N91-22968 Formation of the Terrestrial Planets from Planetesimals GEORGEW. WETHERILL Carnegie Institution of Washington ABSTRACT Previous work on the formation of the terrestrial planets (e.g. Safronov 1969; Nakagawa et al. 1983; Wetherill 1980) involved a stage in which on a time scale of ~ 106years, about 1000 embryos of approximately uniform size (.., 10_5g) formed, and then merged on a 107 - 10' year time scale to form the final planets. Numerical simulations of this final merger showed that this stage of accumulation was marked by giant impacts (10 _7 -- 10_ g) that could be responsible for providing the angular momentum of the Earth-Moon system, removal of Mercury's silicate mantle, and the removal of primordial planetary atmospheres (Hartmann and Davis 1975; Cameron and Ward 1976; Wetherill 1985). Requirements of conservation of angular momentum, energy, and mass required that these embryos be confined to a narrow zone between about 0.7 and 1.0 AU. Failure of embryos to form at 1.5 - 2.0 AU could be attributed to the longer (.-. 107 years) time scale for their initial stage of growth and the opportunity of effects associated with the growth of the giant planets to forestall that growth. More recent work (Wetherfll and Stewart 1988) indicates that the first stage of growth of embryos at 1AU occurs by a rapid runaway on a much shorter ~ 3 x 104 year time scale, as a consequence of dynamical friction, whereby equipartition of energy lowers the random velocities and thus increases the gravitational cross-section of the larger bodies. Formation of embryos at ,-, 2 AU would occur in < 106 years, and it is more difficult to 4 understand how their growth could be truncated byevents in the outer solar system alone. Those physical processes included in this earlier work are not capable of removing the necessary mass, energy, and angular momentum 98 AMERICAN AND SOVIET RESEARCH from the region between the Earth and the asteroid belt, at least on such a short time scale. An investigation has been made of augmentation of outer solar system effects by spiral density waves produced by terrestrial planet embryos in the presence of nebular gas, as discussed by Ward (1986). This can cause removal of angular momentum and mass from the inner solar system. The theoretical numerical coefficients associated with the radial migration and eccentricity damping caused by this effect are at present uncertain. It is found that large values of these coefficients, compression of the planetesimal swarm by density wave drag, followed by resonance effects following the formation of Jupiter and Saturn, "clears" the region between Earth and the asteroid belt, and also leads to the formation of Earth and Venus with approximately their observed sizes and heliocentric distances. For smaller, and probably more plausible values of the coefficients, this mechanism will not solve the angular-momentum-energy problem. The final growth of the Earth on a --. l0s year time scale is punctuated by giant impacts, up to twice the mass of Mars. Smaller bodies similar to Mercury and the Moon are vulnerable to collisional fragmentation. Other possibly important physical phenomena, such as gravitational resonances between the terrestrial planet embryos have not yet been considered. INTRODUCTION This article will describe recent and current development of theories in which the terrestrial planets formed by the accumulation of much smaller (one- to 10-kilometer diameter) planetesimals. The alternative of forming these planets from massive gaseous instabilities in the solar nebula has not received much attention during the past decade, has been discussed by Cameron et al. (1982), and will not be reviewed here. In its qualitative form, the planetesimal, or "meteoric" theory of planet formation dates back at least to Chladni (1794) and was supported by numerous subsequent workers, among the most prominent of which were Chamberlain and Moulton (Chamberlain 1904). Its modern development into a quantitative theory began with the work of O.Yu. Schmidt and his followers, most notably V.S. Safronov. The publication in 1969 of his book "Evolutionary of the Protoplanetary Swarm" (Safronov 1969) and its publication in English translation in 1972 were milestones in the development of this subject, and most work since that time has consisted of extension of problems posed in that work. The formation of the terrestrial planets from planetesimals can be conveniently divided into three stages: 100 PLANETARY SCIENCES (1) The formation of the planetesimals themselves from the dust of the solar nebula. The current