, the larger bodies of the swarm remain of similar size
and constitute a "marching front" that diminishes in number as the mass
of the bodies increases. Masses of ~ 1025g are achieved in ~ 106 years.
An alternative mode of growth was proposed by Greenberg et al.
(1g78~. They found that instead of the orderly "marching front," runaway
growth caused a single body to grow to ~ 1023g in 104 years, at which
time almost all the mass of the system remained in the form of the original
10~6g planetesimals. It is now known (Patterson and Spaute 19~) that the
runaway growth found by Greenberg et al. were the result of an inaccurate
numerical procedure. Nevertheless, as discussed below, it now appears
likely that similar runaways are expected when the problem is treated
using a more complete physical theory and sufficiently accurate numerical
procedures.
This recent development emerged from the work of Stewart and Kaula
(1980) who applied Boltzmann and Fokker-Planck equations to - he problem
of the velocity distribution of a swarm of planetesimals, as determined by
their mutual gravitational and collisional evolution. This work was extended
by Stewart and Wetherill (1988) to develop equations describing the rate of
change of the velocity of a body of mass inland velocity Vat as a result of
collisional and gravitational interaction with a swarm of bodies with masses
m2 and velocities V2. In contrast with earlier work, these equations for the
gravitational interactions contain dynamical friction terms of the form
dot a' (m2V22me Via) .
(~3)
These terms tend to equipartition energy between the larger and smaller
members of the swarm. For equal values of V: and V2, they cause the
velocity of a larger mass m, to decrease with time. In earlier work, the
gravitationally induced "stirring" was always positive-definite, as a result of
using relaxation time expressions that ensured this result.
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103
The dV/dt equations of Stewart and Wetherill have been used to de-
velop a numerical procedure for studying the evolution of the mass and
velocity distribution of a growing swarm of planetesimals at a given helio-
centric distance, including gas drag, as well as gravitational and collisional
interactions (Wetherill and Stewart 1988~.
When approached in this way it becomes clear that the coupled non-
linear equations describing the velocity and size distribution of the swarm
bifuricate into two general Apes of solutions. The first, orderly growth, was
described by the Moscow and Kyoto workers. The second is "runaway"
growth whereby within a local zone of the solar nebula (e.g. 02 AU in width)
a single body grows much faster than its neighbors and causes the mass
distribution to become discontinuous at its upper end. Whether or not the
runaway branch is entered depends on the physical parameters assumed
for the planetesimals. More important however, are the physical processes
included in the equations. In particular, inclusion of the equipartition
of energy terms causes the solutions to enter the runaway branch for a
very broad range of physical parameters and initial conditions. When
these terms are not included, the results of Safronov, Hayashi, and their
coworkers are confirmed (see Figure 1~. On the other hand, when these
terms are included, runaway solutions represent the normal outcome of the
calculations.
The origin of the runaway can be easily understood. For an initial
swarm of planetesimals of equal or nearly equal mass, the mass distribution
will quickly disperse as a result of stochastic differences in the collision rate
and thereby the growth rate of a large number of small bodies. As a result
of the equipartition of energy terms, this will quickly lead to a velocity
dispersion, whereby the larger bodies have velocities, relative to a circular
orbit, significantly lower than that of the more numerous smaller bodies of
the swarm. The velocity of the smaller bodies is actually accelerated by the
same equipartition terms that decrease the velocity of the larger bodies.
A simplified illustration of this effect is shown in Figure 2. This
calculation is simplified in that effects associated with failure of the two-body
approximation at low velocities and with fragmentation are not included.
- After only ~ 3 x 104 years, the velocities of the largest bodies relative
to those of the smaller bodies has dropped by an order of magnitude. These
lower velocities increased the gravitational cross-section of the larger bodies
sufficiently to cause them to grow approximately 100 times larger than those
bodies in which most of the mass of the swarm is located. This "midpoint
mass" (mp), defined by being the mass below which half the mass of the
swarm is located, is indicated on Figure 2. For bodies of this mass the
Safronov number ~ is 0.6 when defined as
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PL4NETARY SCIENCES
1o8
107
co
-
o
m
ILL
o
UJ
i
>
-
~ lo2
Cat
1o6
105
104
103
lop
1
"SAFRONOV" CASE
(No gas drag,
positive definite
gravitational acceleration)
~~ 6 X 104 years
=~ ~~v~n5~
1 1 .
