Making the Terrestrial Planets: N-Body Integrations …extranet.on.br/rodney/curso2010/aula2/chambers-wetherill.pdfICARUS 136, 304–327 (1998) ARTICLE NO.IS986007 Making the Terrestrial

ICARUS 136, 304–327 (1998)ARTICLE NO. IS986007

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

J. E. Chambers

Armagh Observatory, College Hill, Armagh BT61 9DG, United KingdomE-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

grains of dust near the midplane of the protoplanetarynebula accrete one another via low-velocity collisions,We simulate the late stages of terrestrial-planet formation

using N-body integrations, in three dimensions, of disks of up eventually forming 1- to 10-km sized ‘‘planetesimals’’to 56 initially isolated, nearly coplanar planetary embryos, plus (Weidenschilling 1997). These objects are large enough toJupiter and Saturn. Gravitational perturbations between em- possess nonnegligible gravitational fields that increase theirbryos increase their eccentricities, e, until their orbits become collision cross sections, aiding further growth to formcrossing, allowing collisions to occur. Further interactions pro- p3000-km diameter ‘‘planetary embryos.’’ The second actduce large-amplitude oscillations in e and the inclination, i, is characterized by ‘‘runaway growth,’’ in which equiparti-with periods of p105 years. These oscillations are caused by

tion of random orbital energy between planetesimals en-secular resonances between embryos and prevent objects fromsures that the largest objects have orbits with low eccentric-becoming re-isolated during the simulations. The largest objectsities and inclinations—orbits that are most efficient attend to maintain smaller e and i than low-mass bodies, sug-scooping up more material (e.g., Wetherill and Stewartgesting some equipartition of random orbital energy, but accre-

tion proceeds by orderly growth. The simulations typically pro- 1989, Kokubo and Ida 1996). Runaway growth of the big-duce two large planets interior to 2 AU, whose time-averaged gest objects is enhanced by gas drag acting on small colli-e and i are significantly larger than Earth and Venus. The sion fragments, giving them circular, co-planar orbits tooaccretion rate falls off rapidly with heliocentric distance, and (Wetherill and Stewart 1993). In Act Three, planetary em-embryos in the ‘‘Mars zone’’ (1.2 , a , 2 AU) are usually bryos strongly perturb the orbits of their neighbors untilscattered inward and accreted by ‘‘Earth’’ or ‘‘Venus,’’ or scat- they become crossing. Runaway growth now slows or shutstered outward and removed by resonances, before they can

down completely, and the embryos accrete each other inaccrete one another. The asteroid belt (a . 2 AU) is efficientlygiant impacts, leading to a handful of terrestrial planetscleared as objects scatter one another into resonances, whereon widely separated orbits.they are lost via encounters with Jupiter or collisions with

Act Three has been modeled extensively using thethe Sun, leaving, at most, one surviving object. Accretionalevolution is complete after 3 3 108 years in all simulations that Opik–Arnold scheme to follow the dynamical and colli-include Jupiter and Saturn. The number and spacing of the sional evolution of disks of planetary embryos in threefinal planets, in our simulations, is determined by the embryos’ dimensions (e.g., Wetherill 1992, 1994, 1996). This tech-eccentricities, and the amplitude of secular oscillations in e, nique treats individual close encounters and collisions ef-prior to the last few collision events. 1998 Academic Press fectively and uses a simple parameterization of the impor-

Key Words: planetary formation; terrestrial planets; plane- tant effects of the major Jupiter and Saturn resonances intary dynamics; extra-solar planetary systems.the asteroid belt. However, it does not include distantperturbations between embryos or sequences of encoun-ters due to node-crossing events, so the effects of secular1. INTRODUCTIONperturbations and resonances between embryos are be-yond its ability.The planetesimal theory of terrestrial-planet formation

is commonly viewed as a play in three acts. In Act One, The final stage of planetary accretion has also been mod-

3040019-1035/98 $25.00Copyright 1998 by Academic PressAll rights of reproduction in any form reserved.

TERRESTRIAL-PLANET FORMATION 305

eled using N-body integrations in two dimensions, by Lecar tion 4 examines the end products. In Section 5 we discussthe results in comparison to the observed solar system.and Aarseth (1986), and Beauge and Aarseth (1990). In

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 SIMULATIONSof secular and resonant interactions between embryos.

We performed three sets of nine N-body integrations,However, calculations limited to two dimensions artificiallyeach set using a different model for the formation of thedecrease the collisional timescale with respect to the time-terrestrial planets.scale for orbital evolution.

These approximations were chosen because they require Model A. These integrations simulate the evolution ofsubstantially less computer time than more-realistic a disk of planetary embryos that initially spans most ofN-body integrations in three dimensions. Both types of the region currently occupied by the terrestrial planets. Insimulation yielded plausible planetary systems, although this model it is assumed that the giant planets do notthese were not always similar to our own. They also pro- significantly influence the formation of the terrestrial plan-vided 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

Model B. As Model A, but the effects of the giant plan-level of understanding that goes beyond that achievableets are modeled by adding Jupiter and Saturn to the simula-from analytic models.tions after 107 years. The giant planets are assumed to haveRecent improvements in the performance of computertheir current masses and orbital elements.workstations, and the development of a new N-body algo-

rithm, now make it possible to carry out N-body integra- Model C. As Model B, but the initial disk of embryostions, in three dimensions, of several tens of gravitationally is extended to encompass the region that now contains theinteracting bodies for the p108 orbits necessary to form asteroid belt. Jupiter and Saturn are added at 107 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 intosystem of terrestrial planets by integrating the orbits of a batches of three, each batch using different values for thedisk of planetary embryos for p100 million years, subject surface density of solid material at 1 AU, s, and the spacingonly to mutual gravitational interactions, inelastic colli- between embryos, D. One batch each uses (s, D) 5 (6, 7),sions, and external perturbations from the giant planets? (10, 7), and (6, 10), where s is measured in units of gcm22

If it is possible, such simulations should indicate whether and D in mutual Hill radii, RHM , whereterrestrial planets such as our own are inevitable, giventhe size and location of the giant planets, or whether theirformation depends critically on the nature of the disk of RHM 5 Sm1 1 m2