status of this difficult question has been reviewed by Weidenschilling et al. (1988). (2) The local accumulation of these one- to 10-kilometer planetesi- mals into ~ 10_5 - 10_6g "planetary embryos" revolving about the sun in orbits of low eccentricity and inclination. Recent work on this problem has been summarized by Wetherill (1989a), and will be briefly reviewed in this article. (3) The final merger of these embryos into the planets observed today. Fairly recent discussions of this stage of accumulation have been given by Wetherill (1986, 1988). This work needs to be updated in order to be consistent with progress in our understanding of stage (2). Particular attention will be given to that need in the present article. FORMATION OF THE ORIGINAL PLANETESIMALS The original solid material in the solar nebula was most likely con- centrated in the micron size range, either as relic interstellar dust grains, as condensates from a cooling solar nebula, or a mixture of these types of material. The fundamental problem with the growth of larger bodies from such dust grains is their fragility with regard to collisional fragmentation, not only at the approximate kilometers per second sound speed velocities of a turbulent gaseous nebula, but even at the more modest ,-_60 m/sec differential velocities associated with the difference between the gas veloc- ity and the Keplerian velocity of a non-turbulent nebula (Whipple 1973; Adachi et aL 1976; Weidenschilling 1977). Agglomeration under these con- ditions requires processes such as physical "stickiness," the imbedding of high-velocity projectiles into porous targets, or physical coherence of splash products following impact. Despite serious efforts to experimentally or theoretically treat this stage of planetary growth, our poor understanding of physical conditions in the solar nebula and other physical properties of these primordial aggregates make itvery difficult. Because of these difficulties, many workers have been attracted to the possibility that growth of bodies to one- to 10-kilometer diameters could be accomplished by gravitational instabilities in a central dust layer of the solar nebula (Edgeworth 1949; Safronov 1960, Goldreich and Ward 1973). Once bodies reach that size, it is plausible that their subsequent growth would be dominated by their gravitational interactions. Weidenschilling (1984) however has pointed out serious difficulties that are likely to preclude the development of the necessary high concentration and low relative velocity (approximately 10centimeters per second) ina central dust layer. Therefore the question of how the earliest stage of planetesimal growth took place remains an open one that requires close attention. AMERICAN AND SOVIET RESEARCH 101 GROWTH OF PLANETESIMAI.,S INTO PLANETARY EMBRYOS If somehow the primordial dust grains can agglomerate into one- to 10-kilometer diameter planetesimals, it isthen necessary to understand the processes that govern their accumulation into larger bodies. The present mass of the terrestrial planets is~ 102Sg,therefore about 101° 10kin (~ 101Sg) bodies are required for their formation. It is com- pletely out of the question to consider the gravitationally controlled orbital evolution of such a large swarm of bodies byeither the conventional meth- ods of celestial mechanics, or by Monte Carlo approximations to these methods. Therefore all workers have in one way or another treated this second stage of planetary growth by methods based on gas dynamics, partic- ularly by the molecular theory of gases, in which the planetesimals assume the role of the molecules in gas dynamics theory. This approach issimilar to that taken by Chandrasekhar (1942) in stellar dynamics. Nevertheless, the fact that the planetesimais are moving in Keplerian orbits rather than in free space requires some modification of Chandrasekhar's theory. The most simple approach to such a "gas dynamics" theory of plan- etesimals is to simply assume that a planetesimal grows in mass (M) by sweep up of smaller bodies in accordance with a simple growth equation: dM 7rR2p,VFg (1) dt where R is the physical radius of the growing planetesimal, p, is the surface mass density of the material being swept up, V is their relative velocity, and Fa represents the enhancement of the physical cross-section by "gravitational focussing," given in the two-body approximation by r,= (I+2o), O) V _ where 0 is the Safronov number, 0 = V_, and V, is the escape velocity of the growing body. Although it is possible to gain considerable insight into planetesimal growth by simple use of equation (1), its