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The Rate of Planet Formation and the
Solar System's Small Bodies
VIKTO R S . SAFRO NOV
Schmidt Institute of the Physics of the Earth
ABSTRACT
The evolution of random velocities and the mass distribution of pre-
planetary body at the early stage of accumulation are currently under
review. Arguments have been presented for and against the view of an
extremely rapid, runaway growth of the largest bodies at this stage with
parameter values of ~ ~ 103. Difficulties are encountered assuming such
a large O: (a) bodies of the Jovian zone penetrate the asteroid zone too
late and do not have time to hinder the formation of a normal-sized planet
in the astroidal zone and thereby remove a significant portion of the mass
of solid matter and (b) Uranus and Neptune cannot eject bodies from the
solar system into the cometary cloud. Therefore, the values (9 ~ 102 appear
to be preferable.
INTRODUCTION
By the beginning of the twentieth century, Ligondes, Chamberlain,
and Multan had suggested the idea of planet formation via the combining
(accumulation) of solid particles and bodies. However, it long remained
forgotten. It was only by the 1940s that this idea was revived by O.Yu.
Schmidt, the outstanding Soviet scientist and academician (1944, 1957~. He
initiated a systematic study of this problem and laid the groundwork for
contemporary planet formation theory. He also suggested the first formula
for the rate of mass increase of a planet which is absorbing all the bodies
colliding with it. After it was amended and added to, this formula took on
its present form (Safronov 1954, 1969~:
116
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dm = 2 ~ 2GmN
dt ~TrPV(l+ v2 )=
4~r2~(1 + 2(~)
117
_, (1)
where m and r are the mass and radius of the Secreting planet, P is its
period of revolution around the Sun, p and ~ are the volume and surface
density of solid matter in a planet's zone, and ~ is the dimensionless
parameter characterizing random velocities of bodies in a planet's zone (in
relation to the Kepler circular velocity of a preplanetary swarm's rotation):
~ ~ oc r42~-~/24
v = `~ (fir ,
(~2)
However, Schmidt did not consider the increase of a planet's collisional
cross section as a result of focusing orbits via its gravitational field, and the
factor (1 + 20) in his formula was absent. For an independent feeding
zone of a planet, the surface density a at a point in time t is related to the
initial surface density JO by the simple ratio:
= ~o(1—m/Qj,
(3)
where Q is the total mass of matter in a feeding zone of m. It is proportional
to the width of the zone 2ARf and is determined, when ~ is on the order
of several units, by eccentricities of the orbits of bodies, that is, by their
velocities v; when ~ ~ 1, it is determined by the radius of the sphere of
the planet's gravitational influence. In both cases it is proportional to the
radius of an Secreting planet r. (Schmidt took Q to be equal to the present
mass of a planet.)
It is clear from (1) and (2) that relative velocities of bodies in the
planet's zone are the most significant factor determining a planet's growth
rate. The characteristic accumulation time scale is Ma °( (3~1 OC V2. Investi-
gations have shown that velocities of bodies are dependent primarily upon
their distribution by mass. Velocities of bodies in the swarm rotating around
the Sun increase as a result of the gravitational interactions of bodies and
decrease as a result of their inelastic collisions. In a quasistationary state,
opposing factors are balanced out and certain velocities are established in
the system. If the bunk of the mass is concentrated in the larger bodies,
expression (2) for the velocities is applicable for velocities with a param-
eter e on the order of several units. In the extreme case of bodies of
equal mass, e ~ 1. If the bulk of mass is contained in smaller bodies, an
average gravitating mass is considerably less than the mass m of the largest
body. The parameter e in expression (2) then increases significantly. The
velocities of bodies, in turn, influence their mass distribution. Therefore,
we need investigations of the coupled evolution of both distributions to
produce a strict solution to the problem. This problem cannot be solved
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analytically. We thus divided it into two parts: (1) relative velocities of bod-
ies were estimated for a pre-assigned mass distribution, and (2) the mass
spectrum was determined for pre-assigned velocities. At the same time, we
conducted a qualitative study of the coagulation equation for preplanetary
bodies. This effort yielded asymptotic solutions in the form of an inverse
power law with an exponent q:
n(~m) = cm-9,
(4)
(1.5 < q ~ 2) which is valid for all values m except for the largest bodies.
