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ICARUS 136, 304–327 (1998) IS986007 ARTICLE NO. Making the Terrestrial Planets: N-Body Integrations of Planetary Embryos in Three Dimensions J. E. Chambers Armagh Observatory, College Hill, Armagh BT61 9DG, United Kingdom E-mail: [email protected] and G. W. Wetherill Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad Branch Road NW, Washington DC 20015 Received January 2, 1998; revised May 26, 1998 We simulate the late stages of terrestrial-planet formation using N-body integrations, in three dimensions, of disks of up to 56 initially isolated, nearly coplanar planetary embryos, plus Jupiter and Saturn. Gravitational perturbations between embryos increase their eccentricities, e, until their orbits become crossing, allowing collisions to occur. Further interactions produce large-amplitude oscillations in e and the inclination, i, with periods of p105 years. These oscillations are caused by secular resonances between embryos and prevent objects from becoming re-isolated during the simulations. The largest objects tend to maintain smaller e and i than low-mass bodies, suggesting some equipartition of random orbital energy, but accretion proceeds by orderly growth. The simulations typically produce two large planets interior to 2 AU, whose time-averaged e and i are significantly larger than Earth and Venus. The accretion rate falls off rapidly with heliocentric distance, and embryos in the ‘‘Mars zone’’ (1.2 , a , 2 AU) are usually scattered inward and accreted by ‘‘Earth’’ or ‘‘Venus,’’ or scattered outward and removed by resonances, before they can accrete one another. The asteroid belt (a . 2 AU) is efficiently cleared as objects scatter one another into resonances, where they are lost via encounters with Jupiter or collisions with the Sun, leaving, at most, one surviving object. Accretional evolution is complete after 3 3 108 years in all simulations that include Jupiter and Saturn. The number and spacing of the final planets, in our simulations, is determined by the embryos’ eccentricities, and the amplitude of secular oscillations in e, prior to the last few collision events. 1998 Academic Press Key Words: planetary formation; terrestrial planets; planetary dynamics; extra-solar planetary systems. 1. INTRODUCTION The planetesimal theory of terrestrial-planet formation is commonly viewed as a play in three acts. In Act One, 304 0019-1035/98 $25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved. grains of dust near the midplane of the protoplanetary nebula accrete one another via low-velocity collisions, eventually forming 1- to 10-km sized ‘‘planetesimals’’ (Weidenschilling 1997). These objects are large enough to possess nonnegligible gravitational fields that increase their collision cross sections, aiding further growth to form p3000-km diameter ‘‘planetary embryos.’’ The second act is characterized by ‘‘runaway growth,’’ in which equipartition of random orbital energy between planetesimals ensures that the largest objects have orbits with low eccentricities and inclinations—orbits that are most efficient at scooping up more material (e.g., Wetherill and Stewart 1989, Kokubo and Ida 1996). Runaway growth of the biggest objects is enhanced by gas drag acting on small collision fragments, giving them circular, co-planar orbits too (Wetherill and Stewart 1993). In Act Three, planetary embryos strongly perturb the orbits of their neighbors until they become crossing. Runaway growth now slows or shuts down completely, and the embryos accrete each other in giant impacts, leading to a handful of terrestrial planets on widely separated orbits. Act Three has been modeled extensively using the Öpik–Arnold scheme to follow the dynamical and collisional evolution of disks of planetary embryos in three dimensions (e.g., Wetherill 1992, 1994, 1996). This technique treats individual close encounters and collisions effectively and uses a simple parameterization of the important effects of the major Jupiter and Saturn resonances in the asteroid belt. However, it does not include distant perturbations between embryos or sequences of encounters due to node-crossing events, so the effects of secular perturbations and resonances between embryos are beyond its ability. The final stage of planetary accretion has also been mod- 305 TERRESTRIAL-PLANET FORMATION eled using N-body integrations in two dimensions, by Lecar and Aarseth (1986), and Beaugé and Aarseth (1990). In addition, Cox and Lewis (1980) carried out 2D calculations that neglected long-range perturbations between embryos. Numerical integrations automatically include the effects of secular and resonant interactions between embryos. However, calculations limited to two dimensions artificially decrease the collisional timescale with respect to the timescale for orbital evolution. These approximations were chosen because they require substantially less computer time than more-realistic N-body integrations in three dimensions. Both types of simulation yielded plausible planetary systems, although these were not always similar to our own. They also provided insight into the chaotic nature of planet formation that results from the central role of close encounters—a level of understanding that goes beyond that achievable from analytic models. Recent improvements in the performance of computer workstations, and the development of a new N-body algorithm, now make it possible to carry out N-body integrations, in three dimensions, of several tens of gravitationally interacting bodies for the p108 orbits necessary to form the cores of the inner planets. This led us to pose the following question. Is it possible to create a recognizable system of terrestrial planets by integrating the orbits of a disk of planetary embryos for p100 million years, subject only to mutual gravitational interactions, inelastic collisions, and external perturbations from the giant planets? If it is possible, such simulations should indicate whether terrestrial planets such as our own are inevitable, given the size and location of the giant planets, or whether their formation depends critically on the nature of the disk of embryos formed by runaway growth. (Alternatively, it may all be a matter of luck, with the final outcome depending on a few key events that occur essentially at random.) It should also become possible to predict the characteristics of terrestrial planets in extra-solar systems long before we can determine them observationally. Conversely, if N-body simulations involving a few dozen embryos cannot produce something akin to the terrestrial planets, they may at least indicate what extra physics is required to do so. With this in mind we have carried out 27 integrations of disks of planetary embryos, starting with objects on isolated, nearly coplanar orbits, and following their evolution for at least 108 years. In two thirds of the simulations we have also included the effects of the giant planets Jupiter and Saturn, assuming they formed before the accretion of the terrestrial planets was complete. All the integrations were performed on dedicated DEC alpha workstations, requiring p3 years of CPU time. The next section describes the integration method and the initial conditions used in the simulations. Section 3 looks at the evolution of the disks of embryos, whilst Sec- tion 4 examines the end products. In Section 5 we discuss the results in comparison to the observed solar system. Finally, the last section contains a summary. 