dependence on velocity limits its usefulness unless a way isfound to calculate the relative velocity. Safronov (1962) made a major contribution to this problem by recognition that this relative velocity is not a free parameter, but is determined by the mass distribution of bodies. The mass distr_ution is in turn determined by the growth of the bodies, which in turn is dependent on the relative velocities by equation (1). Thus the mass and velocity evolution are coupled. Safronov made use of Chandrasekhar's relaxation time theory to de- velop expressions for the coupled growth of mass and velocity. He showed that a steady-state velocity distr_ution in the swarm was established as 102 PLANETARY SCIENCES a result of the balance between "gravitational stirring" that on the aver- age increased the relative velocity, and collisional damping, that decreased their relative velocity. The result was that the velocity and mass evolution were coupled in such a way that the relative velocity of the bodies was self-regulated to remain in the proper range, i.e. neither too high to pre- vent growth by fragmentation, nor too low to cause premature isolation of the growing bodies as a result of the eccentricity becoming too low. In Safronov's work the effect of gas drag on the bodies was not included. Hayashi and his coworkers (Nakagawa et al. 1983) complemented the work of Safronov and his colleagues by including the effects of gas drag, but did not include collisional damping. Despite these differences, their results are similar. The growth of the planeteslmals to bodies of ~ 1025 - 1026g begins with a steep initial distribution of bodies of nearly equal mass. With the passage of time, the larger bodies of the swarm remain of similar size and constitute a "marching front" that diminishes in number as the mass of the bodies increases. Masses of ,-, 1025g are achieved in ,-, 106 years. An alternative mode of growth was proposed by Greenberg et al. (1978). They found that instead of the orderly "marching front," runaway growth caused a single body to grow to -,, 102ag in 104 years, at which time almost all the mass of the system remained in the form of the original 1016g planeteslmals. It is now known (Patterson and Spaute 1988) that the runaway growth found by Greenberg et al. were the result of an inaccurate numerical procedure. Nevertheless, as discussed below, it now appears likely that similar runaways are expected when the problem is treated using a more complete physical theory and sufficiently accurate numerical procedures. This recent development emerged from the work of Stewart and Kaula (1980) who applied Boltzmann and Fokker-Planck equations to the problem of the velocity distribution of a swarm of planetesimals, as determined by their mutual gravitational and coUisional evolution. This work was extended byStewart and Wetherill (1988) to develop equations describing the rate of change of the velocity of a body of mass mland velocity V_ as a result of collisional and gravitational interaction with a swarm of bodies with masses m2 and velocities V2. In contrast with earlier work, these equations for the gravitational interactions contain dynamical friction terms of the form dVx (m2V] rnxVx2). (3) d---_cx These terms tend to equipartition energy between the larger and smaller members of the swarm. For equal values of Vx and V2, they cause the velocity of a larger mass ml to decrease with time. In earlier work, the gravitationally induced "stirring" was always positive-definite, as a result of using relaxation time expressions that ensured this result. AMERICAN AND SOVIET RESEARCH 103 The dV/dt equations of Stewart and Wetherill have been used to de- velop a numerical procedure for studying the evolution of the mass and velocity distribution of a growing swarm of planetesimals at a given helio- centrie distance, including gas drag, as well as gravitational and collisional interactions (Wetherill and Stewart 1988). When approached in this way it becomes clear that the coupled non- linear equations describing the velocity and size distribution of the swarm bifuricate into two general types of solutions. The first, orderly growth, was described by the Moscow and Kyoto workers. The second is "runaway" growth whereby within a local zone of the solar nebula (e.g. 02AU in width) a single body grows much faster than its neighbors and causes the mass distribution to become discontinuous at its upper end. Whether or not the runaway branch is entered depends on the physical parameters assumed for the planetesimals. More important however, are the physical processes