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Late Stages of Accumulation and
Early Evolution of the Planets
ANDREY V. VITYAZEV AND G.V. PECHERNIKOVA
Schmidt Institute of the Physics of the Earth
ABSTRACT
This article briefly discusses recently developed solutions of problems
that were traditionally considered fundamental in classical solar system
cosmogony: determination of planetary orbit distribution patterns, values
for mean eccentricity and orbital inclinations of the planets, and rotation
periods and rotation axis inclinations of the planets. We will examine
two important cosmochemical aspects of accumulation: the time scale
for gas loss from the terrestrial planet zone, and the composition of the
planets in terms of isotope data. We conclude that the early beginning of
planet differentiation is a function of the heating of protoplanets during
collisions with large (thousands of kilometers) bodies. This paper considers
energetics, heat mass transfer processes, and characteristic time scales of
these processes at the early stage of planet evolution.
INTRODUCTION
Using the theory of preplanetary cloud evolution and planet forma
tion, which is based on the ideas of Schmidt, Gurevich, and Lebedinskiy,
and which was developed in the works of Safronov (1969, 1982), we can
estimate a number of significant parameters for the dynamics of bodies
which accumulate in the planets. However, it long proved impossible to
solve a number of problems in classical solar system cosmogony which
were traditionally considered fundamental. Such problems include the the
oretical derivation of patterns of planetary and satellite orbit distributions
(the socalled TitiusBode law), theoretical estimations of the value for
143
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mean orbital eccentricities and inclinations, planet rotation periods and
rotation axis inclinations, and other characteristics of the present structure
of the Sun's planetary system. In addition, certain consequences of the
theory and, most importantly, a conclusion on the relatively cold initial
Earth and the late beginning of its evolution, clashed with data on geo
and cosmochemistry. These data are evidence of the existence of planet
heating during the process of growth and very early differentiation. This
paper briefly discusses a modified version of the theory which the authors
developed in the 1970's and 1980's. Using this new version we were able
to provide a fundamental solution to several key problems of planetary
cosmogony and generate a number of new findings. The most significant
of these appear to be an estimate of the composition of an accumulating
Earth with data incorporated on oxygen isotopy and a conclusion on the
early beginning of differentiation in growing planets.
FORMATION OF THE PLANETARY SYSTEM
Despite promising data on the existence of circumstellar disks, we
have yet to discover an analogue to a circumsolar gasdust disk. Nor
have calculations (Ruzmaikina and Maeva 1986; Cassen and Summers
1983) produced a satisfactory picture of circumsolar disk formation, or a
reliable estimate of its mass M and characteristic initial dimensions R*.
Nevertheless, a model of a lowmass disk (M < 0.1 Mod, with moderate
turbulence, a hot circumsolar zone and cold periphery, has received the
widest recognition in the works of a majority of authors.
STANDARD ANI) MK DISK MODELS
A reconstruction of surface density distribution a(R) according to
Weidenshilling (1977) is shown in Figure 1. Mass of the disk generated by
adding on to produce the cosmic (solar) composition of presentday planet
matter, is
(0.01  0.07)M,with FIR > Ro = lAU) of (R/Ro)~312
.
It is usually supposed that by the time the Sun achieved main sequence,
its luminosity L* did not greatly differ from the present Lo, and the
temperature in the disk's central plane (z = 0) was on the order of a
black body T ~ 300(Ro/R)~/2K Estimations of the degree of ionization
(Ne/N ~ 10~) and gas conductivity ~ < 103CGSE) in the central plane
are insignificant, and the impact of the magnetic field is usually neglected
in considering subsequent disk evolution. Using the hydrostatic equations
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c;,2/cm2
105 _
104
103
1o2
10
/
i'
;^ —~10
/
~ BY 1J
/
/
/
7 ,'
1 ~ // /
V~\\
~ `\\
1
145
100
R. ate.
FIGURE 1 Surface density distribution a(R) in lowmass, circumstellar gaseous disks.