Stable and unstable solutions were found and disclosed an instability of
solutions where q > 2 was noted. In this case, the system does not evolve
in a steady-state manner. Then the velocities of bodies assumed a power
the law of mass distribution (3) with q < 2. They are most conveniently
expressed in the form (2~. Then ~ ~ 3 . 5 was found for the system
without the gas. In the presence of gas, the parameter 0 is two to three
times greater for relatively small bodies.
The most lengthy stage was the final stage of accumulation at which
the amount of matter left unaccreted was significantly reduced. There was
almost no gas remaining at this stage in the terrestrial planet zone. We
found from Expression (1) that with ~ = 3 the Earth CHOW 10 g/cm2)
accreted about 99~o of its present mass in a ~ 108 year period. The other
terrestrial planets were also formed during approximately the same time
scale. The time scale of this accumulation process has been repeatedly
discussed, revised, and again confirmed for more than 20 years. It may
seem strange, but this figure remains also the most probable at this time.
The situation in the region of the giant planets has proven much more
complex. From eq.(l) we can find an approximate expression for the time
scale T formation of the planet, assuming a ~ aof2. Thus,
T ~ oa ,
(5)
where ~ is the planet density. Current masses of the outer planets (Uranus
and Neptune) correspond to JO ~ 0.3, that is, to a value about 30 times less
than in the Earth's zone. The periods of revolution of these planets around
the Sun P oc R3/2 are two orders of magnitude greater than that of the
Earth's. Therefore, with the same values for 43 (cited above), the growth
time scale of Uranus and Neptune would be unacceptably high: T ~ 10~t
years. Of course, this kind of result is not proof that the theory is invalid.
It does, however, indicate that some important factors have not been taken
into account in that theory. In order to obtain a reasonable value for T.
we must Increase product BOO in (5) several dozen times. More careful
consideration has shown that such an increase of BOO has real grounds.
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119
According to (2), velocities of bodies should increase proportionally to
the radius of the planet. It is easy to estimate that when Uranus and
Neptune grew to about one half of the Earth's mass (i.e., a few percent of
their present masses), velocities of bodies for ~ = const ~ 5 should have
already reached the third cosmic velocity, and the bodies would escape the
solar system. Therefore, further growth of planet mass occurred with v =
const, and consequently, according to (2), ~ must have increased. With the
increase of m, the rate of ejection of bodies increased more rapidly than
the growth rate of the planet. ~ the end of accumulation it exceeded the
latter several times, the parameter ~ begin increased about an order of
magnitude. It also follows from here that the initial amount of matter in
the region of the giant planets (that is, ~0) was several times greater than
the amount entered into these planets. Therefore, the difficulty with the
time scale for the formation of the outer planets proved surmountable (at
least in the first approximation). Furthermore, the very discovery of this
difficulty made it possible to discern an important, characteristic feature
of the giant planet accumulation process: removal of bodies beyond the
boundaries of the planetary system. Since they are not only removed from
the solar system, but also to its outer region primarily, a source of bodies
was thereby discovered which formed the cometary cloud.
The basic possibility of runaway growth, that is "runaway" in terms of
the mass of the largest body from the general distribution of the mass of
the remaining bodies in its feeding zone, has been demonstrated (Safronov
1969~. Collisional cross-sections of the largest gravitating bodies are pro-
portional to the fourth degree of their radii. Therefore, the ratio of the
masses of the first largest body m (planet ~`embryo,,) to the mass me of the
second largest body, which is located in the feeding zone of m, increases
with time. An upper limit for this ratio was found for the case of ~ =
const: lim~m/m,) ~ (243~3.