included in the equations. In particular, inclusion of the equipartition of energy terms causes the solutions to enter the runaway branch for a very broad range of physical parameters and initial conditions. When these terms are not included, the results of Safronov, Hayashi, and their coworkers are confirmed (see Figure 1). On the other hand, when these terms are included, runaway solutions represent the normal outcome of the calculations. The origin of the runaway can be easily understood. For an initial swarm of planetesimals of equal or nearly equal mass, the mass distribution will quickly disperse as a result of stochastic differences in the collision rate and thereby the growth rate of a large number of small bodies. As a result of the equipartition of energy terms, this will quickly lead to a velocity dispersion, whereby the larger bodies have velocities, relative to a circular orbit, significantly lower than that of the more numerous smaller bodies of the swarm. The velocity of the smaller bodies is actually accelerated by the same equipartition terms that decrease the velocity of the larger bodies. A simplified illustration of this effect is shown in Figure 2. This calculation issimplified inthat effects associated with failure of the two-body approximation at low velocities and with fragmentation are not included. After only _, 3 x l0 t years, the velocities of the largest bodies relative to those of the smaller bodies has dropped byan order of magnitude. These lower velocities increased the gravitational cross-section of the larger bodies sufficiently to cause them to grow approximately 100 times larger than those bodies in which most of the mass of the swarm is located. This "midpoint mass" (nap), defined by being the mass below which half the mass of the swarm is located, is indicated on Figure 2. For bodies of this mass the Safronov number 0 is 0.6 when defined as 104 PLANETARY SCIENCES I 108_- "SAFRONOV'" CASE / | (No gasdrag, 107_ positive definite L__ gravitational acceleration) _-mm105__ 1.5X105y ears _=lO4 _L_, = Z_ _L_ 4.4X10"years "=lo2 s.sx10sv.,r, Ittl I I III I I11. [__L 1 1016 1018 1020 1022 1024 1026 1028 MASS(GRAMS) FIGURE la Evolution of the mass distribution of a swarm of planetesimals distributed between 0.99 and 1.01 AU for which the velocity distribution is determined entirely by the balance between positive-definite gravitational "pumping up" o[ velocity and collisional damping. The growth is "orderly," i.e., it does not lead to a runaway, but rather to a mass distribution in which most of the mass is concentrated in 1024 -- 1025g bodi¢_ at the upper end of the mass distribution. 0- V:(m,) (4) 2V2(mp) i.e. its relative velocity issimilar to its own escape velocity. In contrast, the value of 0L calculated using the velocity of this body and that of the largest body of the swarm has a quite high value of 21. At this early stage of evolution the growth isstill orderly and continuous (Figure 3). However, by 1.3 x 10s years, the velocities of the largest bodies have become much lower than their escape velocities (Figure 2), and a bulge has developed at the upper end of the swarm as a result of their growing much faster than the smaller bodies in the swarm. At 2.6 x 105 years, a single discontinuously distributed body with a mass -,,10_6isfound. At this time it has accumulated 13% of the swarm, and the next largest bodies are more than 100 times smaller. This runaway body will quickly capture all the residual material in the original accumulation zone, specified in this case to be 0.02 ALl in width. The orbit of the runaway body willbe nearly circular, and itwill be able AMERICAN AND SOVIET RESEARCH 105 "NAKAGAWA" CASE lOB GAS DRAG, NO COLLISIONAL DAMPING _T-0 107 POSITIVE DEFINITE GRAVITATIONAL ACCELERATION loe I 11,_ 5X 104 yearl 105-- _ _ 1.5 X 105yean; _ 104 11}": 102- 10 - I 1 _ J __ 1 lols lO2O 1_2 lO24 Io20 lO2S 10TM MASS(GRAMS) FIGURE lb Evolution of the mass distribution of a swarm in which the velocity damping is provided by gas drag, rather than by coUisional damping. The resulting distribution is similar to that of Figure la. to capture bodies approaching within several Hill sphere radii (Hill sphere radius = distance to colinear Lagrangian points). Even in the absence of competitors in neighboring zones, the runaway growth will probably self- terminate because additions to its mass (Am) willbe proportional to (AD) 2, where AD is the change in planetesimal diameter, whereas the material available to be accumulated will be proportional to AD. Depending on the initial surface density, runaway growth of this kind can be expected to produce approximately 30 to 200 bodies in the terrestrial planet region with sizes ranging from that of the Moon to that of Mars. There are a number of important physical processes that have not been included in this simplified model. These include the fragmentation of the smaller bodies of the swarm, the failure of the two-body approximation at low velocities, and the failure of the runaway body to be an effective perturber of small bodies that cross the orbit of only one runaway. These conditions are more difficult to model, but those calculations that have been made indicate that they all operate in the direction of increasing the rate of the runaway. 