2. N-BODY SIMULATIONS We performed three sets of nine N-body integrations, each set using a different model for the formation of the terrestrial planets. Model A. These integrations simulate the evolution of a disk of planetary embryos that initially spans most of the region currently occupied by the terrestrial planets. In this model it is assumed that the giant planets do not significantly influence the formation of the terrestrial planets, and hence they are not included in the integrations. Model B. As Model A, but the effects of the giant planets are modeled by adding Jupiter and Saturn to the simulations after 107 years. The giant planets are assumed to have their current masses and orbital elements. Model C. As Model B, but the initial disk of embryos is extended to encompass the region that now contains the asteroid belt. Jupiter and Saturn are added at 107 years, as in Model B. The nine simulations using each model are divided into batches of three, each batch using different values for the surface density of solid material at 1 AU, s, and the spacing between embryos, D. One batch each uses (s, D) 5 (6, 7), (10, 7), and (6, 10), where s is measured in units of gcm22 and D in mutual Hill radii, RHM , where RHM 5 S D S m1 1 m2 3MA 1/3 D a1 1 a2 2 (1) for embryos with masses m1 and m2 , and semi-major axes a1 and a2 . 2.1. Initial Conditions The initial conditions were chosen bearing in mind the form of the present planetary system and the results of simulations of the runaway-growth phase of terrestrialplanet formation (e.g., Wetherill and Stewart (1993). Disk density. In 18/27 simulations we adopt a surface density of solid material, s 5 6 gcm22 at 1 AU. This corresponds to the minimum mass needed to form the current terrestrial planets. We choose a density profile that varies as 1/a—a smaller gradient than used by some authors—in view of the large amount of solid material (s p 10 gcm22) required beyond the ice condensation point to form Jupiter’s core before loss of the nebula gas (Pollack et al. 1996). As a variant on our standard initial conditions we set s 5 10 gcm22 at 1 AU in three of the simulations for each model. 306 CHAMBERS AND WETHERILL Radial extent. The 18 simulations using Models A and B begin with a disk of embryos having semi-major axes 0.55 , a , 1.8 AU, covering most of the terrestrial-planet region. The lower bound is a compromise between making the simulation realistic and avoiding a short integration timestep (and hence a large CPU overhead), which is necessary when some objects have small a. In the simulations using Model C, we extend the outer edge of the disk to 4.0 AU to include embryos that may have formed in the asteroid belt. Models A and B begin with 24–40 embryos, depending on the values of s and D, while Model C begins with 34–56 embryos. The initial disk mass varies between 1.8 and 3.2 M% for Models A and B, and between 5.0 and 8.6 M% for Model C (which includes material in the asteroid belt). Orbital elements. The initial eccentricities, e, and inclinations, i, are 0 and 0.18, respectively, for all embryos. These values are somewhat arbitrary, but they quickly become randomized once close encounters occur, and most objects soon attain much larger values of e and i. However, for the embryos to have formed via runaway growth there must have been an epoch when e and i were small, and we assume that this is still the case at the start of our simulations. The remaining elements for each embryo— the mean longitude and nodal longitude—are chosen randomly. Embryo separations. We assume that the planetary embryos that formed via runaway growth have become isolated from one another prior to the start of our simulations due to mutual accretion events and eccentricity damping via dynamical friction. Once damping forces have been overcome, a system containing more than two embryos becomes unstable with respect to close encounters on a timescale, tc , that depends exponentially on the initial orbital spacing, D, measured in mutual Hill radii (Chambers et al. 1996). In most of our simulations we use D 5 7, which corresponds to an embryo crossing time, tc p 5 3 104 years for the case in which all objects have the same mass. Three integrations per model begin with D 5 10, implying that tc p 107 years in the equal mass case. These two values of tc bracket the probable time required to form embryos in the terrestrial region (Wetherill and Stewart 1993). In practice, tc is somewhat smaller in our integrations because the embryos begin with a range of masses (see Section 3.1). We also note that tc would be reduced if the embryos began with non-zero eccentricities (Yoshinaga et al. 1998). The values of D chosen for our integrations are broadly consistent with the results of N-body integrations of embryo formation by Kokubo and Ida (1998). These authors found that embryos formed from a population of p104 small bodies tend to have orbits spaced by about 10 mutual Hill radii. Embryo masses. Having chosen s and D, the mass of each embryo is uniquely determined. These range from 0.02 M% at the inner edge of the disk to 0.1 M% at the outer edge in Models A and B, with s 5 6 gcm23. Larger objects are present initially in the simulations that have a higher disk density or contain embryos in the asteroid belt. Embryo density. The embryos’ radii are calculated assuming a density of 3 gcm23. This value is also used to determine the new radius of an object following the accretion of a smaller body. 2.2. Integration Method The N-body integrator used in the simulations is based upon a second-order mixed-variable symplectic integrator written by Levison and Duncan (1994), which uses an algorithm described by Wisdom and Holman (1991). One drawback of this fast package is that close encounters between massive bodies are not calculated accurately, so we modified the code to integrate all objects using a Bulirsch–Stoer method (Stoer and Bulirsch 1980), also supplied by Levison and Duncan, whenever the separation between a pair of objects falls below 2 Hill radii. The Bulirsch–Stoer algorithm uses a variable timestep to accurately follow the orbital evolution during an encounter, whilst the symplectic integrator uses a fixed timestep of 10 days. The symplectic algorithm uses Jacobi coordinates, which makes it necessary to periodically re-index the objects in order of increasing semi-major axis. This procedure minimizes numerical error introduced by having different physical and Jacobi ordering of the objects. Combining the symplectic and non-symplectic methods leads to a secular growth in the energy error, typically one part in 102 –103 over 108 years. While less than ideal, we believe this is acceptable given that neglected effects, such as dynamical friction due to residual planetesimals, probably have a comparable or larger effect. Collisions between embryos are assumed to be completely inelastic, forming a single new body whose orbit is determined by conservation of linear momentum. Our model assumes that mass loss due to fragmentation during a collision is negligible. This assumption is necessary in order to avoid a rapid increase in the number of bodies present in the integration, which would make the computational expense prohibitive. 3. EVOLUTION Figures 1–3 show a series of snapshots, in semi-major axis/mass space, taken from three of the simulations described above—one for each model. Each symbol in the figures represents a surviving embryo, with the horizontal bars depicting the perihelion and aphelion distances of its orbit. In these and most of the other integrations the evolution passes through four stages, described below. TERRESTRIAL-PLANET FORMATION 307 FIG. 1. Masses and semi-major axes of the surviving objects at six epochs during a simulation using Model A, with D 5 7 and s 5 6 gcm22. Note how embryo isolation is overcome and collisions occur (b), the disk becomes dynamically excited (c), protoplanets form—see object at 0.9 AU (d), the small objects are swept up (e), leaving the largest surviving objects isolated from one another (f ). 