An independent estimate of m/m based on the present inclinations
of the planetary rotation axes (naturally considered as the result of large
bodies falling at the final stage of accumulation) was in agreement with this
maximum ratio with the values (3 ~ 3 . 5.
These results were a first approximation and, naturally, required further
in-depth analysis. Several years later, workers began to critically review the
results from opposing positions. Levin (1978) drew attention to the fact
that as a runaway m occurred, parameter ~ must increase. The ratio
m/m will correspondingly increase. Assuming me and not m is an effective
perturbing body in expression (3), he concluded that the ratio m/m may
have unlimited growth. Another conclusion was reached in a model of
many planet embryos (Safronov and Ruzmaikina 1978; Vityazev et al. 1978;
Pechernikova and Vityazev 1979~. At an early stage all the bodies were
small The zones of gravitational influence and feeding of the largest
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bodies, proportional to their radii, were narrow. Each zone had its own
leader (a potential planet embryo) and there were many such embryos in
the entire zone of the planet (with the total mass mp): Ne ~ (mp/m)il3 for
low values of (3 and several times greater when (3 was high. Leaders with
no overlapping feeding zones were in relatively similar conditions. Their
runaway growth in relation to the remaining bodies in their own zones was
not runaway in relation to each other. There was only a slight difference
in growth rates that was related to the change of alp with the distances
from the Sun. This difference brought about variation in the masses of
two adjacent embryos in the terrestrial planet region m/m ~ 1 + 2m/mp.
As the leaders grew, their ring zones widened, adjacent zones overlapped,
adjacent leaders appeared in the same zone and the largest of them began
to grow faster than the smaller one which had ceased being the leader.
Masses of the leader grew, but their number was decreased. Normal bodies,
lagging far behind the leader in terms of mass, and former leaders me, ma,
..., the largest of which had masses of ~ 10-im, were located in the zone
of each leader m. Consequently, there was no substantial gap in the mass
distribution of bodies. If the bunk of the mass in this distribution had been
concentrated in the larger bodies, for example, if it had been compatible
with the power law (4) with the exponent q < 2, the relative velocities of
bodies could have been written in the form (2), with values of ~ within the
first 10. The leaders in this model comprised a fraction He of all the matter
in the planet's zone, which for low values of ~ is equal to
lee = Nem/mp ~ (m/mp)213'
(6)
while for large values of 6, it is several times greater. Approximately
the same mass is contained in former leaders. At the early stages of
accumulation m << mp and He ~ 1. Thus, velocities of bodies are not
controlled by leaders and former leaders, but by all the remaining bodies
and are highly dependent upon the mass distribution of these bodies. In
the case of the power law (4) with q < 2, runaway embryo growth leads
only to a moderate increase in (3 to Oman ~ 10 . 20 at Me ~ 10-~' when
control of velocities begins to be shifted to the embryos and ~ decreases
to 1 . 2 at the end of accumulation (Safronov l982~. However, it has not
proven possible, without numerical simulation of the process which takes
into account main physical factors, to find the mass distribution that is
established during the accumulation.
The first model calculation of the coupled evolution of the mass
and velocity distributions of bodies at the early stage of accumulation
(Greenberg e! al. 19783 already led to interesting results. The authors
found that the system, originally consisting of identical, kilometer-sized
bodies, did not produce a steady-state mass distribution, like the inverse
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121
power law with q < ~ Only several large bodies with r ~ 200 kilometers had
formed in it within a brief time scale (2~104 years.), while the predominate
mass of matter continued to be held in small bodies. Therefore, velocities
of bodies also remained low. Hence, ~ and not v increased in expression
(2) as m increased. The algorithm did not allow for further extension of the
calculations. If we approximate the mass distribution contained in Graph
4 of this paper by the power law (this is possible with r > 5 kilometers),
we easily find that the indicator q decreases over time from q > 10 where
t = 15,000 years to ~ 3.5 where t = 22,000 years. It can be expected that
further evolution of the system leads to q < 2.