The solid line indicates models of disks near FS class stem: 1: m/k = 1; 2: m/k = 1/2;
3: m/1: = 1/4; the dashed line indicates models of disks near GO class stem: 4: m/k = 1;
5: m/k =1/2; 6: m/k =1/4; the dotted and dashed line shows disks near GS class stars: 7:
m/k = 1; 8: m/k = 1/2; 9: my = 1/4; the straight lines are critical density distributions for
the corresponding classes of stars: 10: °.Cr(FS); 11: ~Cr(GO); 12: ~Cr(G5~. Disk mass in
models 112 is equal to M = 510 EM. 13 is the standard model.
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and the equation of the state of ideal gas at a temperature which is not
dependent on z, we yield a density distribution for z:
p(z) ~ pO(R)exp(—z2/h2), h2 = 2kTR3/GMep, (1)
where p(= 2.3) is the mean molecular mass, and k is the Boltsman constant.
The model which has been termed standard is derived, by taking into
account cr(R) ~ ~ po(R)h(R)
po(R) Cal R~§,P(R) ~ R',T(R) ~ R~,~ < 1,)~ 3. (2)
Flattening of the disk is high: ~ = h/R ax (R~/2, ~ < 0.1. Disk rotation
is differential and differs little from Kepler's:
= ~)k(1 + C) /, Ok = Vk/R = iGMe/R3, (3)
~ ~ (c2/Vk )(d in P/d in R) < 0.1, C2 = kit/.
Here C5 is the speed of sound. In the standard model Vk2 ~ c,2 ~
VA2, VA is the Alfven velocity. Quasiequilibrium disk models are constructed
by Vityazev and Pechernikova (1982) which do not use the contemporary
distribution cr(R). TheY were called MKmodels since they are only defined
~ ~ _ _ , ~ _ _ ~ ~ _ ~ A ~ ~
by a mass M and a moment K of a disk which is rotating around a star with
mass M* and luminosity L*. Expressions for densities p(R'z) and a(R) were
obtained by resolving the system of hydrodynamic equations with additional
conditions (low level of viscous impulse transport and minimum level of
dissipative function). The distribution of surface density in disks was found
to be:
a(R) ~ 1 55 lO5m ( k ) (M* )
. 3
Rl75exp[—2.8 — — R]g/cm2
(k) (Me)
(4)
where M is m.102M~, K is k 1O5lg cm2s~l, and R is in AU. By varying
m and k, star mass M*, and luminosity L*, we can generate a set of
models of quasiequilibrium circumstellar disks (see Figure 1~. We will
note that the distributions of ~ (R) in the standard and MKmodels are
qualitatively similar. However, there is a noticeable excess of matter in
the remote zone in the first model in comparison to the latter ones. This
excess may be related to diffusion spread of the planetesimal swarm during
planet accumulation. Therefore, present planetary system dimensions may
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Be..
.'
An,' , . . .
i'
I..
~ ~— _
.
. !—Jajce '' , ~ ' _'. ~
IF'
,~
147
( ~ ~~\ i) )
( i> )
FIGURE 2 Evolution of the preplanetary disk. The left side shows flattening of the
dust subdisk and the formation of the swarm of planetesimals. The right side illustrates
plantesimals joining together to form planets (Levin 1964~.
primary stages of its evolution and of planet formation (Figure 2) within
the framework of the aforementioned lowmass disk models.
PI^NETESIMAL MASS SPECTRUM N(M,T) AND
MATTER REDISTRIBUTION
After dust settles on the central plane and dust clusters are formed
due to gravitational instability, there occurs the growth and compacting of
some clusters, and the breakdown and absorption of others. This process
is described in detail by Pechernikova and Vi~razev (1988~. We later briefly
touch upon the specific features of the final stages of accumulation of
sufficiently large bodies, when a stabilization effect develops for the orbits
of the largest bodies (Vityazev et al. 1990~. In the coagulation equation
at = 2 A; A(m , m—ml )n(ml)n(m—ml )dm1—n(m t)
Am
) AmaX (m, ml )n(m1 )d
o
(5)
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the subintegral kernel A(m,mV), describing collision efficiency, must take
into account the less efficient diffusion of large bodies. Let us write
A(m,mV) = A*B/(A* + B), where
A* = (or + ri)2~1 + 2G(`m + mi)/(`r + ri)V2jV = vikrik'
is the usual coefficient which characterizes the collision frequency of gravi
tating bodies m and me with good mixing,
B = [D(m) + D(mV)~/`R(m) + AR(mV)] = DikA]?~tk,
is the coefficient accounting for diffusion, D(m) = e2(m)R2/TE(m), e is the
eccentricity, rE is the characteristic Chandrasekhar relaxation time scale,
AR = eR. Estimates demonstrate that at the initial stages (m << 0.1 met B
~ A* and A ~ A*, while at the later stages (m > 0.1m~) we have B(i ~
k) < A* (i ~ k) and in the limit A(i ~ k) ~ B(i ~ k). Pechernikova (1987)
showed that if A oc Imp + many _ mike then the coagulation formula has
an asymptotic solution that can be expressed as
n(`m) cx m~q, q = 1 + a/2.
(6)
For the initial stages ~ = 2/3  4/3 and q* = 4/3  5/3. For the later stages
B(i ~ k) or mik513 and q* ~ 1/6, q* < q < q*, that is, the gently sloping
power law spectrum for large bodies. With this finding we can understand
the relative regularity of mass distribution in the planetary system. Unlike
the findings of numerical experiments (Isaacman and Sagan 1977), only the
low mass in the asteroid belt and a small Mars is an example of significant
fluctuation. The authors, using numerical integration of equations, such as
ad = ~ ~ tRvR ad] Jr 3 (3 [my (3 Adams ~
(7)
also examined the overall process of solid matter surface density redistri
bution a`(R,t) (a,` ~ 102~) resulting from planetesimal diffusion. The
spread effect of a disk of preplanet bodies proved significant for the later
stages and was primarily manifested for the outer areas (Vityazev e! al.
1990~. The effect is less appreciable for the zone of the terrestrial planet
group, and we shall forego detailed discussion of it here.
RELATIVE VELOCIlY SPECTRUM
Wetherill's numerical calculations for the terrestrial planet zone (1980)
confirmed the order of value of relative velocities that had been estimated
earlier by Safronov (1969~. Similar calculations for the zone of outer planets
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149
are preliminary. In particular, there is vagueness in the growth time scales
for the outer planets. By simplifying the problem for an analytical approach,
we can look separately at the problem of mean relative velocities v (i.e. e,
of the planetesimals and the problem of eccentricities e and inclination
~ of the orbits of accreting planets. The second problem is considered
below. We will discuss here an effect which is important in the area of
giant planets. As planetesimal masses (m) grow, their relative velocities (v)
also increases. At a sufficiently high mean relative velocity (v ~ 1/3 · Vk)
part of the bodies from the "high velocity Maxwell tail" may depart the
system, carrying away a certain amount of energy and momentum. With
this scenario, the formula for the mean relative velocity (Vityazev et al.
1990; Safronov 1969) must appear as follows:
~2
(8)
where Te, Tg and Ts are correspondingly the characteristic Chandrasekhar
relaxation time scales, gas deceleration and the characteristic time scale
between collisions, ,0~1 = 0.05  0.13 (Safronov 1969; Stewart and Kaula
1980), p2 = 0.5 (Safronov 1~9) and HE iS part of the amount of energy
removed by the "rapid particles"
`,E ~ / v2n(v)dv/ / v2n(v)dv = I`(5/2,b)/6~(5/2) ~ (9)
6 Vcr O
A.
(b1/2 + Vat + b1/2 + 3 b312) /3~/;7, b = 3 Vc2r ~ 3 (I—1) Vk
For now e ~ i ~ v/Vk << 1, the usual expressions for v follow from (8), in
particular, with Tg ~ Te, TS in a system of bodies of equal mass m we have
v = x/Gm/Or~where ~ ~ 1.
(10)
Despite continuing growth of the mass, as e approaches ecr ~ 0.3  0.4, HE
becomes comparable to ,0~, the relative velocity v in the system of remaining
bodies discontinues growth. In other words, with e ~ ecr, the parameter
~ in expression (10) grows with m proportionally to m2/3, reaching in the
outer zone values ~ 102. This effect gives us acceptable time scales for
outer planet growth. Because of low surface density, accreting planets in
the terrestrial group zone cannot attain a mass sufficient with a build up of
e to ecr Therefore the mean eccentricities do not exceed the values emus
= 0.2  0.25.
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MASSES OF THE LARGEST BODIES IN
A PLANETS FEEDING ZONE
Studies on accumulation theory previously assumed that in terms of
mass a significant runaway of planet embryos from the remaining bodies
in the future planet's feeding zone occured at an extremely early stage.