3.1. Embryo Isolation Is Overcome (Figs. 1b, 2b, 3b) The initial orbital spacing of the embryos is large enough that no pair of objects is able to undergo close encounters in the absence of external perturbations, due to conservation of energy, E, and angular momentum, h (Gladman 1993). When more than two embryos are present, E and h are no longer conserved within each pair of objects, and their isolation can be overcome. Figure 4 shows the evolution of the perihelion and aphelion distances (q and Q, respectively) of the innermost 12 embryos for the first 20,000 years of a simulation using Model A. Initially the embryos’ semi-major axes remain almost constant, while their eccentricities, e, exhibit erratic oscillations whose amplitude increases over time. After about 12,000 years the eccentricities have increased sufficiently for neighboring embryos to undergo close encounters, and the initial isolation is broken. The time required 308 CHAMBERS AND WETHERILL FIG. 2. As Fig. 1 for a simulation using Model B, with the same values of D and s. In this case the first protoplanet forms at 1.1 AU (d), and all but two objects are eventually swept up (f ). to do this varies from one simulation to another, depending mainly on the initial embryo separation, D. However, in every case the embryos achieve crossing orbits eventually. At this point the first collisions occur, reducing the total number of objects and increasing their mean separation in mutual Hill radii (see Eq. (1)). It is conceivable that the surviving embryos could become isolated once more, returning to a state of affairs similar to, but more stable than, the start of the simulation. In practice this never happens in our simulations because the accretion timescale is always much longer than the timescale for increases in e. Hence, once the first close encounters take place, the embryos remain on crossing orbits for as long as a significant number of objects survive. Re-isolation can occur in simulations that substantially alter the ratio of the accretion timescale to the dynamical timescale by constraining embryos to move in two dimensions or by artificially increasing their physical radii. In TERRESTRIAL-PLANET FORMATION 309 FIG. 3. As Fig. 1 for a simulation using Model C, with the same values of D and s. Here protoplanets first form at 0.7 and 2.2 AU (d), most embryos with a . 2 AU are removed by resonances rather than collisions (e), leaving just two surviving planets (f ). trial integrations with radii enhanced by a factor of 10 we find that re-isolation does occur, producing a final system containing five to eight low-mass planets with very low orbital eccentricities. The time of the first close encounter, tc , depends on the masses and spacing of the embryos. In the simulations with initial spacing D 5 7, the first close encounters typically occur after tc p 104 years, whilst integrations with D 5 10 exhibit encounters after p4 3 105 years. These times are significantly shorter than those found by Chambers et al. (1996) for systems of equally spaced, equal-mass planets (tc p 5 3 104 and 107 years, respectively). However, these authors noted that a spectrum of planetary masses can substantially reduce the orbit-crossing time, and we suggest that this is the cause of the rapid onset of close encounters seen in our integrations. Planetary embryos in different parts of the disk experience their first close encounter at different epochs, usually 310 CHAMBERS AND WETHERILL FIG. 4. Perihelion (q) and aphelion (Q) distances for the 12 innermost embryos during the first 20,000 years of a simulation using Model A. Note the rapid orbital evolution following the first close encounter at p12,000 years. beginning with objects close to the inner edge of the disk. Figure 5 shows the time of first encounter for the embryos in four of the simulations. The times are measured in units of the orbital period, and, given that the embryos are uniformly spaced in mutual Hill radii, one would expect the crossing times to be roughly independent of a. This is true for the outer part of the disk in each case, but in the inner region the time of first encounter generally decreases with increasing a. This suggests that once embryos near to the inner edge of the disk achieve crossing orbits, they significantly influence the orbital evolution of embryos further out, hastening the onset of encounters. However, a particular embryo invariably undergoes its first close approach with an object that was initially in the same part of the disk (rather than an interloper from elsewhere) so the effect is an indirect one—a close approach between one pair of embryos destabilizes neighboring objects, leading to a ‘‘wave’’ of close encounters that sweeps through the inner part of the disk. This effect is generally limited to the region P , 2 years and is particularly marked for the simulations with D 5 10. Chambers and Wetherill (1996) found that the alreadyshort crossing times seen here are reduced still further when a population of smaller objects (mass p 0.01 embryo masses) is present in addition to the embryos. During the earlier runaway growth stage, when planetesimals are accreting each other to form embryos, the eccentricities of the largest objects are damped by dynamical friction and collisions with smaller bodies (Wetherill and Stewart 1993), and presumably the embryos remain isolated from one another. However, as the fraction of solid disk material contained in the embryos increases, their mutual interactions will become stronger. At the same time, the corre- FIG. 5. Times of first close encounter for each embryo in four simulations. The times are divided by the orbital period P, and should be independent of P if embryos are perturbed only by others in the same part of the disk. Here D is the initial embryo separation in mutual Hill radii. TERRESTRIAL-PLANET FORMATION sponding decrease in the total mass of the planetesimals reduces their ability to damp the embryo’s eccentricities. This combination is likely to produce a rapid transition to a situation in which the embryos achieve crossing orbits and the disk becomes dynamically excited. 3.2. The Disk Becomes Dynamically Excited (Figs. 1c, 2c, 3c) Once neighboring embryos have acquired crossing orbits with significant eccentricities, strong orbital interaction can occur due to close encounters and resonances between embryos. This in turn leads to rapid changes in e and i, and the disk becomes dynamically excited. This can be seen in Figs. 1c, 2c, and 3c, where many of the embryos have horizontal bars that overlap, indicating crossing orbits with nonnegligible eccentricities. Figure 6 shows the mass-weighted values of Ï(e2 1 i 2), e, and i, for objects in two simulations using Model A. Each of these quantities increases logarithmically with time; in other words, very quickly at first, and then more slowly at later times. The rate of increase is largest for simulations with high surface densities and thus more massive embryos. The upshot is that gravitational focusing during close encounters will diminish over time, since this is most efficient at small relative velocities. Hence the collision probability for a given pair of embryos will also decrease with time. This is reflected in a short-lived burst of collisions near the start of each simulation, followed by a monotonic decline in the collision rate (see Section 3.3). The embryos’ eccentricities and inclinations undergo large variations throughout an integration. These changes are primarily caused by secular perturbations