i\_ years
~\~ 4.4 X 10~ years
rim
1 ~ 111 1
~)8.5 X 105 years
1016 1018 102° 1022 1024 1026 1028
MASS(GRAMS)
FIGURE la Evolution of the mass distribution of a swarm of planetesimals distributed
between 0.99 and 1.01 AU for which the velocity distribution is determined entirely by
the balance between positive~efinite gravitational "pumping up" of velocity and oollisional
damping. The growth is "orderly," i.e., it does not lead to a runaway, but rather to a mass
distribution in which most of the mass is concentrated in 1024 - 1025g bodies at the upper
end of the mass distribution.
V2(m )
2V2 (mp)
(4)
i.e. its relative velocity is similar to its own escape velocity. In contrast, the
value of 8~; calculated using the velocity of this body and that of the largest
body of the swarm has a quite high value of 21.
At this early stage of evolution the growth is still orderly and continuous
(Figure 3~. However, by 1.3 x 105 years, the velocities of the largest bodies
have become much lower than their escape velocities (Figure 2), and a
bulge has developed at the upper end of the swarm as a result of their
growing much faster than the smaller bodies in the swarm. At 2.6 x 105
years, a single discontinuously distributed body with a mass ~ 1026 is found.
At this time it has accumulated 13% of the swarm, and the next largest
bodies are more than 100 times smaller. This runaway body will quickly
capture all the residual material in the original accumulation zone, specified
in this case to be 0.02 AU in width.
The orbit of the runaway body will be nearly circular, and it will be able
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1o8
107
co
-
o
m
a.
o
cr
LO
m
a
IU
-
~= lo3
6
105
104
2
10
(T=n
"NAKAGAWA" CASE
GAS DRAG, NO COLLISIONAL
DAMPING
POSITIVE DEFINITE GRAVITATIONAL
ACCELERATION
4~ 5 X 104 years
~\~1.5 X 105 years
1 . I L 1,l
1ol6 1ol8 1020 l~2
n x no v~-
~ 1.0 X 1 o6 years
1024 1026 1028
MASS(GRAMS}
105
FIGURE lb Evolution of the mass distribution of a swarm in which the velocity damping
is provided bar gas dog, rather than bar collisional damping. Ice resulting distribution is
similar to that of figure la.
to capture bodies approaching within several Hill sphere radii (Hill sphere
radius = distance to colinear Lagrangian points). Even in the absence of
competitors in neighboring zones, the runaway growth will probably self-
terminate because additions to its mass (Am) will be proportional to (AD)2,
where AD is the change in planetesimal diameter, whereas the material
available to be accumulated will be proportional to /`D. Depending on
the initial surface density, runaway growth of this kind can be expected
to produce approximately 30 to 200 bodies in the terrestrial planet region
with sizes ranging from that of the Moon to that of Mars.
There are a number of important physical processes that have not
been included in this simplified model. These include the fragmentation of
the smaller bodies of the swarm, the failure of the two-body approximation
at low velocities, and the failure of the runaway body to be an effective
perturber of small bodies that cross the orbit of only one runaway. These
conditions are more difficult to model, but those calculations that have
been made indicate that they all operate in the direction of increasing the
rate of the runaway.
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PLANETARY SCIENCES
VELOCITY DISTRIBUTION
~ 3
o
~ 1 o2
can
~ 10
car
o
>
1018 1019 102° 1021 1022 1023 1024
MASS(G RAMS)
mp
.
DYNAMICAL FRICTION
GAS DRAG
NO FRAGMENTATION
NO REDUCTION OF
PERTURBATION BY
RUNAWAY
mp ~
1.3 X 105
years ~
LAX 104 years
GRAVITATIONAL
\` CROSS~ECTION
KNOT ENHANCED FOR
LOW VELOCITY
BODIES
I ~
\ ~2.6X105years
~ '
1 1 1 1 1 1 1 1 1 1
1025 1026 1027
FIGURE 2 Velocity distribution corresponding to inclusion of equipartition of energy
terms. After 3 X 104 years, the velocities of the largest bodies drop well below that of the
midpoint mass mp. This leads to a rapid growth of the largest bodies, and ultimately to a
runaway, as described in the tern.