According to Safronov's wellknown estimates (1969), the mass of the
largest body (after the embryo) me was 102  1o3 of the mass m of the
Secreting planet. Pechernikova and Vityazev (1979) proposed a model for
expanding and overlapping feeding zones, and they considered growth of
the largest bodies. The halfwidth of a body's feeding zone is determined
by mean eccentricity e of orbits of the bulk of bodies at a given distance
from the Sun:
AR(`t`) ~ e(t`)R ~ v(`t~JR/vk,
(11)
The characteristic mixing time scale for R in this zone virtually coincides
with the characteristic time scale for the transfer of regular energy motion
to chaotic energy motion, and the characteristic time scale for energy
exchange between bodies. It is clear from (10) and (11) that
veto oc rote, eft) ot rote, /`R(t) oc rote.
The mass of matter in the expanding feeding zone
tR+^R
QfR, t) = Or / OR d R.
(12)
(13)
~ R—~ R
will also grow with a time scale or rate, while disregarding the difference in
matter diffusion fluxes across zone boundaries. When the mass of a larger
body myth begins to equal an appreciable amount Q(t) with the flow of
time (see Figure 3), the growth of the feeding zone decelerates. At this
stage the larger bodies of bordering zones begin to leave behind, in terms
of their mass, the remaining bodies in their zones. However, (as seen in
Figure 3), this runaway by mass is much less than was supposed in earlier
studies.
MASSES, RELATIVE DISTANCES, AND THE NUMBER OF PLANETS
Mass increase of the largest body in a feeding zone is represented by
the wellknown formula:
dm/dt = ~rr2(1 + 2e))pdv = 2(1 ~ 20)r2Wk~d, (14)
where the surface density of condensed matter ad can be considered a
sufficiently smooth function R and it can be assumed that ad~t=0) =
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mom
MA
o
0.8
0.6
0.4
0.2
o
151
<\~°
it\\
8
out \oONT
2\:
'at
'at\
.
0.3 0.4 0.5 0.6 0.7
0.1 0.2
0.01 0.05 0.1
l l l l l l l l
0.10.2 0.3 0.4 0.5 0.6 0.7 0.8
0.8 0.9 m/Q
0.6 0.7 0.8 0.9 m/M
l
0.2 0.3 0.4 0 5
0.9 ~~e
FIGURE 3 The region of determination and model distributions for ml/m as the ratio
of the mass, m1, of the largest body in the feeding zone of a growing planet to the mass
of a planet m: 1: ml/m = 1  0.62 (m/Q)0 3, which corresponds to the growth of bodies
in the expanding feeding zone; 2: m2/m; 3: m3/m; 4: m/m = 1  m/Q; 5: m/m = (1 
m/Q)~. The circles and dots indicate the results of numerical simulation of the process of
terrestrial planet accumulation (Ipatov 1987; Wetherill 1985~.
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aO(RO/R)~ with its values JO and v in each zone. As bodies precipitate
to the largest body in a given zone m, a growing portion of matter is
concentrated in m and the corresponding decrease in surface density is
written as:
act(t) = Bitt = 0~1—m~t)/Q(t)~. (15)
From (103 through (15) for the growth rate of a planet's radius we have
(with ~ ~ 23:
dr (1 + 2~3)~oWt (Rote
21rb tRJ
it  26(2+v)r3(t)(R/R )u
~ 3~oR2 [1 + e)2+V _ (1 _ e)2+v] J
It follows from (16) that the largest ooay wn~cn Is not ansornea Dy tne other
bodies (a planet) ceases growing when it reaches a certain maximum radius
(mass). The value of this radius is only determined by the parameters of
the preplanetary disk If we put a zero value in the bracket in (16), in the
first approximation for e, we can yield max r, max m, max e, and max /`R.
In particular,
~ ~ ~ . ~ . . ~ . .. ..
(16)
1 Or ~ I
t~e~~Mc3 )
2 ~ 5~tR >` 1/2
max e =  ~—J
· aft (R. O)r5/4, cm;
· max r.
(17)
The growth time scale is an integration (16~. It is close to the one generated
by Safronov (1969) and Wetherill (1980) and is on the order of 6590 million
years for Secreting 8090% of the mass.
