An alternative model for the formation of the asteroids

ICARUS 100, 307-325 (1992)

An Alternative Model for the Formation of the Asteroids GEORGE W . WETHERILL

Carnegie Institution of Washington, Department of Terrestrial Magnetism, 5241 Broad Branch Road, N.W., Washington, DC 20015 Received May 14, 1992; revised September 17, 1992

The growth and orbital evolution of a swarm of -1026-g "planetar), embryos," originally distributed throughout both the terrestrial planet and the asteroidal regions has been simulated using a Monte Carlo technique previously used by the author to study the formation of the terrestrial planets alone. The effects of the giant planets, primarily Jupiter, are simply assumed to be those operative at present: chaotic acceleration in resonance regions and gravitational ejection of objects by encounters with Jupiter at aphelia ~>4.75 AU. It is found that the asteroidal embryos are very effective in scattering one another into resonance regions. The resulting orbital evolution clears the asteroid belt of embryos in about half of the simulations. Embryos accumulate in the terrestrial planet region to form a stochastically determined variety of planetary configurations that are similar in total mass, specific energy, specific angular momentum, planetary size, and orbital elements to our Solar System. The small quantity of material remaining in the asteroid region has a velocity distribution very much like that of the observed asteroids. It is concluded that this model, although certainly imperfect, represents a viable alternative to models in which the initial asteroidal population is limited to bodies the size of the present asteroids. © 1992AcademicPress.Inc.

I. INTRODUCTION Almost all r e c o v e r e d meteorites are produced by collisional fragmentation of asteroidal bodies. The record of Solar System events preserved in these meteorites is of paramount significance. As indicated by their cosmic ray exposure ages, the latest of these collisions took place in relatively recent times. These recent collisions, as well as the effects of colliding with the Earth, have a minor effect on the meteorite, however. The presently observed physical and chemical states of these bodies represent the outc o m e of a long series of previous events, extending from presolar times up to the present. Among these previous events, the earlier temporal extreme is indicated by isotope anomalies that are unequilibrated inhomogenous relics of presolar interstellar grains (Anders 1988, Zinner 1988). At the other extreme is shock metamorphism produced by major asteroidal collisions in the last few hundred million years in the relatively well-

understood present-day Solar System (Turner 1981, 1988). In between we find an exquisitely detailed record of events that took place during the accumulation, metamorphism, igneous differentiation, and collisional reworking that accompanied asteroidal formation in the early Solar System (e.g., Kerridge and Matthews 1988). Use of these observations to test theories of planetary formation requires development of quantitative theoretical models of asteroid formation and evolution that are not only internally consistent, but that are also compatible with models for the formation of the terrestrial planets and the planets of the outer Solar System. A limited, but significant, amount of work has been directed toward this goal (reviewed in Wetherill 1989). The general approach of most recent efforts in this direction derives from the assumption that there never were objects in the asteroid belt significantly larger than the present large asteroids (1023-1024 g). This assumption is supported by the near-invulnerability o f significantly larger bodies to mutual collisional destruction at present asteroidal collision velocities during the 4.5 x 109-year history of the Solar System, together with the inference that if larger bodies had once been present, they would still be there. On the other hand, it is at least plausible that the original solar nebula did not exhibit the gross deficiency in surface density found in the present asteroidal region (Weidenschilling 1977). A fairly conventional way of reconciling these assumptions and observations, using reasonable primordial surface densities, is that planetesimals accumulated to form a very large n u m b e r of objects in the size range of the present asteroids on a time scale of no longer than - 1 0 7 years. It is then proposed that, in one of several conceivable ways, the massive planet Jupiter prevented the formation of larger runaway planetary embryos in the asteroid belt and thereby limited asteroids to their present size. It is also proposed that the same, or other primordial effects associated with the formation of Jupiter (and perhaps Saturn as well) accelerated the asteroids to their present relative velocities, averaging - 5 km/sec, well b e y o n d those achievable by gravitational

307 0019-1035/92 $5.00 Copyright © 1992 by Academic Press, Inc. All rights of reproduction in any form reserved.

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GEORGE w. WETHER1LL

perturbations between the asteroids themselves. Collisions between the primordial asteroidal bodies can then be expected to initiate a fragmentation cascade, leading to the present size distribution of the asteroids, as well as to the loss of almost all the material originally in the asteroid belt by mutual fragmentation and PoyntingRobertson drag. It is quite possible that an adequate theory of asteroidal origin could be developed along these general lines. To do so it will be necessary to develop a quantitative understanding of a number of things; e.g.,

I do not intend to treat the model described earlier as a "straw-person" to be vanquished by the alternative presented here. Rather, the purpose of this work is to extend the range of possibilities, in the hope that doing so will facilitate comparison with observations and thereby further the achievement of a correct understanding of these important events.

(i) The way in which long-range perturbations by the giant planets could, on the necessary short time scale, have prevented runaway growth from forming asteroidal bodies with masses >1025 g. (ii) How external perturbations were capable of increasing asteroid velocities to their present values, and thereby initiate the present destructional regime. (iii) At least in its simpler forms, quantitative development of the above model leads to an asteroid belt in which the initial mass and number of bodies were very large. Reliance on asteroidal collisions to reduce the population of the asteroid belt to that observed today may require the present asteroid belt to be more highly collisionally evolved than it appears to be (Davis et al. 1979, 1988). Furthermore, under these circumstances, the present-day smaller (< 100 km diameter) meteorite sources should represent the residue of a large number of different original parent bodies. This situation is only partially alleviated by derivation of meteorites from restricted asteroidal regions, such as the vicinity of the 3 : 1 Jovian commensurability resonance at 2.5 AU (Wetherill 1968, 1985a, 1987, Wisdom 1985). This abundance of primordial parent bodies must be reconciled with the evidence (Pellas and StOrzer 1981, G6pel et al. 1991) for derivation of entire major classes of ordinary chondrites from single "onionskin" primordial parents.

The basic approach adopted in this investigation is to follow the three-dimensional orbital evolution of about two hundred 1025- to 1027-g bodies, as they gravitationally perturb one another and undergo mutual collisions, leading to merger and growth, or sometimes to fragmentation. The intrinsically stochastic Monte Carlo technique employed is derived from the work of Opik (1951, 1976) and Arnold (1965) and has been used by the author in many simulations of multiplanet accumulation in the terrestrial planet region (Wetherili 1980, 1985b, 1988a,b, 1990, 1991c). In the more recent of these earlier investigations, the initial conditions are those expected to correspond to the rapid (-<105 years) formation of runaway terrestrial planetary embryos at 1 AU, in nearly circular coplanar orbits, as indicated by studies of the early low-velocity stage of planetary growth in which 1- to 10-km planetesimals accumulate to form 3 × 1025-to 3 × 1026-g (approximately lunar- to Mercury-size) bodies (Wetherill and Stewart 1989). A practical difficulty in describing the assumptions, parameters, and results of this study arises from the need to describe in an efficient manner the large number of simulations of multiplanet accumulation that were found to be necessary. Altogether, 434 calculations are reported here. The need for so many calculations comes about in two ways:

In this article an alternative approach to asteroid formation is proposed and explored. The fundamental assumption is that Jupiter failed to prevent runaways in the asteroidal region, and that large -1026-g potential planetary embryos formed in the present asteroid belt prior to the completion of Jupiter's growth. The subsequent evolution of these objects is then followed in a reasonably quantitative but necessarily preliminary manner, with attention directed toward the goal of eventual comparison with observational astronomical and meteoritic data. In this alternative model, the principal mechanisms responsible for clearing the asteroid belt of its original mass are gravitational, rather than collisional. Some results, based on this general model, have been presented in the context of the frequency of occurrence of Earth-like planets in other planetary systems (Wetherill 1991a).