GROWTH OF RUNAWAY PLANETARY EMBRYOS INTO
TERRESTRIAL PINNERS
Because of the depletion of material in their vicinity, it seems most
likely that the runaway bodies described above will only grow to masses in
the range of 6 x 1025g to 6 x 1026g, and further accumulation of a number
of these "planetary embryos" will be required to form bodies of the size of
Earth and Venus.
Both two-dimensional and three-dimensional numerical simulations
of this final accumulation of embryos into terrestrial planets have been
reported. All of these simulations are in some sense "Monte Carlo" cal-
culations, because even in the less demanding two-dimensional case, a
complete numerical integration of several hundred bodies for the required
number of orbital periods is computationally prohibitive. Even if such
calculations were possible, the intrinsically chaotic nature of orbital evo-
lution dominated by close encounters causes the final outcome to be so
exquisite sensitive to the initial conditions that the final outcome is essen-
tially stochastic. l~o~imensional calculations have been reported by Cox
and Lewis (1980~; Wetherill (1980~; Lecar and Aarseth (1986~; and Ipatov
(l9Sla). In some of these two-dimensional cases numerical integration was
carried out during the dose encounter.
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108
oh 1o7
-
o
m
o
ce
m
an
LO
-
106
1o5
1o4
1o3
102
10
~T = 0
1
~3.2 X104
I ~ years
~ 1.3 X 105vears
DYNAMICAL FRICTION
GAS DRAG
NO FRAGMENTATION
NO REDUCTION OF
GRAVITATIONAL
PERTURBATIONS
FOR RUNAWAYS
NO GRAVITATIONAL
ENHANCEMENT OF
LOW VELOCITY BODIES
`2.6X105years
1.3 X 105 years
I ~ 2.6 X 105 years
1 1, \1 - , 1
1016 1018 1020 1022 1024 10 26 1~8
MASS(GRAMS)
107
FIGURE 3 Effect of introducing equipartition of energy terms on the mass distribution.
The tendency toward equipartition of energy results in a velocity dispersion (Figure 23 in
which the velocity (with respect to a circular orbit) of the massive bodies falls below that
of the swarm. After ~ 105 Yeats, a "multiple runaway" appeam as a bulge in the mass
distribution in the mass range 1024 - 1025g. After 2.6 X 105 years, the largest body
has swept up these larger bodies, leading to a runaway in which the mass distribution is
discontinuous. Me largest body has a mass of ~ 1026g, whereas the other remaining
bodies have masses < 1024g.
The three-dimensional calculations (Wetherill 1978, 198Q 1985, 1986,
1988) make use of a Monte Carlo technique based on the work of Opik
(1951) and Arnold (1965~. In both the two- and three-dimensional cal-
culations, the physical processes considered are mutual gravitational per-
turbations, physical collisions, and mergers, and in some cases collisional
fragmentations and tidal disruption (Wetherill 1986, 1988~.
In the work cited above it was necessary to initially confine the initial
swarm to a region smaller than the space presently occupied by the observed
terrestrial planets. This is necessary because a system of this kind nearly
conserves mass, energy, and angular momentum. The terrestrial planets
are so deep in the Sun's gravitational well that very little (< 5%) of the
material is perturbed into hyperbolic solar system escape orbits. The loss
of mass, energy, and angular momentum by this route is therefore small.
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PLANETARY SCIENCES
Angular momentum is strictly conserved by gravitational perturbations and
physical collisions. Some energy is radiated away as heat during collision
and merger of planetesimals. A closed system of bodies, such as a stellar
accretion disk, that conserves angular momentum and loses energy can
only spread, not contract. Therefore the initial system of planetesimals
must occupy a narrower range of heliocentric distance than the range of
the present terrestrial planets. In particular (Wetherill 1978), it can be
shown by simple calculations that a swarm that can evolve into the present
system of the terrestrial planets must be initially confined to a narrow band
extending from about 0.7 to 1.1 AU.
In theories in which planetesimals grow into embryos via the orderly
branch of the bifurcation of the coupled velocity-size distribution equations,
the time scale for growth of ~ 1026g embryos at 1 AU is 1 to 2 x 106 years.
If the surface density of material falls off as a~3/2 beyond 1 AU, the time
scale for similar growth at larger heliocentric distances will vary as a~3, the
additional a~3/2 arising from the variation of orbital encounter frequency
with orbital period. Thus at 2 AU, the comparable time scale for the growth
of planetesimals into embryos would be 1~20 million years. Jupiter and
Saturn must have formed while nebular gas was still abundant. Observations
of pre-main-sequence stars, and theoretical calculations (Lissauer 1987;
Wetherill 1989b) permits one to plausibly hypothesize that Jupiter and
Saturn had already formed by the time terrestrial-type "rocky" planetesimals
formed much beyond 1 AU. In some rather uncertain way it is usually
supposed that the existence of these giant planets then not only cleared
out the asteroid belt, aborted the growth of Mars, and also prevented
the growth of planetesimals into embryos much beyond 1 AU. Interior
to 0.7 AU, it can be hypothesized that high temperatures associated with
proximity to the Sun restrained the formation or growth of planetesimals.
Subject to uncertainties associated with hypotheses of the kind dis-
cussed above, the published simulations of the final stages of planetary
growth, show that an initial collection of several hundred embryos spon-
taneously evolve into two to five bodies in the general mass range of the
present terrestrial planets. In some cases the size and distribution of the
final bodies resemble rather remarkably those observed in the present solar
system (Wetherill 1985~. The process is highly stochastic, however, and
more often an unfamiliar assemblage of final planets is found, e.g. ~ three
bodies, ~ 4 x 1027g of mass at 0.55, 1.0, and 1.4 AU.
Even when the initial embryos are quite small (i.e. as small as 1/6
lunar mass), it is found that the growth of these bodies into planets is
characterized by giant impacts at rather high velocities (> 10 kilometers
per second). In the case of Earth and Venus, these impacting bodies may
exceed the mass of the present planet Mars. These models of planetary
accumulation thereby fit in well with theories of the formation of the Moon
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AMERICAN AND SOVIET RESEARCH
109
accumulation thereby fit in well with theories of the formation of the Moon
by "giant splashes" (Hartmann and Davis 1975; Cameron and Ward 1976),
removal of Earth's atmosphere by giant impacts (Cameron 1983~; and
impact removal of Mercury's silicate mantle (Wetherill 1988; Benz et al.
1988~.
Some rethinking of this discussion is required by the results of Wetherill
and Stewart (1988~. Now that it seems more likely that runaway growth
of embryos at 1 AU took place on a much faster time scale, possibly as
short as 3 x 104 years, it seems much less plausible that the formation
of Jupiter and Saturn can explain the absence of one or more Earth-size
planets beyond 1 AU. Growth rates in the asteroid belt may still be slow
enough to be controlled by giant planet formation, but this will be more
difficult interior to 2 AU.
The studies of terrestrial planet growth discussed earlier in which
the only physical processes included are collisions and merger, are clearly
inadequate to cause an extended swarm of embryos to evolve into the
present compact group of terrestrial planets. Accomplishment of this will
require inclusion of additional physical processes.
Three such processes are known, but their significance requires much
better quantitative evaluation and understanding. These are:
(1) Loss of total and specific negative gravitational binding energy as
well as angular momentum by exchange of these quantities to the gaseous
nebula via spiral density waves (Ward 1986, 1988~.
(2) Loss of material, with its associated energy and angular momen-
tum from the complex of resonances in the vicinity of 2 AU (Figure 4~. All
of these resonances will not be present until after the formation of Jupiter
and Saturn, and therefore are not likely to be able to prevent the formation
of runaway bodies in this region. After approximately one million years
however, they can facilitate removal of material from this region of the
solar system by increasing the eccentricity of planetesimals into terrestrial
planets and Jupiter-crossing orbits. Because bodies with larger semi-major
axes are more vulnerable to being lost in this way, the effect will be to
decrease the specific angular momentum of the swarm as well as cause the
energy of the swarm to become less negative.
