ORBITAL EVOLUTION OF PLANETS EMBEDDED IN...

THE ASTRONOMICAL JOURNAL, 117 :3041È3053, 1999 June1999. The American Astronomical Society. All rights reserved. Printed in U.S.A.(

ORBITAL EVOLUTION OF PLANETS EMBEDDED IN A PLANETESIMAL DISK

JOSEPH M. HAHN AND RENU MALHOTRA

Lunar and Planetary Institute, 3600 Bay Area Boulevard, Houston, TX 77058-1113 ; hahn=lpi.jsc.nasa.gov, renu=lpi.jsc.nasa.govReceived 1998 December 29 ; accepted 1999 February 16

ABSTRACTThe existence of the Oort comet cloud, the Kuiper belt, and plausible inefficiencies in planetary core

formation all suggest that there was once a residual planetesimal disk of mass D10È100 in the vicin-Mîty of the giant planets following their formation. Since removal of this disk requires an exchange of

orbital energy and angular momentum with the planets, signiÐcant planetary migration can ensue. Theplanet migration phenomenon is examined numerically by evolving the orbits of the giant planets whilethey are embedded in a planetesimal disk having a mass of We Ðnd that Saturn,M

D\ 10È200 M

^.

Uranus, and Neptune evolve radially outward as they scatter the planetesimals, while JupiterÏs orbitshrinks as it ejects mass. Higher mass disks result in more rapid and extensive planet migration. Iforbital expansion and resonance trapping by Neptune are invoked to explain the eccentricities of Plutoand its cohort of Kuiper belt objects at NeptuneÏs 3 :2 mean motion resonance, then our simulationssuggest that a disk mass of order is required to expand NeptuneÏs orbit by *a D 7 AU, inM

DD 50 M

ôrder to pump up Plutino eccentricities to eD 0.3. Such planet migration implies that the solar systemwas more compact in the past, with the initial Jupiter-Neptune separation having been smaller by about30%.

We discuss the fate of the remnants of the primordial planetesimal disk. We point out that most of theplanetesimal disk beyond NeptuneÏs 2 :1 resonance should reside in nearly circular, low-inclination orbits,unless there are (or were) additional, unseen, distant perturbers. The planetesimal disk is also the sourceof the Oort cloud of comets. Using the results of our simulations together with a simple treatment ofOort cloud dynamics, we estimate that D12 of disk material was initially deposited in the OortM

^cloud, of which D4 will persist over the age of the solar system. The majority of these comets orig-Mînated from the Saturn-Neptune region of the solar nebula.

Key words : Kuiper belt, Oort cloud È solar system: formation

1. INTRODUCTION

It is generally accepted that a Ðnal distinct stage in theformation of our planetary system consisted of the clearingof a residual planetesimal population by the gravitationalperturbations of fully formed giant planets. The formationof the Oort cloud, which is a spherical swarm of cometsorbiting the Sun at distances of D103È105 AU, is thought tobe a product of this stage of solar system formation. Massestimates for the Oort cloud are in the range D10 to 100

(Weissman 1996). Another remnant of the planetesimalM^disk is the recently discovered population of the Kuiper belt

objects (KBOs) that orbit beyond Neptune. Although themass of the observable portion of the Kuiper belt is onlyD0.26 (Jewitt, Luu, & Trujillo 1998), planetesimal ac-M

^cretion models require an initial mass that is of order D35for the assembly of Pluto and the RD 100 kmÈsizedM

^KBOs in the 30È50 AU zone (Stern & Colwell 1997 ;Kenyon & Luu 1998). Noting also that the solid cores of thegiant planets are of order D10 and that core formationM

^,

is not likely to have been 100% efficient, it is quite plausiblethat there was still a residual planetesimal disk of massD10È100 in the vicinity of the giant planets after theyM

^formed. The eventual removal of this mass via gravitationalscattering by the giant planets could have caused signiÐcantevolution of the planetary orbits, such that the presentlyobserved orbital conÐguration of the Jovian planets isconsiderably altered from that obtained soon after theirformation.

Not only is an early epoch of planet migration plausiblebased upon formation considerations, but recent advances

in our knowledge of the outer solar system provide newmotivation for studying this process. The determination ofreasonably reliable orbits for several dozen KBOs hasrevealed that their distribution is quite nonuniform: there isa near-complete dearth of KBOs having semimajor axes ainterior to NeptuneÏs 3 :2 mean motion resonance at 39.4AU, and there is a prominent concentration of objects at the3 :2 resonance with moderately high eccentricities eD 0.1È0.35 (Jewitt et al. 1998). Several other KBOs orbit at the4 :3, 5 :3, and 2 :1 resonances, all with moderately higheccentricities. There is also a nonresonant KBO populationbeyond the 3 :2 resonance, which extends out to 47 AU;these objects exist in more circular orbits with Thise[ 0.1.peculiar orbital distribution supports the hypothesis thatNeptuneÏs orbit migrated radially outward, sweeping theprimordial Kuiper belt with that planetÏs mean motionresonances and capturing Pluto, as well as a cohort ofKBOs, at those resonances (Malhotra 1993, 1995). Both thesemimajor axes and the eccentricities of captured bodieswould have grown concurrently with the planetÏs orbitalexpansion. The inclinations of KBOs can also becomeexcited when a vertical secular resonance sweeps past(Malhotra 1998). Analysis of the resonance-sweepingmechanism shows that if planet migration were responsiblefor the eccentric and inclined orbits of Pluto and the otherKBOs at the 3 :2 resonance, then NeptuneÏs orbit must haveexpanded by 5 AU on a timescale of orderAU[ *a [ 10

yr or longer (Malhotra 1993, 1998). Malhotraqm

D 107(1999) estimated that the gravitational clearing of a residualplanetesimal disk having a mass of about 35 distributedM

^3041

3042 HAHN & MALHOTRA Vol. 117

in the vicinity of Uranus and Neptune would expandNeptuneÏs orbit by *a D 5 AU. The planet migration/resonance-sweeping hypothesis not only accounts for theabundance of eccentric, inclined KBOs locked in orbitalresonance with Neptune, it also accounts for the lack oflow-eccentricity orbits in the 3 :2 resonance and thedepleted region interior to the 3 :2, as well as the excitedeD 0.1 state of the nonresonant KBOs. It is thus possiblethat the characteristics of the orbital migration history ofNeptune (and, by extension, the other giant planets) is pre-served in the details of the KBO orbital distribution.

Orbital migration was Ðrst noted in the giant-planet ac-cretion models of & Ip (1984), which showedFerna� ndezthat the orbits of a growing proto-Uranus and proto-Neptune could shift by D5È10 AU while embedded in aplanetesimal disk having a mass of order D100 M

^& Ip 1984, 1996). & Ip (1984) argue(Ferna� ndez Ferna� ndezthat Uranus and Neptune would preferentially gain angularmomentum (and thus expand their orbits) by scatteringplanetesimals and lowering their perihelia down to Jupiterand Saturn. Jupiter, being an e†ective planetesimal ejector,would shrink its orbit as a consequence of the resultingenergy and angular momentum losses. However, it shouldbe noted that this argument, while plausible, is not fullysupported by their numerical simulations. Those simula-tions show net orbital expansion by Uranus and Neptune insome instances and orbital decay in others. Individual runsshow considerable to-and-fro motion by Neptune. Suchevolution would only stir up the planetesimals rather thantrapping them at resonances. We note that these simula-tions were not fully self-consistent, since gravitational inter-actions were considered only between planets andplanetesimals that were on crossing orbits ; long-rangeforces between the planets and the planetesimals wereneglected, as were the mutual planet-planet perturbations.Also, the giant planetsÏ gravitational cross sections wereartiÐcially increased in order to speed up the systemÏs evolu-tion. This approximate treatment of planet-planetesimaldynamics was necessitated partly because of the limitedcomputer resources available at the time. A more realistictreatment of disk clearing and planet migration is clearlywarranted.