Lissauer (1987) and Artymovich (1987) later considered the possibility
of rapid protoplanet growth (of the largest body) at very low velocities of
the surrounding bodies. The basic idea behind their research was the rapid
accretion by a protoplanet m of bodies in its zone of gravitational influence.
The width of this zone AR9 = krH is equal to several Hills sphere radii
rH = (m/3M~/3R. As m increases, zone OR, expands and new bodies
appear in it, which, according to the hypothesis, had virtually been in
circular orbits prior to this. According to Artymovich, under the impact of
perturbations of m, these bodies acquire the same random velocities as the
remaining bodies of the zone of m over a time scale of approximately 20
synodical orbital periods. Assuming that any body entering the Hills sphere
of m, falls onto m (or is forever entrapped in this sphere, for example,
by a massive atmosphere or satellite swarm, and only then falls onto m),
Artymovich obtains a very rapid runaway growth of m until the depletion
of matter in zone AR9 within 3 104 years in Earth's zone and 4 105 years in
Neptune's zone. Subsequent slow increase of ARg and growth of m occurs
as a result of the increase in the eccentricity of the orbit of m owing to
perturbations of adjacent protoplanets. Lissauer estimates the growth rate
of m using the usual formula (1~. Assuming random velocities of bodies
until their encounters with a protoplanet to be extremely low, he assumes
that after the encounter they approximate the difference of Kepler circular
velocities at a distance of AR = rH. That is, they are proportional to
R-~/2. From this he obtains (3eff°C R. Assuming further that these rates
are equal to the escape velocity from m at a distance of rH, he finds for R =
1 AU vhe = (r/rH)~12 ~ 1/lS, where ve = (2Gm/r)~/2, and he produces the
"upper limit" of Jeff ~ 400 for this value of v/ye using data from numerical
integration (Wetherill and Cox 1985~. 1b generalize the numerical results
to varying it's, he proposes the expression
Jeff ~ 400Rae (~/4~) i13
(7)
for a "maximum effective accretion cross-section" at the earliest stage of
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it must be noted that the question of the role of runaway growth in
planet formation cannot be considered resolved, despite the great progress
achieved in accumulation theory. It is, therefore, worthwhile to consider
how the existence of asteroids and comets constrains the process.
CONDITIONS OF ASTEROID FORMATION
It is now widely recognized that there have never been normal-sized
planets in the asteroid zone. Schmidt (1954) was convinced that the growth
of preplanetary bodies in this zone originally occurred in the same way
as in other zones, but was interrupted at a rather early stage because
of its proximity to massive Jupiter. Jupiter had succeeded in growing
earlier and its gravitational perturbations increased the relative velocities
of the asteroid bodies. As a result, the process by which bodies merged
in collisions was superseded by an inverse process: their destruction and
breakdown. It was later found that Jovian perturbations may increase
velocities and even expel asteroids from the outermost edge of the belt R
> 3.5 AU, and resonance asteroids from "Kirkwood gaps," whose periods
are commensurate with Jupiter's period of revolution around the Sun. The
gaps are extremely narrow, while the mass of all of the asteroids is only
one thousandth of the Earth's mass. Therefore, removal of 99.9% of the
mass of primary matter from the asteroid zone is a more complex problem
than the increase of relative velocities of the remaining asteroids, (up to
five kilometers per second, on the average).
A higher density of solid matter in the Jovian zone, due to condensa-
tion of volatiles, triggered a more rapid growth of bodies in the zone, and
correspondingly, the growth of random velocities of bodies and eccentrici-
ties of their orbits. With masses of the largest bodies > 1027 g, the smaller
bodies of the Jovian zone (JZB) began penetrating the asteroid zone (AZ)
and "sweeping out" all the asteroidal bodies which stood in its way and
were of significantly smaller size. It has been suggested that the bulk of
bodies was removed from the asteroid zone in this way (Safronov 1969~.