(i) The intrinsically stochastic nature of the simulated, and presumably the actual, accumulation process. It would be easy to be misled into thinking that the differences one finds between the results of calculations is a result of the use of different parameters, when, in fact, these differences are likely to be of purely stochastic origin. For this reason, all calculations were repeated about 15 times using different sequences of the random numbers used in the Monte Carlo procedure. (ii) The present lack of detailed understanding of many of the physical processes involved, as well as of the actual conditions in the early Solar System. This requires consideration of a wide range of parameters in order to identify the extent to which an effect is a general property of accumulating systems of this kind, or if it requires special, and perhaps improbable, conditions for it to be realized.

II. METHOD FOR MODELLING SIMULTANEOUS MULTIPLANET GROWTH IN THE TERRESTRIAL PLANET REGION AND THE ASTEROID BELT

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FORMATION OF THE ASTEROIDS

The latter problem has been handled by the expedient of defining a "nominal model" corresponding to assumptions and initial conditions that are treated in some detail. The sensitivity of the results to the particular nature of the nominal model will then be explored by more limited studies of alternative parameters and assumptions. It will be seen that a number of calculated results are insensitive to these variations. Robust results of this kind, when they are consistent with observation, are of particular interest because they represent good candidates for aspects of this work that are more likely to survive future advances in understanding. Furthermore, recalcitrant differences between calculation and observation can identify the need for conceptual advances that go beyond simple adjustment of parameters. A. Initial Conditions

During the last several years, considerable progress has been made by extending .the validity of calculations of the growth of 1- to 10-kin planetesimals to predict the formation of -1026-g runaway embryos in the inner Solar System. It is now possible to include the size and velocity region in which three-body (Sun, embryo, planetesimal) effects are of major importance (Wetherill and Cox 1984, 1985, Ida and Nakazawa 1988, 1989, Greenzweig and Lissauer 1990, 1992, Ida 1990). Toward the end of the process of embryo growth it seems most likely that, as the supply of residual planetesimals and/or nebular gas density decreases, the long-range positive velocity perturbations between embryos in neighboring concentric, nearly circular, orbits become comparable in magnitude to the negative velocity changes resulting from equipartition of energy. At some point this will permit previously isolated embryos to begin to cross one another's orbits. It is commonly assumed that at this time the embryos were spaced by about 2V/3 mutual Hill sphere radii (RH), where

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and a is the semimajor axis, and M~, M2, and M o are the embryo masses and the solar mass, respectively (Birn 1973). Under the same assumptions, the mass of the embryo will be given by M e = 8 V / 2 x 31/47r3/2a3o-3/2M~ I/2

= 6.21 × lOZ4a3o-3/2grams,

(2)

where o- is the surface mass density of the material available for accumulation into embryos (Lissauer 1987). For the nominal case, the initial spacing and mass of the embryos is obtained from Eqs. (1) and (2) by assuming a surface density of 6.2 g/cm 2 at 1 AU and varying with

heliocentric distance as a -3/2 (Weidenschilling 1977). In order to avoid effects resulting from the possibly nonphysical monotonic nature of the variations of initial embryo mass with semimajor axis given by Eq. (2), this monotonic distribution was assumed to be modulated by a stochastic component, introduced by using for each body a value of the surface density (o-) in Eq. (2) that randomly varies from 0.75 to 1.25 times the calculated value. Starting at 3.8 AU, the first embryo mass is calculated from Eq. (2). The spacing between embryos is then calculated from Eq. (1) as 2V~ RH, thereby defining the position of the next embryo, the mass of which can then be calculated from Eq. (2). This procedure is repeated until the semimajor axis is less than 0.55 AU, after which the surface density is first truncated exponentially, then cut off at 0.45 AU. The outer boundary of the embryo swarm was taken to be 3.8 AU. In this regard these calculations differ greatly from my earlier published work, in which the initial swarm was confined to the immediate region of the present planets Earth and Venus. Initial eccentricities were chosen such that the orbits of neighboring embryos were in shallow orbital crossing, eo = 2.4Rn/a o .....

(3)

where a0 is the averaged initial semimajor axis of an embryo and its nearest neighbor on the low mass side. Initial inclinations were usually assumed to be one-half of the initial eccentricities. It was assumed that the time required for a runaway embryo to form was proportional to its orbital period, i.e., proportional to a 3/2 and to 1/o-. The embryos were assumed to be created after 80,000 years at 1 AU and their time of formation at other heliocentric distances was assumed to vary with a and o- in this manner. The initial embryo swarm used for the nominal model is shown in Fig. 1. The total initial mass is 2.23 × 1028 g. This is about four Earth masses and about twice the total mass of the present planetary system out to 5 AU. In addition, 24 °'test" bodies of dynamically negligible mass were uniformly distributed in the semimajor axis between 2.15 and 3.3 AU, in order to learn something about the velocity distribution of residual small asteroidal bodies that escaped the usual fates of being ejected, totally destroyed by collisions, or captured by a growing embryo. It was assumed that the test bodies did not collide with other bodies, but were subject to gravitational perturbations, including gravitational ejection from the system. B. Gravitational Effects Caused by Jupiter and Saturn

Basically, the effect of these planets on the evolution of the swarm was assumed to be the same as their effects in the present Solar System. Indeed, the present study

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can be thought of as an exploration of how operation of these more familiar mechanisms would have affected the evolution of a population of growing embryos interior to these planets, without assuming any special cosmogonic effects, such as sweeping secular resonances (Ward 1980) or Jupiter-scattered planetesimals (Safronov 1979, Kaula and Bigeleisen 1975, Ip 1987, Wetherill 1989). If, in fact, such cosmogonic effects were also important, they would make an additional contribution to the orbital evolution that can be considered at some future time. The formation of Jupiter was assumed to lag behind the starting times of the asteroids, usually occurring 5 myr. after the early runaways formed in the terrestrial planet region. The formation of Saturn was assumed to occur at 10 myr. The gravitational effects attributable to these planets began at the time they were formed. No noticeable difference was found for calculations where this delay interval was doubled to l0 myr. and 20 myr. The effects of these outer planets were assumed to be twofold: (i) Ejection into hyperbolic Solar System escape orbits of those objects whose aphelia come too close to Jupiter's orbit. Both numerical integrations (Lecar and Franklin 1973) and Monte Carlo investigations (Wetherill 1979) lead to the result that the residual Jupiter-crossing lifetimes of asteroidal bodies with aphelia beyond about 4.75 AU is ~105 years, and only rarely do such bodies return to asteroidal orbits for longer times before ejection from

