Making the Terrestrial Planets: N-Body Integrations of Planetary Embryos in Three Dimensions

Transcription

1 ICARUS 136, (1998) ARTICLE NO. IS 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 and G. W. Wetherill Department of Terrestrial Magnetism, Carnegie Institution of Washington, 5241 Broad Branch Road NW, Washington DC 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 10 5 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, sug- gesting 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 scat- tered 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 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 Academic Press Key Words: planetary formation; terrestrial planets; plane- tary 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, 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 3000-km diameter planetary embryos. The second act is characterized by runaway growth, in which equiparti- tion of random orbital energy between planetesimals en- sures that the largest objects have orbits with low eccentric- ities 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 big- gest objects is enhanced by gas drag acting on small colli- sion fragments, giving them circular, co-planar orbits too (Wetherill and Stewart 1993). In Act Three, planetary em- bryos 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 colli- sional evolution of disks of planetary embryos in three dimensions (e.g., Wetherill 1992, 1994, 1996). This tech- nique treats individual close encounters and collisions ef- fectively 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 /98 $25.00 Copyright 1998 by Academic Press All rights of reproduction in any form reserved. 304

2 TERRESTRIAL-PLANET FORMATION 305 eled using N-body integrations in two dimensions, by Lecar tion 4 examines the end products. In Section 5 we discuss and Aarseth (1986), and Beaugé and Aarseth (1990). In the results in comparison to the observed solar system. addition, Cox and Lewis (1980) carried out 2D calculations Finally, the last section contains a summary. that neglected long-range perturbations between embryos. Numerical integrations automatically include the effects 2. N-BODY SIMULATIONS of secular and resonant interactions between embryos. However, calculations limited to two dimensions artificially We performed three sets of nine N-body integrations, decrease the collisional timescale with respect to the timeterrestrial each set using a different model for the formation of the scale for orbital evolution. planets. These approximations were chosen because they require Model A. These integrations simulate the evolution of substantially less computer time than more-realistic a disk of planetary embryos that initially spans most of N-body integrations in three dimensions. Both types of the region currently occupied by the terrestrial planets. In simulation yielded plausible planetary systems, although this model it is assumed that the giant planets do not these were not always similar to our own. They also prosignificantly influence the formation of the terrestrial planvided insight into the chaotic nature of planet formation ets, and hence they are not included in the integrations. that results from the central role of close encounters a level of understanding that goes beyond that achievable Model B. As Model A, but the effects of the giant planfrom analytic models. ets are modeled by adding Jupiter and Saturn to the simula- Recent improvements in the performance of computer tions after 10 7 years. The giant planets are assumed to have workstations, and the development of a new N-body algorithm, their current masses and orbital elements. now make it possible to carry out N-body integra- Model C. As Model B, but the initial disk of embryos tions, in three dimensions, of several tens of gravitationally is extended to encompass the region that now contains the interacting bodies for the 10 8 orbits necessary to form asteroid belt. Jupiter and Saturn are added at 10 7 years, the cores of the inner planets. This led us to pose the as in Model B. following question. Is it possible to create a recognizable The nine simulations using each model are divided into system of terrestrial planets by integrating the orbits of a batches of three, each batch using different values for the disk of planetary embryos for 100 million years, subject surface density of solid material at 1 AU,, and the spacing only to mutual gravitational interactions, inelastic collibetween embryos,. One batch each uses (, ) (6, 7), sions, and external perturbations from the giant planets? (10, 7), and (6, 10), where is measured in units of gcm If it is possible, such simulations should indicate whether 2 and in mutual Hill radii, R HM, where 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 R HM m 1 m 2 3M 1/3 a 1 a 2 embryos formed by runaway growth. (Alternatively, it may 2 (1) all be a matter of luck, with the final outcome depending on a few key events that occur essentially at random.) It for embryos with masses m 1 and m 2, and semi-major axes should also become possible to predict the characteristics a 1 and a 2. of terrestrial planets in extra-solar systems long before we can determine them observationally. Conversely, if N-body 2.1. Initial Conditions simulations involving a few dozen embryos cannot produce something akin to the terrestrial planets, they may at least The initial conditions were chosen bearing in mind the indicate what extra physics is required to do so. form of the present planetary system and the results of With this in mind we have carried out 27 integrations simulations of the runaway-growth phase of terrestrialof disks of planetary embryos, starting with objects on planet formation (e.g., Wetherill and Stewart (1993). isolated, nearly coplanar orbits, and following their evolu- Disk density. In 18/27 simulations we adopt a surface tion for at least 10 8 years. In two thirds of the simulations density of solid material, 6 gcm 2 at 1 AU. This corre- we have also included the effects of the giant planets Jupi- sponds to the minimum mass needed to form the current ter and Saturn, assuming they formed before the accretion terrestrial planets. We choose a density profile that varies of the terrestrial planets was complete. All the integrations as 1/a a smaller gradient than used by some authors in were performed on dedicated DEC alpha workstations, view of the large amount of solid material ( 10 gcm 2 ) requiring 3 years of CPU time. required beyond the ice condensation point to form Jupiter s The next section describes the integration method and core before loss of the nebula gas (Pollack et al. 1996). the initial conditions used in the simulations. Section 3 As a variant on our standard initial conditions we set 10 looks at the evolution of the disks of embryos, whilst Sec- gcm 2 at 1 AU in three of the simulations for each model.

3 306 CHAMBERS AND WETHERILL Radial extent. The 18 simulations using Models A and Embryo masses. Having chosen and, the mass of B begin with a disk of embryos having semi-major axes each embryo is uniquely determined. These range from 0.55 a 1.8 AU, covering most of the terrestrial-planet 0.02 M at the inner edge of the disk to 0.1 M at the region. The lower bound is a compromise between making outer edge in Models A and B, with 6 gcm 3. Larger the simulation realistic and avoiding a short integration objects are present initially in the simulations that have a timestep (and hence a large CPU overhead), which is necessary higher disk density or contain embryos in the asteroid belt. when some objects have small a. In the simulations Embryo density. The embryos radii are calculated as- using Model C, we extend the outer edge of the disk to suming a density of 3 gcm 3. This value is also used to 4.0 AU to include embryos that may have formed in the determine the new radius of an object following the accreasteroid belt. tion of a smaller body. Models A and B begin with embryos, depending on the values of and, while Model C begins with Integration Method embryos. The initial disk mass varies between 1.8 and 3.2 The N-body integrator used in the simulations is based M for Models A and B, and between 5.0 and 8.6 M for upon a second-order mixed-variable symplectic integrator Model C (which includes material in the asteroid belt). written by Levison and Duncan (1994), which uses an algorithm Orbital elements. The initial eccentricities, e, and incliback described by Wisdom and Holman (1991). One draw- nations, i, are 0 and 0.1, respectively, for all embryos. of this fast package is that close encounters between These values are somewhat arbitrary, but they quickly massive bodies are not calculated accurately, so we modibecome randomized once close encounters occur, and most fied the code to integrate all objects using a Bulirsch Stoer objects soon attain much larger values of e and i. However, method (Stoer and Bulirsch 1980), also supplied by Levison for the embryos to have formed via runaway growth there and Duncan, whenever the separation between a pair of must have been an epoch when e and i were small, and objects falls below 2 Hill radii. The Bulirsch Stoer algo- we assume that this is still the case at the start of our rithm uses a variable timestep to accurately follow the simulations. The remaining elements for each embryo orbital evolution during an encounter, whilst the symplecthe mean longitude and nodal longitude are chosen rantic tic integrator uses a fixed timestep of 10 days. The symplecdomly. algorithm uses Jacobi coordinates, which makes it nec- Embryo separations. We assume that the planetary essary to periodically re-index the objects in order of embryos that formed via runaway growth have become increasing semi-major axis. This procedure minimizes nu- isolated from one another prior to the start of our simulamerical error introduced by having different physical and tions due to mutual accretion events and eccentricity damp- Jacobi ordering of the objects. ing via dynamical friction. Once damping forces have been Combining the symplectic and non-symplectic methods overcome, a system containing more than two embryos leads to a secular growth in the energy error, typically one becomes unstable with respect to close encounters on a part in over 10 8 years. While less than ideal, we timescale, t believe this is acceptable given that neglected effects, such c, that depends exponentially on the initial oras dynamical friction due to residual planetesimals, probabital spacing,, measured in mutual Hill radii (Chambers et al. 1996). bly have a comparable or larger effect. Collisions between embryos are assumed to be com- In most of our simulations we use 7, which correpletely inelastic, forming a single new body whose orbit sponds to an embryo crossing time, t c years for is determined by conservation of linear momentum. Our the case in which all objects have the same mass. Three model assumes that mass loss due to fragmentation during integrations per model begin with 10, implying that a collision is negligible. This assumption is necessary in t c 10 7 years in the equal mass case. These two values of order to avoid a rapid increase in the number of bodies t c bracket the probable time required to form embryos present in the integration, which would make the computain the terrestrial region (Wetherill and Stewart 1993). In tional expense prohibitive. practice, t c is somewhat smaller in our integrations because the embryos begin with a range of masses (see Section 3. EVOLUTION 3.1). We also note that t c would be reduced if the embryos began with non-zero eccentricities (Yoshinaga et al. 1998). Figures 1 3 show a series of snapshots, in semi-major The values of chosen for our integrations are broadly axis/mass space, taken from three of the simulations deconsistent with the results of N-body integrations of em- scribed above one for each model. Each symbol in the bryo formation by Kokubo and Ida (1998). These authors figures represents a surviving embryo, with the horizontal found that embryos formed from a population of 10 4 bars depicting the perihelion and aphelion distances of small bodies tend to have orbits spaced by about 10 mutual its orbit. In these and most of the other integrations the Hill radii. evolution passes through four stages, described below.

4 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 7 and 6 gcm 2. 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) Embryo Isolation Is Overcome (Figs. 1b, 2b, 3b) Figure 4 shows the evolution of the perihelion and aphelion distances (q and Q, respectively) of the innermost The initial orbital spacing of the embryos is large enough 12 embryos for the first 20,000 years of a simulation using that no pair of objects is able to undergo close encounters Model A. Initially the embryos semi-major axes remain in the absence of external perturbations, due to conserva- almost constant, while their eccentricities, e, exhibit erratic tion of energy, E, and angular momentum, h (Gladman oscillations whose amplitude increases over time. After 1993). When more than two embryos are present, E and about 12,000 years the eccentricities have increased suffih are no longer conserved within each pair of objects, and ciently for neighboring embryos to undergo close encounters, their isolation can be overcome. and the initial isolation is broken. The time required

5 308 CHAMBERS AND WETHERILL FIG. 2. As Fig. 1 for a simulation using Model B, with the same values of and. 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,. 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 dimen- sions or by artificially increasing their physical radii. In

6 TERRESTRIAL-PLANET FORMATION 309 FIG. 3. As Fig. 1 for a simulation using Model C, with the same values of and. 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). significantly shorter than those found by Chambers et al. (1996) for systems of equally spaced, equal-mass planets (t c and 10 7 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 experi- ence their first close encounter at different epochs, usually 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, t c, depends on the masses and spacing of the embryos. In the simulations with initial spacing 7, the first close encounters typically occur after t c 10 4 years, whilst integrations with 10 exhibit encounters after years. These times are

7 310 CHAMBERS AND WETHERILL FIG. 4. Perihelion (q) and aphelion (Q) distances for the 12 innermost 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 10. Chambers and Wetherill (1996) found that the already- embryos during the first 20,000 years of a simulation using Model short crossing times seen here are reduced still further A. Note the rapid orbital evolution following the first close encounter when a population of smaller objects (mass 0.01 embryo at 12,000 years. 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 beginning with objects close to the inner edge of the disk. the largest objects are damped by dynamical friction and Figure 5 shows the time of first encounter for the embryos collisions with smaller bodies (Wetherill and Stewart 1993), in four of the simulations. The times are measured in units 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- 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 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 is the initial embryo separation in mutual Hill radii.

8 TERRESTRIAL-PLANET FORMATION 311 sponding decrease in the total mass of the planetesimals embryos, or with Jupiter and Saturn in simulations that reduces their ability to damp the embryo s eccentricities. include the giant planets. This combination is likely to produce a rapid transition to For example, Fig. 8 shows the evolution of e for four a situation in which the embryos achieve crossing orbits embryos from two simulations using Model C. The figure and the disk becomes dynamically excited. also shows the critical argument of a secular resonance involving the longitude of perihelion,, of each object and 3.2. The Disk Becomes Dynamically Excited that of another nearby embryo. Libration of the critical (Figs. 1c, 2c, 3c) argument about 0 indicates that the perihelion directions of the two embryos are aligned, whilst oscillation about Once neighboring embryos have acquired crossing orbits 180 implies that the perihelion directions are antialigned. with significant eccentricities, strong orbital interaction can In each case the changes in e are associated with librations occur due to close encounters and resonances between in the critical angle. At some epochs additional oscillations embryos. This in turn leads to rapid changes in e and i, are apparent, caused by secular interaction with other emand the disk becomes dynamically excited. This can be bryos. An object usually undergoes strong interactions with seen in Figs. 1c, 2c, and 3c, where many of the embryos two or three other embryos simultaneously and typically have horizontal bars that overlap, indicating crossing orbits moves back and forth between several secular resonances with nonnegligible eccentricities. during an integration. Figure 6 shows the mass-weighted values of (e 2 i 2 ), Secular resonances with the giant planets also occur. e, and i, for objects in two simulations using Model A. Each Figure 9 shows two cases in which the orbits of embryos of these quantities increases logarithmically with time; in lie within the 5 resonance, where the longitude of perihe- other words, very quickly at first, and then more slowly at lion of the protoplanet precesses at a similar rate to that later times. The rate of increase is largest for simulations of Jupiter. This causes smooth, long-period changes in the with high surface densities and thus more massive embryos. object s eccentricity. This resonance influences the orbits The upshot is that gravitational focusing during close enespecially the region 0.5 a 0.7 AU, and occasionally of objects with a 2 AU in several of our simulations, counters will diminish over time, since this is most efficient at small relative velocities. Hence the collision probability causes embryos to fall into the Sun. The analogous 6 for a given pair of embryos will also decrease with time. resonance with Saturn plays an important role for objects This is reflected in a short-lived burst of collisions near the with 2 a 2.3 AU. We note that the locations of these two secular resonances will depend to some extent on how start of each simulation, followed by a monotonic decline in much nebula gas is present during the accretion of the the collision rate (see Section 3.3). terrestrial planets (Lecar and Franklin 1997) and also the The embryos eccentricities and inclinations undergo degree to which the giant planets orbits change during large variations throughout an integration. These changes the formation of the planetary system. are primarily caused by secular perturbations rather than The remaining diagrams in Fig. 9 show examples of close encounters. In the early evolution, each episode typianother common situation in which an embryo s longitude cally lasts for 5 20 thousand years (Fig. 7), and changes of perihelion,, becomes almost stationary. Like the Kozai in e are often correlated with the behavior of the argument resonance, this resonance does not directly involve the of perihelion,, due to the Kozai resonance (Kozai 1962). orbits of other bodies. Whilst is almost stationary, the For example, the strong increase in the eccentricity of embryo s nodal longitude and argument of perihelion usu- Planet 11 at t 50,000 years in Fig. 7 is associated with a ally circulate rapidly in opposite directions. This behavior libration of about 90. Kozai oscillations, often seen in is generally accompanied by a monotonic increase in e, the orbits of comets, are due to the long-term interaction which can also cause embryos to fall into the Sun. of an orbit with a planar distribution of matter, which is In contrast to secular perturbations, close encounters, clearly the case here. which produce an abrupt jump in the orbital elements, Secular resonances between pairs of embryos are com- appear to play a minor role in determining changes in e mon in each of the simulations. These produce correlated and i only a handful of examples can be seen in Figs. oscillations in the eccentricities or inclinations of the two 7 9. Close encounters mainly affect e and i by determining objects, and librations of the resonant critical argument, the point in a secular cycle at which an object is scattered with periods 10 5 years. A common example involves a away from one secular resonance and into another. This situation in which the longitudes of perihelion of two emcritical in turn establishes the initial values of e and i and the bryos precess at the same rate, with the two orbits aligned argument for the new resonance. or anti-aligned with one another. Occasionally three, or even four, orbits are temporarily locked together in this 3.3. Protoplanets Are Formed (Figs. 1d, 2d, 3d) way. The resulting high-frequency oscillations in e are fur- After 3 6 million years, one or two objects of ther modulated by secular interaction with other nearby Earth masses have formed interior to 1.5 AU in each of