One can state the following for distances between two Secreting planets:
Rn+~—Rn ~ AnR + An+l R = enRn + en+l Rn+l ~ (~18~)
hence
Rn+~/Rn ~ (~1 + en)/(l—en+~) = b
(19)
In view of (17) the following theoretical estimate can be made for terrestrial
planets: borax e = 0.2  0.25) = 1.51.67. For the zone of outer planets
bitmap e = ecr = 0.32  0.35) = 1.85  2.3. The real values b in the present
solar system are cited in Table 1. The theory that was developed not only
explains the physical meaning of the TitiusBode law, but also provides
a satisfactory estimate of parameter b. The partial overlapping of zones,
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TABLE 1 The real values b in the present solar system
.
153
Venus Earth Mars Asteroids Jupiter Satum Uranus Nep~ne
(Ceres)
Merwry Venus Earth Mars Ceres Jupiter Satwn Uranus
b 1.87 1.38 1.52 1.77 1.88 1.83 2.01 1.57
embryo drift, and the effects of radial redistribution of matter (Vityazev et
al. 1990) complicates the formulae, but this does not greatly influence the
numerical values max m, max e, and b.
An estimate of the number of forming planets can easily be generated
from (19) for a preplanetary disk with moderate mass and distribution BARD
according to the standard or MKmodels with preassigned outer and inner
boundaries:
N— l n(R* /R* ~
l n[( 1 + max e)/( 1  max e)]
(20)
With low values of max e from (20) we have N ~ ln(R*/R*~/2 max e
and yield a natural explanation of the results of the numerical experiment
(Isaacman and Sagan 1977~: N or 1/e. For a circumsolar disk with initial
mass ~ 0.1Me, the theory offers a satisfactory estimate of the number of
planets which formed:
N(R* 10iAU, max e ~ ecr = 1/3) < 10.
DYNAMIC CHARACTERISTICS OF FORMING PLANETS
Workers were long unsuccessful in developing estimates of planet
eccentricities, orbital inclinations, and mean periods of axis rotation, which
were formed during the planets' growth process. In other words, estimates
that fit well with observational data. Perchernikova and Vityazev (1980,
1981) demonstrated that by taking into account the input of large bodies,
the existing discrepancy between theory and observations could be resolved.
When Secreting planets approach and collide with large bodies at the
earliest stages e increases. A rounding off of orbit takes place at the final
stage: by the time accumulation is completed estimated values for e are
close to present "mean" values (Laska 1988~. The same can be said for
orbital inclinations. Vib~razev and Pechernikova (1981) and Vityazev et at
(1990) developed a theory for determining mean axial spin periods and axis
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inclinations. The angular momentum vector for the axial spin of a planet
K, inclined at an angle ~ to axis z (z is perpendicular to the orbital plane),
is equal to the sum of the regular component Kit, which is directed along
axis z and the random component K2, which is inclined at an angle ~ to
z. For Kit in a modified GiuliHarris approximation (for terrestrial planets)
the following was generated:
K 48 ~/~ t2M ~ i14 ~ 3 ~ 5/~2
mpl3F~ (`m/Qj, Fat (`m/Q = 1') = 9.6 102
Dispersion K2 is (q = 11/6~:
DI(2 ~ 4.18 . 1o2p i/3Gmt0/3
(21)
· F2(m/Qj, F2(m/Q = 1) = 0.123. (22)
The authors demonstrated that as planets accumulate what takes place is
essentially direct rotation (< ~ ~ ~ 90°~. Large axis rotation inclinations
and reverse rotation of individual planets are a natural outcome of the
accumulation of bodies of comparable size. It is clear from (21) and (22)
that the theoretical dependence of the specific axis rotation momentum
(or m2/3) approaches what has been observed. It is worth recalling that
this theory does not allow us to determine the direction and velocity at
which a planet, forming at a given distance, will rotate. It only gives us the
corresponding probability (Figure 4~.
COSMOCHEMICAL ASPECTS OF EVOLUTION
Vityazev and Pechernikova (1985) and ViWazev et al. (1989) have
repeatedly discussed the problem of fusing physicomechanical and physico
chemical approaches in planetary cosmogony. We will only mention two
important findings here. Vityazev and Pechernikova (1985, 1987) proposed
a method for estimating the time scale for gas removal from the terrestrial
planet zone. They compared the theory of accumulation and data on
ancient irradiation by solar cosmic rays of the olivine grains and chondrules
of meteorite matter with an absolute age of 4.5 to 4.6 billion years.