the Solar System by major planet perturbations. For this reason, in the present calculations when the aphelion of a planet exceeded 4.75 AU, it was assumed to be lost and was removed from the swarm. It would be possible to explicitly consider the evolution of its Jupiter-crossing orbit. This is an effective allocation of effort, however, only for problems in which the rare capture of Jupiter crossers into asteroidal orbits is important, as is the case for decoupling of present-day Jupiter-family comets (Wetherill 1991b). (ii) Commensurability and secular resonances. The other major effect of Jupiter and Saturn is the acceleration of asteroidai bodies in the vicinity of several regions in which the orbital period of Jupiter is a simple integer fraction of the asteroid's period (commensurability resonances) or is near the secular resonances where the rate of precession of the asteroid's longitude of perihelion or the longitude of its node is in resonance with one of the normal modes of the planetary system (Williams and Faulkner 1981, Knezevic e t a l . 1991). The commensurability resonances are probably the cause of all the strongly depleted observed Kirkwood gaps in the present asteroid belt. In a somewhat similar way, the u s secular resonance defines a depleted region in a, e, and i space in the inner Solar System, whereas the strong v6 resonance, in combination with perturbations by Mars, defines the inner boundary of the main asteroid belt. Perturbations by Saturn make the largest contribution to the v6 resonance. The position and assumed effective widths of the resonances used in these calculations are given in Table I. At the present time, the 3:1 commensurability resonance at 2.5 AU and the v6 secular resonance in the inner asteroid belt appear to be primarily responsible for delivery of asteroidal collision fragments to the vicinity of the Earth in the form of meteorites, as well as of larger Earthapproaching Apollo-Amor objects. After the formation of Jupiter (and Saturn for the v6 resonance), such resonance regions must have played a similar role in the early Solar System. If, for some reason, the semimajor axis of the outer planets, or the normal modes of the planetary system, were somewhat different from those found today, of course the positions of these resonances would also be different. Nevertheless, insofar as these effects are primarily dependent on the existence and strength of the resonances, and not on their precise position, it is believed possible to obtain considerable insight into processes that operated in the early Solar System using data appropriate for the present-day asteroidal resonances. The dynamical importance of the 3 : 1, 5 : 2, 7 : 3, and 2 : 1 Jovian commensurability resonances is demonstrated by the striking near-absence of observed asteroids in their vicinity. With the possible exception of the 2 : 1 resonance at 3.27 AU, theoretical and numerical studies indicate the

FORMATION OF THE A~TEROIDS TABLE I A. Commensurability Resonances Resonance 4:1 3:1 5:2 7:3 2"1 9:5 7:4 5:3 8:5

Position (AU) 2.05 to 2.07 2.48 to 2.52 2.79 to 2.81 2.95 to 2.97 3.26 to 3.28 3.50 to 3.52 3.57 to 3.59 3.68 to 3.72 3.78 to 3.82

B. Secular Resonances /25 /2~ /26 /)16 V16 /216

2.00 2.08 2.20 2.52 3.35 1.49 1.69 1.89

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2.02 2.12 2.48 3.10 3.68 1.51 1.71 1.91

(i < I0 °) (11.5 ° < i < 13.5 °) (18.9 ° < i < 21.2 °) (18.9 ° < i < 21.2 °) (0.20 < e < 0.30) (0.20 < e < 0.30) (e < 0.2)

presence of an orbitally chaotic zone in the vicinity of these resonances, leading to a more or less random walk increase in eccentricity of bodies in the resonance. Although the low order of the 2 : 1 resonances suggests that perturbations in this region should be especially strong, mapping of its chaotic zone indicates that it is weak, or possibly absent. (Wisdom 1983, Froeschl6 and Greenberg 1989, Murray 1986, Yoshikawa 1991). For this reason, the width of this resonance (Table I) was not assumed to be larger than its adjacent higher order neighbors. Williams (1969) has shown that proximity to the /'6 resonance also results in large increases in eccentricity. In addition, there are several resonances (4: 1, 9 : 5 , 7 : 4, 5 : 3, 8 : 5) outside the present boundaries of the asteroid belt whose effects would probably be of comparable prominence, except for the fact that there are very few asteroids in their vicinity, at least partly a result of these resonances. All of these commensurability resonances, as well as the/'6 and/'5 secular resonances, cause large variations in eccentricity, but have little effect on inclination. Eccentricity variations are of primary importance to orbital evolution and planetary growth because the value of the eccentricity determines the positions of the perihelia and the aphelia of the planetary embryos and therefore determines whether or not close encounters or collisions between embryos can take place. The values of eccentricity also

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determine when the aphelia are close enough to Jupiter's orbit to cause their orbits to be unstable with respect to ejection from the Solar System. The only " i n c l i n a t i o n " resonance likely to be of importance to the problems under consideration here is the/'16 resonance near 2.0 AU. It is possible that large amplitude fluctuations in inclination, associated with this resonance, will cause the inclination of some of the embryos to random walk up to values of 15° to 20 ° . These increases in inclination caused by this resonance, together with changes in semimajor axis caused by close encounters, may insert bodies into the otherwise nearly inaccessible high inclination region of orbital element space in which the strong/'6 resonance is located at semimajor axes beyond 2.2 AU (Williams and Faulkner 1981, Knezevic et al. 1991). The importance of eccentricity variations caused by the /'6 resonance throughout most of the asteroid belt could thereby be controlled by the effectiveness of the Vl6 inclination resonance. Present knowledge of perturbations associated with these resonances is insufficient to allow anything approaching detailed simulation of their effects. Furthermore, these effects are likely to be somewhat different from those occurring in the present Solar System because of the occurrence of close encounters between e m b r y o s while one of the embryos is in a resonant region. In these calculations the effects of the resonances are introduced in the following necessarily simplified manner. While a body is in a resonant region (Table I), it was assumed that the eccentricity (inclination for/'16) of the body undergoes a random walk described by Ae = _+V'-~R,

(4)

where a e is the change in eccentricity, D is a characteristic diffusion time associated with the resonance, and tR is the time interval between orbital changes resulting from perturbations of the e m b r y o in the resonance by another embryo, as determined by the Monte Carlo program. This time interval will vary because of the stochastic nature of the close encounters between embryos. The sign of the eccentricity change is chosen at random. The magnitude of the diffusion coefficient D is treated as a free parameter. Detailed results will be given for values of this quantity ranging from D = (1 myr -1) to D = (~) myr -I. It was also assumed that the eccentricities resulting from the resonant perturbations do not b e c o m e less than 0.02 nor larger than 0.9. In the case of the/'16 inclination resonance, a random walk in inclination was calculated in the same way using an inclination diffusion coefficient (DI) of ~o myr-~ and a maximum resonance-induced inclination of 23 ° . It was found that for the most part the important factor is the product of the diffusion coefficient and the width of the resonance, this product representing the

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GEORGE W. WETHERILL

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FIG. 2. An e x a m p l e of the evolution of the eccentricity of a test body initially at 2.55 A U . Its orbital evolution begins when its neighboring e m b r y o s form at 1 myr. Until about 5 myr, the eccentricity changes only as a result of a series of close e n c o u n t e r s with embryos. At that time, after the formation of Jupiter, it is perturbed into a resonance in which a r a n d o m walk in eccentricity c a u s e s rapidly changing excursions in eccentricity, which end when a chance close e n c o u n t e r r e m o v e s the body from a resonance. At about 10 myr, the body is again in a resonance region, intervals of r e s o n a n t and n o n r e s o n a n t acceleration continue until 160 myr, w h e n the aphelion b e c o m e s >4.75 A U and the body is a s s u m e d to be ejected in a hyperbolic orbit, following a close e n c o u n t e r that r e m o v e s the body from the "'protective" effect of the resonance.