3MAD1/3 Sa1 1 a2

2 D (1)embryos formed by runaway growth. (Alternatively, it mayall be a matter of luck, with the final outcome dependingon a few key events that occur essentially at random.) It for embryos with masses m1 and m2 , and semi-major axesshould also become possible to predict the characteristics a1 and a2 .of terrestrial planets in extra-solar systems long before wecan determine them observationally. Conversely, if N-body 2.1. Initial Conditionssimulations involving a few dozen embryos cannot produce

The initial conditions were chosen bearing in mind thesomething akin to the terrestrial planets, they may at leastform of the present planetary system and the results ofindicate what extra physics is required to do so.simulations of the runaway-growth phase of terrestrial-With this in mind we have carried out 27 integrationsplanet formation (e.g., Wetherill and Stewart (1993).of disks of planetary embryos, starting with objects on

isolated, nearly coplanar orbits, and following their evolu- Disk density. In 18/27 simulations we adopt a surfacedensity of solid material, s 5 6 gcm22 at 1 AU. This corre-tion for at least 108 years. In two thirds of the simulations

we have also included the effects of the giant planets Jupi- sponds to the minimum mass needed to form the currentterrestrial planets. We choose a density profile that variester and Saturn, assuming they formed before the accretion

of the terrestrial planets was complete. All the integrations as 1/a—a smaller gradient than used by some authors—inview of the large amount of solid material (s p 10 gcm22)were performed on dedicated DEC alpha workstations,

requiring p3 years of CPU time. required beyond the ice condensation point to form Jupi-ter’s core before loss of the nebula gas (Pollack et al. 1996).The next section describes the integration method and

the initial conditions used in the simulations. Section 3 As a variant on our standard initial conditions we set s 5 10gcm22 at 1 AU in three of the simulations for each model.looks at the evolution of the disks of embryos, whilst Sec-

306 CHAMBERS AND WETHERILL

Radial extent. The 18 simulations using Models A and Embryo masses. Having chosen s and D, the mass ofeach embryo is uniquely determined. These range fromB begin with a disk of embryos having semi-major axes

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 theouter edge in Models A and B, with s 5 6 gcm23. Largerregion. The lower bound is a compromise between making

the simulation realistic and avoiding a short integration objects are present initially in the simulations that have ahigher disk density or contain embryos in the asteroid belt.timestep (and hence a large CPU overhead), which is nec-

essary 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 gcm23. This value is also used to4.0 AU to include embryos that may have formed in the determine the new radius of an object following the accre-asteroid belt. tion of a smaller body.

Models A and B begin with 24–40 embryos, depending2.2. Integration Methodon 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 The N-body integrator used in the simulations is basedM% for Models A and B, and between 5.0 and 8.6 M% for upon a second-order mixed-variable symplectic integratorModel C (which includes material in the asteroid belt). written by Levison and Duncan (1994), which uses an algo-

rithm described by Wisdom and Holman (1991). One draw-Orbital elements. The initial eccentricities, e, and incli-back of this fast package is that close encounters betweennations, i, are 0 and 0.18, respectively, for all embryos.massive bodies are not calculated accurately, so we modi-These values are somewhat arbitrary, but they quicklyfied the code to integrate all objects using a Bulirsch–Stoerbecome randomized once close encounters occur, and mostmethod (Stoer and Bulirsch 1980), also supplied by Levisonobjects soon attain much larger values of e and i. However,and Duncan, whenever the separation between a pair offor the embryos to have formed via runaway growth thereobjects falls below 2 Hill radii. The Bulirsch–Stoer algo-must have been an epoch when e and i were small, andrithm uses a variable timestep to accurately follow thewe assume that this is still the case at the start of ourorbital evolution during an encounter, whilst the symplec-simulations. The remaining elements for each embryo—tic integrator uses a fixed timestep of 10 days. The symplec-the mean longitude and nodal longitude—are chosen ran-tic algorithm uses Jacobi coordinates, which makes it nec-domly.essary to periodically re-index the objects in order ofEmbryo separations. We assume that the planetaryincreasing semi-major axis. This procedure minimizes nu-embryos that formed via runaway growth have becomemerical error introduced by having different physical andisolated from one another prior to the start of our simula-Jacobi ordering of the objects.tions due to mutual accretion events and eccentricity damp-

Combining the symplectic and non-symplectic methodsing via dynamical friction. Once damping forces have beenleads to a secular growth in the energy error, typically oneovercome, a system containing more than two embryospart in 102–103 over 108 years. While less than ideal, webecomes unstable with respect to close encounters on abelieve this is acceptable given that neglected effects, suchtimescale, tc , that depends exponentially on the initial or-as dynamical friction due to residual planetesimals, proba-bital spacing, D, measured in mutual Hill radii (Chambersbly have a comparable or larger effect.et al. 1996).

Collisions between embryos are assumed to be com-In most of our simulations we use D 5 7, which corre- pletely inelastic, forming a single new body whose orbit

sponds to an embryo crossing time, tc p 5 3 104 years for is determined by conservation of linear momentum. Ourthe case in which all objects have the same mass. Three model assumes that mass loss due to fragmentation duringintegrations per model begin with D 5 10, implying that a collision is negligible. This assumption is necessary intc p 107 years in the equal mass case. These two values of order to avoid a rapid increase in the number of bodiestc bracket the probable time required to form embryos present in the integration, which would make the computa-in the terrestrial region (Wetherill and Stewart 1993). In tional expense prohibitive.practice, tc is somewhat smaller in our integrations becausethe embryos begin with a range of masses (see Section 3. EVOLUTION3.1). We also note that tc would be reduced if the embryosbegan with non-zero eccentricities (Yoshinaga et al. 1998). Figures 1–3 show a series of snapshots, in semi-major

axis/mass space, taken from three of the simulations de-The values of D chosen for our integrations are broadlyconsistent with the results of N-body integrations of em- scribed above—one for each model. Each symbol in the

figures represents a surviving embryo, with the horizontalbryo formation by Kokubo and Ida (1998). These authorsfound that embryos formed from a population of p104 bars depicting the perihelion and aphelion distances of

its orbit. In these and most of the other integrations thesmall bodies tend to have orbits spaced by about 10 mutualHill radii. evolution passes through four stages, described below.