The present work revisits the planet migration scenariousing more sophisticated numerical techniques to evolve asystem of giant planets that is embedded in a disk composedof many low-mass particles. In ° 2, we describe the modeland test results in detail. Primary results on the giantplanetsÏ orbital evolution with disks having di†erent initialmasses are given in ° 3. Since the planetesimal disk is thesource of the Oort cloud, we discuss the implications of ourresults for the formation of the Oort cloud in ° 4. Particlesthat manage to avoid ejection will reside in what are some-times called the classical and the scattered disks, which arecharacterized in ° 5. The role of unmodeled e†ects such asdisk self-gravity and the role of spiral density waves arediscussed in ° 6, and conclusions are given in ° 7.

2. NUMERICAL METHOD

Our model consists of the Sun, the four giant planetsJupiter, Saturn, Uranus, and Neptune, and a population ofnumerous (N) low-mass particles initially distributed in adisk. Ideally, one would like to have N D 1010 in order tofollow the evolution of this system with a completely self-consistent calculation of the mutual gravitational pertur-

bations and also include the external forcing due to theGalactic tide. However, such a calculation exceeds currentcomputational resources. Instead, as we describe below, weuse a fast orbit integrator together with several simpliÐca-tions to obtain an approximation to the ideal within a rea-sonable amount of computing time.

In our simulations, all bodies are treated as point masses ;thus, the possibility of collisions or accretion is neglected.This approximation is justiÐed during the late stages ofgiant-planet formation, when the frequency of planet-particle collisions is smaller than the scattering frequencyby a factor of the square of the ratio of the planetÏs physicalradius to its Hill radius, which ranges from 10~6 for Jupiterto 10~8 for Neptune.

The model includes the mutual gravitational forcesexerted between the planets, as well as the forces betweenthe planets and the low-mass particles, but the forces amongthe particles themselves are neglected. This approximationis employed to reduce the computational expense, and italso eliminates the unphysical self-stirring that would other-wise occur in the model disks considered below, which havemasses of distributed among justM

D\ 10È200 M

^N \ 1000 particles. Had the particle-particle interactionsbeen included, the relatively small number of N massivedisk particles would quickly stir themselves up so much thatthe resulting system would hardly resemble a planet-forming particle disk. Although this difficulty is avoided byneglecting the particle-particle forces, this approximationalso precludes the possibility that the planets might gener-ate spiral density waves in the disk (see ° 6).

Dynamical models have shown that particles scatteredinto wide orbits with semimajor axes AU becomea Z 3000decoupled from the planets as a result of the Galactic tide(Duncan, Quinn, & Tremaine 1987 ; see also ° 4). Thesedistant bodies are deposited in the Oort cloud, which ise†ected here by simply removing particles from the systemwhen a [ 3000 AU.

A second-order mixed variable symplectic (MVS)mapping is used to rapidly advance the heliocentric posi-tions and velocities of the planets and low-mass particles asthey interact in the SunÏs gravitational Ðeld. Our implemen-tation follows the algorithm of Wisdom & Holman (1991),with the improvements of Saha & Tremaine (1992). A Ðxedstep size of *t \ 0.4 yr is used, which is sufficiently short toresolve the orbits of particles that evolve down to periheliaof q ^ 3.5 AU, as well as most encounters between particlesand planets. For the latter, the dynamical timescale of atypical scattering event is1

T * ^83S 2q3

GMp

, (1)

where is the scattering planetÏs mass and q is the par-MpticleÏs closest approach distance. Evaluating T * with q set

to each planetÏs Hill sphere radius shows that T * ^ 4.1 yr atJupiter, 10 yr at Saturn, 29 yr at Uranus, and 57 yr atNeptune. Thus the integration time step of *t \ 0.4 yr isat worst of the dynamical time for orbits that graze110JupiterÏs Hill sphere. However, an MVS algorithm that

ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ1 This encounter time is deÐned as the time to change the particleÏs true

anomaly from [n/2 to n/2 for a planet-centered parabolic orbit having apericenter distance q from the planet.

No. 6, 1999 PLANETS EMBEDDED IN A PLANETESIMAL DISK 3043

employs a Ðxed step size will fail to resolve very closeplanet-particle encounters that penetrate well within aplanetÏs Hill sphere and will also fail to correctly evolve avery eccentric orbit when the perihelion passage is not welltime-sampled (Rauch & Holman 1999). These difficultiesare mitigated by the using the following approximations :

1. During a close encounter between a particle and aplanet, two-body trajectories are adopted for their relativemotion whenever a particle passes sufficiently close to aplanet. Experimentation shows that a ““ sufficiently close ÏÏencounter is one that changes the particleÏs fractional dis-tance from the planet by more than 50%, or varies itsplanet-centered angular coordinate by more than 90¡,during the time step *t. For particles that are initially inlow-eccentricity heliocentric orbits, the two-body approx-imation is triggered only when a particle approaches withinabout 10% of the planetÏs Hill sphere. For more distantencounters, particle trajectories are in fact evolved withgreater accuracy using the standard MVS mapping.

2. When a particle evolves into an orbit of sufficientlyhigh eccentricity, its motion during perihelion passage maybe poorly time-sampled and the MVS mapping canproduce unphysical evolution. For the step size employedhere, signiÐcant errors accrue in orbits having perihelia q [

AU.2 Proper treatment of such orbits is not of minor3.5import, since approximately 30% of the disk particles cyclethrough the inner solar system with heliocentric distances ofr \ 3.5 AU. A simple, but impractical, solution is to use astep size small enough to resolve all perihelion passages.However, a more practical approach consists of turning o†all planetary perturbations while a particle travels interiorto a heliocentric distance of r \ 3.5 AU; this results in apiecewise Keplerian trajectory that is approximatelycorrect. With this approximation, the particleÏs pertur-bations upon the planets are still not fully time-sampled,which causes slow drifts in the systemÏs total energy andangular momentum. Numerical experimentation has shownthat the most economical and robust solution to thisproblem requires turning o† the particleÏs perturbationsupon the planets while its perihelion distance is q \ 3.5 AU.

The remainder of this section describes a few critical testsof the algorithm in order to demonstrate that the approx-imations made here do not introduce any artiÐcial orbitalevolution of the planets and the particles that is of anysigniÐcance.

The quality of the orbit integrations may be illustratedwith the restricted three-body problem that consists of amassless particle perturbed by a planet on a circular orbitabout the Sun. For this system the particleÏs Jacobi integral

is conserved, where E is the particleÏs energy,J \E[ Lz)

pis its component of angular momentum perpendicular toLzthe planetÏs orbit, and is the planetÏs mean motion.)

pFigure 1 shows the mean fractional variations *J/J thatresult when integrating 100 particles for 104 yr with aJupiter-mass planet at AU. Squares are shown fora

p\ 5

ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ2 It should be noted that the symplectic integrators of Levison &

Duncan (1994) and Duncan, Levison, & Lee (1998) do not su†er thisinstability when massless test particles achieve low-perihelion orbits (H.Levison 1998, personal communication). However, the interventiondescribed above is still required since there is no fully symplectic algorithmthat can compute low-perihelion trajectories when the particles carry mass.