"strength" of the resonance. For this reason only one of these parameters was varied. An example of the variation of eccentricity of a body as a function of time is shown in Fig. 2. The times at which the body is in a resonance region are characterized by large amplitude fluctuations in eccentricity. These will continue until a sufficiently close encounter with another embryo removes the body from the resonance. After such an encounter the eccentricity remains nearly constant for relatively long periods of time, changing abruptly following a close encounter. From time to time, the body is perturbed back into a resonance, initiating another period of large amplitude excursions in eccentricity. Eventually (in this case at 160 myr) the increase in eccentricity caused the aphelion of the body to exceed 4.75 AU, making it vulnerable to ejection from the Solar System by Jupiter. In these calculations ejection was not permitted while the body was actually in the resonance, because of the probable protection from close encounters with Jupiter by a libration condition (Milani et al. 1989). Following removal of the body from the resonance, a perturbation by another embryo while the aphelion was still beyond 4.75 AU resulted in ejection from the Solar System. It was also found that ignoring this protection mechanism had no noticeable effect on the results of the calculations, however.

Bodies distributed in mass and semimajor axis, as shown in Fig. I, were permitted to spontaneously perturb and collide with one another, following their " c r e a t i o n " at a time that varies with semimajor axis and surface density as described in Section IIB. These perturbations, collisions, and resulting orbital evolution were followed by a Monte Carlo technique described previously (Wetherill 1986). The probability of each body making a close encounter with each of the other bodies during the first time step is calculated by use of the collision formula of Opik (1951), including the effect of gravitational focusing, designating the body with the lowest eccentricity the "target" in Opik's expression. These probabilities were augmented by the procedure for including encounters within 10 two-body Tisserand spheres of influence radii, as described by Arnold (1965). If this probability was greater than a random number between 0 and 1, an actual encounter is assumed to occur. If no encounter occurs, the orbits are left unchanged. On the other hand, if an encounter does take place, the minimum unperturbed separation between the bodies, in units of the gravitational collision radius, is then chosen at random, appropriately geometrically weighted so that the probability of an encounter in the interval Ar was proportional to r, where r is the minimum separation distance in these units. If the separation distance is less than the gravitational radius, an impact is scored. The mass of the larger body is augmented by that of the smaller body (identified arbitrarily if the two bodies are of equal mass). The smaller body is then removed from the swarm. A new orbit is calculated for the larger body by use of angular momentum conservation in the reference frame of the center of mass of the two bodies. If an impact does not occur, the mutual two-body scattering of the encountering bodies is calculated using the expressions used by Opik (1951) and Arnold (1965), assuming encounter at the selected distance and at a random azimuth on a target circle fixed on one of the bodies and oriented perpendicular to their relative-velocity vector. The resulting changes in the three relative-velocity components are then calculated in a reference frame in which the center of mass moves on a circular Keplerian orbit at the heliocentric distance of the encounter. These relative velocities are then converted into new heliocentric velocities, and new perturbed orbital elements are calculated for both of the bodies. The validity of calculating close encounter perturbations in this way has been investigated by numerical integration of the three-body equations of motion (Wetherill and Cox 1984). As expected from Safronov's analytical theory, almost all of the encounters are in the range of relative velocity/escape velocity > 0.35, for which the two-body algorithm is a good approximation. Minor modifications, directed toward more appropriate

FORMATION OF THE ASTEROIDS

313

treatment of the rare values of V/Ve < 0.1, were to increase the probability of impact by a factor of 4 when V/Ve is between 0.1 and 0.03 and to limit the maximum two-body gravitational cross-section enhancement to a rarely achieved factor of 3000 (corresponding to V/Ve = 0.018). After all the encounters found to occur during the first time step were calculated, new encounter probabilities with all the other bodies were calculated for those bodies whose orbital elements had changed. The above procedure was then repeated for a second time step. The duration of the time steps were automatically adjusted so that for the pair of bodies with the highest encounter probability during the time step, the probability of two encounters remained about 20%. For almost all the bodies, the probability of two encounters was very much less than this. The procedure was repeated until all the remaining bodies were in noncrossing orbits. It sometimes happens that the kinetic energy of the impact is sufficiently high that disruption of the larger body, rather than its growth, is the most likely outcome of the encounter. A simple criterion for identifying catastrophic collisions was used. Fragmentation was assumed to occur when the impact energy in the center of mass frame of the two bodies exceeded the sum of the gravitational binding energy of a body having the combined mass of the projectile and target and an additional energy of 3 × 10 ~° erg/g, representing the energy that went into comminuting, heating, melting, and partially vaporizing the two bodies. When it is found that the conditions for breakup are met, the body is split into four equal-size pieces with nearly the same orbits. In order to limit the number of bodies whose orbital evolution had to be followed, objects were assumed to be totally disrupted and lost from the swarm when they became smaller than 8 x 10 25 g. The total mass lost in this way is typically only - 2 % of the mass of the system and has a negligible effect on the overall growth of the planets. Of course it is not claimed that this is a physically realistic mode of breakup. It would be more acceptable to break the bodies into some kind of a power-law size spectrum or in the case of superfragmentation of the sort studied by Cameron et al. (1988) into a single residual body of considerably smaller mass. The simplified breakup model was chosen in order to:

tion, without introducing any more fragments than necessary. This permits learning something about the orbital evolution of small fragments of embryos. When more than four fragments are assumed, a divergent fragmentation cascade is found to develop. Even when this cascade is arbitrarily terminated at e.g., 5 x 10 24 g, the time required for each Monte Carlo calculation increases by a factor of about 20 because of the large number of bodies whose orbital evolution must be followed. For an initial investigation of the sort reported here, this would preclude carrying out the number of calculations required to avoid being misled by the stochasticity of the outcome of the calculations. A study in which such a cascade does develop is underway at present.

(i) Avoid destructive collisions being treated as a contribution to planetary growth. (ii) Mark bodies that are collisional fragments, so that in cases where they remain as "final planets," appropriate reservations can be made concerning how seriously one should take the calculated mass of such final planets. (iii) Introduce into the calculation an accretion-fragmentation hierarchy at the lower end of the mass distribu-

As mentioned earlier, the initial swarm used for this study differs greatly from that used in my earlier calculations. In the earlier calculations, the mass, specific energy, and specific angular momentum were set nearly equal to their present values. In the present calculations the initial angular momentum is significantly greater than that observed at present. The initial energy is considerably higher (less negative) and the initial mass is larger. In

III. RESULTS OF CALCULATIONS USING THE NOMINAL MODEL

A. General Comments A total of 115 Monte Carlo calculations were made using the parameters and assumptions of the "nominal model" described in the previous section. Calculations were made for five values of the diffusion parameter D (Eq. (4)) of 10, 5, 2.5, 1.25, and 0.61 x 10 -7 year i using resonance positions and widths given in Table I. In addition, sets of calculations were made with the initial inclination equal to only 11o the eccentricity, and for the case where the inclination diffusion coefficient for the vt6 resonance was equal to (~o) m y r - ~, rather than its usual value of (~0) myr -I. Attention was directed toward four aspects of the results: (i) The degree to which the specific energy, specific angular momentum, and mass of the swarm spontaneously evolve toward the observed present-day values of these quantities. (ii) The orbital distribution, number, and mass of the final planets. (iii) The degree to which the assumed model is successful in clearing the asteroidal region between 2.17 and 3.3 AU of residual embryos. (iv) The velocity and semimajor axis distribution of the few test bodies and collision fragments that still remain in the asteroid belt at the end of the calculation.

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G E O R G E W. W E T H E R I L L

the initial value of 2.25 x 10z~ g (Fig. 1) and is similar to the total present mass of the terrestrial planet and asteroidal region (1.18 × 1028 g). In my earlier simulations of the formation of the terrestrial planets, all the present mass of the terrestrial planets was assumed to initially be within 1.1 AU. In the present work, only 65% of the initial mass was within I AU. The new physical mechanisms included in the present work are found to be adequate to provide the necessary migration of mass, both outward and inward, and to redistribute the energy and angular momentum, along with the mass, in an acceptable manner.

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The distribution of the "final planets" found in these simulations is shown in Fig. 5 and can be compared with the initial distribution (Fig. 1). The principal feature demonstrated by this figure is the concentration of >1027-g planets in the region interior to the present asteroid belt. The largest bodies are found between 0.5 and 1.2 AU. On the average, 4.0 final planets with masses greater than [026 g are found interior to 2 AU. The average number of final planets of mass greater than 3.5 x 1027 g is 1.6 per simulation. 50% of the simulations contain a " M e r c u r y , " i.e., a final planet of m a s s < 1 0 2 7 g interior to 0.48 AU. Most of these small bodies with small heliocentric distances are collision fragments. In 94% of the simulations a final planet larger than 1026 g appears in the position of " M a r s . " This final planet tends to be larger than our planet Mars. On the average, per simulation, 0.52 final bodies greater than twice the mass of Mars are found in

order for this model to be a viable candidate for further consideration, it is necessary that the system spontaneously evolve toward the observed values.

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the outer terrestrial planet region between 1.3 and 2.15 AU. No obvious differences are seen between the final planetary distribution resulting from the alternative versions of the nominal model, indicated by the different symbols in Figs. 3 and 5. In a general way the calculated distribution of terrestrial planets is similar to the present distribution of terrestrial planets. In detail, it differs from that observed. For the most part, this can be considered a consequence of the numerous stochastic fluctuations one would expect during the "random walk" orbital evolution from the very different initial distribution of mass, angular momentum, and energy illustrated by comparing Figs. 1 and 5 and by Fig. 3. These final configurations are similar to those reported earlier, in which the initial distribution was confined to a region more narrow than the present range of the terrestrial planets (Wetherill 1986, 1988a). The principal difference is the more frequent occurrence in the present study of a body more than twice the mass of Mars as the outermost terrestrial planet. It is possible that this simply represents a statistical fluctuation in the formation of our Solar System, because about half of the simulated planetary systems do not contain such an object. In that case the answer to the question " W h y isn't Mars more similar in size to Earth and Venus?" would be simply that it was unlucky. On the other hand, this difference might be caused by some more important process that has been

315

ignored by the oversimplicity of the model. If so, the problem is not solved by use of the variations in the nominal model to be described in the next section. The fundamental cause for the frequent presence of a fairly large planet in this region is the assumed absence of major resonances that can produce large scale random walks in eccentricity at semimajor axes interior to 2 AU, together with the fact that even a very high eccentricity body with a semimajor axis in that region will have its aphelion comfortably inside the orbit of Jupiter. As pointed out earlier (Wetherill 1991a), these facts cause the terrestrial planet region to be a safe haven for planet formation, and the presence of planets resembling Earth in size and heliocentric distance is likely to be a common feature of planetary systems associated with solar-mass stars. E. Final Eccentricities and Inclinations

The final eccentricities and inclinations with respect to the plane of the Solar System of the larger (>3.5 × l027 g) simulated terrestrial planets, with final semimajor axes between 0.6 and 1.3 AU, are shown in Fig. 6. Low values of these quantities, as observed for Earth and Venus, are common, but higher values also sometimes are found. It is possible that the occurrence of these higher values should not be considered a prediction of the general model under discussion. It may be an artifact of the oversimplified way in which fragmentation was introduced. Work in progress, in which a full collisional fragmentation cascade is produced, suggests that large eccentricities and inclinations should be much more rare. These large values are produced by the large stochastic impulses produced by

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FIG. 7. "'Giant" impacts (>10Z6 g) onto bodies larger than 5 x 1027g found in one of the s u b s e t s (DR = ] m y r ~) of the nominal model calculations. T h e results are typical of all the subsets. About one such impact more m a s s i v e than the o b s e r v e d planet Mars is found for those calculations in which an " E a r t h - V e n u s " - s i z e d final planet is found. In addition, an average of two impacts are found intermediate in mass b e t w e e n Mercury and Mars, and about eight impacts of bodies between 1027 g (1.4 lunar m a s s e s ) and the m a s s of Mercury. The larger impacts o c c u r mostly b e t w e e n 50 and 100 myr.

giant impacts. In the presence of a larger proportion of collision fragments, it may be expected that these fluctuations will be damped by the tendency toward equipartition of energy between the largest and smaller bodies. F. Giant Impacts As in my earlier investigations, it is found that the accumulation process is punctuated by giant impacts in which major portions of the final mass of an Earth-size planet are contributed by discrete accumulation events. The o c c u r r e n c e of these giant impacts in the present study is illustrated in Fig. 7. The points plotted represent all the impacts of bodies larger than 10 27 g onto bodies of final mass greater than 5 × 1027 g that were found for a subset of 10 of the 19 calculations made with a resonance diffusion coefficient of (¼) myr : (pentagons in Figs. 3 and 5). The subset was chosen by exclusion of simulations for which no bodies as large as 5 × 10 27 g were formed, or for which the accumulation histories of such large bodies were complicated by giant impacts that caused fragmentation and reassemblage of these bodies during their growth. This latter p h e n o m e n o n may be of importance, but it is too sensitive to uncertainties in fragmentation modeling to pursue at this time. No significant differences were

found between the distribution of giant impacts shown in Fig. 7 and those found for the other six variants of the nominal model. The frequency and size of these giant impacts is similar to those reported earlier (e.g., Wetherill 1990). On the average about one impact more massive than Mars and about two with mass intermediate between Mercury and Mars occur per simulation. A major difference between the present investigation and my earlier results is found in the temporal distribution of the giant impacts. In the present work the larger impacts are distributed in time principally between about 20 and 200 myr, whereas in the earlier work these massive impacts almost always took place within 50 myr. This difference is a consequence of the greater importance, in the present study, of bodies with larger semimajor axes and therefore longer orbital periods. Because these large impacts contribute a major portion of the planet's final mass, this has the consequence that, in some cases, the growth of Earth and Venus required about 200 myr for its completion, whereas in the earlier work the growth of such large terrestrial planets was very nearly complete by 100 myr. G. Residual Material in the Asteroid Belt An indication of the predicted final velocity distribution of residual material in the asteroid belt can be obtained from Fig. 8. It may be expected that, within the context of the model under consideration, three kinds of residual bodies should be found in the asteroid belt after the principal period of planetary growth. These are: (i) Residual planetary embryos, which in some cases have grown beyond their original mass by the accumulation of smaller bodies. It is found that "asteroidal planets" occur in about half (60 out of 115) of these simulations. In cases where an e m b r y o remains in the asteroid belt, with three exceptions, only one such body is found. This can be understood to be a consequence of the process by which embryos are removed from the belt. Mutual perturbations between embryos cause them to perturb one another into resonant regions, initiating an orbital evolution that eventually leads to capture by a terrestrial planet or ejection into the influence of Jupiter. How is the last such e m b r y o to be removed? Often this is accomplished by perturbation into a resonance by a body that is itself vulnerable to removal; for example, if it is crossing the orbit of a terrestrial planet. Jupiter-crossing planetesimals, if they exist, could accomplish the same result. In other cases no scavenger of this kind is available, and a small planet becomes stranded in the asteroid belt. Stranded asteroidal planets are clearly of no relevance to our own asteroid belt, however, because no bodies of this size are