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) Figure 4 shows the evolution of the perihelion and aph-elion 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 usingthat no pair of objects is able to undergo close encounters Model A. Initially the embryos’ semi-major axes remainin the absence of external perturbations, due to conserva- almost constant, while their eccentricities, e, exhibit erratiction of energy, E, and angular momentum, h (Gladman oscillations whose amplitude increases over time. After1993). When more than two embryos are present, E and about 12,000 years the eccentricities have increased suffi-h are no longer conserved within each pair of objects, and ciently for neighboring embryos to undergo close encoun-

ters, and the initial isolation is broken. The time requiredtheir isolation can be overcome.


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), andall but two objects are eventually swept up (f).

to do this varies from one simulation to another, depending happens in our simulations because the accretion timescaleis always much longer than the timescale for increases inmainly on the initial embryo separation, D. However, in

every case the embryos achieve crossing orbits eventually. e. Hence, once the first close encounters take place, theembryos remain on crossing orbits for as long as a signifi-At this point the first collisions occur, reducing the total

number of objects and increasing their mean separation cant number of objects survive.Re-isolation can occur in simulations that substantiallyin mutual Hill radii (see Eq. (1)). It is conceivable that

the surviving embryos could become isolated once more, alter the ratio of the accretion timescale to the dynamicaltimescale by constraining embryos to move in two dimen-returning to a state of affairs similar to, but more stable

than, the start of the simulation. In practice this never sions or by artificially increasing their physical radii. In


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), mostembryos 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 significantly shorter than those found by Chambers et al.(1996) for systems of equally spaced, equal-mass planetsfind that re-isolation does occur, producing a final system

containing five to eight low-mass planets with very low (tc p 5 3 104 and 107 years, respectively). However, theseauthors noted that a spectrum of planetary masses canorbital eccentricities.

The time of the first close encounter, tc , depends on the substantially reduce the orbit-crossing time, and we suggestthat this is the cause of the rapid onset of close encountersmasses and spacing of the embryos. In the simulations with

initial spacing D 5 7, the first close encounters typically seen in our integrations.Planetary embryos in different parts of the disk experi-occur after tc p 104 years, whilst integrations with D 5 10

exhibit encounters after p4 3 105 years. These times are ence their first close encounter at different epochs, usually


for the outer part of the disk in each case, but in the innerregion the time of first encounter generally decreases withincreasing a.

This suggests that once embryos near to the inner edgeof the disk achieve crossing orbits, they significantly influ-ence the orbital evolution of embryos further out, has-tening the onset of encounters. However, a particular em-bryo invariably undergoes its first close approach with anobject that was initially in the same part of the disk (ratherthan an interloper from elsewhere) so the effect is an indi-rect one—a close approach between one pair of embryosdestabilizes neighboring objects, leading to a ‘‘wave’’ ofclose encounters that sweeps through the inner part of thedisk. This effect is generally limited to the region P , 2years and is particularly marked for the simulations withD 5 10.

Chambers and Wetherill (1996) found that the already-FIG. 4. Perihelion (q) and aphelion (Q) distances for the 12 inner-most embryos during the first 20,000 years of a simulation using Model short crossing times seen here are reduced still furtherA. Note the rapid orbital evolution following the first close encounter when a population of smaller objects (mass p 0.01 embryoat p12,000 years. masses) is present in addition to the embryos. During the

earlier runaway growth stage, when planetesimals are ac-creting 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 andFigure 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 oneof the orbital period, and, given that the embryos are uni- another. However, as the fraction of solid disk materialformly spaced in mutual Hill radii, one would expect the contained in the embryos increases, their mutual interac-

tions will become stronger. At the same time, the corre-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 beindependent 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.


sponding decrease in the total mass of the planetesimals embryos, or with Jupiter and Saturn in simulations thatinclude the giant planets.reduces their ability to damp the embryo’s eccentricities.

For example, Fig. 8 shows the evolution of e for fourThis combination is likely to produce a rapid transition toembryos from two simulations using Model C. The figurea situation in which the embryos achieve crossing orbitsalso shows the critical argument of a secular resonanceand the disk becomes dynamically excited.involving the longitude of perihelion, f, of each object andthat of another nearby embryo. Libration of the critical3.2. The Disk Becomes Dynamically Excitedargument about 0 indicates that the perihelion directions(Figs. 1c, 2c, 3c)of the two embryos are aligned, whilst oscillation about

Once neighboring embryos have acquired crossing orbits 1808 implies that the perihelion directions are antialigned.with significant eccentricities, strong orbital interaction can In each case the changes in e are associated with librationsoccur due to close encounters and resonances between in the critical angle. At some epochs additional oscillationsembryos. This in turn leads to rapid changes in e and i, are apparent, caused by secular interaction with other em-and the disk becomes dynamically excited. This can be bryos. An object usually undergoes strong interactions withseen in Figs. 1c, 2c, and 3c, where many of the embryos two or three other embryos simultaneously and typicallyhave horizontal bars that overlap, indicating crossing orbits moves back and forth between several secular resonances

during an integration.with nonnegligible eccentricities.Secular resonances with the giant planets also occur.Figure 6 shows the mass-weighted values of Ï(e2 1 i2),

Figure 9 shows two cases in which the orbits of embryose, and i, for objects in two simulations using Model A. Eachlie within the n5 resonance, where the longitude of perihe-of these quantities increases logarithmically with time; inlion of the protoplanet precesses at a similar rate to thatother words, very quickly at first, and then more slowly atof Jupiter. This causes smooth, long-period changes in thelater times. The rate of increase is largest for simulationsobject’s eccentricity. This resonance influences the orbitswith high surface densities and thus more massive embryos.of objects with a , 2 AU in several of our simulations,The upshot is that gravitational focusing during close en-especially the region 0.5 , a , 0.7 AU, and occasionallycounters will diminish over time, since this is most efficientcauses embryos to fall into the Sun. The analogous n6at small relative velocities. Hence the collision probabilityresonance with Saturn plays an important role for objectsfor a given pair of embryos will also decrease with time.with 2 , a , 2.3 AU. We note that the locations of theseThis is reflected in a short-lived burst of collisions near thetwo secular resonances will depend to some extent on howstart of each simulation, followed by a monotonic decline inmuch nebula gas is present during the accretion of thethe collision rate (see Section 3.3).terrestrial planets (Lecar and Franklin 1997) and also theThe embryos’ eccentricities and inclinations undergodegree 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 ofclose encounters. In the early evolution, each episode typi- another common situation in which an embryo’s longitudecally lasts for 5–20 thousand years (Fig. 7), and changes of perihelion, f, becomes almost stationary. Like the Kozaiin e are often correlated with the behavior of the argument resonance, this resonance does not directly involve theof perihelion, g, due to the Kozai resonance (Kozai 1962). orbits of other bodies. Whilst f is almost stationary, theFor example, the strong increase in the eccentricity of embryo’s nodal longitude and argument of perihelion usu-Planet 11 at t p 50,000 years in Fig. 7 is associated with a ally circulate rapidly in opposite directions. This behaviorlibration of g about 908. 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 emon in each of the simulations. These produce correlated and i—only a handful of examples can be seen in Figs.