9 312 CHAMBERS AND WETHERILL FIG. 6. Mass-weighted mean of (e 2 i 2 ) 1/2 (upper) versus time for two of the simulations using Model A, with 6 and 10 gcm 2, 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- case yielding larger values. the simulations. Such objects are apparent at 0.9, 1.1, and tricities and inclinations present at the start of the integrations 0.7 AU in Figs. 1d, 2d, and 3d, respectively. In simulations are conducive to gravitational focusing between using Model C, where the disk of embryos extends into embryos a necessary prerequisite for runaway growth. the asteroid belt, additional large objects are present beyond However, as e and i increase this situation will change. We 1.5 AU. However, each of these is the result of only can see how the effects of gravitational focusing vary during one or two collisional events, and their large mass is due a simulation by looking at the distribution of close-encoun- to the high initial masses of the embryos in this region. ter distances. In the absence of focusing, the number of Do these protoplanets form by a continuation of runaway encounters, N, with minimum separation, D, is given by growth or via more orderly growth? The low eccen- N D, while in the limit of strong focusing N D 1/2.

10 TERRESTRIAL-PLANET FORMATION 313 FIG. 7. Eccentricity and argument of perihelion,, versus time for four embryos for the first 10 5 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, indicating orbital evolution controlled by the Kozai resonance. Figure 10 shows the close-encounter distribution at three of objects present probably makes it difficult for runaway epochs of a simulation using Model A, where we have growth to get going before the supply of embryos is ex- combined the close-approach data for all the embryos pres- hausted. ent. A 2 test indicates that we can reject the hypothesis There is some indication that equipartition of random that focusing is absent at the first epoch ( years) at orbital energy ( dynamical friction ) takes place. For example, the 99.5% confidence level. Conversely, the data for the Fig. 12 shows the eccentricities of all surviving obthe third epoch (9 10 million years) are consistent with a lack jects in Model C after 10 million years as a function of of gravitational focusing, while those for second epoch their mass. The largest objects tend to have lower values of (3 3.5 million years) are ambiguous the probability that e than the smaller bodies. Conversely, the smallest embryos the distribution is uniform is roughly 10%. generally do not have eccentricities close to zero. Given that gravitational focusing is probably significant A quantitative measure of the importance of dynamical during at least a part of the simulations, does this friction comes from examining the values of e and i for lead to runaway growth? Figure 11 shows the mass each pair of objects just before they collide. Consider first distributions at four epochs, combining the data for all the larger of the two objects in each collision. In Model the embryos with a 1.5 AU in integrations using C, 69% of these objects have inclinations below the mean Model C. As the embryos accrete one another the mass value for that epoch, while for e the figure is 56%. The distribution flattens, except for the largest objects, which corresponding numbers for the smaller colliding body are march steadily toward higher masses i.e., orderly growth 59 and 45%, respectively. In other words, weak equiparti- occurs. Apparently, gravitational focusing during the tion of random energy takes place for objects undergoing early stages of the simulations is not enough to allow collisions a necessary prerequisite for runaway growth. runaway growth to occur. In addition, the small number Incidentally, the preference for low inclinations over low

11 314 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,, of each object and another nearby embryo. Librations about 0/180 indicate that the perihelion directions of the two objects are aligned/anti-aligned. dn dt p/(1 e) a a p /(1 e) n(a) P(a) P node(a p, a, e, i) P conj (a p, a, e, i) da, than the planet per unit semimajor axis. In addition, P node 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 P node 2 2R p f grav 2ae sin i, 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 de- crease 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 a p. The rate at which the planet accretes the smaller embryos in its vicinity is where R p (a p ) is the radius of the planet, and f grav (e, i) is the gravitational focusing factor. Finally, P conj is the probability that the planet is in the correct part of its orbit at conjunction for a collision to take place, given by P conj 2R p f grav 2 a p. If we assume that the embryo masses, m, and their mean spacing are comparable to their initial values, then n(a) 1/a 3/2, and m(a) a 3/2, which also implies that the planewhere a, e, i, and P are the orbital elements and period of an embryo, and n(a) is the number of embryos smaller tary radius is given by R p ap 1/2. Provided that embryos

12 TERRESTRIAL-PLANET FORMATION 315 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 5 resonance between the longitudes of perhelion, and J, of the embryo and Jupiter. In the lower diagrams, the change in e is associated with epochs when 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 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 dt f 2 grav e sin i p/(1 e) a a p /(1 e) 1 a da f 2 grav(e, i) 4 a 3 p sin i. (2) dn N(N 1). (3) dt Note the steep dependence of the collision rate on a p and the lack of dependence on e outside the gravitational- focusing 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 3 p 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 7. The solid lines show the actual fraction remaining, whilst the dashed lines indicate the fraction expected according to a decay 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 propor- tional 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 de- crease as the mean inclination of the embryos rises (see Eq. (2)) 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 10 7 years, with their current masses and orbital elements. The immediate effect

13 316 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 20 million years after the giant FIG. 11. Cumulative mass distribution for all the simulations using FIG. 13. Number of collisions, as a function of semi-major axis, a, Model C, at four epochs during the integrations. Note that only objects of the larger impactor, for all integrations using Model C. The curve with a 1.5 AU are included in the distributions. shows the predicted distribution following a 1/a 3 law.

14 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 N(N 1). TERRESTRIAL-PLANET FORMATION 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 50 million years due to interactions with a much smaller body. A series of close encounters with the smaller embryo nudges the protoplanet into the 5 resonance, producing a rapid drop in e (Fig. 16). The residual embryo is subsequently removed, leaving the larger body on a low-eccen- tricity orbit. This late-stage orbital damping occurs in sev- planets are added, most of the embryos with a 2AU have been removed (Fig. 3e). Thus, 30 million years into a simulation, a combination of rapid accretion in the inner eral of our simulations, but does little to alter the semi- part of the disk and partial clearing of the asteroid belt by major axes of the final planets, these being determined resonances leaves three to six large protoplanets containing earlier in the evolution. most of the remaining mass, together with a compa- rable number of smaller unaccreted embryos. 4. THE FINAL STATE OF THE SIMULATIONS The transition from this state of affairs to a system of isolated, final planets depends principally on the ampli- Figures show the final states of the integrations tude of the secular oscillations in the eccentricities of the using Models A, B, and C, respectively. The figures indicate surviving protoplanets. These oscillations have two the osculating orbit of each surviving planet, with the sources. First, secular resonances between neighboring planet itself represented by a filled symbol whose radius planets on crossing or nearly crossing orbits, which produce is proportional to the radius of the body. The same data short-period ( 10 5 year) cycles. Second, secular perturbations are given in Tables I, II, and III, except that the values of and/or resonances with Jupiter and Saturn, having e and i have been averaged over the last 10 6 years of the periods of years. In general the former are dominant integration. The column headed last event refers to the except for a 2.1 AU, where the 6 resonance causes time at which the last accretion or ejection took place. large changes in e and occasionally the region a 0.6 AU, Where close encounters are still taking place, it is assumed where the 5 resonance can play an important role (these that the last event time will be greater than the length of values of a assume that there is no longer a significant the integration. amount of nebula gas present and that the giant planets The results of simulations using Model A are given in have their modern orbits). Fig. 17. In each case 10 8 years have elapsed since the start The secular oscillations in e cause the perihelion and of the calculation. In most cases the evolution is incomplete aphelion distances (q and Q, respectively) to change on in the sense that several objects are still able to undergo timescales that are short compared to the collision time- close encounters with one another. However, the collision scale. In order to avoid collisions with one another, protoplanets rate has slowed to almost zero, so it is not apparent how must be spaced so that the maximum value of much longer would be required to achieve a set of isolated Q for the innermost object is less than the minimum value objects. In many of the simulations the innermost objects of q for the second object, and so on. Further evolution have achieved non-crossing orbits. These planets tend to takes place until this situation is achieved, with surplus have smaller i than objects further from the Sun, and they protoplanets being thrown back and forth until they merge usually contain most of the mass. However, it is quite likely with another or are scattered beyond 2 AU and removed that the configuration of these objects will change due to by a resonance. subsequent interaction with objects at larger a. The one

15 318 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 10 7 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 As with Model A, the planets closest to the Sun tend to contains five approximately equal-mass planets, with a be the largest. In addition, there is considerable scatter in roughly geometric orbital spacing. the mean values of e and i from one integration to another. The simulations using Model B were continued for 10 8 Simulations using Model C (shown in Fig. 19) usually years or until close encounters had ceased, whichever was produced only one or two planets interior to 2 AU. Several longer. The final state of each simulation is shown in Fig. of the simulations also contain an object in the asteroid 18. Generally there are two objects with a 1.7 AU, while belt, although these are invariably very large compared to in several cases a third object lies further from the Sun. a typical asteroid, due to the large embryo masses used in In the two simulations that yielded three planets, these our calculations. It is possible that some or all of these have roughly geometric orbital spacings. This is also true objects have orbits that are unstable over the age of the for the three outer planets in simulation 7B, although the solar system. In only one of the two simulations that ended two inner planets lie closer together than a geometric law with 2 isolated planets do the objects have geometrically would predict. In each case the mean spacing is somewhat spaced orbits. In almost all the simulations the orbital larger than the planets we observe in the inner solar system. spacing is larger than in the inner part of the solar system,

16 FIG. 16. Time evolution of the semi-major axis, a, eccentricity, e, and critical argument for the 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 50.8 million years. TERRESTRIAL-PLANET FORMATION 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 mate- 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 rial remains at the end of simulations using Model A. massive as Earth. The addition of the giant planets, and their associated In general, the surviving objects in our simulations have resonances, reduces the number of surviving objects to time-averaged values of e and i that are substantially larger 85% in Model B a fraction which is independent of than those of Earth and Venus (e 0.03 and i 2 ). Earth- initial semi-major axis, a 0. The majority of the remainder sized planets with eccentricities of 0.2 are not uncommon are lost via hyperbolic ejection. in our simulations. The large mean values of e occur early In Model C the fraction of survivors with a 0 1AUis in each integration and lead to correspondingly large maximum the same as in Model B. Exterior to 1 AU the proportion values, e max, during each secular oscillation. This in of objects that survive decreases monotonically with in- turn requires that protoplanets remain widely spaced to creasing distance from the Sun. Clearly, the presence of avoid scattering or accreting one another. In general, e max material beyond 2 AU reduces the fraction of embryos is too large to permit more than two, or occasionally three, with 1 a 0 2 AU that survive until the end of the simula- final planets to form in the simulations that contain Jupiter tion (57% in Model C compared to 83% in Model B). The and Saturn. Thus, the small number of terrestrial planets additional embryos in Model C remove many objects with

17 320 CHAMBERS AND WETHERILL TABLE I Surviving Objects for Each of the Simulations Using Model A Simulation Last event a i Mass Component code (10 6 year) (AU) e (deg.) (Earth 1) embryos 1A A A A A A A A A

18 TERRESTRIAL-PLANET FORMATION 321 TABLE II Surviving Objects for Each of the Simulations Using Model B Simulation Last event a i Mass Component code (10 6 year) (AU) e (deg.) (Earth 1) embryos 1B B B B B B B B B a 0 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 Last event a i Mass Component code (10 6 year) (AU) e (deg.) (Earth 1) embryos 1C C C C C C C C C