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observed. On the other hand, the frequent occurrence of such objects in other planetary systems is a prediction of this model. (ii) Residual partly grown ~<1024-g planetesimals that had not been accumulated by embryos at the time the accumulation process under consideration here began. The stage of growth under consideration is thought to follow an earlier stage in which - 1- to 10-km-radius planetesimals accumulated at very low relative velocities to form the -1026-g planetary embryos that constitute the original swarm chosen as the initial state of the present model (Fig. 1). An earlier study of this low velocity stage of accumulation (Wetherill and Stewart 1989) is being extended to include a number of relevant physical phenomena that were not included before (Williams et al. 1992). It seems likely that the low velocity stage ended when the long range perturbations between neighboring embryos became sufficiently strong to permit their orbits to cross, causing them to accelerate one another to velocities in the vicinity of the mutual escape velocities. It is not at all clear that this "liberation of the embryos" was necessarily delayed until all the nonrunaway 10 ]6- to 1023-g planetesimals were accumulated by embryos. Rather, it now seems quite possible that reduction of the "equipartition drag" that offsets the mutual acceleration of the embryos took place while a significant proportion, e.g., 10 to 30%, of the planetesimals remained in the system. During the final stage of planetary growth considered in the present study, these residual bodies would be subjected to the same gravitational effects experienced by the embryos, and in addition, for actual bodies, they would be largely destroyed by mutual collisions. These processes may be expected to remove almost all of an original population of such asteroid-size bodies, but those few that survived may be identified with some portion of the present asteroidal population, perhaps undifferentiated asteroids of both the C- and S-classes, or perhaps of only the C class. The expected final distribution of surviving planetesimals of this kind can be estimated from the final positions

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FIG. 8. Calculated and observed final positions and noncircular components of velocities for "indestructible" test bodies (see Section 2A) and for collision fragments that are found in the asteroid belt at the end of the calculation. (a) On the average, 3% of the original test bodies survived gravitational removal from the asteroid belt. The nominal case is represented by the crosses. The solid squares represent calculations

in which it was assumed that all the e m b r y o s were formed at the same time, the open circles represent cases where the vt6 secular resonance and the high inclination portion of the v 6 secular resonance were ignored, and the open squares represent calculations in which fragmentation was not included. These variant models are discussed later in the text. Although their original distribution was uniform, the surviving objects tend to concentrate in the outer portion of the asteroid belt. (b) Collision fragments that remain in the asteroid belt. The symbols have the same meanings as they have in Fig. 8a. Of course, in this case, no open squares are formed because they correspond to calculations in which the effect of collision was not included. (c) Observed velocities of asteroids classified as C (solid squares) and S (open squares) types by T e d e s c o et a/. (1989). The calculated distribution is quite similar to those observed.

318

GEORGE W. W E T H E R I L L

of surviving 2 x 1024-g test bodies, 24 of which were included in the original swarm, as described in Section IIA. On the average 3% of these bodies remain. These bodies were assumed to be immune to loss by collision or capture. Thus, the gravitational clearing alone appears to be about 97% efficient. Those still remaining at the end of these calculations are protected surrogates, representing the o u t c o m e of an extensive gravitational and collisional winnowing process. The velocity of these bodies relative to a circular orbit at their heliocentric distance is proportional to the quantity (e 2 + sin 2 i) V2, where e and i are the final eccentricities and inclinations, respectively. The distribution of this quantity as a function of position in the asteroid belt is shown in Fig. 8a. The crosses ( x ) represent the surviving residual test bodies for the 115 calculations carried out for the nominal model. These results can be compared with the observed velocity distribution of C and S asteroids plotted in Fig. 8c (Tedesco et al. 1989, Knezevic and Milani 1989). As may be seen by this comparison, this model provides an asteroidal population with a velocity distribution quite similar to that observed. It may also be noted that, although the test bodies were originally uniformly distributed in semimajor axis, the final distribution is concentrated toward the outer asteroid belt. Residual test bodies from the simulations that correspond to modifications o f the nominal model are represented by the remaining symbols in Fig. 8a. These alternatives will be discussed in the next section. It may be seen, however, that there is no obvious difference between the nominal model and its alternatives in this regard. (iii) Residual fragments in the asteroid belt. A third type of material that should be found in the asteroid belt is e m b r y o fragments that survived the gravitational and collisional processes that r e m o v e d most of them from the system. Because of the oversimplified way in which fragmentation was considered, no quantitative statements can be made about the absolute mass or the size distribution of these bodies. On the other hand, the orbital evolution of small, asteroid-size fragments is independent of their mass, and the heliocentric distances and velocities of such bodies should be similar to those of the fragments that remain at the end of these simulations. The distribution of these fragment velocities with semimajor axis produced by the nominal model are shown by the ( x ) symbol in Fig. 8b, together with fragments made with additional simulations to be discussed in the next section. The velocity distribution is very similar to that of the residual test bodies, but they tend to be concentrated toward the inner asteroid belt, in contrast to the concentration of the residual test bodies toward the outer asteroid belt.

IV. MODIFICATIONS OF THE NOMINAL MODEL As discussed in Section II, our present understanding of the basic physical processes, as well as of the p r o p e r initial conditions in the early Solar System, is insufficient to clearly define the appropriate assumptions and parameters that should be used. In this section a range of alternatives will be considered in order to shed some light on the sensitivity of the results discussed in the previous section to these uncertainties. In order to conserve space, not all the details presented for the nominal model are given. In the absence of specific remarks, it may be assumed that no significant differences were found. These results have been grouped as follows:

A. Simulations in Which the Inclination Secular Resonance, ut6, and the High Inclination Portion of the Eccentricity Secular Resonance 126Are Ignored At present no quantitative data are available about the strength of the v j6 resonance. The importance of the high inclination portion of the v6 resonance is highly dependent on the ability of the b,16resonance to place bodies into high inclination orbits. For this reason the effect of ignoring these resonances will be considered together. Fifty-five calculations have been made in which the strength of these secular resonances is set to zero, using four different values of the diffusion coefficient for the commensurability resonances. The evolution of the specific energy and angular m o m e n t a are shown in Fig. 9, and the final distribution of planetary bodies is shown in Fig. 10. No essential difference exists between the results of these calculations and those made using the nominal model. The similarity of the positions and velocities of residual test bodies and surviving fragments for this case has already been illustrated in Figs. 8a and 8b. No significant differences are found when these are compared with the data discussed in Section III.