7–9. Close encounters mainly affect e and i by determiningoscillations in the eccentricities or inclinations of the twothe point in a secular cycle at which an object is scatteredobjects, and librations of the resonant critical argument,away from one secular resonance and into another. Thiswith periods p105 years. A common example involves ain turn establishes the initial values of e and i and thesituation in which the longitudes of perihelion of two em-critical argument for the new resonance.bryos precess at the same rate, with the two orbits aligned

or anti-aligned with one another. Occasionally three, or3.3. Protoplanets Are Formed (Figs. 1d, 2d, 3d)even four, orbits are temporarily locked together in this

way. The resulting high-frequency oscillations in e are fur- After 3–6 million years, one or two objects of 0.3–0.5Earth masses have formed interior to 1.5 AU in each ofther modulated by secular interaction with other nearby


FIG. 6. Mass-weighted mean of (e2 1 i2)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 increaseapproximately linearly in log time, with the high-s 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 integra-tions are conducive to gravitational focusing between0.7 AU in Figs. 1d, 2d, and 3d, respectively. In simulations

using Model C, where the disk of embryos extends into embryos—a necessary prerequisite for runaway growth.However, as e and i increase this situation will change. Wethe asteroid belt, additional large objects are present be-

yond 1.5 AU. However, each of these is the result of only can see how the effects of gravitational focusing vary duringa simulation by looking at the distribution of close-encoun-one or two collisional events, and their large mass is due

to the high initial masses of the embryos in this region. ter distances. In the absence of focusing, the number ofencounters, N, with minimum separation, D, is given byDo these ‘‘protoplanets’’ form by a continuation of run-

away growth or via more orderly growth? The low eccen- N Y D, while in the limit of strong focusing N Y D1/2.


FIG. 7. Eccentricity and argument of perihelion, g, versus time for four embryos for the first 105 years of a simulation using Model A. Notethe 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 of objects present probably makes it difficult for runawaygrowth to get going before the supply of embryos is ex-epochs of a simulation using Model A, where we have

combined the close-approach data for all the embryos pres- hausted.There is some indication that equipartition of randoment. A x 2 test indicates that we can reject the hypothesis

that focusing is absent at the first epoch (0–105 years) at orbital energy (‘‘dynamical friction’’) takes place. For ex-ample, Fig. 12 shows the eccentricities of all surviving ob-the 99.5% confidence level. Conversely, the data for the

third epoch (9–10 million years) are consistent with a lack jects in Model C after 10 million years as a function oftheir mass. The largest objects tend to have lower values ofof gravitational focusing, while those for second epoch

(3–3.5 million years) are ambiguous—the probability that e than the smaller bodies. Conversely, the smallest embryosgenerally do not have eccentricities close to zero.the distribution is uniform is roughly 10%.

Given that gravitational focusing is probably significant A quantitative measure of the importance of dynamicalfriction comes from examining the values of e and i forduring at least a part of the simulations, does this

lead to runaway growth? Figure 11 shows the mass each pair of objects just before they collide. Consider firstthe larger of the two objects in each collision. In Modeldistributions at four epochs, combining the data for all

the embryos with a , 1.5 AU in integrations using C, 69% of these objects have inclinations below the meanvalue for that epoch, while for e the figure is 56%. TheModel C. As the embryos accrete one another the mass

distribution flattens, except for the largest objects, which corresponding numbers for the smaller colliding body are59 and 45%, respectively. In other words, weak equiparti-march steadily toward higher masses—i.e., orderly growth

occurs. Apparently, gravitational focusing during the tion of random energy takes place for objects undergoingcollisions—a necessary prerequisite for runaway growth.early stages of the simulations is not enough to allow

runaway growth to occur. In addition, the small number Incidentally, the preference for low inclinations over low


FIG. 8. Eccentricity versus time for four embryos during simulations using Model C. Also shown are the critical arguments for a secularresonance between the longitudes of perihelion, f, of each object and another nearby embryo. Librations about 0/1808 indicate that the periheliondirections of the two objects are aligned/anti-aligned.

eccentricities can be understood following the discussion than the planet per unit semimajor axis. In addition, Pnode

is the probability that one of the nodes of an embryo’sbelow (see Eq. (2)).Figure 13 shows how the number of collisions varies orbit is close enough to the planet’s orbit to permit colli-

sions, given bydepending on the semi-major axis of the larger of the twocolliding bodies just prior to impact (for all the simulationsusing Model C). The collision rate decreases rapidly with

Pnode Q 2 32Rp fgrav

2ae sin i,increasing heliocentric distance, after a peak at about

0.7 AU, so that very few collisional events occur beyond2 AU. Note that this is not simply a reflection of the de-

where Rp(ap) is the radius of the planet, and fgrav(e, i) is thecrease in the number density of objects with increasing a,gravitational focusing factor. Finally, Pconj is the probabilitysince this falls off more slowly.that the planet is in the correct part of its orbit at conjunc-To see how this effect might arise, consider a planet thattion for a collision to take place, given byis 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 whichthe planet accretes the smaller embryos in its vicinity is Pconj Q

2Rp fgrav

2fap.

dNdt

Q Eap/(12e)

ap/(11e)

n(a)P(a)