106 PLANETARY SCIENCES VELOCITY DISTRIBUTION DYNAMICAL FRICTION GASDRAG 103 Rip NOFRAGMENTATION =E v I- PERTURBATION BY irr"a _la_a_la_lmm_m_ NROUNRAEDWUCATYION OF o er mp _ % GRAVITATIONAL 5 102 - / _ CROSS-SECTION to 13X105"_ %_NOT ENHANCED FOR n.. ¢.J years / --_ LOWVELOCITY ft. / ,_ BODIES n- mp _3.2XlO4yem _ " , ..2.6XlO5yem 10 ,g 1 I [ I I I I I I 101e 1019 lo20 1021 1022 1023 1024 1025 lo2e 1027 MASS(GRAMS} FIGURE 2 Velocity distribution corresponding to inclusion of equipartition of energy terms. Alter 3 X 104 years, the velocities of the largest bodies drop well below that of the midpoint mass mr,. This leads to a rapid growth of the largest bodies, and ultimately to a runaway, as described in the text. GROWTH OF RUNAWAY PLANETARY EMBRYOS INTO TERRESTRIAL PLANETS Because of the depletion of material in their vicinity, it seems most likely that the runaway bodies described above will only grow to masses in the range of 6 x 1025g to 6 x 1026g, and further accumulation of a number of these "planetary embryos" will be required to form bodies of the size of Earth and Venus. Both two-dimensional and three-dimensional numerical simulations of this final accumulation of embryos into terrestrial planets have been reported. All of these simulations are in some sense "Monte Carlo" cal- culations, because even in the less demanding two-dimensional case, a complete numerical integration of several hundred bodies for the required number of orbital periods is computationally prohibitive. Even if such calculations were possible, the intrinsically chaotic nature of orbital evo- lution dominated by close encounters causes the final outcome to be so exquisitely sensitive to the initial conditions that the final outcome isessen- tially stochastic. Two-dimensional calculations have been reported by Cox and Lewis (1980); Wetherill (1980); Lecar and Aarseth (1986); and lpatov (1981a). In some of these two-dimensional cases numerical integration was carded out during the dose encounter. AMERICAN AND SOVIET RESEARCH 10'7 T=0 GASDRAG NOFRAGMENTATION 1101__ DNYONRAEMDICUACLTIOFNRICOTFION _, GRAVITATIONAL __. _ _-3.2X 10" PERTURBATIONS " _,A _' FO,RUNAWAYS =o,o6 1.3xlO%,,. lOS- I _ NOGRAVITATIONAL "_ | _k ENHANCEMENT OF _ 104 - 2.6 X105year= lo2- I I t 1.3xlOSyea. t ___L__II_ ,, II, tl _ , I__ 10TM 10TM 1020 1022 1024 1026 1028 MASS(GRAMS) FIGURE 3 Effect of introducing equipartition of energy terms on the mass distribution. The tendency toward equipartition of energy results in a velocity dispersion (Figure 2) in which the velocity (with respect to a circular orbit) of the massive bodies falls below that of the swarm. After ,'_ 105 years, a "multiple runaway" appears as a bulge in the mass distribution in the mass range 1024 -- 1025g. After 2.6 X 105 years, the largest body has swept up these larger bodies, leading to a runaway in which the mass distribution is discontinuous. The largest body has a mass of ,"-, 1026g, whereas the other remaining bodies have masses < 1024g. The three-dimensional calculations (Wetherill 1978, 1980, 1985, 1986, 1988) make use of a Monte Carlo technique based on the work of 6pik (1951) and Arnold (1965). In both the two- and three-dimensional cal- culations, the physical processes considered are mutual gravitational per- turbations, physical collisions, and mergers, and in some cases collisional fragmentations and tidal disruption (Wetherill 1986, 1988). In the work cited above it was necessary to initially confine the initial swarm to aregion smaller than the space presently occupied bythe observed terrestrial planets. This is necessary because a system of this kind nearly conserves mass, energy, and angular momentum. The terrestrial planets are so deep in the Sun's gravitational well that very little (< 5%) of the material is perturbed into hyperbolic solar system escape orbits. The loss of mass, energy, and angular momentum by this route is therefore small.