B. No Fragmentation As discussed in the introduction, the way in which fragmentation was introduced into the nominal model was of limited value. Nevertheless, it would seem even more unphysical to ignore fragmentation and assume that bodies can merge and grow even when they collide at such high velocities that they should fragment one another and be dispersed. The results of 71 calculations, for which fragmentation was completely ignored, are given in Figs. 11 and 12. Five different values of the resonance diffusion coefficient were used. These results include 26 calculations for which the/~'16 and high inclination v6 resonance was included, and 45 calculations in which these resonances were not introduced. Again, omission of these resonances had no noticeable effect on the results. A1-

319

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though it is certainly preferable to include fragmentation, it turns out that its inclusion is not essential to the results shown in these figures. When fragmentation is not included, there is some difference in the proportion of calculations for which the asteroid belt was completely cleared of embryos at the end of the calculation. In the absence of fragmentation,

a clear asteroid belt was obtained in only 32% of the simulations, whereas in the nominal case, a clear asteroid belt was found in 48% of the cases. For the most part, this difference is attributable to allowing collisions to result in growth in all cases, whereas some of them should have led to fragmentation. As a result, bodies survived that otherwise would have been lost by continued fragmentation or because small bodies are more vulnerable to ejec-

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C. All the Embryos Formed at the Same Time In the calculations reported up to this point, it was a s s u m e d that the formation of e m b r y o s was delayed at larger heliocentric distance, as discussed in Section IA. This is believed to be appropriate because the time required for an e m b r y o to grow will increase when its orbital period increases, as well as when the surface density decreases. Although introduction of this delay seems the p r o p e r thing to do, one can speculate about circumstances in which it might not be the way things happened. For example, it is conceivable that e m b r y o s remained locked in nearly circular orbits until the nebular gas was simultaneously r e m o v e d throughout both the terrestrial planet and asteroidal regions. Eighteen calculations were made in which it was assumed that all the e m b r y o s were formed simultaneously. The resulting final states are shown in Figs. 13 and 14. A value of (¼) m y r i was used for the resonance diffusion coefficient (Eq. (4)), and the v~6 and high inclination z,6 r e s o n a n c e s were not included. The asteroid belt was cleared of e m b r y o s in 7 out of the 18 cases. The distributions of residual test bodies and asteroidal fragments for this case have already been presented (Figs. 8a and 8b). Although the n u m b e r of calculations is not large, it is clear

using 1. In using cases well.

(1) Surfilce density varies as l/a. An initial swarm was calculated in the same way that is described in Section IIA, except that it was a s s u m e d that the initial surface density varies as 1/a instead of a ( I / a ) 1-5. As a consequence, Eq. (2) then gives larger r u n a w a y e m b r y o s at large heliocentric distances (Fig. 15). The total mass of this s w a r m is 3.01 × 102~ g, 35% higher than that of the s w a r m used in the previous three subsections. A n u m b e r of the initial bodies are larger than the present planet Mars. The results of 31 calculations of the orbital evolution of this s w a r m are given in Fig. 16 for three different values of the eccentricity diffusion coefficient. Again, the final configurations are not significantly different. In 16 out of the 31 cases, the asteroid belt was cleared of e m b r y o s . (2) Distributions with smaller initial bodies. For swarms in which the initial surface density varies as a 3/2 (Fig. 1) or as l/a (Fig. 15), the m a s s of the e m b r y o s in the outer portion of the asteroid belt is considerably larger than the mass of those in the terrestrial planet region. An even more marked increase in e m b r y o size with heliocentric distance was a s s u m e d by L i s s a u e r (1987) and Wetherill (1989) to obtain a - 1 0 E a r t h - m a s s Jupiter core from a single runaway e m b r y o . The primordial actual variation

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321

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o f surface density with heliocentric distance is not known. An even more rapid decrease in surface density with heliocentric distance is quite possible. For example, in a model residual nebula calculated by Boss (1989), the surface density falls off as a - 2 between I and 5 AU. Another assumption that also leads to smaller asteroidal embryos is changing the initial gravitational range of the e m b r y o from 2~¢/3 (Birn 1973) to smaller values.

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The consequences of these two possibilities have been explored by use of the initial swarms shown in Fig. 17. The open squares in Fig. 17 represent the initial masses of embryos for which the surface density at 1 AU remained at its previous value of 6.2 g/cm 2, but decreases as a - z with heliocentric distance. The open circles in Fig. 17 represent a swarm identical to the nominal case (Fig. l) except that the spacing factor is assumed to be 1.5V~ instead of 2 . 0 V ~ b e y o n d 2 AU. In both cases the difference in e m b r y o size between the asteroid belt and the terrestrial planet region is reduced. The results of calculations using these initial swarms are given in Fig. 18. In both cases the resonance diffusion coefficient was set equal to (¼) myr-~. These final distributions are similar to those found when the initial e m b r y o s were more massive. For the case in which the spacing p a r a m e t e r equalled 1.5V~ (open circles), the asteroid belt was cleared in 54% of the 16 calculations. The average final mass was 1.217 × 1028 g, in comparison with the observed value of 1.18 × 1028 g. The mean deviation from this average was 0.050 × 1028 g. For the case in which the surface density varied as a 2 (open squares), embryos in the asteroid belt were in the lunar size range. The asteroid belt was cleared in only 38% of the calculations. Although the difference may be of statistical origin, this result could also indicate an incipient

322

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inability of small e m b r y o s to clear the asteroid belt. The average final mass for this case was 1.265 × 1028 g with a mean deviation of 0.059 × 10~ g. E. O t h e r Variations (1) N o r e s o n a n c e s b e y o n d the 2 ." 1 K i r k w o o d gap at 3.27 A U . In all the previous cases it was assumed that

major commensurability resonances exist in the outer belt, even though Kirkwood gaps are not observed, probably as a result of widespread orbital instability in this region. In order to judge the importance of this assumption, 66 calculations were made in which these outer resonances were ignored. Some other minor modifications were made: The 4 : 1 and v, resonances were assumed to be present from 2.00 to 2.08 AU, and loss to Jupiter ejection was assumed for bodies with aphelia beyond 4.5 AU, rather than 4.75 AU. The v~ and high inclination b~ resonances were ignored. Three values of the resonance diffusion coefficient were used: 1.0 myr-~, ~ m y r - ' , and myr 1 The results of these calculations are shown in Fig. 19 and are essentially the same as those in which the outer asteroidal resonances were included. The asteroid belt was cleared of e m b r y o s in 61% of the cases. (2) R a n d o m ' ' j u m p s " in r e s o n a n c e s rather than r a n d o m walk. In all the calculations reported up until this point,

it was assumed that random walks in eccentricity (and inclination for v~6) occurred in accordance with Eq. (4). Examination of the results of Wisdom (1983) for the 3 • 1 gap suggests that it might be just as reasonable to suppose that sudden random changes in eccentricity occurred on a time scale of 1 to 2 × l0 s years, rather than changes by a more gradual diffusion. Assumption of eccentricity changing in jumps was made in calculations reported in a

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different context, i.e., presenting calculations supporting the idea that planets similar in mass and position to Earth may be rather frequent companions of solar mass stars (Wetherill 1991a). The results given in Fig. 20 represent 47 calculations in which the eccentricity of a body was assigned a random value between 0.2 and 0.8 at every time step in which that body was found to be in a resonance region, with the proviso that a minimum delay of 2 × l0 s years (solid

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FORMATION OF THE ASTEROIDS

squares in Fig. 20) or 1 x 105 years (open circles) was required between successive "jumps." For these data points, a swarm in which surface density varies as (I/a), extending out to only 3.3 AU, was used. For the points indicated by squares and circles, the spacing parameter (see Section IIA) was 2X/3 mutual Hill sphere radii, whereas the crosses represent a spacing parameter of 5 mutual Hill sphere radii. Commensurability resonances beyond the 2 : 1 resonance, as well as the v16 resonance and the high inclination v6 resonance, were not included. The final planetary distribution resembles all the others. The asteroid belt is cleared in 66% of the calculations. More details of these particular calculations, as well as related calculations in which resonances and Jupiter ejection were ignored, are given in Wetherill (1991a). V. C O N C L U D I N G R E M A R K S