3 Pnode(ap , a, e, i) 3 Pconj(ap , a, e, i) da,If we assume that the embryo masses, m, and their meanspacing are comparable to their initial values, then n(a) p1/a3/2, and m(a) p a3/2, which also implies that the plane-where 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 Rp p a1/2p . Provided that embryos


FIG. 9. Eccentricity versus time for four embryos during simulations using Model C. Also shown in the upper diagrams are the critical argumentsfor the n5 resonance between the longitudes of perhelion, f and fJ , of the embryo and Jupiter. In the lower diagrams, the change in e is associatedwith epochs when f itself is almost stationary.

in each part of the disk are subject to accretion by roughly law in which the collision probability, for a given embryo,depends on the number of surviving objects, N, so that thethe same number of planets, the overall accretion rate

will be total population varies as

dNdt

Yf 2

grav

e sin iEap/(12e)

ap/(11e)

1a4 da Y

f 2grav(e, i)a3

p sin i. (2) dN

dtY 2N(N 2 1). (3)

Note the steep dependence of the collision rate on ap Clearly the slopes of the curves in Fig. 14 are shallowerand the lack of dependence on e outside the gravitational- than the simple theory predicts, and for simulations usingfocusing regime. Despite the crude assumptions used to Model C the number of collisions is approximately propor-derive Eq. (2), the fit with the collision rate observed in tional to log t. This disagreement is not too surprising sinceour simulations is quite good, as the 1/a3

p curve in Fig. 13 Eq. (3) ignores changes in the orbital element distributionsindicates. The discrepancy between theory and simulation of the embryos—in particular the collision rate will de-for a , 0.7 AU presumably reflects the effects of the disk crease as the mean inclination of the embryos rises (seetruncation at 0.55 AU. Eq. (2)).

The next obvious question is how the collision rate varieswith time? Figure 14 shows the fraction of the initial objects

3.4. The Planets Become Isolated (Figs. 1f, 2f, 3f)that remain, versus time t, averaged over all the simulationsthat began with an embryo separation D 5 7. The solid In the simulations using Models B and C, the giant plan-

ets Jupiter and Saturn are added at 107 years, with theirlines show the actual fraction remaining, whilst the dashedlines indicate the fraction expected according to a decay current masses and orbital elements. The immediate effect


FIG. 12. Eccentricity versus mass for all objects remaining at 10million years in the simulations using Model C.

is to introduce a number of strong mean-motion and secu-lar resonances into the region a . 2 AU. Embryos inthe vicinity of these resonances undergo large increases ineccentricity until they are removed by an impact on theSun, by a collision with another embryo inside 2 AU, orby hyperbolic ejection following a close approach to Jupi-ter. This mechanism for clearing material from the asteroidbelt still operates today, but it is more efficient in oursimulations since the embryos are large enough that a closeencounter between a pair of embryos can often scatter oneobject into a resonance, where it is quickly lost beforeFIG. 10. Number of close encounters, N, versus distance of closestanother encounter can scatter it out of the resonance again.approach, D, at three epochs of a simulation using Model A. Note that

the close-encounter data for all embryos has been combined. The net result is that p20 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,of the larger impactor, for all integrations using Model C. The curveModel C, at four epochs during the integrations. Note that only objects

with a , 1.5 AU are included in the distributions. shows the predicted distribution following a 1/a3 law.


For example, Fig. 15 shows the time evolution of q andQ 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 andQ for a single object. Also shown, in blue, are q and Qfor one of the largest objects that does not survive. In eachcase the protoplanets destined to remain at the end of thesimulation rarely approach one another, if at all. However,the eccentricities of their orbits are such that there is noroom left for the ‘‘blue’’ protoplanet. Consequently, thisexcess object is either accreted or ejected.

Note that the final spacing of the planets is determinedby the values of e while there are still ‘‘too many’’ proto-planets present. The eccentricities are often subsequentlyreduced by interactions with residual small embryos, whichare then accreted or ejected. For example, the ‘‘green’’planet in Fig. 15c undergoes a substantial decrease in e atFIG. 14. The fraction of surviving objects versus time for all simula-

tions using Models A and C. The dashed lines show the expected fraction p50 million years due to interactions with a much smallerdue to a decay law of the form dN/dt Y 2N(N 2 1). body. A series of close encounters with the smaller embryo

nudges the protoplanet into the n5 resonance, producinga rapid drop in e (Fig. 16). The residual embryo is subse-quently removed, leaving the larger body on a low-eccen-planets are added, most of the embryos with a . 2 AU

have been removed (Fig. 3e). Thus, 30 million years into tricity orbit. This late-stage orbital damping occurs in sev-eral of our simulations, but does little to alter the semi-a simulation, a combination of rapid accretion in the inner

part of the disk and partial clearing of the asteroid belt by major axes of the final planets, these being determinedearlier in the evolution.resonances leaves three to six large ‘‘protoplanets’’ con-

taining 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 ofisolated, ‘‘final’’ planets depends principally on the ampli- Figures 17–19 show the final states of the integrations

using Models A, B, and C, respectively. The figures indicatetude of the secular oscillations in the eccentricities of thesurviving protoplanets. These oscillations have two the osculating orbit of each surviving planet, with the

planet itself represented by a filled symbol whose radiussources. First, secular resonances between neighboringplanets on crossing or nearly crossing orbits, which produce is proportional to the radius of the body. The same data

are given in Tables I, II, and III, except that the values ofshort-period (p105 year) cycles. Second, secular perturba-tions and/or resonances with Jupiter and Saturn, having e and i have been averaged over the last 106 years of the

integration. The column headed ‘‘last event’’ refers to theperiods of 106–107 years. In general the former are domi-nant except for a p 2.1 AU, where the n6 resonance causes time at which the last accretion or ejection took place.