rather than close encounters. In the early evolution, each episode typically lasts for 5–20 thousand years (Fig. 7), and changes in e are often correlated with the behavior of the argument of perihelion, g, due to the Kozai resonance (Kozai 1962). For example, the strong increase in the eccentricity of Planet 11 at t p 50,000 years in Fig. 7 is associated with a libration of g about 908. Kozai oscillations, often seen in the orbits of comets, are due to the long-term interaction of an orbit with a planar distribution of matter, which is clearly the case here. Secular resonances between pairs of embryos are common in each of the simulations. These produce correlated oscillations in the eccentricities or inclinations of the two objects, and librations of the resonant critical argument, with periods p105 years. A common example involves a situation in which the longitudes of perihelion of two embryos precess at the same rate, with the two orbits aligned or anti-aligned with one another. Occasionally three, or even four, orbits are temporarily locked together in this way. The resulting high-frequency oscillations in e are further modulated by secular interaction with other nearby 311 embryos, or with Jupiter and Saturn in simulations that include the giant planets. For example, Fig. 8 shows the evolution of e for four embryos from two simulations using Model C. The figure also shows the critical argument of a secular resonance involving the longitude of perihelion, f, of each object and that of another nearby embryo. Libration of the critical argument about 0 indicates that the perihelion directions of the two embryos are aligned, whilst oscillation about 1808 implies that the perihelion directions are antialigned. In each case the changes in e are associated with librations in the critical angle. At some epochs additional oscillations are apparent, caused by secular interaction with other embryos. An object usually undergoes strong interactions with two or three other embryos simultaneously and typically moves back and forth between several secular resonances during an integration. Secular resonances with the giant planets also occur. Figure 9 shows two cases in which the orbits of embryos lie within the n5 resonance, where the longitude of perihelion of the protoplanet precesses at a similar rate to that of Jupiter. This causes smooth, long-period changes in the object’s eccentricity. This resonance influences the orbits of objects with a , 2 AU in several of our simulations, especially the region 0.5 , a , 0.7 AU, and occasionally causes embryos to fall into the Sun. The analogous n6 resonance with Saturn plays an important role for objects with 2 , a , 2.3 AU. We note that the locations of these two secular resonances will depend to some extent on how much nebula gas is present during the accretion of the terrestrial planets (Lecar and Franklin 1997) and also the degree to which the giant planets’ orbits change during the formation of the planetary system. The remaining diagrams in Fig. 9 show examples of another common situation in which an embryo’s longitude of perihelion, f, becomes almost stationary. Like the Kozai resonance, this resonance does not directly involve the orbits of other bodies. Whilst f is almost stationary, the embryo’s nodal longitude and argument of perihelion usually circulate rapidly in opposite directions. This behavior is generally accompanied by a monotonic increase in e, which can also cause embryos to fall into the Sun. In contrast to secular perturbations, close encounters, which produce an abrupt jump in the orbital elements, appear to play a minor role in determining changes in e and i—only a handful of examples can be seen in Figs. 7–9. Close encounters mainly affect e and i by determining the point in a secular cycle at which an object is scattered away from one secular resonance and into another. This in turn establishes the initial values of e and i and the critical argument for the new resonance. 3.3. Protoplanets Are Formed (Figs. 1d, 2d, 3d) After 3–6 million years, one or two objects of 0.3–0.5 Earth masses have formed interior to 1.5 AU in each of 312 CHAMBERS AND WETHERILL FIG. 6. Mass-weighted mean of (e2 1 i 2)1/2 (upper) versus time for two of the simulations using Model A, with s 5 6 and 10 gcm22, respectively. Also shown are the mass-weighted means of e (middle) and i (lower). Here i is measured in radians. Note that all these quantities increase approximately linearly in log time, with the high-s case yielding larger values. the simulations. Such objects are apparent at 0.9, 1.1, and 0.7 AU in Figs. 1d, 2d, and 3d, respectively. In simulations using Model C, where the disk of embryos extends into the asteroid belt, additional large objects are present beyond 1.5 AU. However, each of these is the result of only one or two collisional events, and their large mass is due to the high initial masses of the embryos in this region. Do these ‘‘protoplanets’’ form by a continuation of runaway growth or via more orderly growth? The low eccen- tricities and inclinations present at the start of the integrations are conducive to gravitational focusing between embryos—a necessary prerequisite for runaway growth. However, as e and i increase this situation will change. We can see how the effects of gravitational focusing vary during a simulation by looking at the distribution of close-encounter distances. In the absence of focusing, the number of encounters, N, with minimum separation, D, is given by N Y D, while in the limit of strong focusing N Y D 1/2. TERRESTRIAL-PLANET FORMATION 313 FIG. 7. Eccentricity and argument of perihelion, g, versus time for four embryos for the first 105 years of a simulation using Model A. Note the general increase in e, the scarcity of sudden jumps associated with close encounters, and the correlation between the behavior of e and g, indicating orbital evolution controlled by the Kozai resonance. Figure 10 shows the close-encounter distribution at three epochs of a simulation using Model A, where we have combined the close-approach data for all the embryos present. A x 2 test indicates that we can reject the hypothesis that focusing is absent at the first epoch (0–105 years) at the 99.5% confidence level. Conversely, the data for the third epoch (9–10 million years) are consistent with a lack of gravitational focusing, while those for second epoch (3–3.5 million years) are ambiguous—the probability that the distribution is uniform is roughly 10%. Given that gravitational focusing is probably significant during at least a part of the simulations, does this lead to runaway growth? Figure 11 shows the mass distributions at four epochs, combining the data for all the embryos with a , 1.5 AU in integrations using Model C. As the embryos accrete one another the mass distribution flattens, except for the largest objects, which march steadily toward higher masses—i.e., orderly growth occurs. Apparently, gravitational focusing during the early stages of the simulations is not enough to allow runaway growth to occur. In addition, the small number of objects present probably makes it difficult for runaway growth to get going before the supply of embryos is exhausted. There is some indication that equipartition of random orbital energy (‘‘dynamical friction’’) takes place. For example, Fig. 12 shows the eccentricities of all surviving objects in Model C after 10 million years as a function of their mass. The largest objects tend to have lower values of e than the smaller bodies. Conversely, the smallest embryos generally do not have eccentricities close to zero. A quantitative measure of the importance of dynamical friction comes from examining