(3) Resonant interactions between the growing embryos. Studies of
the orbital evolution of Earth-crossing asteroids (Milan) et al. 1988) show
that although the long-term orbital evolution of these bodies is likely to be
dominated by close planetary encounters, more distant resonant interactions
are also prominent. Similar phenomena are to be expected during the
growth of embryos into planets. These have not been considered in a
detailed way in the context of the mode of planetary growth outlined here,
but relevant studies of resonant phenomena during planetary growth have
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PI~NETARY SCIENCES
~ I I
RESONANCES NEAR 2.0 A.U.
4oo
in
o Boo
~ - 20°
loo
~ _
1+ 4:}
al
_ ~
1! ~
-1
-
a)- no
i I /
I I /
91601 ~
. 1111 1 1 1
1 _
2.0 2.5 3 0 3 5
SEMI MAJOR AXIS ( A.U. )
1
-1
FIGURE 4 Complex of resonances in the present solar system in the vicinity of 2 AU.
These resonances are likely responsible for forming a chaotic "giant Kirlewood gap" that
defines the inner edge of the main asteroid belt.
been published (Ipatov 1981b; Weidenschilling and Davis 1985; Patterson
19~).
All of these phenomena represent real effects that undoubtedly were
present in the earlier solar system and must be taken into consideration in
any complete theory of terrestrial planet formation. Quantitative evaluation
of their effect however, is difficult at present, and there is no good reason
to believe they are adequate to the task
A preliminary evaluation of the effect of the first two phenomena,
spiral density waves and Jupiter~aturn resonances near 2 AU have been
carried out. Some of the result of this investigation are shown in Figures 5
and 6.
In Figure S the point marked "initial swarm" corresponds to the
specific energy and angular momentum of an extended swarm of runaway
planetesimals extending from 0.45 to Z35 AU, with a surface density falling
off as 1/a. The size of the runaway embryos is prescribed by the condition
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AMERICAN AND SOVIET RESEARClI
-8
_ _
U1
_
P::
~ -6
-
o
-
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To
~3
x
~3
111
1 1 1 ' 1 ' ' ' ' 1 '
, 1 1 1 1 ~ ~ ' 1 1-
ENERGY AND ANGULAR MOMENTUM EVOLUTION-
DURING GROWTH OF PLANETS _
~ ~ 'l"
PRESENT PLANETS xxx,~x
DENSITY WAVES ONLY
RESONAN CE ONLY
NO CHANGE
2 1 , , , 1 1 1 1 , 1 1 1 1 ,
0 am
o
-
INITIAL SWARM
3.5 4 4.5 5 1 5.5
ANGULAR MOMENTUM/g ( 10 cgs)
FIGURE ~ Energy and angular momentum evolution of an initial swarm with runaway
planetesimal and gas surface density as 1/a between the 0.7 and 2.20 AU. Interior of 0.7
AU, it is assumed that the gas density is greatly reduced, possibly in an association bipolar
outflow producing a "hole" in the center of the solar nebula. The surface density falls off
exponentially between 2.20 and 2.35 AU and between 0.45 and 0.7 AU. The total initial
mass of the swarm is 1.407 1028g. The open circles represent simulations in which only
gravitational perturbations and collisional damping are included. The crosses are simulations
in which mass, angular momentum, and negative energy are lost by means of the resonances
shown in Figure 3. The solid squares represent simulations in which angular momentum
and losses result from inclusion of spiral density wave damping, as described by Ward (1986,
1988~.
that they be separated by 4 Hill sphere radii from one another. As a result,
the mass of the embryos is 2.3 x 1026g at 2 AU, 0.8 x 1026g at 1 AU,
and 0.5 x 1026g at 0.7 AU. The specific energy and angular momentum
of the present solar system is indicated by the point so marked near the
upper left of the Figure. The question posed here is whether inclusion
of 2 AU resonances and Ward's equations for changes in semi-major axis
and eccentricity of the swarm can cause the system to evolve from the
initial point to the `'goal" representing the present solar system. It is found
that this is possible for sufficiently large values of the relevant parameters.
More recent work (Ward, private communication 1989) indicates that the
published parameters may be an order of magnitude too large. If so,
this will greatly diminish the importance of this mechanism for angular
momentum removal.