FIG. 1.ÈFractional variations in a particleÏs Jacobi integral J as a func-tion of a particleÏs perihelion distance q averaged over 104 yr. Jupiter lieson a circular orbit at 5 AU, and the integrator step size is *t \ 0.4 yr.Squares indicate particles having initial eccentricities of e\ 0.1, inclina-tions i\ 3¡, and semimajor axes 4 AU\ a \ 50 AU, while crosses andcircles are for eccentric particles having initial perihelia 0.05 AU\ q \ 6AU, a \ 6 AU, and 0¡ \ i\ 10¡. Filled circles and squares indicate par-ticles that approached within a Hill radius, or 0.35 AU, of Jupiter.

those particles that start in rather circular orbits near andbeyond the planet ; these experience a that*J/J [ 10~6decreases with distance provided they have not alreadymade a close approach to Jupiter. Filled squares indicateparticles that approach closer than a Hill distance toJupiter ; these experience larger variations in J as a conse-quence of the close-encounter approximation used here.The remaining particles (crosses and circles) are all onJupiter-crossing orbits, with their perihelia ranging over0.05 AU\ q \ 6 AU. Crosses are shown for particles thatdo not encounter Jupiter, whereas Ðlled circles representparticles that do encounter the planet. The crosses showthat *J/J grows inward of JupiterÏs orbit as a result of thepoor time sampling of the particlesÏ perihelion passages.However, the error *J/J levels o† where the integratorturns o† the planetÏs perturbation, and the particle motionsare temporarily Keplerian. Failure to implement this pro-cedure would have instead yielded disastrous results inwardof about q ^ 1 AU. Shrinking the step size *t would ofcourse reduce the error in J, but at the expense of preciouscomputer cycles.

An alternate scheme that avoids the growth of numericalerrors among low-q particles is to simply remove themwhen they drop down to a heliocentric distance of r \ 3.5AU. Although this procedure inhibits error growth, it is lessthan desirable since it introduces an unphysical (and veryhungry) mass sink into the system. When applying this pro-cedure to the simulations of planet migration described in° 3 (which employ the Keplerian approximation wheneverr \ 3.5 AU), we Ðnd that the orbital migration of Saturn,Uranus, and Neptune proceeds radially outward at similarrates. However, the removal of r \ 3.5 AU particles results


in an artiÐcial sunward mass Ñux that causes JupiterÏs orbitto expand rather than shrink. Nonetheless, the fact that theorbital evolution of the outer giant planets is rather insensi-tive to the treatment of low-q particles adds conÐdence thatthe Keplerian approximation employed here is not drivingthe planet migration reported in ° 3. However, enforcing theparticlesÏ Keplerian motion inside of r \ 3.5 AU is stillnecessary in order to accurately model JupiterÏs orbitalhistory.

In the absence of any close encounters between particlesand planets, the MVS integrator preserves the systemÏs inte-grals in the expected manner. However, Figure 1 shows thatthe close-encounter algorithm used here will not preserve aparticleÏs J to better than one part in 103 should it passwithin a Hill distance of a planet. This numerical errorresults in an unphysical di†usion of particle trajectories.This is a concern since the planet migration phenomenonis sensitive to the Ñux of particles that are scattered andexchanged among the planets & Ip 1984 ; Mal-(Ferna� ndezhotra 1993).

To judge the e†ects of numerical di†usion, the restrictedthree-body problem is again integrated for a system of 100massless particles that orbit in the vicinity of a Neptune-mass planet in a circular orbit at AU. The particlesÏa

p\ 30

initial semimajor axes range over 28.5 AU\ a \ 31.5 AUand have low initial eccentricities e and inclinations i. Forthis system, each particleÏs dimensionless Jacobi integral,

J \ apa

] 2S a

ap

(1[ e2) cos i] O(k) , (2)

is a conserved quantity. Here a particleÏs semimajor axis a isin units of and k \ 5.15] 10~5 is NeptuneÏs mass ina

psolar units. In this test run, the particles initially have amean Jacobi integral SJT ^ 2.998 with a dispersion p

J\

5 ] 10~4. To obtain a physically meaningful measure ofthe numerical di†usion that occurs in the model, deÐne athreshold value which corresponds to an orbitJU \ 2.989,having a perihelion at 20 AU (near UranusÏs orbit) and anaphelion at 30 AU (near NeptuneÏs orbit), so that a/a

p\ 56and Although this orbit is dynamically forbidden toe\ 15.the test particles, numerical di†usion might allow particles

to cross UranusÏs orbit, which would result in an unphysicalexchange of particles between Neptune and Uranus. Thesystem is integrated for 5 ] 107 yr, and at the end of therun, 11 particles have di†used into forbidden Uranus-crossing orbits that have The timescale to di†useJ \JU.into crossing orbits, is obtained from Figure 2, whicht

d,

shows the particlesÏ average J and their dispersion versuspJtime. The characteristic di†usion timescale for particles

to di†use across the Jacobi ““ gap,ÏÏ i.e., the time whenis yr.o J[ p

Jo\JU, t

d^ 2.5] 107

However, it should be noted that a particleÏs J is notconserved in a multiplanet system. For example, adding aJupiter at 5 AU to the above simulation will drive theseparticles into Uranus-crossing orbits having at aJ \JUrate that is about 10 times faster. Since the numerical di†u-sion rate is considerably slower than the dynamical dif-fusion that occurs in a multiplanet system, we concludethat the transport of particles between the planets due tonumerical di†usion is not signiÐcant in these simulations.

As already noted, it is the close encounters between theplanets and the planetesimals, and their concomitantexchange of angular momentum, that drive the planet

FIG. 2.ÈA system consisting of Neptune on a circular orbit at a \ 30AU and 100 massless particles having initial semimajor axes 28.5AU\ a \ 31.5 AU, eccentricities e\ 0.05, inclinations i\ 0.025 rad, andJacobi integrals 2.997\ J \ 2.999, evolved for 5] 107 yr. The solid curvegives the swarmÏs average J, with vertical bars indicating the swarmÏsstandard deviation The characteristic timescale to di†use into Uranus-p

J.

crossing orbits having that are dynamically forbidden isJ \ JU tdD

2.5] 107 yr.

migration process. In our simulations, the planet-particlerelative motion during very close encounters is not com-puted exactly, so any small error in the angular momentumexchange is also reÑected in the recoiling planetÏs orbit. TheÐnal test discussed below veriÐes that the errors in thisangular momentum exchange are in fact too small to drivethe planet migration described in ° 3.

The motion of Neptune, as well as several hundred par-ticles of inÐnitesimal mass m, is integrated at the usual stepsize of *t \ 0.4 yr for 200 yr. The same experiment is thenrepeated using a step size 50 times smaller. Since integrationerrors decrease with step size, this latter run may be regard-ed as a much more exact representation of the particle tra-jectories. Upon di†erencing the two runs, the error in eachparticleÏs velocity relative to the planet, dV, is calculated asthe particle exits the planetÏs Hill sphere. Figure 3a showsthat most encounters occur at the periphery of the planetÏsHill sphere, which result in rather small relative velocityerrors. However, the very close encounters having periapse

which account for 6% of these encounters, resultq [ 0.1RH,in sizable velocity errors that are of order dV /V D 0.1. Theerror in the speciÐc angular momentum exchangedl

pbetween the planet and a scattered particle is dlp\ [mr

pwhere is the planetÏs heliocentric coordi-Â dV/(Mp] m), r

pnate and is its mass. Figure 3b shows a histogram of theMpz-component of the errors which are distributed aboutdl

p,

zero. The net speciÐc angular momentum exchangedbetween the planet and the particle swarm is the sum L \

and its root mean square error iso; lpo , dL \ (; dl

p2)1@2.