Where close encounters are still taking place, it is assumedlarge changes in e and occasionally the region a p 0.6 AU,where the n5 resonance can play an important role (these that the last event time will be greater than the length of

the integration.values of a assume that there is no longer a significantamount of nebula gas present and that the giant planets The results of simulations using Model A are given in

Fig. 17. In each case 108 years have elapsed since the starthave their modern orbits).The secular oscillations in e cause the perihelion and of the calculation. In most cases the evolution is incomplete

in the sense that several objects are still able to undergoaphelion distances (q and Q, respectively) to change ontimescales that are short compared to the collision time- close encounters with one another. However, the collision

rate has slowed to almost zero, so it is not apparent howscale. In order to avoid collisions with one another, pro-toplanets must be spaced so that the maximum value of much longer would be required to achieve a set of isolated

objects. In many of the simulations the innermost objectsQ for the innermost object is less than the minimum valueof q for the second object, and so on. Further evolution have achieved non-crossing orbits. These planets tend to

have smaller i than objects further from the Sun, and theytakes place until this situation is achieved, with surplusprotoplanets being thrown back and forth until they merge usually contain most of the mass. However, it is quite likely

that the configuration of these objects will change due towith another or are scattered beyond 2 AU and removedby a resonance. subsequent interaction with objects at larger a. The one


FIG. 15. Time evolution of the aphelion, Q, and perihelion, q, distances (same color for each body) for each of the objects that survives untilthe 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 largeobject 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 tobe the largest. In addition, there is considerable scatter incontains five approximately equal-mass planets, with a

roughly geometric orbital spacing. the mean values of e and i from one integration to another.Simulations using Model C (shown in Fig. 19) usuallyThe simulations using Model B were continued for 108

years or until close encounters had ceased, whichever was produced only one or two planets interior to 2 AU. Severalof the simulations also contain an object in the asteroidlonger. The final state of each simulation is shown in Fig.

18. Generally there are two objects with a , 1.7 AU, while belt, although these are invariably very large compared toa typical asteroid, due to the large embryo masses used inin several cases a third object lies further from the Sun.

In the two simulations that yielded three planets, these our calculations. It is possible that some or all of theseobjects have orbits that are unstable over the age of thehave roughly geometric orbital spacings. This is also true

for the three outer planets in simulation 7B, although the solar system. In only one of the two simulations that endedwith .2 isolated planets do the objects have geometricallytwo inner planets lie closer together than a geometric law

would predict. In each case the mean spacing is somewhat spaced orbits. In almost all the simulations the orbitalspacing is larger than in the inner part of the solar system,larger than the planets we observe in the inner solar system.


seen in our calculations, and their large eccentricities, aredirectly related to one another.

Approximately two thirds of the ‘‘completed’’ simula-tions contain a large (.0.5M%) planet lying within the Sun’shabitable zone. This is the region in which a geologicallyactive planet can support liquid water. The conservativeestimate for the habitable zone adopted here is 0.95 ,a , 1.37 AU (Kasting et al. 1993). It is quite likely thatthe region is somewhat larger, in which case a greaterproportion of our simulations would contain a habitableplanet.

Looking now at the overall distribution of the survivingobjects, Fig. 20 shows the masses and semi-major axes ofall the remaining objects at the end of the integrations.In Model A, the survivors encompass a broad range ofheliocentric distances, with the largest objects clusteredaround 1 AU and an extended tail of smaller bodies outto 3 AU. The inner edge of the distribution is quite sharpand almost identical to the inner cutoff of the initial diskof embryos at 0.55 AU.

In Model B the remaining objects occupy a narrowerrange of heliocentric distance, with only one survivor hav-ing a . 2 AU. The few objects that entered this regionduring each integration were either scattered back tosmaller values of a or removed by resonances with thegiant planets. The largest bodies lie closer to the Sun thanin Model A, in a cluster centered on 0.6 AU, with a secondgroup between about 1.0 and 1.4 AU. All the survivingobjects with a . 1.5 AU are of less than one third of anEarth mass.

FIG. 16. Time evolution of the semi-major axis, a, eccentricity, e, In Model C the picture is different again. Now most ofand critical argument for the n5 resonance for the ‘‘green’’ planet in Fig.

the large objects are confined to a region a , 1.2 AU, with15. Note the rapid drop in e when the critical argument starts to circulatea tail of smaller objects extending outward. The differenceslowly, following a close encounter at p50.8 million years.between the two regions is clear: most of the planets inte-rior to 1.2 AU have masses greater than 1 M% , whilst mostof the objects with a . 1.2 AU are less massive than Earth.and the planets are also typically spaced more widely than

those from simulations using Model B. Once again, the Table IV shows the fates of the embryos according totheir initial location in the disk, giving the fraction thatobjects closest to the Sun tend to be the largest, and these

usually have smaller eccentricities than their ‘‘asteroidal’’ are incorporated into surviving planets versus those thatwere ejected or collided with the Sun. All the initial mate-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.The addition of the giant planets, and their associatedmassive as Earth.

In general, the surviving objects in our simulations have resonances, reduces the number of surviving objects top85% in Model B—a fraction which is independent oftime-averaged values of e and i that are substantially larger

than those of Earth and Venus (e p 0.03 and i p 28). Earth- initial semi-major axis, a0 . The majority of the remainderare lost via hyperbolic ejection.sized planets with eccentricities of 0.2 are not uncommon

in our simulations. The large mean values of e occur early In Model C the fraction of survivors with a0 , 1 AU isthe same as in Model B. Exterior to 1 AU the proportionin each integration and lead to correspondingly large maxi-

mum values, emax , during each secular oscillation. This in of objects that survive decreases monotonically with in-creasing distance from the Sun. Clearly, the presence ofturn requires that protoplanets remain widely spaced to

avoid scattering or accreting one another. In general, emax material beyond 2 AU reduces the fraction of embryoswith 1 , a0 , 2 AU that survive until the end of the simula-is too large to permit more than two, or occasionally three,

final planets to form in the simulations that contain Jupiter tion (57% in Model C compared to 83% in Model B). Theadditional embryos in Model C remove many objects withand Saturn. Thus, the small number of terrestrial planets


TABLE ISurviving Objects for Each of the Simulations Using Model A

Simulation Last event a i Mass Componentcode (106 year) (AU) e (deg.) (Earth 5 1) embryos

1A .100 0.63 0.09 2 0.54 170.96 0.05 3 0.39 91.23 0.05 3 0.30 41.57 0.13 5 0.08 11.81 0.13 5 0.03 11.97 0.04 2 0.49 8

2A .100 0.58 0.08 4 0.31 90.77 0.08 4 0.26 101.05 0.08 3 0.45 81.45 0.04 2 0.48 61.73 0.08 5 0.13 42.22 0.16 6 0.07 12.35 0.08 7 0.13 2