Subsequent estimates have shown that the JZB could only have removed
about one half of the initial mass of AZ matter (Safronov 1979~. In 1973,
Cameron and Pine proposed a resonance mechanism by which resonances
scan the AZ during the dissipation of gas from solar nebulae. However, it
was demonstrated ~rbett and Smoluchowski 1980) that for this to be true,
a lost mass of gas must have exceeded the mass of the Sun. In a model of
low-mass solar nebulae ~ ~ 0.1M~), resonance displacement could have
been more effective as Jupiter's distance from the Sun varied both during
its accretion of gas and its removal of bodies from the solar system.
We can make a comparison of the growth rates of asteroids and JZB
using (1), if we express it as
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dra _ (l + 2~3a~aai; {Rj >3/2 ~ (8
drj (i + 243j)(Jj~a (RaJ
where the indices a and j, respectively, denote the zones under consid-
eration. Assuming 8)j ~ USA, [j ~ 2ba/3, ttj/aa ~ 3(Rj/Ra)-n, thatis,
a threefold density increase in the Jovian zone due to the condensation
of volatiles, we have dra/drj ~ (Rj/Ra)3/2+n/9 ~ 0.1 23/2+n. It is clear
from this that the generally accepted value for the density a"(R) of a
gaseous solar nebula of n = 3/2 yields dra ~ drj, that is, it does not cause
asteroid growth to lag behind JZB. In order for JZB's to have effectively
swept bodies out of the AZ, there would have to have been a slower drop
of crg(R) with n ~ 1/2 (Ruzmaikin et al. 1989) and, correspondingly, for
solid matter ~rj ~ 15 . 20 g/cm2. This condition is conserved when there
is runaway growth of the largest bodies in both zones. Expression (8)
is now formulated not for the largest, but for the second largest bodies.
lithe ratio ej ~ 2Oa is conserved. However, a large body rapidly grows
in the asteroidal zone, creating the problem of how it is to be removed.
The second condition for effective AZ purging is the timely appearance
of JZB's in it. Eccentricities of their orbits must increase to 0.3 - 0.4,
while random velocities of v ~ ewR must rise to two to three kilometers
per second. According to (2), the mass of Jupiter's "embryo" must have
increased to a value of m'j ~ m~0.1~3/2. It is clear from this that JZB
penetrating the asteroidal zone at the stage of rapid runaway growth of
Jupiter's core, according to Lissauer, (with (3eff ~ 103 to mc ~ l5m~3) is
ruled out entirely. In view of these considerations, the values of ~j < 20
. 30 and ea < 10 . 15 are more preferable. But the time scale for the
growth of Jupiter's core to its accretion of gas then reaches 107 years.
THE REMOVAL OF BODIES FROM THE SOI^R SYSTEM AND THE
FORMATION OF THE COMETARY CLOUD
Large masses of giant planets inevitably lead to high velocities of
bodies at the final stage of accumulation and, consequently, to the removal
of bodies from the solar system at this stage. It follows from this that the
initial mass of solid matter in the region of giant planets was significantly
greater than the mass which these planets now contain. It also follows that
part of the ejecta remained on the outskirts of the solar system and formed
the cloud of comets. The condition, that the toter angular momentum
of all matter be conserved, imposes a constraint upon the expelled mass
(~102m~3~. Since the overwhelming majority of comets unquestionably
belongs to the solar system and could not have been captured from outside,
and since u' sz~ comet formation at distances of more than 100 AU from the
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AMERICAN AND SOVIET RESEARCH
125
Sun are only possible with an unacceptably large mass of the solar nebula,
the removal of bodies by giant planets appears to be a more realistic way
of forming a comet cloud. The condition for this removal is of the same
kind as the condition for the ejection of bodies into the asteroidal zone. It
is more stringent for Jupiter: velocities of bodies must be mice as great.
For the outer planets, the ejection condition can be expressed as mp/m~ >
4 (~/Rae)3/2. It follows from this that Neptune's removal of bodies is only
possible where (3 < 102; for Uranus it is only possible for an even lower (3.
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Representative terms from entire chapter:
solid matter