For the encounters shown in Figure 3, the fractional errorin the angular momentum exchanged is only dL /L \ 0.016.The above procedure is also repeated for encounters atJupiter, which are less well time-resolved, and yields a largerfractional error dL /L \ 0.12. It should be noted that these


FIG. 3.È(a) Fractional error in the relative velocities dV /V vs. periapseq (in units of NeptuneÏs Hill radius of 500 particles after scattering o†RH)Neptune. The particlesÏ initial heliocentric orbits were a ^ 30 AU,e\ 0.05, and i\ 0.025 rad. Particles that trigger the two-body close-encounter algorithm are indicated by circles. (b) Histogram of the z-component of the speciÐc angular momentum errors for each scatteringdl

pevent. The total speciÐc angular momentum exchanged between the planetand the particle swarm is AU2 yr~1, which has an rmsL \ o; l

po\ 1.27

sum \ 0.64 AU2 yr~1 and an rms error dL \ 0.020 AU2 yr~1.(; lp2)1@2

fractional errors decrease for higher relative velocities,which reÑects the fact that the two-body approximation,which neglects the SunÏs gravity, becomes more accuratewith faster encounters. Thus the error in the total angularmomentum exchanged between the planet and neighboringparticles will steadily decrease as the planet heats up theparticle disk. Since these fractional errors are small, we con-clude that the close-encounter approximation used heredoes not drive the planet migration described below.

3. SIMULATIONS OF PLANET MIGRATION

3.1. Initial ConditionsIn the simulations reported here, we adopted initial plan-

etary orbits similar to those used in previous investigationsof planet migration, in which the semimajor axes of Jupiter,Saturn, Uranus, and Neptune are displaced from theirpresent orbits by respective amounts *a \ ]0.2, [0.8,[3.0, and [7.0 AU (Malhotra 1995). The planets areassumed to have their present masses. The initial planetesi-mal disk is composed of 1000 equal-mass particles distrib-uted in orbits of 10 AU \ a \ 50 AU such that the diskÏsinner edge lies just exterior to Saturn and the outer edge liesjust beyond the present location of NeptuneÏs 2 :1 reso-nance. The disk surface density p varies as a~1, p(a) \ 4.0

AU2)][(1 AU)/a], where is the total disk] 10~3[MD/(1 M

Dmass. Thus, approximately two-thirds of the disk startsexterior to NeptuneÏs initial orbit. Four separate simula-tions are presented below, in which the diskÏs initial mass is

50, 100, and 200 and the individual diskMD

\ 10, M^particles have masses m\ 0.01, 0.05, 0.1, and 0.2 AnM

^.

additional 50 massless particles are also distributed between

50 and 100 AU in order to assess the degree of perturbationof a hypothetical part of the Kuiper belt extending well pastthat which is presently observable.

It should be noted that accretion models advocate aninitial disk containing several tens of Earth masses in the30È50 AU zone in order to form Pluto and QB1-typeKuiper belt objects (Stern & Colwell 1997 ; Kenyon & Luu1998), a scenario that is bracketed by the models exploredhere.

For those particles that initially lie far from any planets,initial eccentricities of and inclinationse

d\ 0.01 i

d\ e

d/2

are adopted. However, particles having a semimajor axisnear a planet will already have experienced a history ofstirring that results in particle dispersion velocities of thev

dorder of the planetÏs escape velocity at its Hill sphere radius(Ida & Makino 1993), whereRH

vdD J2GM/RH , RH \ [M/(3 M

_)]1@3a , (3)

with G the gravitational constant, M the planetÏs mass, andthe solar mass. Assuming that the particlesÏ inclinationsM

_are half their eccentricities, the particlesÏ dispersion velocityis which cor-v

d^ [(e

d2 ] i

d2)GM

_/a]1@2 D e

d(5GM

_/4a)1@2,

responds to an eccentricity of WithedD (24/5)1@2(RH/a).

for both Uranus and Neptune, the particlesRH/a ^ 0.025starting in each planetÏs heated zone initially have e

d\ 0.06

and The adopted half-width of each planetÏsid\ 1¡.7.

heated zone is simply each planetÏs ““ feeding zone ÏÏ of *a \(e.g., Ida & Makino 1993), such that *a \ 1.4 AU2J3RHfor Uranus and *a \ 2.0 AU for Neptune when evaluated

at their initial heliocentric distances.For a planet that is embedded in a swarm of identical

particles of mass m, dynamical friction will tend to seek anequipartition of energies in the systemÏs epicyclic motions.Again assuming that a planetÏs inclination obeys i

p\ e

p/2,

its initial eccentricity is then where areep\ e

d(m/M)1@2, e

dthe eccentricities of particles in the heated zone. SinceJupiter and Saturn start interior to the particle disk, initiallycircular and coplanar orbits are adopted for these twoplanets.

3.2. ResultsFigure 4 shows the orbital histories of the four giant

planets as they scatter the surrounding disk particles in eachof the four simulations. We Ðnd that the lowest mass disk,

yields little evolution in the planetsÏ orbits,MD

\ 10 M^

,but the higher mass disks result in signiÐcant radial dis-placements of the planets during the 30 Myr runs. Thesesimulations conÐrm the expectation that Saturn, Uranus,and Neptune migrate radially outward while Jupitermigrates slightly inward. For a given disk mass, the magni-tude of radial migration is largest for Neptune and suc-cessively less so for the interior planets. As might beexpected, planetesimal disks of greater mass result in planetmigration that is larger in magnitude and more rapid, andalso more stochastic because of the individual particlesÏgreater mass. Although the planetsÏ orbital eccentricitiesand inclinations remain small, there is a clear trend towardhigher eÏs and iÏs at larger disk masses. Since their Ðnal eÏsand iÏs are largely determined by the numbers and masses ofdisk particles used in the simulations, their Ðnal state havelittle relation to the giant planetsÏ current e and i conÐgu-rations. It is worth noting that, in the high-mass disk simu-lations, the planets pass through a few mutual low-order


FIG. 4.ÈSemimajor axes of the giant planets while embedded in planetesimal disks of mass 50, 100, and 200 The boundaries of the grayMD

\ 10, M^

.regions denote the planetsÏ perihelion and aphelion distances.

mean motion resonances, but the planetsÏ orbits do notpersist in any resonance-locked conÐgurations. This is notentirely surprising, as the general trend in the orbital evolu-tion is such that the planets are driven toward greatermutual orbital separation, which is not conducive to main-taining a resonance libration (Dermott, Malhotra, &Murray 1988).

Figure 5 shows the state of the system atMD

\ 50 M^logarithmic time intervals. Large circles indicate the

planetsÏ eccentricities and semimajor axes, while smallcircles and crosses denote particles that have and have nothad a close approach to a planet, respectively. Those par-ticles scattered by the planets yet still bound to the Sun tendto have perihelia between the orbits of Saturn and Neptune,as indicated by the two curves. It should be noted that in allof the simulations reported here, the orbital migration ofthe planets has not ceased by the end of the runs, and thatfurther planet migration will continue (albeit more slowly)past 30 Myr. The simulation of the disk isM

D\ 50 M

^extended to 50 Myr in Figure 6, which shows Neptuneslowly expanding its orbit out to 30 AU. These resultssuggest that in order to actually ““ park ÏÏ Neptune at a \ 30

AU requires an adjustment of parameters toward a slightlylower disk mass, a steeper gradient in the diskÏs surfacedensity proÐle, and/or an outer disk edge that lies closerthan 50 AU.