the values of e and i for each pair of objects just before they collide. Consider first the larger of the two objects in each collision. In Model C, 69% of these objects have inclinations below the mean value for that epoch, while for e the figure is 56%. The corresponding numbers for the smaller colliding body are 59 and 45%, respectively. In other words, weak equipartition of random energy takes place for objects undergoing collisions—a necessary prerequisite for runaway growth. Incidentally, the preference for low inclinations over low 314 CHAMBERS AND WETHERILL FIG. 8. Eccentricity versus time for four embryos during simulations using Model C. Also shown are the critical arguments for a secular resonance between the longitudes of perihelion, f, of each object and another nearby embryo. Librations about 0/1808 indicate that the perihelion directions of the two objects are aligned/anti-aligned. eccentricities can be understood following the discussion below (see Eq. (2)). Figure 13 shows how the number of collisions varies depending on the semi-major axis of the larger of the two colliding bodies just prior to impact (for all the simulations using Model C). The collision rate decreases rapidly with increasing heliocentric distance, after a peak at about 0.7 AU, so that very few collisional events occur beyond 2 AU. Note that this is not simply a reflection of the decrease in the number density of objects with increasing a, since this falls off more slowly. To see how this effect might arise, consider a planet that is the largest object in its part of the disk, moving on a low-e, low-i orbit, with semi-major axis ap . The rate at which the planet accretes the smaller embryos in its vicinity is dN Q dt E ap /(12e) ap /(11e) n(a) 3 Pnode(ap , a, e, i) 3 Pconj(ap , a, e, i) da, P(a) where a, e, i, and P are the orbital elements and period of an embryo, and n(a) is the number of embryos smaller than the planet per unit semimajor axis. In addition, Pnode is the probability that one of the nodes of an embryo’s orbit is close enough to the planet’s orbit to permit collisions, given by Pnode Q 2 3 2Rp fgrav , 2ae sin i where Rp(ap) is the radius of the planet, and fgrav(e, i) is the gravitational focusing factor. Finally, Pconj is the probability that the planet is in the correct part of its orbit at conjunction for a collision to take place, given by Pconj Q 2Rp fgrav . 2fap If we assume that the embryo masses, m, and their mean spacing are comparable to their initial values, then n(a) p 1/a 3/2, and m(a) p a 3/2, which also implies that the planetary radius is given by Rp p a p1/2. Provided that embryos 315 TERRESTRIAL-PLANET FORMATION FIG. 9. Eccentricity versus time for four embryos during simulations using Model C. Also shown in the upper diagrams are the critical arguments for the n 5 resonance between the longitudes of perhelion, f and fJ , of the embryo and Jupiter. In the lower diagrams, the change in e is associated with epochs when f itself is almost stationary. in each part of the disk are subject to accretion by roughly the same number of planets, the overall accretion rate will be f 2grav dN Y dt e sin i E f 2grav(e, i) 1 . da Y ap /(11e) a 4 a 3p sin i ap /(12e) (2) Note the steep dependence of the collision rate on ap and the lack of dependence on e outside the gravitationalfocusing regime. Despite the crude assumptions used to derive Eq. (2), the fit with the collision rate observed in our simulations is quite good, as the 1/a 3p curve in Fig. 13 indicates. The discrepancy between theory and simulation for a , 0.7 AU presumably reflects the effects of the disk truncation at 0.55 AU. The next obvious question is how the collision rate varies with time? Figure 14 shows the fraction of the initial objects that remain, versus time t, averaged over all the simulations that began with an embryo separation D 5 7. The solid lines show the actual fraction remaining, whilst the dashed lines indicate the fraction expected according to a decay law in which the collision probability, for a given embryo, depends on the number of surviving objects, N, so that the total population varies as dN Y 2N(N 2 1). dt (3) Clearly the slopes of the curves in Fig. 14 are shallower than the simple theory predicts, and for simulations using Model C the number of collisions is approximately proportional to log t. This disagreement is not too surprising since Eq. (3) ignores changes in the orbital element distributions of the embryos—in particular the collision rate will decrease as the mean inclination of the embryos rises (see Eq. (2)). 3.4. The Planets Become Isolated (Figs. 1f, 2f, 3f ) In the simulations using Models B and C, the giant planets Jupiter and Saturn are added at 107 years, with their current masses and orbital elements. The immediate effect 316 CHAMBERS AND WETHERILL FIG. 12. Eccentricity versus mass for all objects remaining at 10 million years in the simulations using Model C. FIG. 10. Number of close encounters, N, versus distance of closest approach, D, at three epochs of a simulation using Model A. Note that the close-encounter data for all embryos has been combined. is to introduce a number of strong mean-motion and secular resonances into the region a . 2 AU. Embryos in the vicinity of these resonances undergo large increases in eccentricity until they are removed by an impact on the Sun, by a collision with another embryo inside 2 AU, or by hyperbolic ejection following a close approach to Jupiter. This mechanism for clearing material from the asteroid belt still operates today, but it is more efficient in our simulations since the embryos are large enough that a close encounter between a pair of embryos can often scatter one object into a resonance, where it is quickly lost before another encounter can scatter it out of the resonance again. The net result is that p20 million years after the giant FIG. 11. Cumulative mass distribution for all the simulations using Model C, at four epochs during the integrations. Note that only objects with a , 1.5 AU are included in the distributions. FIG. 13. Number of collisions, as a function of semi-major axis, a, of the larger impactor, for all integrations using Model C. The curve shows the predicted distribution following a 1/a3 law. TERRESTRIAL-PLANET FORMATION FIG. 14. The fraction of surviving objects versus time for all simulations using Models A and C. The dashed lines show the expected fraction due to a decay law of the form dN/dt Y 2N(N 2 1). planets are added, most of the embryos with a . 2 AU have been removed (Fig. 3e). Thus, 30 million years into a simulation, a combination of rapid accretion in the inner part of the disk and partial clearing of the asteroid belt by resonances leaves three to six large ‘‘protoplanets’’ containing most of the remaining mass, together with a comparable number of smaller unaccreted embryos. The transition from this state of affairs to a system of isolated, ‘‘final’’ planets depends principally on the amplitude of the secular oscillations in the eccentricities of the surviving protoplanets. These oscillations have two sources. First, secular resonances between neighboring planets on crossing or nearly crossing orbits, which produce short-period (p105 year) cycles. Second, secular perturbations and/or resonances with Jupiter and Saturn, having periods of 106 –107 years. In general the former are dominant except for a p 2.1 AU, where the n6 resonance causes large changes in e and occasionally the region a p 0.6 AU, where the n5 resonance can play an important role (these values of a assume that there is no longer a significant amount of nebula gas present and that the giant planets have their modern orbits). The secular oscillations in e cause the perihelion and aphelion distances (q and Q, respectively) to change on timescales that are short compared to the collision timescale. In order to avoid collisions with one another, protoplanets must be spaced so that the maximum value of Q for the innermost object is less than the minimum value of q for the second object, and so on. Further evolution takes place until this situation is achieved, with surplus protoplanets being thrown back and forth until they merge with another or are scattered beyond 2 AU and removed by a resonance. 