It is concluded that the nominal model fulfills quite well the requirements set forth at the beginning of Section III. Even though the initial mass, specific angular momentum, and specific energy of the initial embryo swarm differed considerably from that observed today, the values of these quantities spontaneously evolved to the vicinity of their present values. Although subject to large stochastic fluctuations, the calculated range in orbital distribution, number, and mass of the final planetary distributions includes our observed Solar System. In about half the calculations the asteroid belt was cleared of embryos. In the other half of the calculations a small planetary body remained in the asteroidal region. Only rarely (3 out of 115 cases) were two such bodies found. It was also found that the velocity distribution of residual material located in the asteroid belt spontaneously evolved to a distribution similar to that observed at present. Giant impacts on Earth-size planets were found to occur in similar numbers to those found in my earlier calculations in which the initial swarm was confined to a limited portion of the terrestrial planet region. The distribution in the time of these impacts was noticeably different, however. In the present calculations, giant impacts were nearly uniformly distributed between 20 and 200 myr, whereas in the earlier calculations these large impacts rarely occurred more than 50 myr after the beginning of planetary growth. This delay in the time scale is a consequence of the greater importance of bodies from larger initial heliocentric distances when the initial swarm includes the asteroidal region. All of the above conclusions appear to be insensitive to details of the way in which resonance perturbations are introduced, to variations in the mass distribution of the embryo swarm, and to the effects of collisional fragmentation. The most noticeable difference between the result of these calculations and the observed Solar System is the

323

frequent appearance of a body more than twice as massive as Mars between about 1.3 AU and the present asteroidal region. This result is a consequence of the absence of mechanisms that inhibit orbital stability in this region. Although the small mass of Mars is often attributed to effects caused by Jupiter, it is not clear that such effective Jovian effects existed. For example, Jupiter-scattered planetesimals, even if they existed, do not seem to be effective interior to 2 AU (Wetherill 1989, Ip 1987). At present, there is no basis for assuming that Jovian resonances inhibit planetary formation in this region. If the small size of Mars is not simply a stochastic peculiarity of our Solar System, then some new additional physical mechanism appears to be required. It is possible to suggest such mechanisms, for example, gravitational interactions between nebular gas and planet-size bodies (Ward 1986, 1988, Takeda et al. 1985, Takeda 1988, Wetherill 1991c, Artymowicz 1992) or resonances associated with the terrestrial planets themselves, perhaps as suggested by the results of Laskar et al. (1992) concerning a secular resonance between Earth and Mars. Resolution of this matter will require improved quantitative understanding of these and other conceivable mechanisms. Although it seems likely that the general conclusions stated above are not sensitive to the introduction of collisional fragmentation, there is another important aspect of the general problem to which a much superior understanding of fragmentation is crucial. A principal goal of a model in which the primordial evolution of the terrestrial planet and the asteroidal region are unified must be to achieve a comparison between theory and observation of asteroidal bodies, particularly the detailed record of early Solar System events provided by asteroidal meteorites. At present, achievement of that goal is limited by the inability to make quantitative inferences regarding the provenance and thermal history of the small quantity of residual material left in the asteroidal region following the growth of the planets and the nearly complete clearing of the asteroidal region. Qualitatively, it is very likely that this material represents an assemblage of residual < 1024-g planetesimals from both the terrestrial planet and asteroidal regions as well as similar size collision fragments of embryos from both these regions. Progress toward understanding how many planetesimals remain at the time the embryos begin to cross one another's orbits in this regard will require improved understanding of the transition between the low velocity stage of embryo growth and the later high velocity stage in which planets form by the collision of embryos. This must be sufficiently quantitative to permit inferences regarding the number, size, and heliocentric distances of residual planetesimals. Evaluation of the contribution of the embryo population to the introduction of fragments in the asteroid complex will require more understanding of

324

GEORGE W. WETHERILL

the outcomes of impacts between large bodies, and possibly of their tidal encounters as well. In the absence of this required understanding, a wide range of speculation is possible. For example, the present carbonaceous asteroids and meteorites might be thought to be the relics of primordial indigenous asteroidai planetesimals that were not incorporated into embryos, whereas unmetamorphosed ordinary chondrites may be derived from planetesimals formed at smaller heliocentric distances and transferred to the asteroid belt. The concentration of surviving asteroidal test bodies toward the outer asteroid belt together with their velocity distribution supports this suggestion (Fig. 8a). Metamorphosed and igneous meteorites may be fragments from a relatively small number of embryos that lost a portion of their mass by collisions or tidal encounters (Sridar and Tremaine 1992, Boss e t al. 1991) prior to gravitational removal of their sources by mutual perturbations and acceleration by resonances as found in these calculations. Such fragments are found to be concentrated in the inner asteroid belt. Speculations of this kind must be replaced by serious quantitative investigation. To do so will require better understanding of the number of planetesimals remaining at the time when the embryos cross one another's orbits, the collisional evolution of these planetesimals, and the dynamics of fracturing for bodies with masses of between ~102~ and several times 1026 g.

LING, AND A. W. HARRIS 1979. Collisional evolution of asteroids, populations, rotations, and velocities. In Asteroids (T. Gehrels, Ed.), pp. 528-557. Univ. of Arizona Press, Tucson. DAVIS, D. R., S. J. WEIDENSCHILEING, P. FARINELLA, P. PAOLICCHI, AND R. P. BINZEL 1988. Asteroidal collisional history: Effects on sizes and spins. In Asteroids H (R. Binzel, T. Gehrels, and M. S. Manhews, Eds.), pp. 805-826. Univ. of Arizona Press, Tucson. FROESCHE~, CL., AND R. GREENBERG 1989. Mean motion resonances. In Asteroids 11 (R. P. Binzel, T. Gehrels, and M. S. Manhews, Eds.), pp. 827-844. Univ. of Arizona Press, Tucson. GOPEL. C., G. MANHES, AND C. J. ALLEGRE 1991. Constraints on the time of accretion and thermal evolution of chondrite parent bodies by precise U - P b dating of phosphates. Meteoritics 26, 338. GREENZWEIG, Y.. AND J. LISSAUER 1990. Accretion cross sections of planetesimals, h'arus 87, 40-77.

GREENZWEIG, Y., AND J. J. LISSAUER1992. Accretion rates of protoplanets. 11. Gaussian distribution of planetesimal velocities. Preprint. IDA, S. 1990. Stirring and dynamical friction rates of planetesimals in the solar gravitational field. Icarus 88, 129-145. IDA, S., AND K. NAI
ACKNOWLEDGMENTS

KERRIDGE, J. ~'., AND M. S. MATTHEWS (Eds.) 1988. Meteorites and the Early Solar System. Univ. of Arizona Press, Tucson.

1 thank Jay Melosh, Stan Dermott, and Renu Malhotra for valuable conversations. I am also appreciative of the contributions made by Michael Acierno in keeping our network of computers alive, and of Jan Dunlap and Mary Coder for transforming this manuscript into something presentable.

KNEZEVIC, Z., AND A. M1LANI 1989. Asteroid proper elements from an analytical second order theory. In Asteroids H (R. P. Binzel, T. Gehrels, and M. S. Matthews, Eds.), pp. 1073-1089. Univ. of Arizona Press, Tucson. KNEZEVIC, Z., A. MILANI, P. FARINELLA, CH. FROESCHLI~, AND CL. FROESCHLf~ 1991. Secular resonances from 2 to 50 AU. h'arus 93, 316-330. LASKAR, J., T. QUINN, AND S. TREMAINE 1992. Configuration of resonant structure in the Solar System. Icarus 95, 148-152.

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