3.3. DiscussionThe disk particles used in all of these simulations are

sufficiently massive that their perturbations upon Neptuneresult in nonadiabatic expansion of that planetÏs orbit.However, it is evident in Figure 4 that planet migration issmoother when the disk particles have lower mass. Thus, amore realistic simulation employing larger numbers of par-ticles will better resolve the disk-planet perturbations. Weexpect that such simulations will exhibit orbital migrationthat proceeds more smoothly. Nearly adiabatic orbitalmigration is in fact required if Neptune is to efficientlycapture particles at its exterior mean motion resonances.Since smooth outward expansion was not realized in thesesimulations, Neptune did not capture any particles at itsmean motion resonances. Thus, the simulated diskÏs endstate cannot be directly compared with the delicate reso-nant structure that is observed in the Kuiper belt. Neverthe-


FIG. 5.ÈEccentricities e vs. semimajor axes a at logarithmic time intervals for the system. Small circles indicate scattered particles that haveMD

\ 50 M^passed within a Hill radius of a planet, and crosses indicate particles that have not encountered a planet. Large circles denote the planets, and the vertical

dashes indicate the location of NeptuneÏs outer four mean motion resonances. Orbits lying above the left curve have perihelia inside of SaturnÏs orbit, andthose above the right curve have perihelia interior to Neptune.

less, these simulations do provide useful constraints on thelikely mass of the initial debris disk that may have beenpresent during an early epoch of planet migration.

Figure 7 reports the radial displacement *a experiencedby each planet after t \ 3 ] 107 yr as a function of theinitial disk mass As noted above, Jupiter migratesM

D.

sunward while the other planets migrate outward in eachsimulation. The time-averaged torque thatT0 \ *L /tdrives Neptune by increasing its angular momentum by *Lduring the run time t is shown in Figure 8. This torque, aswell as the displacements *a of Figure 7, should be regardedas an upper limit, since resonance trapping did not occurin these simulations. Had resonance capture been realizedhere, an opposing torque on the planet would have devel-oped, since the planet must transfer angular momentum toexpand the orbits of particles trapped at resonances.

The back-torque due to a ring of mass m that is capturedat an exterior ( j] 1) : j mean motion resonance can be cal-culated from the rate of change of its angular momentum,

Since the resonant torque fromLm

\m[GM_

a(1[ e2)][email protected] planet expands the ringÏs semimajor axis at the rate

and also pumps up eccentricities at the ratea5 /a \ a5p/a

p

(Malhotra 1993), the rate of changede2/dt \ (a5p/a

p)/( j] 1)

of isLm

dLm

dt^ a

mm

pT , (4)

where is the ratio of the semi-a \ ap/a \ [ j/( j ] 1)]2@3

major axes of the planet and the resonance site, is thempplanetÏs mass, and is the netT \ m

p(GM

_ap)1@2a5

p/2a

ptorque on the planet, which simpliÐes to T \ T0Thus the back-torque from the[ dLm/dt ^ T0/(1 ] am/m

p).

resonance-captured particles will signiÐcantly slow theplanetÏs orbital expansion if the mass trapped at resonanceis in excess of the planetÏs mass, i.e., In a genericmZ m

p/a.

system, the strongest exterior mean motion resonance is the2 :1 ( j \ 1, a \ 0.63), which can be expected to capture themost planetesimal mass. If the capture efficiency is v, thenwe can estimate that planet migration will slow if

vMD,extZ 1.6m

p, (5)

where is the mass of the disk exterior to the planetÏsMD,extinitial orbit.


FIG. 6.ÈSimulation of the system extended out toMD

\ 50 M^t \ 5 ] 107 yr. Gray indicates perihelion and aphelion distances. Note

that Uranus and Neptune pass through a 2 :1 mean motion resonance att \ 3.05] 107 yr, which results in a brief eccentricity excitation.

If resonance capture is rather inefficient, then a high-massdisk is simply fuel for planet migration. But if resonancetrapping is e†ective, say, vD 50%, then the planetÏs migra-tion will slow after the planet has captured a mass m com-parable to its own at its exterior resonances. In the diskmodels considered here (with this wouldM

D,extB 23MD),

occur only in a high-mass disk with MD

Z 80 M^

.If migration by Neptune across a distance *a D 5È10 AU

is to explain the sculpted appearance of the Kuiper belt,then Figures 4 and 7 suggest that the initial planetesimaldisk must have had a mass in excess of 10 (since aM

DM

^lower mass disk results in insufficient planet migration that

FIG. 7.ÈRadial displacement *a vs. disk mass for each planet afterMDt \ 3 ] 107 yr.

FIG. 8.ÈTime-averaged torque on Neptune (solid curve), in units ofT0that planetÏs current angular momentum/orbital period ratio L /P, plottedvs. the disk mass The dotted curves indicate sums of the resonant diskM

D.

torque contributions (eq. [6]).

proceeds too slowly) but likely less than D100 (sinceM^such a high-mass disk would likely produce additional giant

planets). A more precise disk mass estimate requiresdetailed knowledge of the diskÏs radial extent and the par-ticulars of the disk surface density variations p(a).

4. THE OORT CLOUD MASS

The observed Ñux of long-period comets provides a con-straint on the present mass of the Oort cloud. In thissection, we use the results of our numerical simulations toestimate the mass of its progenitor, the primordial plan-etesimal disk in the outer solar system.

After adjustment for the efficiency of comet detections, itis estimated that D63 long-period comets per year passthrough the inner solar system with perihelia q \ 4 AU(Everhart 1967). This Ñux stems entirely from the Oortcloud ; objects originating in the Kuiper belt evolve insteadinto the short-period Jupiter-family comets (Levison 1996).Dynamical models of the Oort cloud, when adjusted tomatch the Ñux of long-period comets, require a reservoir of

bodies (Heisler 1990).3 Multiplying byNOC D 1.6] 1013the typical comet mass yields the total mass of the Oortcloud. Based upon an admittedly uncertain relationshipbetween cometary brightness and size, Weissman (1996)concludes that the mean comet mass is of order D1016 g,indicating an Oort cloud mass of order MOC D 27 M

^.

However, this mass estimate should be regarded as uncer-tain by at least an order of magnitude, since it relies upon ahost of uncertain quantities such as the efficiency of cometdetections and the comet brightness-mass relationship, as

ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ3 The Heisler (1990) Oort cloud model shows that a reservoir of 1011

comets will produce a Ñux of 0.2 comets per year having q \ 2 AU. Tomatch the observed Ñux of comets with q \ 4 AU, multiply this populationby 63/(2 ] 0.2).


well as uncertainties in the Oort cloud perturbations (e.g.,the strength of the Galactic tidal Ðeld and the frequency ofstellar encounters).

The dynamics of the Oort cloud is succinctly summarizedby Duncan et al. (1987). Unless a particle is otherwiseejected from the system, planetary perturbations cause itssemimajor axis to di†use both inward and outward whilekeeping its perihelion locked in the giant-planet region ofthe solar system. However, a more distant particle is suscep-tible to perturbations by the Galactic tide and passing stars,which cause its perihelion to di†use on a timescale thatvaries as Those particles reaching 3] 103t

qP a~2. AU[

AU that have had their perihelia raised wella [ 2 ] 104beyond the orbit of Neptune are thus decoupled from theplanets and are usually identiÐed as inner Oort cloudcomets. However, the perihelia of more distant bodiesdi†use more rapidly, and those with 2 ] 104 AU[ a [ 105AU that reside in the outer Oort cloud are in fact morelikely to di†use back into the inner solar system andbecome observable as new long-period comets. It is this Ñuxof new comets that provides an important constraint on theamount of mass driven from the initial disk that hasmanaged to avoid ejection during the last 4.5 Gyr.