317 For example, Fig. 15 shows the time evolution of q and Q for each surviving object in four of the simulations, starting from the point at which the giant planets are added. Each pair of lines of a particular color represents q and Q for a single object. Also shown, in blue, are q and Q for one of the largest objects that does not survive. In each case the protoplanets destined to remain at the end of the simulation rarely approach one another, if at all. However, the eccentricities of their orbits are such that there is no room left for the ‘‘blue’’ protoplanet. Consequently, this excess object is either accreted or ejected. Note that the final spacing of the planets is determined by the values of e while there are still ‘‘too many’’ protoplanets present. The eccentricities are often subsequently reduced by interactions with residual small embryos, which are then accreted or ejected. For example, the ‘‘green’’ planet in Fig. 15c undergoes a substantial decrease in e at p50 million years due to interactions with a much smaller body. A series of close encounters with the smaller embryo nudges the protoplanet into the n5 resonance, producing a rapid drop in e (Fig. 16). The residual embryo is subsequently removed, leaving the larger body on a low-eccentricity orbit. This late-stage orbital damping occurs in several of our simulations, but does little to alter the semimajor axes of the final planets, these being determined earlier in the evolution. 4. THE FINAL STATE OF THE SIMULATIONS Figures 17–19 show the final states of the integrations using Models A, B, and C, respectively. The figures indicate the osculating orbit of each surviving planet, with the planet itself represented by a filled symbol whose radius is proportional to the radius of the body. The same data are given in Tables I, II, and III, except that the values of e and i have been averaged over the last 106 years of the integration. The column headed ‘‘last event’’ refers to the time at which the last accretion or ejection took place. Where close encounters are still taking place, it is assumed that the last event time will be greater than the length of the integration. The results of simulations using Model A are given in Fig. 17. In each case 108 years have elapsed since the start of the calculation. In most cases the evolution is incomplete in the sense that several objects are still able to undergo close encounters with one another. However, the collision rate has slowed to almost zero, so it is not apparent how much longer would be required to achieve a set of isolated objects. In many of the simulations the innermost objects have achieved non-crossing orbits. These planets tend to have smaller i than objects further from the Sun, and they usually contain most of the mass. However, it is quite likely that the configuration of these objects will change due to subsequent interaction with objects at larger a. The one 318 CHAMBERS AND WETHERILL FIG. 15. Time evolution of the aphelion, Q, and perihelion, q, distances (same color for each body) for each of the objects that survives until the end of four simulations. Jupiter and Saturn are added to the simulation at 107 years. In each case, the blue lines indicate Q and q for a large object that is accreted or ejected before the end of the simulation. example of a ‘‘completed’’ simulation—number 5A— contains five approximately equal-mass planets, with a roughly geometric orbital spacing. The simulations using Model B were continued for 108 years or until close encounters had ceased, whichever was longer. The final state of each simulation is shown in Fig. 18. Generally there are two objects with a , 1.7 AU, while in several cases a third object lies further from the Sun. In the two simulations that yielded three planets, these have roughly geometric orbital spacings. This is also true for the three outer planets in simulation 7B, although the two inner planets lie closer together than a geometric law would predict. In each case the mean spacing is somewhat larger than the planets we observe in the inner solar system. As with Model A, the planets closest to the Sun tend to be the largest. In addition, there is considerable scatter in the mean values of e and i from one integration to another. Simulations using Model C (shown in Fig. 19) usually produced only one or two planets interior to 2 AU. Several of the simulations also contain an object in the asteroid belt, although these are invariably very large compared to a typical asteroid, due to the large embryo masses used in our calculations. It is possible that some or all of these objects have orbits that are unstable over the age of the solar system. In only one of the two simulations that ended with .2 isolated planets do the objects have geometrically spaced orbits. In almost all the simulations the orbital spacing is larger than in the inner part of the solar system, TERRESTRIAL-PLANET FORMATION FIG. 16. Time evolution of the semi-major axis, a, eccentricity, e, and critical argument for the n 5 resonance for the ‘‘green’’ planet in Fig. 15. Note the rapid drop in e when the critical argument starts to circulate slowly, following a close encounter at p50.8 million years. and the planets are also typically spaced more widely than those from simulations using Model B. Once again, the objects closest to the Sun tend to be the largest, and these usually have smaller eccentricities than their ‘‘asteroidal’’ cousins. Two of the simulations produced only a single, giant terrestrial planet, roughly two and a half times as massive as Earth. In general, the surviving objects in our simulations have time-averaged values of e and i that are substantially larger than those of Earth and Venus (e p 0.03 and i p 28). Earthsized planets with eccentricities of 0.2 are not uncommon in our simulations. The large mean values of e occur early in each integration and lead to correspondingly large maximum values, emax , during each secular oscillation. This in turn requires that protoplanets remain widely spaced to avoid scattering or accreting one another. In general, emax is too large to permit more than two, or occasionally three, final planets to form in the simulations that contain Jupiter and Saturn. Thus, the small number of terrestrial planets 319 seen in our calculations, and their large eccentricities, are directly related to one another. Approximately two thirds of the ‘‘completed’’ simulations contain a large (.0.5M%) planet lying within the Sun’s habitable zone. This is the region in which a geologically active planet can support liquid water. The conservative estimate for the habitable zone adopted here is 0.95 , a , 1.37 AU (Kasting et al. 1993). It is quite likely that the region is somewhat larger, in which case a greater proportion of our simulations would contain a habitable planet. Looking now at the overall distribution of the surviving objects, Fig. 20 shows the masses and semi-major axes of all the remaining objects at the end of the integrations. In Model A, the survivors encompass a broad range of heliocentric distances, with the largest objects clustered around 1 AU and an extended tail of smaller bodies out to 3 AU. The inner edge of the distribution is quite sharp and almost identical to the inner cutoff of the initial disk of embryos at 0.55 AU. In Model B the remaining objects occupy a narrower range of heliocentric distance, with only one survivor having a . 