The open circles near the initial point show the results of five sim-
ulations in which neither of these effects were included. Some evolution
toward the goal is achieved, nevertheless. This results from the more
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112
PLANETARY SCIENCES
_8
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v
to -6
~ -4
A
-2
I I I I ~ row I I I I I I I I, I ~ I
ENERGY AND ANGULAR MOMENTUM EVOLUTION-
DURING GROWTH OF PLANETS _
o o
_ ~~ o
PRESENT PLANETS
- BOTH DENSITY WAVES AND RESONANCE
I, I I I I, I I I I, I I I I,,, I I l
~ TIC 1/o~
INITL\L SWARM
3.5
4 4.5 5 1 5.5
ANGULAR MOMENTUM/g ( 10 cgs)
FIGURE 6 Energy and angular momentum loss when both the effects of resonances and
spiral density are included. For the choice of parameters described in the text, the system
evolves into a distribution matching that parameters described in the text, the system evolves
into a distribution matching that observed.
distant members of extended swarm being more loosely bound than that
employed in earlier studies, and consequently relatively more loss of bodies
with higher angular momentum and with less negative energy.
This effect is enhanced when the resonances shown in Figure 4 are
included (crosses in Figure 5~. The effect of the resonances is introduced
in a very approximate manner. If after a perturbation, the semi-major axis
of a body is between 2.0 and 2.1 AU, its eccentricity is assigned a random
value between 0.2 and 0.8. A similar displacement toward the specific
angular momentum and energy of the present solar system is reached when
da/dt and de/dt terms of the form given by Ward (1986, 1988) are included.
The open squares in Figure 5 result from use of a coefficient having a value
of 29 in Ward's equation for da/dt, and a value of 1 for de/dt. The value
for da/dt is about twice that originally estimated by Ward (1986~.
The effect of including both the resonances and the same values of
spiral density wave damping are shown in Figure 6. The points lie quite
near the values found for the terrestrial planets. Thus if one assumes
appropriate values for these two phenomena, an initial swarm can evolve
into one with specific energy and angular momentum about that found in
the present solar system. The initial surface density can be adjusted to
match the present total mass of the terrestrial planets, without disturbing
the agreement with the observed energy and angular momentum. Similar
agreement has been obtained in calculations in which both the gas and
embryo surface densities varied as a-3t2, instead of a-i as used for the
OCR for page 113
AMERICAN AND SOYlET RESEARCH
113
data of Figure 6. Satisfactory matching has also been found for a swarm in
which the gas surface density fell off as a~3/2 whereas the embryo surface
density decreased as am.
Matching the mass, angular momentum, and energy of the present
terrestrial planet system is a necessary, but not sufficient condition, for
a proper model for the formation of the terrestrial planets. It is also
necessary that to some degree the configuration (i.e. the number, position,
mass, eccentricity, and inclination) of the bodies resemble those observed.
Because a model of this kind is highly stochastic, and we have only one
terrestrial planet system to observe, it is hard to know how exactly the
configuration should match. As in the earlier work "good" matches are
sometimes found, more often the total number of final planets with masses
> 1/4 Earth mass is three, rather than the two observed bodies. It is
possible this is a stochastic effect, but the author suspects it more likely
that the differences result from the model being too simple. Neglecting
such factors as the resonant interactions between the embryos as well as
less obvious phenomena may be important.
Like the previous models in which the swann was initially much more
localized, the final stage of accumulation of these planets from embryos
involves giant impacts. Typically, at least one body larger than the present
planet Mars impacts the simulated "Earth," and impacts twice as large are
not uncommon. Therefore, all of the effects related to such giant impacts,
formation of the Moon, fragmentation of smaller planets, and impact
loss of atmospheres are to be expected for terrestrial planet systems arising
from the more extended initial embryo swarms of the kind considered here.
Furthermore, the inward radial migration associated with density wave drag,
as well as the acceleration in eccentricity caused by the resonances, augment
the tendency for a widespread provenance of the embryos responsible for
the chemical composition of the final planets.
ACKNOWLEDGMENTS
The author wishes to thank W.S. Ward for helpful discussions of
density wave damping, and Janice Dunlap for assistance in preparing this
manuscript. This work was supported by NASA grant NSG 7347 and was
part of a more general program at DTM supported by grant NAGW 398.
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Representative terms from entire chapter:
terrestrial planets
