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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 so-called Titius-Bode law), theoretical estimations of the value for 143

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144 PLANETARY SCIENCES 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 gas-dust 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 low-mass 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 present-day 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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AMERICAN AND SOVIET RESEARCH 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 low-mass, 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 1-12 is equal to M = 510 EM. 13 is the standard model.

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146 PLANETARY SCIENCES 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 MK-models 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.10-2M~, 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 MK-models 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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AMERICAN AND SOVIET RESEARCH 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 low-mass 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 , mml )n(ml)n(mml )dm1n(m t) Am ) AmaX (m, ml )n(m1 )d o (5)

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148 PLANETARY SCIENCES 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 mik-513 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,` ~ 10-2~) 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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AMERICAN AND SOVIET RESEARCH 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 (I1) 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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150 PLANETARY SCIENCES 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 well-known estimates (1969), the mass of the largest body (after the embryo) me was 10-2 - 1o-3 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 half-width 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 well-known 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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AMERICAN AND SOVIET RESEARCH 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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152 PLANETARY SCIENCES 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 = 0~1m~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 65-90 million years for Secreting 80-90% 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)/(len+~) = b (19) In view of (17) the following theoretical estimate can be made for terrestrial planets: borax e = 0.2 - 0.25) = 1.5-1.67. 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 Titius-Bode law, but also provides a satisfactory estimate of parameter b. The partial overlapping of zones,

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AMERICAN AND SOVIET RESEARCH 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 MK-models with pre-assigned 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* 10-iAU, 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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154 PLANETARY SCIENCES 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 Giuli-Harris approximation (for terrestrial planets) the following was generated: K 48 ~/~ t2M ~ i14 ~ 3 ~ 5/~2 mpl3F~ (`m/Qj, Fat (`m/Q = 1') = 9.6 10-2 Dispersion K2 is (q = 11/6~: DI(2 ~ 4.18 . 1o-2p- 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 physico-mechanical 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 10-2 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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A1UERICAN AND SOYIET RESEARCH K1 + K6Z 20t / 155 K1 + K6Z _ ~ 10 K1 5 ~K \ W/K6X . K ~ ~/ K OCR for page 143
l 156 PLANETARY SCIENCES solar cosmic rays (high-energy 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 1000-kilometer 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 5-10% 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 gas-free accumulation of the terrestrial planets at the later stages and repudiates the hypothesis of an accretion-induced 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 chondrite-type 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 rock-forming elements in the entire zone 0.5-1.5 AU. EARLY EVOLUTION OF THE PLANETS Vityazev (1982) and Safronov and Vityazev (1983) showed that by the stage where 1000-kilometer 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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AMERICAN AND SOVIET RESEARCH 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 ~ 10-M. 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 Earth-Moon system (A 1037ergs) and the energy of chemical trans- formations ~ 1037ergs). However, these factors are not usually taken into