Estimates of the Oort cloud mass may be inferred fromthe data given in Table 1. The quantity is the fraction off3kthe dynamically active disk that di†uses into the Oort cloudat a [ 3 ] 103 AU during each simulation. The dynami-cally active disk is that part of the disk where particles arelikely to be perturbed into Neptune-crossing orbits over theage of the solar system. Long-term integrations of Kuiperbelt orbits show that particles in the active disk have semi-major axes where is NeptuneÏs semimajor axisa [ 1.4aN, aN(Duncan, Levison, & Budd 1995). The mass of the dynami-cally active disk, (Table 1), is deÐned here as the totalMadmass of ejected particles plus all survivors having a [ 1.4aN,which are presumably in unstable orbits. Also given is f

h,

which is the fraction of the active disk that has been ejectedfrom the system. Figure 9 displays and versus time forf3k f

hthe run. What is most striking is that at theMD

\ 100 M^end of all four runs both and vary little among thef3k f

hdi†erent simulations (Table 1). This indicates that the totalmass deposited in the Oort cloud depends only on the massof the disk that lies within the planetsÏ gravitational reachand is not very sensitive to the orbital histories of themigrating planets.

Ultimately, all particles starting in the dynamically activedisk are either ejected, deposited in the Oort cloud, or insome instances accreted by the planets. Although the latteroutcome is not modeled here, impacts may be assessed expost facto using the collision probabilities of (1951).O� pikAfter summing the probability of each particle striking eachplanet, we Ðnd that D15 of the 1000 particles would havestruck the giant planets during each simulation, with

TABLE 1

OORT CLOUD MASSES

MD

Mad MOC(M

^) f3k f

hf 3k@ f

h@ (M

^) (M

^)

10 . . . . . . . 0.097 0.355 0.215 0.785 5.7 0.4150 . . . . . . . 0.143 0.364 0.282 0.718 35 3.3100 . . . . . . 0.142 0.327 0.303 0.697 99 10200 . . . . . . 0.141 0.369 0.276 0.724 200 18

FIG. 9.ÈFraction of the active disk that is ejected into hyperbolicorbits, vs. time t for the simulation. The disk fraction thatf

h, M

D\ 100 M

^is scattered into the Oort cloud with a [ 3000 AU is and is thef3k, fq:3.5instantaneous disk fraction having perihelia q \ 3.5 AU. Note the near-

constancy of the ratio Curves for the other runs are quitef3k/fh^ 0.4.similar.

roughly half these impactors striking Jupiter. Such impactswould have contributed no more than D2% to any planetÏsmass, so the neglect of particle-planet collisions is justiÐed.It is also worth noting that a few percent of the disk passesthrough the terrestrial zone, as is indicated by the f

q:3.5curve of Figure 9, which shows the instantaneous disk frac-tion having perihelia inside of 3.5 AU. In these simulationstypically D30% of all disk particles have brief episodes withq \ 3.5 AU. These Ðndings are consistent with earlierstudies showing that planetesimals scattered during theepoch of disk clearing and planet migration may have con-tributed signiÐcant numbers of impactors during the late,heavy bombardment of the terrestrial planets (Wetherill1975 ; Shoemaker & Wolfe 1984).

Table 1 shows that in the four simulations the activedisks are depleted by a factor afterf 4 f3k] f

h^ 50%

t \ 3 ] 107 yr. It should be noted that the family of f (t)curves is described well by a power law f P t0.44. This indi-cates that a fully evolved system having f ] 1 requires anintegration lasting the duration of the diskÏs dynamical life-time, yr, which is beyond our computationalq

dD 1.5] 108

means. However, it is straightforward to extrapolate theformation of the Oort cloud from the simulations at hand.Figure 9 shows that the ratio remains relativelyf3k/fhD 0.4constant during the bulk of the run. This relation permitsan extrapolation to a fully evolved state of the systemhaving where the primes denote Ðnal extrapo-f 3k@ ] f

h@ ] 1,

lated values. Assuming over the age of thef 3k@ /fh@ \ f3k/fhsolar system, the extrapolated fractions become f

h@ \

and these values(1] f3k/fh)~1 ^ 0.73 f 3k@ \ 1 [ fh@ ^ 0.27 ;

are also given in Table 1. Since these disk fractions are bothnumber as well as mass fractions, the extrapolated plan-etesimal mass that is initially deposited in the inner Oortcloud is However, the Galactic tide will subse-f 3k@ Mad.quently strip away comets that di†use past AU, anda Z 105passing stars will eject others. Numerical studies show that


these external perturbations acting over the age of the solarsystem will reduce the Oort cloud mass to about a third ofits initial value (Duncan et al. 1987). Thus the Ðnal Oortcloud mass reported in Table 1 is and it isMOC\ f 3k@ Mda/3,also displayed as the solid curve in Figure 10 as a functionof the initial disk mass M

D.

The nebula origin of the Oort cloud is given by the solidcurve in Figure 11, which shows a histogram of the Oortcloud particlesÏ initial semimajor axes for the M

D\ 50 M

^disk ; after t \ 3 ] 107 yr, the planetary conÐguration inthis system most resembles the solar system. Since the inneredge of our model disks is truncated at 10 AU, any Oortcloud mass originating in interior orbits is still unaccountedfor. We estimate this contribution after the fact by evolvinga system of four giant planets in their present conÐgurationplus an annulus of 153 massless test particles having aninitial p P a~1 surface number density a between 4 and 10AU. Their Oort cloud contribution is given by the dashedcurves in Figures 10 and 11. Evidently, all parts of the giant-planet domain contribute mass to the Oort cloud. TheseÐndings are in agreement with those of Weissman &Levison (1997), who reported that a small fraction of Oortcloud bodies can originate from orbits interior to Jupiterand thus have asteroidal rather than cometary composi-tions.

If the initial disk had a mass 10 M^

[MD

[ 100 M^(° 3), then Figure 10 indicates that the resulting Oort cloud

mass is 0.5 This mass estimate mustM^

[ MOC[ 11 M^

.be qualiÐed for two reasons. For the lower mass disks,

the mass estimate as given in Figure 10MD

¹ 50 M^

, MOCis likely an underestimate because in deriving it we have notaccounted for the possibility that Neptune can migratedeeper into the disk as the system evolves further on time-

FIG. 10.ÈExtrapolated mass of the Oort cloud mass as a functionMOCof the initial disk mass The dashed curve includes contributions byMD.

massless test particles originating in the 4 AU \ a \ 10 AU part of thedisk that is modeled separately. However, each one of these particles, whendeposited in the Oort cloud, is assumed to contribute the same individualmass as its compatriot particles that are employed in the simulations ofFig. 4.

FIG. 11.ÈLocal disk fraction deposited in the Oort cloud vs. initialsemimajor axis a for the simulation. The data are obtainedM

D\ 50 M

^at time t \ 3 ] 107 yr and then extrapolated to a fully evolved state bymultiplying by The gray bars show the extent of radialf 3k@ /f3k \ 1.97.migration by each planet. The 4 AU\ a \ 10 AU component (dashedcurve) is obtained from a separate integration, and errors are assumed.JN

scales longer than D50 Myr, thus allowing additionalmaterial to be injected into the Oort cloud. (This does nota†ect the higher disk mass simulations, since NeptuneÏsgravitational reach has already swept across the entire diskin the higher mass runs.) For the higher mass disks, isMOClikely overestimated in Figure 10 because resonance trap-ping tends to slow planet migration and may reduce themass encountering the planets and, thus, the mass depositedin the Oort cloud. We note that an Oort cloud having atotal mass and a populationMOCD 5 M

^NOCD 1.6]

1013 comets (Heisler 1990) suggests that a characteristicOort cloud comet has a radius of D1 km for a density ofD0.5 g cm~3.