TIME SCALE FOR THE REMOVAL OF GAS FROM THE
TERRESTRIAL PLANET ZONE
It is easy to demonstrate that if gas with a density of at least 102
of the original amount remains during the formation of meteorite parent
bodies in the preplanetary disk between the asteroid belt and the Sun, then
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K1 + K6Z
20t
/
155
K1 + K6Z
_ ~
10
K1
5 ~K \
W/K6X
. K ~
~/ K
l
156
PLANETARY SCIENCES
solar cosmic rays (highenergy nuclei of iron and other elements) could not
have irradiated meteorite matter grains. Vityazev and Pechernikova (1985,
1987) showed that irradiation occurred when 100 to 1000kilometer bodies
appeared. The reasoning was that prior to that time nontransparency of
the swarm of bodies was still sufficiently high, while less matter would have
been irradiated at a later stage than the 510% that has been discovered
experimentally,. The conclusion then follows that there was virtually no
gas as early as the primary stage of terrestrial planet accumulation. This
conclusion is important for Earth science and comparative planetology
because it is evidence in favor of the view of gasfree accumulation of the
terrestrial planets at the later stages and repudiates the hypothesis of an
accretioninduced atmosphere (see, for example the works of the Hayashi
school).
ON THE COMPOSITION OF THE TERRESTRIAL PLANETS
According to current thinking, the Earth (and other planets) was
formed from bodies of differing mass and composition. It is supposed that
the composition of these bodies is, at least partially, similar to meteorites.
Several constraints on the possible model composition of primordial Earth
(generated by the conventional mixing procedure) can be obtained from a
comparison of data on the location of meteorite groups and mafic bedrock
on Earth on the diagram cri70—~80 and density data. Pechernikova and
Vityazev (1989) found constraints from above on Earth's initial composi
tion with various combinations of different meteorite groups: the portion
of carbonaceous chondritetype matter for Earth was < 10%, chondrite
(H. L' L`L, EH, EL) < 70~O and achondrite (Euc) ~ iron < 80%. They
proposed a method which can be used to determine multicomponent m~x
tures of Earth's model composition. The composition of Mars can also
be determined from the hypothesis of the Martian origin of shergottites.
Confirmation of the authors' hypothesis on the removal of the silicate shell
from prot~Mercu~y (Vityazev and Pechernikova 1985; Pechernikova and
Vityazev 1987) would mean that there is an approximately homogeneous
composition for primary rockforming elements in the entire zone 0.51.5
AU.
EARLY EVOLUTION OF THE PLANETS
Vityazev (1982) and Safronov and Vityazev (1983) showed that by the
stage where 1000kilometer bodies are formed, there commences a moder
ate, and subsequently, increasingly intensive process of impact processing,
heating metamorphism, melting, and degassing of the matter of colliding
bodies. It has been concluded that > 90% of the matter of bodies which
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Tim
0.8
0.6
0.4
0.2
o
157
i' \ ~
~ ,,// jI
~ A ,
f ''
i,'
~/ .


"
fit,:


X
~ ,

, , , , ,
0.2 0.4 0.6 0.8 R/Re
FIGURE 5 Estimates of the Earth's initial temperature: 1 is heating by small bodies
(Safronov 1959~; 2 is heating by large bodies (Safronov 1969~; 3 is heating by large bodies
(Safronov 1982~; and 4 is heating by large bodies (Kaula 1980~. The arrow indicates
modifications in the estimates of Tim since the 1950s, including the authors' latest
estimates.
went into forming the terrestrial planets had already passed the swarm
through repeated metamorphism and melting, both at the surface and in
the cores of bodies analogous to parent bodies of meteorites.
INITIAL EARTH TEMPERATURE AND ENERGY SOURCES
If we take into account the collisions of accreting planets with 1000
kilometer bodies, we conclude that there were extremely heated interiors,
beginning with protoplanetary masses ~ 10M. The ratio of the earliest
and current estimates for primordial Earth are given in Figure 5. The latest
temperature estimates indicate the possibility that differentiation began in
the planet cores long before they attained their current dimensions.