3A .100 0.87 0.04 6 1.15 301.50 0.08 6 0.13 21.81 0.07 6 0.26 41.85 0.12 12 0.09 12.06 0.16 5 0.08 12.36 0.12 5 0.11 2

4A .100 0.52 0.40 8 0.22 50.87 0.49 34 0.19 11.04 0.10 2 2.20 172.25 0.24 32 0.10 12.50 0.14 5 0.45 72.57 0.31 11 0.08 1

5A 37 0.61 0.08 5 0.69 120.82 0.08 4 0.79 71.14 0.06 2 0.65 61.63 0.06 6 0.35 22.28 0.07 2 0.76 5

6A .100 0.58 0.06 3 0.96 141.14 0.04 2 1.29 112.02 0.05 6 0.58 32.25 0.05 8 0.41 4

7A .100 0.59 0.30 7 0.49 111.17 0.18 6 0.87 92.21 0.34 6 0.30 22.80 0.11 16 0.14 13.15 0.15 33 0.07 1

8A .100 0.64 0.32 12 0.58 120.99 0.14 11 0.34 31.48 0.18 10 0.41 42.01 0.16 9 0.27 32.57 0.21 25 0.16 12.58 0.10 10 0.12 1

9A .100 0.57 0.09 5 0.29 70.82 0.10 3 0.39 61.12 0.08 3 0.67 71.81 0.28 10 0.11 11.92 0.11 5 0.28 22.47 0.04 9 0.14 1


TABLE IISurviving Objects for Each of the Simulations Using Model B


1B 127 0.71 0.10 5 0.89 241.08 0.08 8 0.59 121.74 0.14 14 0.25 3

2B 83 0.50 0.16 2 1.05 281.39 0.24 5 0.45 8

3B 185 0.76 0.16 4 1.31 331.76 0.22 13 0.29 4

4B 86 0.68 0.13 2 1.85 211.27 0.09 6 1.01 9

5B 50 0.31 0.12 15 1.18 161.27 0.65 17 0.51 5

6B 72 0.60 0.17 2 1.59 211.09 0.29 6 0.70 4

7B 88 0.59 0.08 6 0.46 50.79 0.10 7 0.49 111.40 0.04 4 0.68 62.35 0.07 6 0.18 1

8B 259 0.61 0.37 17 0.94 131.37 0.18 20 0.40 7

9B 73 0.50 0.07 32 0.24 60.98 0.26 3 1.11 121.88 0.21 5 0.28 2

a0 , 2 AU by scattering them into the asteroid belt where the simulations using Model C. Final planets with 0 , a, 1 AU tend to be composed mainly of embryos from thethey are lost via resonances with the giant planets.

Figure 21 shows the composition of the surviving objects inner part of the disk—the region between 0 and 2 AU.Planets with 1 , a , 2 AU contain only a small amountin terms of the initial location of the embryos incorporated

into each object. The graph combines the data from all of material from closer to the Sun, but a substantial fraction

TABLE IIISurviving Objects for Each of the Simulations Using Model C


1C 153 0.90 0.20 4 1.67 272.83 0.36 35 0.05 1

2C 67 0.68 0.17 5 1.33 271.51 0.03 23 0.49 3

3C 229 0.70 0.13 9 1.52 331.31 0.16 11 0.22 1

4C 61 0.60 0.29 7 2.04 231.06 0.19 5 0.87 42.49 0.11 17 0.81 2

5C 266 0.45 0.14 6 1.23 151.61 0.02 8 1.09 5

6C 86 0.53 0.24 15 1.44 131.13 0.15 9 1.53 112.54 0.41 39 0.07 1

7C 124 0.81 0.34 7 2.74 278C 239 0.98 0.15 9 1.37 14

1.56 0.14 18 0.39 39C 68 1.08 0.18 8 2.65 19


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 symbolwhose radius is proportional to the radius of the object.

of their mass comes from the asteroid belt. Surviving ob- the same row of each figure (e.g., 1B, 2B, and 3B) haveidentical initial conditions, except for the values of thejects in the asteroid belt are almost entirely composed ofembryo’s angular elements. The evolution in each simula-material from this region, although we note that the smalltion is highly stochastic, making it difficult to predict thenumber of such objects in our simulations makes this con-kind of planetary system that a particular protoplanetaryclusion somewhat uncertain.disk will produce.

One thing is clear however: each of our simulations5. DISCUSSIONproduces a planetary system that is qualitatively different

The final states of the 27 simulations in Figs. 17–19 vary from our own. On the plus side, the simulations yield asmall number of terrestrial planets moving on isolated or-considerably. This despite the fact that the simulations in


FIG. 18. As Fig. 17 for Model B.

bits. In addition, they frequently produce two planets with (a , 2 AU) is generally much more massive than Mars(p0.1M%).semi-major axes and masses comparable to Earth and

Venus. However, the results differ from the inner solar 5. No analogues of Mercury are present.system in a number of respects:

Some of these points are easier to address than orders.The absence of objects with orbits similar to Mercury1. There are generally too few terrestrial planets.

2. The planets have orbits that are too widely spaced. (a p 0.4 AU) is presumably due to our decision to truncatethe inner disk at 0.55 AU. The steep gravitational potential3. The orbits have values of e and i substantially larger

than Earth and Venus. gradient in this region limits the extent to which objectscan be scattered inward from the inner edge of the disk.4. The outermost object in the terrestrial-planet region


FIG. 19. As Fig. 17 for Model C.

This explains why only two surviving objects have a , collided to make up that planet. For example, if a planetis the result of the accumulation of 20 embryos, and 400.5 AU.

The high mass of ‘‘Mars’’ in our simulations may be were present initially, then the planet’s size is 0.5. Thedifference between the inner and outer terrestrial region isdue to the large masses of the embryos that were initially

present in this region (say 1.2 , a , 2.0 AU). The rapid now more marked than in Fig. 20, especially for simulationsthat include the giant planets. Surviving objects with a .falloff in accretion rate with distance from the Sun (see

Section 3.3) implies that the mass of a planet that forms 1.2 are often composed of only one or two embryos. Incontrast, planets closer to the Sun usually contain .10in the vicinity of Mars largely depends on the masses of

the embryos which began there. To illustrate this, Fig. 22 embryos. Hence, if the embryos in the Mars region wereactually smaller than those used in our simulations, theshows the sizes of all the surviving objects, where ‘‘size’’

indicates the fraction of the embryos in the simulation that low mass of the red planet becomes easier to understand.