2 AU. The few objects that entered this region during each integration were either scattered back to smaller values of a or removed by resonances with the giant planets. The largest bodies lie closer to the Sun than in Model A, in a cluster centered on 0.6 AU, with a second group between about 1.0 and 1.4 AU. All the surviving objects with a . 1.5 AU are of less than one third of an Earth mass. In Model C the picture is different again. Now most of the large objects are confined to a region a , 1.2 AU, with a tail of smaller objects extending outward. The difference between the two regions is clear: most of the planets interior to 1.2 AU have masses greater than 1 M% , whilst most of the objects with a . 1.2 AU are less massive than Earth. Table IV shows the fates of the embryos according to their initial location in the disk, giving the fraction that are incorporated into surviving planets versus those that were ejected or collided with the Sun. All the initial material remains at the end of simulations using Model A. The addition of the giant planets, and their associated resonances, reduces the number of surviving objects to p85% in Model B—a fraction which is independent of initial semi-major axis, a0 . The majority of the remainder are lost via hyperbolic ejection. In Model C the fraction of survivors with a0 , 1 AU is the same as in Model B. Exterior to 1 AU the proportion of objects that survive decreases monotonically with increasing distance from the Sun. Clearly, the presence of material beyond 2 AU reduces the fraction of embryos with 1 , a0 , 2 AU that survive until the end of the simulation (57% in Model C compared to 83% in Model B). The additional embryos in Model C remove many objects with 320 CHAMBERS AND WETHERILL TABLE I Surviving Objects for Each of the Simulations Using Model A Simulation code Last event (106 year) a (AU) e i (deg.) Mass (Earth 5 1) Component embryos 1A .100 2A .100 3A .100 4A .100 5A 37 6A .100 7A .100 8A .100 9A .100 0.63 0.96 1.23 1.57 1.81 1.97 0.58 0.77 1.05 1.45 1.73 2.22 2.35 0.87 1.50 1.81 1.85 2.06 2.36 0.52 0.87 1.04 2.25 2.50 2.57 0.61 0.82 1.14 1.63 2.28 0.58 1.14 2.02 2.25 0.59 1.17 2.21 2.80 3.15 0.64 0.99 1.48 2.01 2.57 2.58 0.57 0.82 1.12 1.81 1.92 2.47 0.09 0.05 0.05 0.13 0.13 0.04 0.08 0.08 0.08 0.04 0.08 0.16 0.08 0.04 0.08 0.07 0.12 0.16 0.12 0.40 0.49 0.10 0.24 0.14 0.31 0.08 0.08 0.06 0.06 0.07 0.06 0.04 0.05 0.05 0.30 0.18 0.34 0.11 0.15 0.32 0.14 0.18 0.16 0.21 0.10 0.09 0.10 0.08 0.28 0.11 0.04 2 3 3 5 5 2 4 4 3 2 5 6 7 6 6 6 12 5 5 8 34 2 32 5 11 5 4 2 6 2 3 2 6 8 7 6 6 16 33 12 11 10 9 25 10 5 3 3 10 5 9 0.54 0.39 0.30 0.08 0.03 0.49 0.31 0.26 0.45 0.48 0.13 0.07 0.13 1.15 0.13 0.26 0.09 0.08 0.11 0.22 0.19 2.20 0.10 0.45 0.08 0.69 0.79 0.65 0.35 0.76 0.96 1.29 0.58 0.41 0.49 0.87 0.30 0.14 0.07 0.58 0.34 0.41 0.27 0.16 0.12 0.29 0.39 0.67 0.11 0.28 0.14 17 9 4 1 1 8 9 10 8 6 4 1 2 30 2 4 1 1 2 5 1 17 1 7 1 12 7 6 2 5 14 11 3 4 11 9 2 1 1 12 3 4 3 1 1 7 6 7 1 2 1 321 TERRESTRIAL-PLANET FORMATION TABLE II Surviving Objects for Each of the Simulations Using Model B Simulation code Last event (106 year) a (AU) e i (deg.) Mass (Earth 5 1) Component embryos 1B 127 2B 83 3B 185 4B 86 5B 50 6B 72 7B 88 8B 259 9B 73 0.71 1.08 1.74 0.50 1.39 0.76 1.76 0.68 1.27 0.31 1.27 0.60 1.09 0.59 0.79 1.40 2.35 0.61 1.37 0.50 0.98 1.88 0.10 0.08 0.14 0.16 0.24 0.16 0.22 0.13 0.09 0.12 0.65 0.17 0.29 0.08 0.10 0.04 0.07 0.37 0.18 0.07 0.26 0.21 5 8 14 2 5 4 13 2 6 15 17 2 6 6 7 4 6 17 20 32 3 5 0.89 0.59 0.25 1.05 0.45 1.31 0.29 1.85 1.01 1.18 0.51 1.59 0.70 0.46 0.49 0.68 0.18 0.94 0.40 0.24 1.11 0.28 24 12 3 28 8 33 4 21 9 16 5 21 4 5 11 6 1 13 7 6 12 2 a0 , 2 AU by scattering them into the asteroid belt where they are lost via resonances with the giant planets. Figure 21 shows the composition of the surviving objects in terms of the initial location of the embryos incorporated into each object. The graph combines the data from all the simulations using Model C. Final planets with 0 , a , 1 AU tend to be composed mainly of embryos from the inner part of the disk—the region between 0 and 2 AU. Planets with 1 , a , 2 AU contain only a small amount of material from closer to the Sun, but a substantial fraction TABLE III Surviving Objects for Each of the Simulations Using Model C Simulation code Last event (106 year) a (AU) e i (deg.) Mass (Earth 5 1) Component embryos 1C 153 2C 67 3C 229 4C 61 5C 266 6C 86 7C 8C 124 239 9C 68 0.90 2.83 0.68 1.51 0.70 1.31 0.60 1.06 2.49 0.45 1.61 0.53 1.13 2.54 0.81 0.98 1.56 1.08 0.20 0.36 0.17 0.03 0.13 0.16 0.29 0.19 0.11 0.14 0.02 0.24 0.15 0.41 0.34 0.15 0.14 0.18 4 35 5 23 9 11 7 5 17 6 8 15 9 39 7 9 18 8 1.67 0.05 1.33 0.49 1.52 0.22 2.04 0.87 0.81 1.23 1.09 1.44 1.53 0.07 2.74 1.37 0.39 2.65 27 1 27 3 33 1 23 4 2 15 5 13 11 1 27 14 3 19 322 CHAMBERS AND WETHERILL FIG. 17. The osculating orbits of each surviving object at the end of the nine simulations using Model A. Each body is represented by a symbol whose radius is proportional to the radius of the object. of their mass comes from the asteroid belt. Surviving objects in the asteroid belt are almost entirely composed of material from this region, although we note that the small number of such objects in our simulations makes this conclusion somewhat uncertain. 5. DISCUSSION The final states of the 27 simulations in Figs. 17–19 vary considerably. This despite the fact that the simulations in the same row of each figure (e.g., 1B, 2B, and 3B) have identical initial conditions, except for the values of the embryo’s angular elements. The evolution in each simulation is highly stochastic, making it difficult to predict the kind of planetary system that a particular protoplanetary disk will produce. One thing is clear however: each of our simulations produces a planetary system that is qualitatively different from our own. On the plus side, the simulations yield a small number of terrestrial planets moving on isolated or- TERRESTRIAL-PLANET FORMATION 323 FIG. 18. As Fig. 17 for Model B. bits. In addition, they frequently produce two planets with semi-major axes and masses comparable to Earth and Venus. However, the results differ from the inner solar system in a number of respects: 1. 2. 3. than 4. There are generally too few terrestrial planets. The planets have orbits that are too widely spaced. The orbits have values of e and i substantially larger Earth and Venus. The outermost object in the terrestrial-planet region (a , 2 AU) is generally much more massive than Mars (p0.1M%). 5. No analogues of Mercury are present. Some of these points are easier to address than orders. The absence of objects with orbits similar to Mercury (a p 0.4 AU) is presumably due to our decision to truncate the inner disk at 0.55 AU. The steep gravitational potential gradient in this region limits the extent to which objects can be scattered inward from the inner edge of the disk. 324 CHAMBERS AND WETHERILL FIG. 19. As Fig. 17 for Model C. This explains why only two surviving objects have a , 0.5 AU. The high mass of ‘‘Mars’’ in our simulations may be due to the large masses of the embryos that were initially present in this region (say 1.2 , a , 2.0 AU). The rapid falloff in accretion rate with distance from the Sun (see Section 3.3) implies that the mass of a planet that forms in the vicinity of Mars largely depends on the masses of the embryos which began there. To illustrate this, Fig. 22 shows the sizes of all the surviving objects, where ‘‘size’’ indicates the fraction of the embryos in the simulation that collided to make up that planet. For example, if a planet is the result of the accumulation of 20 embryos, and 40 were present initially, then the planet’s size is 0.5. The difference between the inner and outer terrestrial region is now more marked than in Fig. 20, especially for simulations that include the giant planets. Surviving objects with a . 1.2 are often composed of only one or two embryos. In contrast, planets closer to the Sun usually contain .10 embryos. Hence, if the embryos in the Mars region were actually smaller than those used in our simulations, the low mass of the red planet becomes easier to understand. TERRESTRIAL-PLANET FORMATION 325 FIG. 21. Fraction of the mass of surviving planets that comes from different regions of the initial disk. The three curves show the mass fractions for final planets with 0 , a , 1, 1 , a , 2 and 2 , a , 3 AU. FIG. 20. The masses and semi-major axes of all surviving objects at the end of the simulations. This raises the interesting possibility that Mars represents a leftover planetary embryo that underwent little or no further accretion after the cessation of the earlier runaway-growth phase of planet formation. The accretion rate 1.5 AU from the Sun may have been so slow that most of the embryos that began here were scattered inward and swept up by Earth and Venus or scattered outward TABLE IV Fraction of the Initial Population of Embryos That Are Incorporated into Surviving Objects, as a Function of the Embryos’ Initial Semi-major Axes Initial a (AU) Model A Model B Model C 0–1 1–2 2–3 3–4 1.00 1.00 — — 0.84 0.83 — — 0.84 0.57 0.31 0.18 FIG. 22. The sizes and semi-major axes of all surviving objects. Here size is defined as the fraction of the embryos initially present in a simulation that accumulated to form each planet. 326 CHAMBERS AND WETHERILL and removed via resonances in the asteroid belt, before they could accrete one another. In this scenario, Mars is the sole embryo that managed to avoid these fates. Once all its competitors were removed there was no longer a mechanism to scatter Mars out of its orbit, and its status changed to that of a final planet. The problem of the large eccentricities and inclinations arises at an early stage in our simulations, as mutual perturbations between embryos rapidly increase the mean values of e and i (Fig. 6). Collisions between objects often diminish these quantities during the final stages of accretion, but the effect is insufficient to yield the essentially circular orbits observed for Earth and Venus. This result is intimately connected with the small number of planets produced by our simulations and their wide spacing. The large eccentricities of the embryos, compounded by the large secular oscillations about the mean values, imply that neighboring protoplanets must be widely spaced to avoid scattering or accreting one another. This in turn limits the number of final planets, since they are restricted to the region between the inner edge of the disk and the point at which strong resonances with the giant planets occur (a p 2 AU). The large values of e and i are especially marked in Model C, which may be due to the large masses of the embryos in the asteroid belt. It it quite possible that objects this large never formed here, which would go some way toward remedying the problem. However, the values of e and i are also too large in simulations using Model B, where there are initially no embryos beyond 1.8 AU. Clearly something else is wrong, and with hindsight we suggest that it is the initial mass distribution used in our calculations. Earlier, during the runaway growth phase of the evolution, equipartition of random energy (dynamical friction) between planetary embryos and planetesimals ensures that the largest objects have orbits with small e and i. This effect is sufficient to overcome perturbations between embryos, which will tend to excite these quantities. However, once the embryos contain a substantial fraction of the solid disk material, dynamical friction will become too weak to damp down inter-embryo perturbations unless these perturbations are also weak. This can only occur if the remaining embryos have orbits which are tens of Hill radii apart, which in turn implies that only a handful of embryos are present at this stage. Hence, a more likely configuration at the start of the final stage of terrestrial-planet formation is a handful (,10) of widely separated protoplanets (with masses of say 0.03– 0.3 M%) and a few tens of smaller ‘‘failed embryos.’’ These objects would presumably be accompanied by a population of residual planetesimals containing a small fraction of the total mass. In this scenario, many of the protoplanets are destined to become true planets and so the final locations of the planets are already determined, to some extent, during the runaway growth phase. We plan to test the scenario outlined here in subsequent simulations. 6. SUMMARY The principal conclusions drawn from our simulations are: • A disk of initially isolated planetary embryos quickly becomes dynamically excited once mutual perturbations allow embryos to achieve crossing orbits. At this point e and i increase rapidly. • Once isolation is overcome, the evolution of e and i is driven largely by secular perturbations rather than close encounters. Secular resonances between pairs of embryos are common, and the n 5 (a p 0.6 AU) and n 6 (a p 2.1 AU) resonances also play a role in some simulations. • Orderly growth takes place despite the presence of gravitational focusing early in the integrations. However, weak dynamical friction is apparent even though only a few tens of objects are present. • Accretion occurs most rapidly in the inner part of the disk, with the collision rate decreasing sharply as a increases. Almost no collisions occur beyond 2 AU. Instead, most material with a . 2 AU is removed by meanmotion and secular resonances with the giant planets. • Embryos with 1.2 , a , 2 AU tend to be scattered outward and removed by resonances or scattered inward and accreted by planets closer to the Sun, before they can accrete one another. Thus, Mars may be an unaccreted embryo that remained in this region. • Accretion is complete after 108 years in 60% of simulations that include Jupiter and Saturn, and in all such simulations after 3 3 108 years. Simulations that neglect the giant planets still have several objects with crossing orbits at 108 years. • The most common outcome is a pair of large planets interior to 2 AU. Occasionally a smaller object remains further from the Sun. The surviving planets tend to have time-averaged eccentricities and inclinations that are substantially larger than those of Earth and Venus. • The number and spacing of the planets in our simulations are largely determined by the mean values of e, and the amplitude of secular oscillations in e, during the final stages of accretion. ACKNOWLEDGMENTS We are grateful to Alan Boss, Martin Duncan, Hal Levison, Jack Lissauer, and Derek Richardson for fruitful discussions during the preparation of this paper. Thanks also to Conel Alexander, Geoff Coxhead, Sandy Keiser, and Martin Murphy for computer support, and to Hal Levison and Martin Duncan for providing the routines that formed the basis of our computer code. This work was partly supported by NASA Grants NAG5-3656, NAG5-4285, and NAG5-4386—thank you. TERRESTRIAL-PLANET FORMATION REFERENCES Beaugé, C., and S. J. Aarseth 1990. N-body simulations of planetary formation. Mon. Not. R. Astron. Soc. 245, 30–39. Chambers, J. E., and G. W. 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