The primary energy sources in the planets are known to us. In addition
to energy released during the impact processes of accumulation, the most
significant sources for Earth are: energy from gravitational differentiation
released during stratification into the core and mantle ~ ~1.5 1038ergs) and
energy from radioactive decay ~ ~ 11038ergs). It is important for specific
zones to account for the energy of rotation released during tidal evolution
of the EarthMoon system (A 1037ergs) and the energy of chemical trans
formations ~ 1037ergs). However, these factors are not usually taken into
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PLANETARY SCIENCES
account in global models. The energy from radioactive decay at the initial
stages plays a subordinate role. However, the power of this source may
locally exceed by several times the mean value cr(U,Th,K) ~ 10~6erg/cm3s
(in the case of early differentiation and concentration of U. Th, and K
in nearsurface shells). The power of the shock mechanism is significantly
greater even on the average: limp ~ 104  10~5erg/cm3s. Intermediate
values are generated for the energy of gravitational differentiation EGD ~
105  10~6erg/cm3s (the time scale for core formation is 0.1 to one billion
years) and the source of equivalent adiabatic heating during collapse of ~
10~6erg/cm3s.
HEAT MASS TRANSFER PROCESSES
Heatmass transfer calculations in planetary evolution models are cur
rent~ made using various procedures to parameterize the entire system
of viscous liquid hydrodynamic equations for a binary or even single
component medium. This issue was explored (Safronov and Vityazev 1986;
Vityazev et al. 1990) in relation to primordial Earth. We will merely note
here the order of values for effective temperature conductivities, which
were used for thermal computations in spherically symmetrical models with
a heat conductivity equations such as:
.~ = 1/R2~'R(R2EKj .'R) + E6i/PCp.
(23)
The value which is normally assumed for aggregate temperature conduc
tivity is a function of molecular mechanisms Km = 10~2cm2/s. For shock
mixing during crater formation, the effective mean is K.mp ~ 110 cm2/s.
Similar values have effective values for thermal convection (K~ ~ 10~10
cm2/s) and gravitational differentiation (~D ~ 1lO cm2/s). It is clear from
these estimates that the energy processes in primordial Earth exceeded by
two orders and more contemporary values in terms of intensity.
INITIAL COMPOSITION INHOMOGENEITIES
The planets were formed from bodies with slightly differing compo
sitions, and which accumulated at various distances. Variations in their
composition and density were, on the average, on the same order as the
adjacent planets:
l5Col/C=lipol/p
AMERICAN AND SOVIET RESEARCH
I5POI ~ I[CI = I5COI /< ~ 103  104
159
t
(25)
where (~~102  103) is the ratio of mass removed from the crater to the
mass of the fallen body. Using the distribution of bodies by mass (6),
we can estimate the distribution of composition and density fluctuations
both for a value (§p) and for linear scales (l). For a fixed lip, spaces
occupied by smallscale and largescale fluctuations are comparable. It can
be demonstrated that during planet growth, relaxation of inhomogeneities
already begins (floating of light objects and sinking of heavy ones). In the
order of magnitude, velocity v, relaxation time scale T. effective temperature
conductivity K and energy release rates ~ for one scale inhomogeneities
occupying a part of the volume c, are obtained from the expressions:
Ivy = ~[p~gl/5n, T it R~/vl
K cx cv 1, e ~ ~ [p~gc~v ~1/R~,
(~26~)
where g is the acceleration of gravity, and ~ is the viscosity coefficient.
Where g ~ 103cm/s2, ,7 ~ 102° poise, [pip ~ 103  104, c ~ 0.1,
we have hi/ ~ 1Os _ 10~6cm/s, K 1  102cm2/s, T ~ (1  1O)106
years, e ~ Or (U,Th,K). Estimates show that, owing to intensive heat transfer
during relaxation of such inhomogeneities, the Secreting planet establishes
a positive temperature gradient (central areas are hotter than the external).
This runs contrary to previous assessments (Safronov, 1959, 1969, 1982;
Kaula 1980~. The second important conclusion is that heavy component
enrichment may occur towards the center, which is sufficient for closing off
largescale thermal convection. This is due to relaxation of the composition
fluctuations during planet growth. Both of these conclusions require further
verification in more detailed computations.