5. THE RESONANT, STIRRED, AND SCATTERED

KUIPER BELT COMPONENTS

Although the giant planets scatter planetesimals through-out the entire solar system, a large gap in orbital phasespace will develop, as illustrated by Figure 12. This gap iseasily explained via the restricted three-body problem,which shows that a planet on a circular orbit will scatter amassless body in a manner that preserves the particleÏsJacobi integral J (eq. [2]). Particles originating in a colddisk in the vicinity of the planet have semimajor axes a ^ a

pand J ^ 3. Once scattered, these particles will have eccen-tricities deÐned by the curve (Fig. 12), assuming i\ 0.e

J/3It has also been shown that a planet of low eccentricity willrepeatedly scatter particles along a curve that approxi-mately preserves J ^ 3 (Ida & Makino 1993). Thus, whenseveral planets are present, those interior to Neptune canloft particles into high-eccentricity Neptune-crossing orbitsand ultimately Ðll the region of phase space. Thosee[ e

J/3particles in the high-eccentricity orbits of FigureeZ eJ/312 are referred to as the scattered disk (e.g., Duncan &

Levison 1997). The region occupied by bodies of lowereccentricity beyond D35 AU is sometimes referred to inrecent literature as the ““ classical ÏÏ Kuiper belt ; the eccen-


FIG. 12.ÈEccentricity e vs. semimajor axis a at time t \ 5 ] 107 yr forthe simulation. Large circles indicate Uranus and Neptune,M

D\ 50 M

^small circles indicate scattered particles that have passed within a Hilldistance of a planet, and crosses denote particles that have not had aclose planetary encounter. The dashed lines indicate NeptuneÏs four outer-most mean motion resonances, and the curve satisÐes eq. (2) with i \e

J/30. Boxes denote observed KBOs having well-determined orbits (fromMarsden 1999), and the scattered object 1996 is indicated.TL66

tric, resonant objects (including the so-called Plutinos atNeptuneÏs 3 :2 resonance) may be considered a distinctdynamical subclass of the latter population.

Although Figure 12 shows the scattered and classicaldisks after only 5] 107 yr, Duncan & Levison (1997) haveintegrated test particle orbits for the age of the solar system.They Ðnd that while most particles in the scattered disk areremoved in less than the age of the solar system, about 1%of the scattered particles survive for longer times (perhapsby acquiring protection via the Kozai mechanism [Kozai1962] or by sticking near mean motion resonances). In theabsence of disk stirring by any other large, distant per-turbers, we note that the phase-space gap between the clas-sical and scattered disk will persist over the age of the solarsystem. This gap should become evident as deeper obser-vations begin to peer beyond NeptuneÏs 2 :1 resonance.

Until very recently, no KBOs were known to orbit atNeptuneÏs 2 :1 resonance, in apparent conÑict with the pre-diction of the planet migration/resonance-sweeping theorythat KBO populations at the 2 :1 and 3 :2 should be simi-larly abundant and have comparable eccentricities(Malhotra 1995). As this paper was being written, welearned that two KBOs, 1997 and 1996 haveSZ10 TR66,been identiÐed as librating in the 2 :1 Neptune resonance(Marsden 1999).4 We note that these two KBOs were for-merly identiÐed in NeptuneÏs 5 :3 and 3 :2 resonances, andthat their orbits were revised to the 2 :1 resonance only afterobservations spanning two and three oppositions, respec-tively. Clearly, orbit-Ðtting biases and observational incom-

ÈÈÈÈÈÈÈÈÈÈÈÈÈÈÈ4 See http ://cfa-www.harvard.edu/iau/lists/TNOs.html.

pleteness remain in the current census of the Kuiper belt.A robust test of the planet migration/resonance-sweepingtheory requires a larger observational sample of reliableKBO orbits.

Morbidelli & Valsecchi (1997) o†er an alternative expla-nation for the structure in the Kuiper belt. They suggestthat if a stationary Neptune had scattered a couple ofEarth-mass planetesimals (i.e., LNSPs, or ““ large Neptune-scattered planetesimals ÏÏ) outward, these massive bodiescould have stirred up KBO eccentricities to eD 0.2, similarto those observed for KBOs at NeptuneÏs 3 :2 resonance.Gravitational scattering can indeed insert KBOs into, aswell as remove objects from, mean motion resonances, butscattered objects tend to librate at resonance with suchlarge amplitudes that close encounters with Neptunebecome possible and long-term orbital stability is precluded(Levison & Stern 1995). Therefore, an additional sequenceof collisions and/or scattering events is required in orderfor particles to di†use to stable, small-amplitude libratingorbits. Only the fortunate few would survive this process, sothe yield of KBOs scattered into stable eD 0.2 orbits at the3 :2 resonance would be much smaller than that whichmight otherwise be acquired by means of adiabatic orbitexpansion and resonance sweeping by Neptune. Morbidelli& Valsecchi (1997) also point to the eD 0.1 KBOs thatreside between the 3 :2 and 2 :1 resonances as additionalevidence for LNSPs. But resonance capture is not entirelyefficient, and similar eccentricities are also achieved asNeptuneÏs orbit expands and its 2 :1 resonance sweepsacross the disk and stirs up eccentricities (Fig. 12).

A census of KBOs beyond NeptuneÏs 2 :1 resonance (at 48AU) would permit an evaluation of possible stirring byhypothetical LNSPs. If the natal planetesimal disk doesextend past 48 AU, then Figure 12 shows that planet migra-tion will produce a stirred zone interior to the 2 :1 yet leavethe disk exterior to the 2 :1 relatively undisturbed. However,an abundant population of eccentric KBOs beyond the 2 :1would suggest a history of additional stirring by otherunseen perturbers (though this would not preclude anepisode of planet migration).

6. DISK SELF-GRAVITY AND THE ROLE OF SPIRAL

DENSITY WAVES

As noted earlier, our simulations neglect the planetesimaldiskÏs self-gravity in order to inhibit an unphysical degree ofself-stirring. The consequences of disk self-gravity could bebetter studied only by simulating disks composed of manymore lower mass particles. In this section, we discuss thepossible consequences of the diskÏs self-gravity.

If the local disk mass exceeds the mass of a nearby planet(as is the case for Neptune in most of the simulations con-sidered here), it is the diskÏs gravity that can control therates at which the perihelia and nodes of both the planetand the disk particles precess and/or regress (Ward 1981).Secular resonances are sites in the disk where a planetÏsperihelion/node varies at the same rate as the disk particlesÏ,and high eccentricities and inclinations can be excited atthese resonances. As planets sculpt the disk and cause itssurface density to evolve over time, the location andstrength of secular resonances will shift. Although radialdrifts in the location of secular resonances might alter thedetails of how a planet depletes the disk as it excites par-ticles into crossing orbits, this issue is likely of lesser impor-tance when compared with the mean motion resonances.