FIG. 21. Fraction of the mass of surviving planets that comes fromdifferent regions of the initial disk. The three curves show the massfractions 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 atthe end of the simulations.

This raises the interesting possibility that Mars repre-sents a leftover planetary embryo that underwent littleor no further accretion after the cessation of the earlierrunaway-growth phase of planet formation. The accretionrate 1.5 AU from the Sun may have been so slow thatmost of the embryos that began here were scattered inwardand swept up by Earth and Venus or scattered outward

TABLE IVFraction of the Initial Population of Embryos That Are Incor-

porated into Surviving Objects, as a Function of the Embryos’Initial Semi-major Axes

Initial a (AU) Model A Model B Model C

0–1 1.00 0.84 0.841–2 1.00 0.83 0.57

FIG. 22. The sizes and semi-major axes of all surviving objects. Here2–3 — — 0.31size is defined as the fraction of the embryos initially present in a simula-3–4 — — 0.18tion that accumulated to form each planet.


and removed via resonances in the asteroid belt, before during the runaway growth phase. We plan to test thescenario outlined here in subsequent simulations.they could accrete one another. In this scenario, Mars is

the sole embryo that managed to avoid these fates. Onceall its competitors were removed there was no longer a 6. SUMMARYmechanism to scatter Mars out of its orbit, and its statuschanged to that of a final planet. The principal conclusions drawn from our simulations

The problem of the large eccentricities and inclinations are:arises at an early stage in our simulations, as mutual pertur-

• A disk of initially isolated planetary embryos quicklybations between embryos rapidly increase the mean valuesbecomes dynamically excited once mutual perturbationsof e and i (Fig. 6). Collisions between objects often diminishallow embryos to achieve crossing orbits. At this point ethese quantities during the final stages of accretion, butand i increase rapidly.the effect is insufficient to yield the essentially circular

• Once isolation is overcome, the evolution of e and iorbits observed for Earth and Venus.is driven largely by secular perturbations rather than closeThis result is intimately connected with the small numberencounters. Secular resonances between pairs of embryosof planets produced by our simulations and their wideare common, and the n5 (a p 0.6 AU) and n6 (a p 2.1spacing. The large eccentricities of the embryos, com-AU) resonances also play a role in some simulations.pounded by the large secular oscillations about the mean

• Orderly growth takes place despite the presence ofvalues, imply that neighboring protoplanets must be widelygravitational focusing early in the integrations. However,spaced to avoid scattering or accreting one another. Thisweak dynamical friction is apparent even though only ain turn limits the number of final planets, since they arefew tens of objects are present.restricted to the region between the inner edge of the disk

• Accretion occurs most rapidly in the inner part ofand the point at which strong resonances with the giantthe disk, with the collision rate decreasing sharply as aplanets occur (a p 2 AU).increases. Almost no collisions occur beyond 2 AU. In-The large values of e and i are especially marked instead, most material with a . 2 AU is removed by mean-Model C, which may be due to the large masses of themotion and secular resonances with the giant planets.embryos in the asteroid belt. It it quite possible that objects

• Embryos with 1.2 , a , 2 AU tend to be scatteredthis large never formed here, which would go some wayoutward and removed by resonances or scattered inwardtoward remedying the problem. However, the values of eand accreted by planets closer to the Sun, before they canand i are also too large in simulations using Model B,accrete one another. Thus, Mars may be an unaccretedwhere there are initially no embryos beyond 1.8 AU.embryo that remained in this region.Clearly something else is wrong, and with hindsight we

• Accretion is complete after 108 years in 60% of simula-suggest that it is the initial mass distribution used in ourtions that include Jupiter and Saturn, and in all such simula-calculations. Earlier, during the runaway growth phase oftions after 3 3 108 years. Simulations that neglect the giantthe evolution, equipartition of random energy (dynamicalplanets still have several objects with crossing orbits atfriction) between planetary embryos and planetesimals en-108 years.sures that the largest objects have orbits with small e and i.

• The most common outcome is a pair of large planetsThis effect is sufficient to overcome perturbations betweeninterior to 2 AU. Occasionally a smaller object remainsembryos, which will tend to excite these quantities. How-further from the Sun. The surviving planets tend to haveever, once the embryos contain a substantial fraction oftime-averaged eccentricities and inclinations that are sub-the solid disk material, dynamical friction will become toostantially larger than those of Earth and Venus.weak to damp down inter-embryo perturbations unless

• The number and spacing of the planets in our simula-these perturbations are also weak. This can only occur iftions are largely determined by the mean values of e, andthe remaining embryos have orbits which are tens of Hillthe amplitude of secular oscillations in e, during the finalradii apart, which in turn implies that only a handful ofstages of accretion.embryos are present at this stage.

Hence, a more likely configuration at the start of thefinal stage of terrestrial-planet formation is a handful (,10) ACKNOWLEDGMENTSof widely separated protoplanets (with masses of say 0.03–0.3 M%) and a few tens of smaller ‘‘failed embryos.’’ These We are grateful to Alan Boss, Martin Duncan, Hal Levison, Jack

Lissauer, and Derek Richardson for fruitful discussions during the prepa-objects would presumably be accompanied by a populationration of this paper. Thanks also to Conel Alexander, Geoff Coxhead,of residual planetesimals containing a small fraction of theSandy Keiser, and Martin Murphy for computer support, and to Hal

total mass. In this scenario, many of the protoplanets are Levison and Martin Duncan for providing the routines that formed thedestined to become true planets and so the final locations basis of our computer code. This work was partly supported by NASA

Grants NAG5-3656, NAG5-4285, and NAG5-4386—thank you.of the planets are already determined, to some extent,


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Making the Terrestrial Planets: N-Body Integrations …extranet.on.br/rodney/curso2010/aula2/chambers-wetherill.pdfICARUS 136, 304–327 (1998) ARTICLE NO.IS986007 Making the Terrestrial