``THERMAL EXPLOSIONS" IN PRIMORDIAL EARTH
Peak heat releases, as these relatively large bodies fall, exceed by many
orders the values for (imp which are listed above. Entombed melt sites seek
to cool, giving off heat to the enclosing medium. However, densitybased
differentiation, triggering a separation of the heavy (Ferich) component
from silicates, may deliver enough energy for the melt area to expand.
Foregoing the details (Vityazev et al. 1990), let us determine the critical
dimensions of such an area. Let us write (23) in nondimensional form:
.~ = .~1 + PeOe0/n),~~ ~3+ I' ee +rr,
where
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160
PLANETARY SCIENCES
a=
(T—Tm)E'r _ /\pgch2~0E r _ ~rh2E p _ ~ ,,0,.
RT2 4>R~2 ~ r—4ART2 ' °— ~ K J '
is the difference between heavy and light component densities, c is the
heavy component portion in terms of volume, h is the characteristic size
of the area (layer here), E is the energy for activation in the expression
for the viscosity coefficient, ~ is the heat conductivity coefficient, veto =
0) is the Stokes' velocity, or filtration rate, whose numerical value is found
from the condition fir = (GD. It can be demonstrated that for ~ > Per
(Ec = 0.88 for the layer and Car = 3.22 for the sphere) and ~ > (3cr
(0cr = 1.2 for the layer and (3cr = 1.6 for the sphere) with ~ = 0 at
the area boundaries, conditions are maintained which promote a "thermal
explosion." The emerging differentiation process can release enough energy
to develop the process in space and accelerate it in time. For /\p = 4.5
g/cm3, pc = 0.2, pep = 108erg/cm3 K, RT2 /E = 50K, ae= 10~2cm2/s, and a
Peclet number ~ 1, critical dimensions of the area (her) are on the order
of several hundred kilometers.
CHARACTERISTIC TIME SCALES FOR THE EARLY
DIFFERENTIATION OF INTERIORS
Experimental data on meteorite material melt is too meager to make
reliable judgements as to the composition of phases which are seeking to
divide in the field of gravitational pull. Classical views, hypothesizing that
the heavy (Ferich) component separates from silicates and sinks, via the
filtration mechanism or as a large diapiere structures in the convecting shell,
have only recently been expressed as hydrodynamic models. Complications
with an estimate of the characteristic time scales for separation are, first
of all, related to highly ambiguous data on numerical viscosity values for
matter in the interiors. Variations in the temperature and content of fluids
on the order of several percent, close to liquidussolidus curves, alter
viscosity numerical values by orders of magnitude. It is clear from this that
even for Stokes' (slowed) flows, velocity v or n~i and characteristic time
scales r or n are uncertain. Secondly, existing laboratory data point to the
coexistence of several phases (components) with sharply varied rheology.
This further complicates the separation picture. Nevertheless, a certain
overall mechanism which is weakly dependent on concrete viscosity values
and density differences, was clearly functioning with interiors differentiation.
A number of indirect signs are evidence of this; they indicate the veIy early
and concurrent differentiation of all terrestrial planets, including the Moon.
The following are time scale estimates of the formation of the Earth's
core:
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AMERICAN AND SOVIET RESEARCH
161
1) From paleomagnetic data, the core existed 2.8 to 3.5 billion years
ago;
2) Based on uraniumlead data, it was formed in the first 100 to 300
million years.
Me following do not clash with items 1 and 2 listed above).
3) Data on the formation of a protective, ionosphere or magne
tosphere screen for 20Ne and 36Ar prior to the first 700 million years,
and
4) Data on intensive degassing in the first 100 million years for IXe.
CONCLUSION
The solution (generated in the 1970's80's at least in principle) to the
primary problems of classical planetary cosmogony has made it possible to
move towards a synthesis of the dynamic and cosmochemical approaches.
The initial findings appear to be promising. However, they require con
firmation. Clearly, there is no longer any doubt that intensive processes
which promoted the formation of their initial shells occurred at the later
stages of planet formation. At the same time, if we are to make significant
progress in this area, we will need to conduct experimental studies on
localized matter separation and laborintensive numerical modeling to sim
ulate largescale processes of fractioning and differentiation of the matter
of planetary cores.
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