In particular, a planet that is embedded in a self-gravitating disk can launch spiral density waves at its meanmotion resonances. Numerous examples of this phenome-non exist in SaturnÏs rings, which exhibit density wavesdriven by orbiting satellites. The gaps in these rings reveal ahistory of angular momentum exchange between ringmaterial and satellites, and similar exchanges are expectedof planet-forming systems. When Neptune launches densitywaves at an exterior ( j] 1) : j mean motion resonance, thedisk exerts the torque on the planet, which opposes itsT

jradial migration :

Tj\ jn2pt2

r dD/dr^ [ 64nj3

75( j] 1)kp2 k

dM

_(a

p)

p)2 (6)

(Goldreich & Tremaine 1980). Here is thet^ [8jkp(r))2/5

planetÏs forcing function, is the planetÏs mass in solarkpunits, ) is the diskÏs mean motion, D is the frequency di†er-

ence from exact resonance, which has the gradient r dD/dr ^ [3( j] 1))2, and the preceding quantities areevaluated at resonance with and ther \ (1 ] j~1)2@3a

pap

)pplanetÏs semimajor axis and mean motion (Hahn, Ward, &

Rettig 1995). The dimensionless ““ disk mass ÏÏ is kd4

for the disknp(ap)a

p2/(1 M

_)^ 1.1] 10~6[M

D/(1 M

^)]

simulations that have AU2). Fig-p(ap)\ 1.3] 10~4M

D/(1

ure 8 sums the torques due to NeptuneÏs outer j\ 1È5Tjresonances for comparison with the nonresonant torque T0that drives NeptuneÏs orbit expansion.

Evidently, the resonant disk torques can inhibitNeptuneÏs outward migration if the disk admits a waveresponse at the resonances. It should be noted thatjZ 3these torques are also operative in nonÈself-gravitatingdisks (Lissauer & Espresate 1998), and similar torques alsoslow orbit expansion when resonance trapping is e†ective(° 3.3). But when trapping is not of concern, the resonanttorques in a nongravitating disk shut o† once particleeccentricities become excited (Lissauer & Espresate 1998).However, in a self-gravitating particle disk, density wavestransport the planetÏs forced disturbance downstream of theresonance in the direction of the planetÏs orbit. As long aswave action is sustained, particles at resonance maintainlow eccentricities and the resonant disk torque can opposeplanet migration.

There are two ways in which the propagation of densitywaves might be inhibited in particle disks. In a disk that ispopulated by comet-sized (or larger) planetesimals, dissi-pative forces such as gas drag or viscosity due to inter-particle collisions are insufficient to damp out density waves(e.g., Hahn et al. 1995). In this case, density waves reÑect at aQ-barrier in the disk and return to the launch zone, wherethey are reabsorbed by the particles at resonance (Toomre1969). This absorption of the returning wavesÏ energy willsteadily heat the disk and can eventually shut o† sub-sequent wave generation. A second way to defeat waves isvia stirring by larger bodies, which also heats the disk andinhibits wave propagation. Consequently, the resonanttorque that the particle disk exerts on Neptune will delaythe onset of orbit expansion until the disk becomes toostirred to sustain density waves at its resonances.jZ 3

7. CONCLUSIONS

The existence of the Oort comet cloud, as well as theKuiper belt, suggests that there was once a residual plan-etesimal disk of mass D10È100 in the vicinity of theM

^

giant planets following their formation. Further, any ineffi-ciencies in the formation of the giant planetsÏ cores impliesadditional disk mass. The eventual clearing of this plan-etesimal population involves a substantial exchange oforbital energy and angular momentum with the planets,implying that the present locations of the giant planets arenot necessarily their formative ones. We have numericallysimulated the evolution of a system of four giant planetsembedded in a planetesimal disk of mass ranging from 10to 200 Our numerical simulations show a gradualM

^.

increase in the mutual separation of the planetsÏ orbits asthe disk is dispersed via gravitational scattering by theplanets. Higher disk masses yield planetary orbital migra-tion that is faster and larger in magnitude. If planet migra-tion and resonance trapping are invoked to explain theeccentricities of Pluto and its cohort of Kuiper belt objectsat NeptuneÏs 3 :2 mean motion resonance, then these simu-lations show that a disk mass of order isM

DD 50 M

^required to expand NeptuneÏs orbit the requisite distance of*a D 7 AU to pump up Plutino eccentricities to eD 0.3.Such an episode of planet migration implies that the solarsystem was more compact in the past, with the initialJupiter-Neptune separation having been smaller by about30%. This Ðnding also conÐrms the disk mass estimate pre-viously obtained by Malhotra (1999).

Our model disk-planet systems behave similarly to otherdisk systems that experience a gravitational or viscoustorque (e.g., Lynden-Bell & Kalnajs 1972 ; Lynden-Bell &Pringle 1974), which causes angular momentum to becarried radially outward (in this application, by the outerthree planets) while disk particles deliver mass radiallyinward. However, these particles tend to be ejected uponreaching JupiterÏs orbit, which accounts for that planetÏsslight orbital decay.

Since our simulations neglected the diskÏs self-gravity,collective e†ects such as density waves are precluded. Aplanet embedded in a self-gravitating planetesimal disk willtend to launch spiral density waves at its resonances. Thetorque due to wave generation is sufficient to opposeNeptuneÏs orbital expansion as long as the disk remainsdynamically cold enough to admit a wave response from itsj ^ 3 or higher resonances. Such an episode of wave gener-ation will delay the onset of planet migration until the diskÏswave response is defeated.

The bulk of the disk particles deposited in the Oort cloudoriginate in the vicinity of the Saturn-Neptune region of thesolar nebula. Assuming that Galactic tides and passing starsdecouple particles from the planetary system when theyachieve a semimajor axis of a [ 3000 AU, and that theseperturbations also remove about two-thirds of the Oortcloud over the age of the solar system (Duncan et al. 1987),we estimate that about of theMOC D 12 M

^M

D\ 50 M

^disk is initially emplaced in the Oort cloud, of which D4will persist to the present age of the solar system.M

^Because of the fact that the disks simulated here weresparsely populated by particles having masses m\ 0.01È0.2

their vigorous scattering caused the planetsÏ orbits toM^

,evolve nonadiabatically such that resonance trapping ofKBOs was inhibited. However, previous studies haveshown that adiabatic orbit expansion by Neptune canaccount for the abundance of eccentric KBOs that areknown to orbit at NeptuneÏs 4 :3, 3 :2, 5 :3, and 2 :1 meanmotion resonances (Malhotra 1995). Unless there are (orwere) additional unseen, distant perturbers, any primordial


KBOs beyond NeptuneÏs 2 :1 resonance should reside innearly circular, low-inclination orbits.

The planet migration/resonance-trapping phenomenonmight also have applications in extrasolar planetarysystems. The most visible component of an extrasolar plan-etary system is likely its dust, which should be most abun-dant when planets and planetesimals are colliding,accreting, and eroding. Dusty circumstellar disks and ringsare known to orbit the stars b Pictoris, Formalhaut, HR4796A, 55 Cancri, and v Eridani (Smith & Terrile 1984 ;Greaves et al. 1998 ; Trilling & Brown 1998 ; Koerner et al.1998 ; Holland et al. 1998 ; Smith et al. 1998). In some ofthese systems, collisions and/or radiation forces will removethe observed dust on a timescale shorter than the age of theparent star. The presence of dust thus suggests an addi-tional sourceÈperhaps dust generation due to collisions byunseen planetesimals that reside in the disk. Any planetsthat might form within this environment will deplete the

disk region that lies within their gravitational reach, whichcould account for these disksÏ central gaps. However, anepisode of planetesimal disk clearing would also driveplanet migration, which can concentrate planetesimals atthe outermost planetÏs exterior mean motion resonances.Since the collision frequency and, hence, the dust generationrate varies as the square of the planetesimal density, onemight speculate that this mechanism is also responsible forthe formation of dust rings observed around v Eri, For-malhaut, and HR 4796A.

The authors thank Derek Richardson for a careful reviewof this paper. This paper is contribution 966 from the Lunarand Planetary Institute, which is operated by the Uni-versities Space Research Association under NASA contractNASW-4574. This research was supported in part byNASAÏs Origins of Solar Systems research program.

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