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158 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 near-surface shells). The power of the shock mechanism is significantly greater even on the average: limp ~ 10-4 - 10~5erg/cm3s. Intermediate values are generated for the energy of gravitational differentiation EGD ~ 10-5 - 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 Heat-mass 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 ~ 1-10 cm2/s. Similar values have effective values for thermal convection (K~ ~ 10-~-10 cm2/s) and gravitational differentiation (~D ~ 1-lO 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 OCR for page 143
AMERICAN AND SOVIET RESEARCH I5POI ~ I[CI = I5COI /< ~ 10-3 - 10-4 159 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 small-scale and large-scale 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 ~ 10-3 - 10-4, c ~ 0.1, we have hi/ ~ 1O-s _ 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 large-scale 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, density-based differentiation, triggering a separation of the heavy (Fe-rich) 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= (TTm)E'r _ /\pgch2~0E r _ ~rh2E p _ ~ ,,0,.- RT2 4>R~2 ~ r4ART2 ' ~ 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 (Fe-rich) 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 liquidus-solidus 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 uranium-lead 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 I-Xe. CONCLUSION The solution (generated in the 1970's-80'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 labor-intensive numerical modeling to sim- ulate large-scale processes of fractioning and differentiation of the matter of planetary cores. REFERENCES Cassen, P.M., and A. Summers. 1983. Models of the formation of the solar nebula. Icarus 53(1):2640. Ipatov, S.I. 1987. Solid-state accumulation of the terrestrial planets. Astron. vestn. 21(3):207-215. Isaacman, R., and C. Sagan. 1977. Computer simulations of planetary accretion dynamics: sensitivity to initial conditions. Icarus 31: 510-533. Kaula, W.M. 1980. The beginning of the Earth's thermal evolution. The continental crust and its mineral deposits. N.Y. pp. 25-31. Laskar, J. 1988. Secular evolution of the Solar System over ten million years. Astron. Astrophys. 198(1/2):341-362. Levin, B.Yu. 1964. The origin of the Earth and the planets. Nauka, Moscow. Pechernikova, G.V. 1987. On the interim asymptotics of mass spectrum in the system of coagulating particles. Kinematics and Physics of Heavenly Bodies 3~5~:85-87. Pechernikova, G.V., and ~V. V'tyazev. 1979. Masses of the largest bodies and velocity dispersion during planet accumulation. Lettem to Astron. J. 5~1~:54-59; Sov. Astron. Lett. 5:31-34. Pechernikova, G.V, and A V. V'tyazev. 1980. The evolution of planetary orbit eccentricities in the process of planet formation. Astron. J. 57~4~:799-811; Sov. Astron. 27:460467. Pechernikova, G.V., and A.V. Vityazov. 1987. Erosion of Mercury silicate shell during its accumulation. Pages 770-771. In: XVIII Lunar and Planetary Sci. Conf., Houston. Pechernikova, G.V., and A.V. V~tyazov. 1988. The evolution of dust clustem in the preplaneta~y disk. Astron. J. 65~1~:58-72.

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162 PLANETARY SCIENCES Pechernikova, G.V., and AV. V~tyazov. 1989. ~170-trl8O - p constraints on the composi- tion of planetesimals forming the Earth. Theses XII of the National Symposium on stable isotopes in geochemistry. Moscow. Ruzmaikina, T.V., and S.V. Maeva. 1986. Study of the process of protoplanetaty disk formation. Astron. vestn. 20~3~:212-2Z7. Safronov, V.S. 1959. On Earth's initial temperature. AS USSR Publ., Geophys. Series 1:139-143. Safronov, V.S. 1969. The evolution of the preplaneta~y cloud and the formation of the Earth and the planets. Nauka, Moscow. Safronov, V.S. 1982. Ike current state of the theory of the origin of the Earth. AS USSR Publ. Earth Physics 6:5-24. Safronov, V.S., and AV. Vityazov. 1983. The origin of the solar system. Conclusions of science and technology. Pages 5-93. In: Astronomiya, vol. 24. VINITI, Moscow. Safronov, V.S., and A.V. Vityazev 1986. The origin and early evolution of the terrestrial planets. Adv. Ph. Geochem. 6:1-29. Stewart, G.R., and W.M. Kaula. 1980. A gravitational kinetic theory for planetesimals. Icarus 44:154-171. Vityazev, A-V. 1gS2. Fractioning of matter during the formation and evolution of the Earth. AS USSR Publ. Earth Physics 6:52~8. Vityazev, A.V., and G.V. Pechernikova. 1981. A solution to the problem of planet rotation within the framework of the statistical theory of accumulation. Astron. J. 58(4~:869-878. Vityazov, AV., and G.V. Pechernikova. 198Z Models of protoplaneta~y disks near F - G stars Letters to Astron. J. 6:371-377; Sov. Astron. Lett. 8:201-208. Vityazov, AU, and G.V. Pechernikova. 1985. Towards a synthesis of cosmochemical and dynamic approaches in planetary cosmogony. Meteoritika 44:3-20. Vityazev, A.V., and Pechernikova G.V., 1987. When the gas was removed from the zone of terrestrial planets? Pages 1044-1045. In: XVIII Lunar and Planetary Sci. Conf., Houston. Vityazev, TV., G.V. Pechernikova and V.S. Safronov. 1990. The Terrestrial Planets Origin and Early evolution. Nauka, Moscow. Weidenschilling, SJ. 19T7. Ike distribution of mass in the protoplaneta~y system and solar nebula. Astron. Space Sci. 51:153-158. Wetherill, G.W. 1980. Formation of the terrestrial planets. Numerical calculations. Pages 3-24. In: The Continent, Crust and Its Mineral Deposits. Geol. Assoc. of Canada. Wetherill, G.W. 1985. Occurence of Giant Impacts during the growth of the terrestrial planets. Science =: