Planetary migration in a disk why did

stop at AU

Ro dney S Gomes

GEAOVUFRJ and ONMCT

Ladeira do Pedro Antonio Centro Rio de Janeiro RJ Brazil

Tel

Fax

Email ro dneyovufrjbr

Alessandro Morbidelli

Observatoire de la Cote dAzur Nice France

Harold F Levison

Southwest Research Institute Boulder Colorado USA

pages gures

Running head Migration in a planetesimal disk

Send corresp ondence to

Ro dney S Gomes

Observatorio do Valongo

Ladeira do Pedro Antonio Centro Rio de Janeiro RJ Brazil

Received accepted

ABSTRACT

We study planetary migration in a gasfree disk of In the case

of our weshow that Neptune could have had either a damp ed

migration limited to a few AUs or a forced migration up to the disks edge

dep ending on the disks mass densityWe also study the p ossibility of runaway

migration of isolated in very massive disk whichmight b e relevant

for extrasolar systems Weinvestigate the problem of the mass depletion of

the Kuip er b elt in the light of planetary migration and conclude that the b elt

lost its pristine mass well b efore that Neptune reached its current p osition

Therefore Neptune eectively hit the outer edge of the protoplanetary disk

We also investigate the dynamics of massive planetary embryos emb edded in

the planetesimal disk We conclude that the elimination of mass or

Marsmass embryos originally placed outside the initial lo cation of Neptune also

requires the existence of a disk edge near AU

Subject headings Kuip er Belt resonance Solar System formation

Intro duction

Planet migration in forming planetary systems o ccurs in two stages The rst one

happ ens due to the interaction of the with the gaseous disk Ward Masset

After the gas disk dissipates the energy and exchange b etween

remaining planetesimals and the planets induce the second stage of planetary migration

This phenomenon was rst brought to lightbyFernandez and Ip

It is now b elieved that planetary migration substantially sculpted the Kuip er b elt

generating most of the features that are nowobserved Malhotra rst showed that

the resonant eccentric orbit of can b e the result of the resonance sweeping through

the protoplanetary disk during Neptunes migration Similarly the same scenario explains

the existence of a signicant fraction of Kuip er b elt b o dies in the ma jor mean mean motion

resonances with Neptune and their wide range of orbital eccentricities

Malhotra Gomes showed that the origin of the so called hot classical Kuip er

b elt a p opulation of nonresonant b o dies with inclinations larger than degrees can

also b e explained as a result of Neptunes migration whichallowed a small p ortion of the

scattered disk p opulation to b e trapp ed on stable orbits with smallmo derate eccentricities

More recently Levison and Morbidelli prop osed that Neptunes migration also

generated the cold classical Kuip er b elt the p opulation of nonresonant b o dies with

inclinations smaller than degrees Brown the memb ers of this p opulation would

have b een transp orted to their current lo cation from a much smaller helio centric distance

through a mechanism that invokes temp orary trapping into the mean motion resonance

The prop erties of the Kuip er b elt are not the only indications of planetary migration

Levison and Stewart showed that the insitu formation of and Neptune is

unlikely suggesting that these planets formed much closer to and where

the growth timescales were dramatically shorter Thommes et al Thommes et al

prop osed a radical dierent view in which Uranus and Neptune formed b etween

Jupiter and Saturn and were scattered outwards where the interactions with the disk of

planetesimals damp ed their eccentricities and inclinations

Despite the imp ortance of planetary migration not muchwork has b een done up to

now to study the migration pro cess per se After the pioneering work of Fernandez and Ip

Hahn and Malhotra tried to b etter characterize planetary migration with

a series of direct numerical integrations In their work the planets initially in a more

compact conguration were emb edded in a planetesimal disk with total mass ranging

from to Earth masses M and with a surface density decaying as the inverse of



the helio centric distance r Because of computational limitations the authors were forced

to simulate the disk with only ob jects which exerted a gravitational inuence on

the planets but not among themselves The authors found that a M disk could bring



Neptune from its initial p osition p ostulated at AU to its quasinal p osition at AU

in million years and therefore concluded that this was the most likely mass of the

planetesimal disk after planetary formation An imp ortantpoint observed in Hahn and

Malhotra is that migration pro ceeded in a nonadiabatic way so that no resonance

trapping of the planetesimals was observed The authors conjectured that if the disk

were comp osed of a larger numb er of smaller planetesimals Neptunes migration would b e

smo other and consequently the resonance trapping phenomenon would o ccur This they

argued could also slow Neptunes outward motion b ecause the resonant particles would

eectively increase Neptunes inertial mass as they need to b e moved together with the

planet

Gomes simulated Neptunes migration using a disk of massive

planetesimals As exp ected he observed a much smo other migration than in Hahn and

Malhotra with many resonant captures However despite the captures with a disk



similar to that of Hahn and Malhotra Earth masses b etween and AU with a r

surface density prole Neptune migrated to AUin y The fact that this

result was so dierent from the one by Hahn and Malhotra shows the necessity of a deep er

understanding of the phenomenon of planetary migration which is precisely the goal of the

present pap er

A detailed study of the general migration pro cess would require the exploration of

ahuge parameter space and thus is b eyond our currenttechnical abilityThus welimit

ourselves to explore the cases that we b elieve might b e the most instructive to understand

the primordial evolution of our Solar System

We start in Section with a simple analytical mo del that stresses the exp onential

character of the migration pro cess This will b e useful to interpret the results of the

numerical simulations presented in the next sections In Section we discuss migration

in largemass disks In Section we consider the case of lowmass disks and discuss how

the resolution of the simulation numb er of massive planetesimals used to mo del the disk

aects the simulation results Section addresses the issue of the depletion of the primordial

mass of the Kuip er b elt and its eects on Neptunes migration We rule out the p ossibility

that the b elt was depleted by some dynamical mechanism that moved most Kuip er b elt

b o dies to Neptunecrossing orbit We also argue that the Kuip er b elt could not have lost

its mass by collisional grinding after that the planet reached AU We therefore conclude

that Neptune stopp ed at its current lo cation b ecause it encountered an eective edge of

the massive protoplanetary disk Then in section we discuss in detail Neptunes

migration in truncated disks and deduce the range of plausible disk masses and sizes that

are compatible with the current p osition of Neptune Wealsoinvestigate the implications

for the Thommes et al scenario Section discusses what would have b een the

dynamical evolution of planetary embryos if they existed in the disk b eyond Neptunes

primordial p osition Our conclusions will b e recollected in section The app endix rep orts

the details on the integration metho ds that wehave used

A simple analytic insight in the migration pro cess

In this section we develop a backoftheenvelop e analytic theory for migration in

planetesimal disks Our goal is to presentanintuitive easy to understand toy mo del

intended to b e a guide for interpreting the range of b ehaviors observed in our numerical

simulations We refer the reader to Ida et al b for a more develop ed analytic theory

The consequences of the encounter b etween two b o dies in orbit around the can b e

eectively computed in most of the cases using an impulse approximation Opik In

this approximation the eect of the encounter is an instantaneous rotation of the orbital

velo cityvectors of the two b o dies computed using the well known Rutherford twob o dy

scattering formul Using this approach it is easy to compute Valsecchi and Manara

that on average that is averaged on all impact parameters and relative orientations

the planetesimals that cause an outward migration to a planet on a circular orbit are those

q

whose z comp onent of the angular momentum H a e cosi is larger than that of

the planet H The opp osite is true for the planetesimals with HH In these formul

p p

a e and i are the semima jor axis eccentricity and inclination of the planetesimal This

is due to the fact that when encountering the planet the particles with HH haveon

p

average a velo city comp onent in the direction tangential to the planets motion that is

larger than the orbital velo city of the planet Thus they accelerate the planet The opp osite

is true of the particles with HH This result applies also if the planet has a mo derate

p

eccentricity

The direction of migration of the planet is therefore determined by the relative

p opulations of planetcrossing planetesimals with HH and HHThismaybe

p p

dierent from case to case Some general trends however can b e outlined For instance in

the case of two planets the inner planet partially depletes the p opulation of planetesimals

with H smaller than the angular momentum of the outer planet so that the latter tends

to migrate outwards Similarly the outer planet partially depletes the p opulation of

planetesimals with H larger than the angular momentum of the inner planet so that the

latter tends to migrate inwards

In our Solar System migration should have had a general trend with Jupiter moving

inward and Neptune moving outward Figure shows an example of semima jor axis vs

eccentricity distribution of the planets and the planetesimals during the migration pro cess

For each planet the solid curves show the b oundaries of the planetcrossing regions and

the dotted curves corresp ond to the condition H H for i Theoverlapping of the

p

Neptunecrossing and Uranuscrossing regions implies a gradual depletion of the ob jects

with HH relative to those with HH The consequence of this imbalance

Neptune Neptune

induces the outward migration of Neptune The same reasoning can b e applied to the other

planets except for Jupiter Jupiter is so massive that it rapidly ejects to the interstellar

space most of the planetesimals that come close to its orbit or sends a small fraction

into the so that it must moveinwards this mechanism has b een prop osed

for the origin of the hot in extrasolar system by Murray et al Notice

however that the situation mighthave b een temp orarily dierent if the planets encountered

discontinuities in the surface density distribution of the planetesimal disk suchasgaps

edges or density clumps p ossibly caused by the migration itself which in some cases

could cause reversals in the direction of migration see x for examples

For a b etter understanding of the numerical simulations presented next we rst

develop an analytic toy mo del of the migration pro cess



During migration the fractional rate of change of the planets semima jor axis da dt

p



where da daa is prop ortional to the ratio of amount of material in orbits that

p

cross the orbit of the planet M t to the mass of the planet M a function k of the

p

distribution of those orbits for example the distribution of H describ ed ab ove and the

timescale b etween close encounters b etween small particles and the planet which in turn is

prop ortional to P where P is the orbital p erio d of the planet P a Therefore

p

da k M t

p

p

dt M a

p p

Note that most of the dynamics of the system is hidden in the parameter k The evolution

of M t can b e approximated by the equation

M tM t a ja j a

p p p

where the rst term in the rhs represents the decay of the planetesimal p opulation

due to the planetesimals nite dynamical lifetime and the second term stands for the

planetesimals that b ecause of the change in the planets p osition enter for the rst time the

region where they can b e scattered by the planet In a is the surface density of the

p

virgin ie not yet scattered planetesimal disk at helio centric distance a Substituting

p

into weget

p



a a M M t M t jk j

p p p

p



Lets assume for simplicity that the term jk j a a M do es not signicantly

p p p

change with time an approximation valid for small migrations but which evidently

lo oses its validity when the migration covers a macroscopic range Then b ecomes

an exp onential equation with solution M tM exp t If is negative then

M t decays exp onentially to and the planet from eq stops migrating In this

case the loss of planetesimals due to their nite dynamical lifetime is not comp ensated

by the acquisition of new planetesimals in the scattering region b ecause the migration

sp eed is to o slow Therefore the planet runs out of fuel We call this migration mo de

damped migration Converselyif is p ositive M t grows exp onentially and the planet

exp onentially accelerates Ida et al b We call this migration mo de forcedmigration

In this case the acquisition of new planetesimals due to the migration exceeds the losses

due to the nite dynamical lifetime and the migration is selfsustained

The description of migration through eqs and is necessarily crude In reality

the migration can pass from damp ed to a forced mo de and viceversa as the surface density

the decay time and the relative planetesimal distribution k change with the planets

lo cation a and planetary migration ratea The changes of and k along the migration

p p

cannot b e estimated a priori Also ifa b ecomes large enough disk particles can cross

p

the planets zone of inuence in a timescale short compared to and thus leave the zone

from the opp osite edge This eect intro duces a new negative term in the rhs of that

also cannot b e evaluated a priori b ecause it dep ends on the details on the interactions

between the particles and the planet Planetesimals can also b e trapp ed in mean motion

resonances which eectively increases the inertial mass of the planet thus causing a

decrease of k On the other hand planetesimals exterior to the planets zone of inuence

may b e dynamically excited to planet crossing orbits by resonances This increases the

delivery of fresh mass to the planet compared to the term a ja j a in Moreover

p p p

the relative orbital distribution of the planetesimals in the planet crossing zone maychange

during the evolution causing a change in k Finally the width of the planet crossing region

changes linearly with a also mo difying the rhs of

p

Therefore this system of equations cannot b e eectively used to simulate the migration

pro cess Indeed wearenotaware of any theory on planetary migration in planetesimal

disks that can substitute for numerical simulations However our toy mo del shows the

intrinsic exp onential nature of the migration pro cess and therefore will b e very useful to

interpret the results of the direct simulations of the migration pro cess that will b e presented

in the next sections

Migration in large mass disks

We presenttwo simulations of migration due to the presence of a massive

disk In b oth cases the initial semima jor axes of Jupiter Saturn Uranus and Neptune are

AU AU AU and AU resp ectively and their initial eccentricities are

In the rst case the disk extends from AUtoAU Following the idea that

the disk should b e strongly depleted in the planetary region due to the of the

planets we assume a disk mass of M inside AU and of M outside AU

 



with a surface density decaying as r in each subregion The disk is simulated using

equalmass particles In the second case the disk extends up to AU and contains



M outside AU with a surface densitydecaying as r while the mass inside AU



is again equal to M It is simulated using equal mass particles These surface



density proles are those typically assumed for the Hayashi

Hahn and Malhotra In these and all other simulations presented in this pap er the

disk particles resp onded to the planets but not to each other We are aware that this

approximation imp osed by the necessitytokeep the computing time within reasonable

limits intro duces some artifacts The frequencies of secular precession of the disk particle

orbits are not correct which misplaces the lo cation of the secular resonances with the

planets Also collective eects are not repro duced which suppresses a torque that would

subtract angular momentum from the planets Goldreich and Tremaine We think

that these artifacts do not havesevere consequences Unlike mean motion resonances see

b elow secular resonances do not play an imp ortant role in planetary migration except for

p ossibly providing additional distant sources of planetesimals to the planet crossing region

Collective eects b ecome unimp ortant as so on as the planetesimal disk b ecomes mo derately

excited Ward Hahn a b

Figure shows the migration of the planets for the two planetesimal disks dened

ab ove For the reasons explained in section Saturn Uranus and Neptune on average

migrate outward while Jupiter migrates inward Neptune undergo es forced migration

b ecause these disks are massive Consequently the planet eventually migrates to the edge of

the disk and in fact go es slightly b eyond it and it can come to a rest only when the disk

has b een mostly depleted which o ccurs at ab out years The fact that Neptunes

real p osition is at AUobviously rules out the idea that a similar extended massivedisk

was present in our Solar System at early times

However other planetary systems mighthave had in the past disks of comparable mass

and even larger radial extent and therefore migration mayhave brought planets to large

distances from their parent stars Such planets have b een p ostulated to explain features

observed in the disks around Pictoris Wahha j et al and Eridani Ozernoy

et al If the observational evidence for their existence is substantiated we b elieve

that forced migration in a massive planetesimal disk mightbea valid explanation of their

origin However the migration pro cess as describ ed here requires that the planet is much

less massive than the disk and is incapable of ejecting most planetesimals to hyp erb olic

orbit thus it only applies to planets like Uranus and Neptune rather then Jupiter and

Saturn

Another imp ortant result shown in Fig is that Neptunes migration is not monotonic

In the case of the highmass disk top panel Neptune reaches AU in less than My

and then comes back to within AU almost equally fast Similar episo des of acceleration

and return although less pronounced are also visible in the lowmass disk case at and

My This b ehavior is due to a a selfsustaining migration pro cess describ ed in Ida et

al b that wecallrunaway migration

Under normal conditions the planetesimals in Neptunecrossing orbit that have

HH are depleted byUranus Therefore there is never a large number of

Neptune

HH particles that could drive Neptune inward so that Neptunes outer migration

Neptune

is irreversible But when Neptune migrates fast or gets far from Uranus it can get in a

mo de where it do es not scatter ob jects into Uranuscrossing orbits These ob jects are

therefore left b ehind in an excited disk as Neptune moves forward compare Fig to

Fig However planets outward migration continues as long as the planetesimals in the

Neptunecrossing zone with HH dominate over those with HH Thetwo

Neptune Neptune

p opulations do not rapidly equilibrate b ecause of the migration itself whichcontinuously

supplies new planetesimals with HH to the Neptune crossing region Ida et al

Neptune

b However when Neptune reaches the edge of the disk and thus the number of

ob jects with HH drops Fig B the remaining ob jects interior to Neptune

Neptune

pull the planet inward Thus Neptune reverses direction and starts a runaway inward

migration The same argument describ ed ab ove applies so that this migration ends only

when the region of the disk partially depleted by Uranus is encountered again

To demonstrate that Uranus has no role in the runaway migration of Neptune or in its

reversal we p erform the following exp eriment At t Mywe remove all planets except

Neptune and extend the disks outer edge to AUfollowing the original surface density

distribution Then we continue the integration with only Neptune and the planetesimal

disk Fig compares Neptunes evolution in the new simulation to its previous one As

exp ected the twoevolutions showessentially the same b ehavior b efore MyHowever

in the new integration Neptune continues its migration until it reaches the new edge In a

third integration we extend the disk up to AU Again Neptune continues its migration

up to the new edge This series of simulations show that once started runaway migration

pro ceeds without the help of the other planets and that hitting the disks edge causes the

migration to b e reversed

However Neptunes b ehavior suddenly changes when we extend the disk further In

the integrations shown by the top curves in the gure we extend the disk up to AU

Neptune migrates much further than in the previous cases but surprisingly it but do es not

reach the new edge In all integrations that wehave made in total shown in Fig

Neptune reverses its migration at AU We note that Neptune is not more likely

to eject planetesimals from the Solar System when it is further from the Sun b ecause for a

particle encountering the planet the probability to b e ejected to hyp erb olic orbit dep ends

exclusively on the Tisserand parameter and the latter is indep endent of the semi ma jor axis

units Therefore the reversion of Neptunes migration requires a more subtle explanation

In order to understand the reversal in Neptunes migration we rst must understand

whytheentire migration pro cess seems to pro ceed with a quasip erio dic alternation of

accelerations and slowdowns or even stops see Fig Referring back to Equation

webelieve that this is due to combination of two eects as the migration pro ceeds a slow

decrease in k and an increase in M Aswe describ ed in x a planet migrates outward if

more of the disk particles encountering it have HH than have HH which implies

p p

that k or it migrates inward if the opp osite is true and k Fig shows the

density of Neptune crossing particles at three timesteps that corresp ond to the b eginning

the middle and the end of a fast migration episo de The gure clearly shows that initially

particles cluster in the HH region but progressively movetowards the HH region

p p

as the migration pro ceeds This shift in the distribution of the particles happ ens b ecause

when the planet migrates suciently fast the timescale for encountering the planet b ecomes

comparable to or longer than that for passing through the planetcrossing region due to

the migration of the planet itself So in a co ordinate system that moves with the planet

most particles simply drift through a signicant fraction of this region b efore suering an

encounter Thus when the planet sees the particle for the rst time its H is signicantly

smaller than the value that characterized the particle when it rst b ecame planetcrosser

and can even b e smaller than H The net result is that k slowly decreases with time

p

At the same time we nd that the amount of mass in the planetcrossing region M

increases with time The value of M changes as a a where is the width of the

p

planetcrossing region rememb er that in this case there is no dynamical depletion of the

planetcrossing particles Because a and in this problem the surface densityofthe

p

disk is prop ortional to the inverse helio centric distance M a

p

p

So M is increasing while k is decreasing Sincea k M a ifk decreases with time

p p

p

a the magnitude ofa actually increases with time This happ ens more slowly than

p p

until k b ecomes equal to at whichpoint migration abruptly stops From Fig it seems

that this phenomenon b ecomes somewhat more pronounced as the planet gets further from

the Sun In fact the timescale for encounters grows as a ie faster than the width

p

of the planetcrossing region prop ortional to a so that it b ecomes easier to shift the

p

distribution of the planet crossing particles as it happ ens in Fig and reduce k

Now that wehave understo o d why Neptunes migration rep eatedly stops we can now

discuss the migration reversal seen at a AU Atevery stopping episo de Neptune

nds itself in an unstable situation If the planet stays at rest for a long enough time

the excitation of the outer cold disk due to secular and resonant p erturbations eventually

brings new material into the planetcrossing region with HH so that the planet starts

p

migrating outward again This happ ens every time that a new acceleration of the migration

is pro duced in Fig But if the excited disk interior to Neptune which is made up of

HH particles slightly overp owers the particles from the outer disk the planet starts

p

to migrate inward This is enough to trigger a runawayinward migration b ecause the

planet nds a massive excited disk inside its orbit ready to rell the planetcrossing region

while the cold outer disk is left b ehind Wehave not b een able to identify any dynamical

reason for why in some cases Neptune sometimes reverses direction Thus we b elieve

it is a matter of chance If so this whole eect may b e the result of the fact that our

simulations contain a relatively small numb er of massive b o dies compared to the real early

Solar System Perhaps an ideal system with a nearly innite numb er of planetesimals with

innitesimal mass would b ehave dierentlyWe will address this issue again in future work

The p ossibility that Neptune mayhave had a p erio d of inward migration if it were

emb edded in a massive disk suggests a new mechanism for the excitation of the classical

Kuip er b elt Neptune mighthave crossed the b elt and then returned to AU dynamically

exciting the Kuip er b elt in its wake Unfortunately this scenario cannot work In the

simulations that we p erformed of this pro cess see Fig after its inward migration

Neptune always reverses its migration once again and eventually reaches the original outer

edge of the disk Therefore Neptune could not have stopp ed at AU but would have

reached a nal p osition beyond the Kuip er b elt Moreover if Neptune had ended its travels

immediately after a p erio d of inward migration the p opulation of the would

probably not havesurvived In fact during a p erio d of inward migration the particles

in exterior mean motion resonances exp erience a decrease in eccentricityuntil they are

eventually released from the resonance

Migration in low mass disks

Wenowinvestigate the migration pro cess for disk masses smaller than M In all



our simulations the initial lo cations of Jupiter Saturn Uranus and Neptune are

and AU resp ectively The disk extends from AUtoAU and has a surface



densityvariation as r It is simulated using eectively equal mass particles

although we employ a computational trick to decrease the amount of CPU time the runs

require see App endix

Figure shows Neptunes migration for disk masses equal to and M The



rst two cases are examples of damp ed migration see sect Neptunes outward motion

rapidly slows down and the planet reaches after y a quasiasymptotic distance that

is well within the outer edge of the disk The part of the disk outside a few AUs b eyond

Neptune preserves its original mass while the part within this distance is completely

depleted These results are qualitatively equivalent to those obtained by Hahn and Malhotra

for disks of and M In our case Neptune stops at and AU



resp ectively but it started more than AU closer to the Sun than in Hahn and Malhotras

simulations

When we increase the disk mass to M we observea change of b ehavior Neptunes



outward migration rst slows down then stays approximately linear b etween and

My and nally accelerates towards the disks edge This evolution suggests that the

surface density of this disk approximately corresp onds to the critical one that separates

dump ed migration from forced migration see sect We b elieve that the acceleration

of Neptunes migration seen after My is due to the following In these simulations

as Neptune migrates outward some of the particles in the external disk b ecome trapp ed

in Neptunes mean motion resonances Malhotra These particles are dragged

outward with Neptunes migration but their eccentricities are pump ed up during this

pro cess The resonant particles eect migration b ecause they eectively increase Neptunes

inertial mass If the migration rate is slow enough so that the changes that the particles

see are adiabatic they stay in the resonance until they reach some critical eccentricityat

which p oint they are released During adiabatic migration the numb er of particles in the

resonances is roughly constant as long as the resonance is still in the disk

In this run Neptune accelerates as its mean motion resonance moves out of the

disk This is most likely due to the fact that the numb er of ob jects in the resonance drops

b ecause ob jects in the resonance are leaving as their eccentricity grows but new particles

are not b eing captured As the resonance is depleted the eective inertial mass of Neptune

decreases which in turn we b elieve causes Neptune to accelerate We also b elieve that

this acceleration is then amplied by the fact that Neptune starts moving quickly enough

so that its migration b ecomes nonadiabatic So the resonant capture eciency drops for

all the resonances and thus the total numb er of ob jects in resonances decreases This b elief

is supp orted by the fact that we observed a decrease in the numb er of ob jects in Neptunes

resonance that starts so on after the drop in the We should p oint out that this last

eect could have happ ened even if the did not hit the end of the disk

The fact that our result for a M disk is qualitatively dierent from that of



Hahn and Malhotra should not b e a concern This mass is close to the threshold for the

transition from damp ed to forced migration As is usually the case when a physical system

is studied close to a threshold small quantitative dierences in the simulations can lead

to qualitatively dierent results The ma jor dierence b etween our simulations and Hahn

and Malhotas is the dierentnumb er of particles used to represent the planetesimal disk

particle sin our case in Hahn and Malhotras case To illustrate how the

numb er of particles matters wehave redone the simulations using only particles

in the disks as in Hahn and Malhotra The results are shown in Fig Two

simulations are done for each disks mass with dierent but equivalent initial conditions

for the disks particles In all cases we notice a large variability of the results In particular

for the disks with and M in one simulation Neptune stops its migration inside



AU as in Fig but in the other case it migrates towards the edge of the disk In the

M case b oth simulations lead Neptune to AU but the evolution paths are quite



dierent Unfortunatelywe cannot prove the same problem do es not exists in our

particle runs illustrated in Fig However the fact that in Fig the nal p osition of

Neptune shows a regular progression with the disk mass makes us think that sto chasticity

of Neptunes migration should b e much less prominent

We can understand this sto chastic b ehavior of Neptunes migration in low resolution

disks on the basis of the analytic insight of sect If the disks surface density is close to

the critical value that separates damp ed migration from forced migration the evolution

b ecomes very sensitive to the density uctuations If the disk is mo deled by a small number

of massive particles the density uctuations are more pronounced and sto chastic while

if the disk is mo deled with a larger numb er of smaller particles the density uctuations

are more eectively averaged out in space and time In particular the encounters of

Neptune with planetesimals with to o large a mass inhibit the resonance trapping pro cess

as p ointed out by Hahn and Malhotra thus changing the orbital distribution of the

planetesimals that drive Neptunes migration Also in the case of a smaller numb er of more

massive planetesimals Neptunes eccentricity is larger on average roughly compared

to for the particles runs whichchanges the dynamics in three ways

Neptunes crossing region is larger so it is easier for particles to b ecome Neptunecrossing

Neptunes resonances b ecome stronger so that the external disk is more easily excited

and Neptune can more easily change a particles Tisserand parameter so that dynamical

evolution o ccurs more quickly

Fig also shows two examples of migration obtained with disks of M but surface



 

densities decaying as r and r Here particles are used to mo del the disk in

b oth cases A steep er surface density implies more mass in the inner part of the disk and

less mass in the outer part Therefore the migration starts faster than in the case of the



r surface density but when the planet reaches the outer part of the disk the lo cally low

surface density puts it in a damp ed migration mo de These two eects a faster initial

migration and a slower nal migration combine in sucha way that the resulting total

migration of the planet b ecomes smaller as the surface density distribution of the disk gets

steep er for equal total disk masses In fact the total angular momentum of the disk

decreases with steep er surface density proles Thusforagiven total disk mass there

must b e a steep enough surface density distribution that makes Neptune stop at AU

For a disk of M between and AU simulations show that the required density





prole is approximately r The problem with this steep prole is that if true the mass

originally in the AU region would b e M an order of magnitude smaller than



that required to grow in situ the large Kuip er b elt ob jects that are observed Stern and

Colwell a Kenyon and Luu

Neptunes p osition and the mass depletion of the Kuip er b elt

If we assume that the primordial Kuip er b elt must havecontained at least M



between AU in order to growobjectscurrently observed the results of the previous

section suggest a scenario simular to the one prop osed by Hahn and Malhotra

the surface density of the disk was shallow exp onent the disk contained M



of material b etween and AU and Neptune started at AU The initial lo cation

of Neptune whichis AU further outward than in the simulations of Fig was chosen

so that it would stop migrating at AU after a damp ed migration Notice that this

scenario is in conict with the conclusions of Levison and Stewart in which Uranus

and Neptune had to form signicantly closer than AU from the Sun Our understanding

of planetary formation is not yet secure enough to condently rule out that that Neptune

formed b eyond AU However the scenario sketched ab ove has another problem If

Neptune stopp ed at AU b ecause its migration was damp ed then the disk b eyond

AUwould have preserved its original large mass The current mass of the region

inferred from the observations Jewitt et al Chiang and Brown Trujillo et al

Gladman et al is now less than M Could the Kuip er b elt lo ose most of



its mass without substantially mo difying Neptunes nal lo cation

Two general mechanisms have b een prop osed for the mass depletion of the Kuip er b elt

the dynamical excitation of most b o dies to the Neptunecrossing orbits after which they

were ejected and the collisional comminution of most of the mass of the Kuip er b elt into

dust

The dynamical depletion mechanism was rst prop osed by Morbidelli and Valsecchi

and Petit et al In their scenario a planetary embryo with mass comparable

to that of or of the Earth was scattered by Neptune onto an elliptic orbit that crossed

the Kuip er b elt for y The rep eated passage of the embryo through the Kuip er b elt

excited the eccentricities of the Kuip er b elt b o dies The vast ma jority of these b ecame

Neptune crossers and were subsequently dynamically removed In the Petit et al

integrations that studied this scenario however the Kuip er b elt b o dies were treated as

massless test particles and therefore their ejection did not alter the p osition of Neptune

Thus wehave redone a Petit et allike simulation in the framework of a more

selfconsistent mo del where Jupiter Saturn Uranus and Neptune are initially at

and AU resp ectively an Earthmass embryo is on a circular orbit at AU

and the disk has M between and AU with a surface density prole decaying as





r The result is shown in Fig The discussion of the dynamical evolution of the embryo

is p ostp oned to sect Here we fo cus on the result that despite the low mass of the disk

only M between and AU at the end of the integration Neptune has migrated



well b eyond AU Indeed in this simulation of the disk particles are still in the

system at the end so we do not get enough dynamical depletion In order to determine

howmuch Neptune would migrate if we removed all of the particles we continued this

simulation and placed another Earthmass embryos outside of Neptune at AU In this

integration Neptune reaches AU after Gy while of the disk particles are still in

the system

In the ab ove simulations Neptune migrates further than it normally would without the

embryos b ecause the embryos dynamically excite the disk exterior to Neptune and feed this

extra mass to it Thus Neptune interacts not only with the p ortion of the disk in its lo cal

neighb orho o d but with the entire mass of the disk at the same time Therefore a M



disk which in absence of the embryowould allow Neptune to migrate only few AUs in a

negative feedback mo de brings Neptune well b eyond its current p osition Wehave done

other numerical exp eriments with a set up equivalent to that of the simulation rep orted

in Fig but dierent disk masses If one requires that Neptune stopp ed at AU the

disk in the AU range should contain only M of planetesimals the exact values



dep ending on the initial lo cation of the planet This disk mass and density prole however

would imply that only M of material originally existed in the Kuip er b elt b etween



and AU which is far less than the mass required M by the mo dels of accretion



of Kuip er b elt b o dies Stern and Colwell a Kenyon and Luu Therefore we

b elieve that we can rule out the Petit et al scenario for dynamical depletion of the Kuip er

b elt

Although wehave only studied the Petit et al scenario we b elieve that our results

can b e applied to al l dynamical depletion scenarios This is b ecause Neptunes resp onse

to the mass once it leaves the Kuip er b elt is unlikely to dep end on whether Kuip er b elt

ob jects are excited to Neptunecrossing orbits by a planetary embryoorby some other

mechanism such as the primordial secular resonance sweeping Nagasawa and Ida

Our results simply imply that Neptune never encountered the missing planetesimals of the

Kuiper beltThus the only typ e of dynamical deletion mechanism that could work is one in

which the Kuip er b elt ob jects were kicked directly to hyp erb olic or Jupitercrossing orbit

and consequently were eliminated without interacting with Neptune Only the passage of

a star through the Kuip er b elt is capable in principle of such an extreme excitation Ida

et al Kobayashi and Ida However a simple mo del in which Neptune is at

already at AU the Kuip er b elt ob jects are fully formed and a passing star causes the

mass depletion and the dynamical excitation of the Kuip er b elt can probably b e ruled out

b ecause it is unlikely to pro duce a Kuip er b elt that is consistent with other observational

constraints Brown Morbidelli

An alternativemechanism for removing the mass from the early Kuip er b elt is the

col lisional grinding scenario prop osed by Stern and Colwell b and Davis and Farinella

A massive Kuip er b elt with large eccentricities and inclinations would undergo

avery intense collisional activity Consequently most of the mass originally incorp orated in

b o dies smaller than km in size could b e comminuted into dust and then evacuated

by radiation pressure and PoyntingRob ertson drag This would cause a substantial mass

depletion provided that the b o dies larger than km which cannot b e eciently destroyed

by collisions initially represented only a small fraction of the total mass

The collisional grinding of the Kuip er b elt would not have b een without consequences

for Neptunes migration The calculations of collisional grinding thus far p erformed have

b een sophisticated particleinab oxsimulations that handle the evolving sizedistribution

of the Kuip er b elt in a narrowannulus ab out the Sun by p opulating an array of mass bins

They then followhow the numb er of ob jects in eachbinchanges due to collisions The

smallest bin in the array is called the dust bin and any mass put in this bin is subsequently

ignored But what really happ ens to this dust This dep ends on the size shap e and

comp osition of the particles whichmaybeallcharacterized by a single parameter the

ratio of the strength of radiation pressure to the strength of gravity see Gustafson

for a review Particles with are blown directly from the Solar System and thus do

not interact with any other ob ject However dust particles spiral inward due to

PoyntingRob ertson PR drag For Kuip er b elt dust this means it will encounter Neptune

see Liou Zo ok Dermott Liou Zo ok MoroMart n Malhotra

If the dust created during the collisional grinding of the Kuip er b elt has a sizedistribution

b

similar to that of the zo diacal cloud N R R where R is particle radius and b

Grogan Dermott Durda then m uch more mass will b e found in the large particles

than in the small particles Thus most of the dust by mass generated will spiral inward

Indeed if we assume a particle with R mhas Gustafson and that only

particles with R m resp ond to radiative forces this is a very conservative upp er

limit but cho osing a larger one strengthens our case then of the mass in dust

will spiral toward the Sun and encounter Neptune If we assume that the particles followa

collisional cascade size distribution b this fraction is In either case the role

of blowo is negligible and almost all the dust will spiral inward

So the natural question is Howwould Neptune resp ond to tens of Earthmasses

of dust sailing by during the collisional grinding phase of the disk Would it migrate

outward as if it were interacting with larger particles If so collisional grinding could not

b e resp onsible for the mass depletion b ecause Neptune would have migrated to o far as our

earlier simulations haveshown

However the resp onse of Neptune is not obvious b ecause in part as the dust particles

migrate inward they get temp orarily trapp ed in mean motion resonances MMRs with

Neptune Liou Zo ok and MoroMart n Malhotra In an MMR the

inward drift is halted b ecause the energy loss due the radiation eects is balanced bythe

resonantinteraction with the planet The net result is that energy is extracted from the planets orbit so that the particles in the resonance try to drag Neptune in with them This

could b e signicantgiven that total mass of the dust generated during the collisional phase

is comparable with the mass of Neptune However at the same time these particles would

have b een slowly leaking out of the resonances and subsequently encountering Neptune

Like their larger brethren this dust would have tried to push Neptune outward

Thus it is not clear exactly how Neptune will resp ond to the dust We are currently

studying this issue but it is very unlikely that the two eects cancel out and Neptunes

semi ma jor axis remains approximately unchanged

Away around this problem is that the dust is collisionally comminutedvery quickly

down to a size at whichitisblown awayby radiation pressure In this case it would not

sp end a signicantly long time in Neptune crossing orbit or in a mean motion resonance

with the planet Kenyon private communication However even in this case collisional

grinding would indirectly aect Neptunes orbit due to the evolution of the secular

resonance during the mass depletion phase As the disk grinds down the secular

resonance most likely will b egin to feed material to Neptune which will then migrate The

secular resonance o ccurs when the p eriapse precession of a Kuip er b elt ob ject matches

that of Neptune This resonance is very p owerful and any ob ject in it suers an increase in

eccentricityuntil it can b e removed from the Kuip er b elt by a close encounter with Neptune

Holman and Wisdom Duncan et al It is currentlyatAU However the

presence of a massive disk or annulus aects the orbital precession frequencies of b oth

Neptune and the disk particles As the disks mass grinds down the precession frequencies

change Consequently secular resonances move p otentially sweeping through the disk and

exciting ob jects to Neptunecrossing orbits

For example assuming that when Neptune reaches AU the disk has already b een

depleted inside AU but is still massive in the AU region wehave computed the

lo cation of the secular resonance as a function of the remaining disks mass Fig

using the following semianalytic mo del The lo cation of the secular resonance is simply

the lo cation where precession rate of the disk particles is the same as that of the dominate

frequency of Neptune For the disk particles on lowinclination nearly circular orbits the

precession frequency can b e determined using the epicyclic approximation cf Section of

Binney T remaine In particular it is very nearly equal to the dierence b etween

the radial oscillation frequency and the angular orbital frequency The latter quantities

can b e computed from the rst and second radial derivatives of the total gravitational

p otential pro duced by the massive Kuip er b elt and the giant planets For our mo del we

approximated the orbits of the planets by individual rings and the density distribution of

the disk by a series of rings spread b etween and AU We determined the rst

and second derivatives of the p otential of the rings numericallyWe calculate the dominate

presession frequency of Neptune by developing a full secular theory of a system containing

the four giant planets and seven ctitious planets with masses and semima jor axes chosen

so that they approximate the disk

As the gure shows initially the resonance is at the inner edge of the disk However

as the disks mass decreases b elow M the secular resonance starts sweeping through



the disk The resonance will b egin to excite disk particles to Neptunecrossing orbits

Because the disk still contains a lot of mass ab out M of material assuming that



This is only true if the disk is to o excited to supp ort a form of a spiral densitywaveknown

as an apsidal wave which can b e generated by the Ward Hahn a b W aves

suchasthiswould not allow the eccentricity of the individual particles to grow signicantly



However waves will only b e generated in disks with e i Hahn which

is much smaller than that required for collisional grinding to b e p owerfull enough to deplete



of the mass e i Stern and Colwell b So that in a collisional

grinding regime the collective resp onse of the disk can b e ignored

the secular resonance is AUwide would start to haveencounters with Neptune forcing

the planet to migrate outward This in turn would move the resonance to a fresh lo cation

in the disk further from the Sun whichinturnwould feed more particles to Neptune In

short our guess is that an instabilitywould b e triggered whichwould feed the remaining

disk particles to Neptune and thusaswe showed for the dynamical depletion mechanisms

ab ove Neptune would b e driven b eyond AU

In conclusion we tend to exclude the p ossibility that collisional grinding depleted

the mass of the Kuip er b elt after that Neptune ended its damp ed migration at AU

Of course all the arguments discussed ab ove can b e circumvented if collisional grinding

o ccurred during Neptunes migration in particular when Neptune was still far from AU

We cannot exclude this p ossibility from the p oint of view of planetary migration However

we remind the reader that there are several other arguments against collisional grinding in

general i the orbital excitation of the cold classical Kuip er b elt do es not seem to b e large

enough compared to that required in the mo del by Stern and Colwell ii most of

the wide binaries in the cold p opulation would not have survived the collisional grinding

phase Petit and Mousis iii if all conditions for the collisional grinding were met in

the Kuip er b elt it is likely that they were met also in the AU region thus preventing

the formation of a massive enough Oort cloud of Stern and Weissman

Charnoz private communication

Therefore we b elieve that the current lo cation of Neptune and the mass deciency of

the Kuip er b elt imply that the protoplanetary disk p ossessed an edge at ab out AU

There are at least vemechanisms that could have truncated the disk at small helio centric

distance prior to planetary accretion A passing star tidally strips the Kuip er b elt after

the observed Kuip er b elt ob jects formed Ida et al Kobayashi Ida An

edge formed prior to planetesimal formation due to aero dynamic drag Youdin Sh u

An edge formed during planet accretion due to sizedep endent radial migration caused

by gas drag Weidenschilling Nearbyearlytyp e stars photoevap orated the outer

regions of the solar nebula b efore planetesimals could form Hollenbach Adams

Magnetohydro dynamic instabilities in the outer regions of the disk prevented the formation

of planetesimals in these regions Stone et al We stress that the truncation of the

disk at AUis not in contradiction with the existence of the Kuip er b elt b eyond AU

In fact the entire Kuip er b elt could have b een pushed out from within AU during

Neptunes migration following the mechanisms discussed by Malhotra Gomes

and Levison and Morbidelli

Migration in a truncated disk

The presence of an edge in a massive disk do es not imply that a migrating planet will

stop at the edge Indeed since angular momentum must b e conserved during the migration

pro cess the nal lo cation of the planets dep ends more on the total angular momentum

in the disk than on the lo cation of the edge To illustrate this Fig shows Neptunes

migration in disks that are initially spread b etween and AU but with masses



varying from to M all with surface density prole prop ortional to r The



initial lo cation of Neptune was at AU The disk with M has a sub critical surface



density Neptune exhibits a dump ed migration and stalls well within the disk Therefore

a massiveannulus is preserved b etween a few AUbeyond the planets lo cation and the

original outer edge of the disk The M disk also app ears to b e initially sub critical and



In order to compare the results of the new integrations with those of Fig the reader

should remind that for a given total mass the surface density is times higher in the new

narrower disks

after a fast start the migration starts to slow However a little b efore y there is

a brief burst of migration that o ccurs when Neptunes mean motion resonance leaves

the disk We b elieve that this burst of migration is due to particles leaving the resonance as

explain ab ove By y there are very few particles left in the disk although Neptune

only reached AU Interestingly roughly of the particles can b e found in orbits that

are decoupled from Neptune b eyond AU Most of these are in mean motion resonances

but some were delivered to this region by the Gomes mechanism

The disk with M has a surface density close to the critical value The planet



migrates outwards in an almost linear wayfor MyWhenitreaches AU the

unstable region of the disk which extends up to a distance of ab out th of the planets

semima jor axis Duncan et al reaches the edge of the disk The planet starts to feel

the disk truncation and its migration is rapidly damp ed The nal lo cation is AU inside

the original disk edge but the entire region b eyond the planet has b een depleted

More massivediskshave sup ercritical densities In the case of M the planet stops



almost exactly at the disks edge while in the other cases it go es several AUs b eyond it We

stress that at the end of all our simulations except the one with M the original disk



was destroyed despite the fact that the Neptunes nal lo cation varied byAU Therefore

for an observer lo oking at the nal planetary conguration there would b e no waytotell

where was the original disks edge and whichwas the original mass of the disk Given a

nal p osition of Neptune there is a one parameter family of solutions for the disks size

and mass that is compatible with the result assuming a given initial p osition of the planet

the situation is even more complicated if also the initial p osition is considered as a free

parameter This is precisely the situation that we are currently facing when we lo ok at our

Solar System Among the family of p ossible solutions for the disks parameters that are compatible

with Neptunes lo cation at AUwe tend to prefer a mass density close to the critical

value of M AU and an outer edge close to AU This is due to the fact that the



smaller the disk the harder it is to push the observed Kuip er b elt ob jects out to their

current lo cations by the mechanisms of Malhotra and particularly by Levison

and Morbidelli

Anarrow low mass disk has several implications concerning events in the inner Solar

System Levison et al prop osed that a late formation or a late outward migration

of Uranus and Neptune triggered the socalled of the Mo on

To explain the delivery to the Mo on of g of material a constraint deduced from

mo dels of impact basin formation they had to p ostulate that the disk in UranusNeptune

region contained M of planetesimals As we describ ed ab ove a disk this massiveis



inconsistent with the current lo cation of Neptune However there are large uncertainties

in the total mass of the basinforming pro jectiles of at least a factor of a few see the

discussion in Levison et al so a disk of M might still b e compatible with the



Late Heavy Bombardment but it is denitely on the lowend

And nally the constraints that wehave presented in this section on the mass and

extent of the original protoplanetary disk has implications for the Solar System formation

mo dels presented in Thommes et al These authors presented a series of mo dels

where the giant planets formed in a very compact conguration that either during or

sometime after Jupiter and Saturn accreted their gaseous envelop es suered a dynamical

instability that scattered Uranus and Neptune outward Uranus Neptune and p erhaps

the core of Saturn then had their orbits circularized by the gravitational interaction ie

with the external protoplanetary disk In their most extreme mo del

Jupiter and Saturn were fully formed and Uranus and Neptune were b etween them b efore

the instabilityWehave p erformed a series of simulations of this extreme case but where

the disk was truncated In the case were the disk contained M between and AU



we found that the probability that b oth Uranus and Neptune b ecame decoupled from

Jupiter and Saturn is smaller than we did simulations and always lost at least one

planet With a M disk we obtained one case out of three simulations where b oth



Uranus and Neptune decoupled from Jupiter and Saturn However as the results presented

earlier in this section suggest the outermost planet ended up at AU to o far from the

Sun We caution that in our simulations the disk was represented by only particles

and p erhaps the results would b e dierent if the disk was b etter resolved although we

b elieve that this is unlikelyThus the most extreme version of the Thommes et al scenario

can most likely b e ruled out

Migration of planetary embryos

Recall that in x we p erformed a study of the Petit et al scenario for the mass

depletion of the Kuip er b elt where we initially placed an Earthmass embryo on a circular

orbit outside the orbit of Neptune at AU Jupiter Saturn Uranus and Neptune

were initially at and AU resp ectively In addition we included a



M disk b etween and AU with a surface density prole decaying as r Naively



we exp ected that the Earthmass embryowould have b een caughtby Neptune during

the planets migration and subsequently b ehaved as a scattered disk b o dy Levison and

Duncan undergoing rep eated close encounters with Neptune clearing out the

Kuip er b elt and eventually b eing ejected by the giant planets That surprisinglyisnot

what happ ened

Fig shows that the embryo migrates much faster than Neptune In this simulation

the planet migrates very fast to the edge of a disk in a runaway migration that leaves the

disk b ehind almost undepleted Then the embryo reverses the migration returning to

AU and nally it turns around again reaching a nal p osition that is well b eyond the

initial edge of the disk AU The embryos nal eccentricity and inclinations are



and resp ectively

Fig shows another example of interesting embryo dynamics Here the initial

conditions were the same as the simulation shown in Fig except the the embryos mass

reduced to that of Mars The Marsmass embryo is less mobile than the Earthmass one

and thus it is more susceptible to b e trapp ed in mean motion resonances In this case its

eccentricity is rst excited to by the mean motion resonance whichwas initially

close byAt this value of the eccentricity the resonance overlaps with the stronger

resonance which captures the embryoatMy The transition to the new resonance

causes the embryos eccentricitytojumpto Once in the resonance two comp eting

eects dominate the embryos dynamics the outward migration of the resonance tends to

increase its eccentricity while the dynamical friction exerted by the disk tends to reduce it

In this case the dynamical friction slightly dominates so that the embryos eccentricityis

slowly reduced At t My the embryo nally leaves the resonance and consequently its

eccentricity falls dramatically to less than The embryo is therefore stabilized outside

Neptunes p osition

Wehave p erformed six simulations like those ab ovevarying the numb er of particles in

the disk the disk mass and the embryo mass In some runs runaway migration is imp ortant

while in others it is not In all cases Neptune stopp ed b efore reaching the embryo Thus

contrary to our exp ectations an Earthmass planetary embryo initially in a lowmassdisk

just outside Neptunes orbit would not have b een scattered by Neptune but would have

migrated ahead of Neptune until or somewhat b eyond the disks edge The embryo

would still b e present in the Solar System with a low eccentricitylow inclination orbit

whichwould have not escap ed detection in the numerous ecliptic surveys that havebeen

p erformed for the detection of Kuip er b elt b o dies

The reason that the embryos in these simulations could nd a stable p osition well

beyond Neptune is that the disk extended to AU If the disk were truncated at a smaller

helio centric distance so that Neptune could reach the outer edge the situation would

b e drastically dierent We p erformed four simulations to study this situation In these

integrations we truncated the disk at AU to insure that Neptune will stop near its

current lo cation In half of the cases Neptune eventually scatters the embryotowards the

inner Solar System where it is ejected from the Solar System by the gas giants Fig In

the remaining cases Neptune scatters the embryooutwards where the dynamical friction

exerted by the other scattered disk planetesimals damp its eccentricity The embryo

therefore survives on a loweccentricity orbit outside the p osition of Neptune and b eyond

the original edge of the disk

In all of the simulations thus far explored the system consisted of the four giant

planets an embryo and a disk Wehave not studied systems with multiple embryos

b ecause this case has already b een ruled out by Morbidelli et al These authors

studied the evolution of a system of multiple embryos initially outside Neptunes orbit

and demonstrated that even neglecting planetary migration and dynamical friction there

are always embryos left on stable orbits b eyond Neptune In these mo dels the surviving

embryos were decoupled from Neptune b ecause of dynamical encounters with other embryos

The inclusion of a transNeptunian disk should make the survival of embryos even more

likely

In conclusion the existence at early ep o chs of numerous Mars to Earthmass embryos

outside the primordial p osition of Neptune seems unlikely If one suchembryo existed

its elimination requires that the primordial massivediskwas truncated not far from the

current Neptunes p osition at AU

Conclusions and Discussion

In this pap er wehaveinvestigated in detail the phenomenon of planetary migration

due to the scattering of disk planetesimals Although our explorations cover a muchwider

parameter space than Hahn and Malhotra in the region of overlap our results are

consistent with theirs

In the case of the giant planets in our Solar System wehave found that dep ending

on the mass density of the disk Neptune could have either exp erienced damp ed migration

in which case it would havemoved only a few AU or forced migration whichwould have

driven it to the edge of the disk However we also argue that if Neptune exp erienced

damp ed migration that left a massive Kuip er b elt b eyond its nal p osition as prop osed by

Hahn and Malhtora it is dicult to remove this mass as the currentKuiperbelt

observations demand without causing Neptune to migrate to o far from the Sun

Thus we conclude that the primordial protoplanetary disk was most likely truncated

near AU b efore Neptune arrived on the scene The exact lo cation of the outer disk

edge cannot b e determined b ecause it dep ends on the disks mass density Indeed in the

exp eriments that we ran with a disk edge at AU and in which the disk was totally

depleted Neptune stopp ed migrating at distances ranging b etween and AU For

anumb er of reasons explained ab ove our preference is for a disk that extended up to

AU with a linear mass densityofabout M AU



Wehave shown that in very massive disks an isolated Neptunemass planet can

exp erience a runaway migration that can transp ort it over very large distances This pro cess

do es not require the existence of multiple planets and is selfsustaining ie it o ccurs b ecause

the migration itself feeds particles to the planet that continues to drive the migration It

also can o ccur in either direction This phenomenon may b e relevant for extrasolar planets

Wehave also concluded that Earth or Marsmass embryos could not have existed in

the planetary disk exterior to Neptune unless the disk was truncated This is additional

supp ort for our conclusion that in order to see the that we see to daythe

protoplanetary disk initially must have had an edge at AU

So far in this pap er wehave fo cused on the evolution of Neptune Unfortunatelywe

nd that wehave a signicant problem with Uranus In all simulations starting from a

compact planetary conguration where Neptune is initially inside AU Uranus always

stopp ed well b efore its current lo cation at AU This is b ecause in these cases the

planetesimals scattered by Neptune interact with Saturn almost at the same time as they

interact with Uranus so that Uranus eectively sees only a small p ortion of the total disks

mass This may indicate that Uranus and Neptune formed at AU and AU

resp ectively see Hahn and Malhotra despite of the apparent diculty of accreting

planets at large helio centric distances Levison and Stewart Thommes et al

Alternativelyitmay indicate that the migration pro cess was triggered by some instability

in the originally compact planetary system something similar to what was prop osed by

Thommes et al This will b e the sub ject of future investigations

App endix Integration metho ds

The simulations presented in this pap er have b een p erformed with two dierent

numerical integrators In b oth integration schemes the disk planetesimals interact with the

planets but not with eachother which signicantly reduces the simulation time

The simulations illustrated in sect and have used the MERCURYintegrator

Chamb ers with a timestep of one year and a relative accuracy during encounters



equal to Disk particles were discarded when they reached a helio centric distance of

AU For the simulations in sect a trick was used that decreased CPU time while

retaining the resolution of a particle disk This trick is based on the idea that at

any time in our simulations a particle resolution is required only for the fraction

of the disk that is suering encounters with Neptune A lower resolution is adequate for

the region of the disk that has yet to get close to Neptune Thus our disks are initially

made of only ob jects A particle is replaced by ten ob jects clones with one tenth

of the original b o dys mass and slightly dierent co ordinates when it rst evolves onto an

orbit with a semima jor axis less than a R where a and R are the Neptunes

N H N H

semima jor axis and the Hills radius Because in general the dynamics of b o dies in this

typ e of orbit are chaotic the orbital evolution of the clones rapidly diverged Thus the

clones exp erience totally indep endent close encounters with Neptune With this trickwe

could simulate Neptunes migration with a resolution of a particles byintegrating

at most planetesimals at anytime

The simulations rep orted in sections and have b een done using the integrator

SyMBA Duncan et al We used particles to simulate the disk in the

simulations of Neptunes migration in presence of an embryo presented in section The

same has b een done for the simulation of the dynamics of the Martianmass embryoin

section The simulations in section on Neptunes migration in presence of a disks

edge have b een done with disk particles SyMBA integrator has also b een used

to repro duce the simulations of runaway migration discussed in section leading to the

reversal of Neptunes motion

In the SYMBA integrations we discarded disk particles when they b ecame closer than

AU to the Sun This has b een done to sp eed up the simulation use a larger timestep

and avoid the problem of the accuracy of the integration of particles with small p erihelion

distance Levison and Duncan The elimination of a signicantnumb er of particles

as they enter the inner Solar System signicantly aects the migration of Jupiter which

consequently pro ceeds outwards This artifact however presumably has no impact on the

migration of Neptune on which this workisfocussed

RG thanks CNPq and FAPERJ for supp orting grants He is also indebted to R Vieira

Martins for allowing the use his p ersonal computers for the numerical integrations AM

and HFL are grateful to CNRS and NSF for supp orting HFLs sabbatical at the Nice

Observatory HFL also thanks NASAs Origins program for supp orting his involvementin

this research

REFERENCES

Bate MR Bonnell IA and Bromm V The formation of a star cluster predicting

the prop erties of stars and brown dwarfs MNRAS

BinneyJandSTremaine Galactic dynamics Princeton NJ Princeton University

Press p

Brown M The Inclination Distribution of the Kuip er Belt Astron J

Chamb ers J E A hybrid symplectic integrator that p ermits close encounters b etween

massive b o dies MNRAS

Chiang EI Brown ME Keck p encilb eam survey for faint Kuip er b elt ob jects

Astron J

Davis D R Farinella P Collisional Evolution of EdgeworthKuip er Belt Ob jects

Icarus

Davis D R Farinella P Collisional Erosion of a Massive EdgeworthKuip er Belt

Constraints on the Initial Population In Lunar Planet Science Conf

Duncan M J Levison H F Budd S M The longterm stability of orbits in the

Kuip er b elt Astron J

Duncan MJ Levison HF Lee MH A multiple time step symplectic algorithm for

integrating close encounters Astron J

Fernandez J A and Ip W H Some dynamical asp ects of the accretion of Uranus

and Neptune the exchange of orbital angular momentum with planetesimals Icarus

Gladman B Kavelaars JJ Petit JM Morbidelli A Holman MJ Loredo Y

The structure of the Kuip er b elt Size distribution and radial extent Astron J

Goldreich P and S Tremaine Disksatellite interactions Astrophysical Journal

Gomes R S The Origin of the Kuip er Belt High Inclination Population Icarus

HahnJM and Malhotra R Orbital evolution of planets emb edded in a planetesimal

disk Astron J

Hahn JM The secular evolution of the Kuip er b elt Astroph J submitted

Hayashi C Structure of the solar nebula growth and decay of magnetic elds and

eects of magnetic and turbulent viscosities on the nebula Prog Theor Phys

Suppl

Hollenbach D and Adams FC Disp ersal of Disks Around Young Stars Constraints

on Kuip er Belt Formation To app ear in Debris Disks and the Formation of Planets

eds L Caro and D Backman San Francisco ASP

Holman M J and Wisdom J Dynamical stability in the outer solar system and the

delivery of short p erio d comets Astron J

Ida S Larwo o d J Burkert A Evidence for Early Stellar Encounters in the Orbital

Distribution of EdgeworthKuip er Belt Ob jects Astroph J

Ida S Bryden G Lin DNC Tanaka H b Orbital migration of Neptune and orbital

distribution of transNeptunian ob jects Astroph J

Jewitt D Luu J Chen J The MaunaKeaCerroTotlolo MKCTKuiperbeltand

Centaur survey Astron J

Kenyon SJ Luu JX Accretion in the early outer solar system Astrophys J

Kobayashi H Ida S The Eects of a Stellar Encounter on a Planetesimal Disk

Icarus

Levison HF Duncan M J From the Kuip er Belt to JupiterFamily Comets The

Spatial Distribution of Ecliptic Comets Icarus

Levison HF Duncan M J Symplectically Integrating Close Encounters with the

Sun Aston J

Levison H F and Stewart G R Remarks on Mo deling the Formation of Uranus

and Neptune Icarus

Levison HF Dones L Chapman CR Stern SA Duncan MJ and Zhanle K

Could the lunar Late Heavy Bombardmenthave b een triggered by the formation of

Uranus and Neptune Icarus

Levison HF Morbidelli A Pushing out the Kuip er b elt Nature submitted

Liou J and H A Zo ok Signatures of the Giant Planets Imprinted on the

EdgeworthKuip er Belt Dust Disk Astronomical Journal

Liou J H A Zo ok and S F Dermott Kuip er Belt Dust Grains as a Source of

Interplanetary Dust Particles Icarus

Malhotra R The origin of Plutos Peculiar orbit Nature

Malhotra R The origin of Plutos orbit Implications for the Solar System b eyond

Neptune Astron J

Masset F S On the Coorbital Corotation Torque in a Viscous Disk and Its Impact

on Planetary Migration Astroph J

Morbidelli A Valsecchi G B Neptune scattered planetesimals could have sculpted

the primordial EdgeworthKuip er b elt Icarus

Morbidelli A Jacob C and Petit JM Planetary embryos never formed in the

Kuip er b elt Icarus

MoroMartn A and R Malhotra Dynamical Mo dels of Kuip er Belt Dust in the

Inner and Outer Solar System Astronomical Journal

Murray N B Hansen M Holman and S Tremaine Migrating Planets Science

Nagasawa M Ida S Sweeping Secular Resonances in the Kuip er Belt Caused by

Depletion of the Solar Nebula Astron J

Opick EJ Interplanetary encounters Elsevier New York

Ozernoy L M Gorkavyi N N Mather J C Taidakova T A Signatures of

Exosolar Planets in Dust Debris Disks Astroph J LL

Petit J M Morbidelli A Valsecchi G B Large scattered planetesimals and the

excitation of the small b o dy b elts Icarus

Petit J M Mousis O KBO binaries are they really primordial Icarus submitted

Stern S A Colwell J E a Accretion in the EdgeworthKuip er Belt Forming

KM Radius Bo dies at AUandBeyond Astron J

Stern S A Colwell J E b Collisional Erosion in the Primordial EdgeworthKuip er

Belt and the Generation of the AU Kuip er Gap Astroph J

Stern S A and PRWeissman Rapid collisional evolution of comets during the

formation of the Oort cloud Nature

Stone JM Gammie CF Balbus SA and Hawley JF Transp ort Pro cesses in

Protostellar Disks In and Planets IV eds V Mannings AP Boss and

SS Russell

Thommes E W Duncan M J and Levison H F The formation of Uranus and

Neptune in the JupiterSaturn region of the Solar System Nature

Thommes E W Duncan M J and Levison H F Oligarchic growth of giant

planets icarus

Trujillo C A Jewitt D C Luu J X Prop erties of the TransNeptunian Belt

Statistics from the CanadaFranceHawaii Telescop e Survey Astron J

Valsecchi Aand Manara G B Dynamics of comets in the outer planetary region

I I Enhanced planetary masses and orbital evolutionary paths Astron Astrophys

Wahha j Z Ko erner D W Ressler M E Werner M W Backman D E Sargent A

I he Inner Rings of Pictoris Astroph J LL

Ward WR Migration by Nebula Tides Icarus

Weidenschilling S Formation of PlanetesimalsCometesimals in the Solar nebula in

Comet IIFestou et al eds University Arizona Press Tucson Az

Ward WR and Hahn JM a Neptunes Eccentricity and the Nature of the Kuip er

Belt Science

Ward WR and Hahn JM b Dynamics of the TransNeptune Region Apsidal

Waves in the Kuip er Belt Astron J

Youdin A N and F H Shu Planetesimal Formation by Gravitational Instability

Astrophysical Journal

A

This manuscript was prepared with the AAS L T X macros v E

Fig Semima jor axis and eccentricity of the planets lled dots and of the planetesimals

p oints at time t from the simulation with a Earth masses disk initially in

the AU region presented in section The solid lines dene the limits for planetary

crossing orbits while the dotted lines showwhereH H for zero inclination orbits

p

Fig Evolution of the semima jor axes of the four giant planets due to a planetesimal

disk of M initially b etween and AU top and M initially b etween

 

and AU b ottom A low mass disk of M is assumed in b oth cases in the AU



range

Fig Semima jor axis and eccentricity distribution of the planetesimals at t

panel A and t y panel B for the simulation presented in the top panel fo

Fig The lines dene the b oundaries of the planetcrossing regions

Fig Neptunes migration in the simulation presented in the top panel of Fig is

shown here on a magnied timescale Other curves show Neptunes migration in new

simulations in which Jupiter Saturn and Uranus are discarded at My and the disk is

extended b eyond AU The similaritybetween the previous and the new evolutions up to

My demonstrates that the other planets play an inessential role in Neptunes runaway

migration In the simulations with the outer edge of the disk at and AU the planet

migrates up to the edge and then reverses migration In the simulations with the disks

edge at AU the inversion o ccurs well b efore the edge is reached

Fig Greyscaleco ded density of planetesimals in Neptunecrossing regions at three

dierent times corresp onding to the b eginning of a fast migration episo de top panel the

middle middle panel and the end of it b ottom panel The migration is then reversed

The semima jor axis is expressed in units of the current semima jor axis of the planet in

order to highlight the dierences among the planetesimals distributions The continuous

light grey curves mark the b orders of Neptunecrossing region and the dashed curvethe

condition H H for i

p

Fig Neptunes semima jor axis evolution for planetesimals disks with several surface

density distributions and total masses Eachdiskwas mo deled using particles In



the cases lab eled M e M e and M e the disk has a surface densitydecaying as r

and total masses of M between and AU resp ectively In the cases lab eled





r and r the disk has a total mass of M and a surface densitydecaying as r





and r resp ectively

Fig Neptunes semima jor axis evolution in a pair of simulations with disks of



b ottom middle and M top The surface density of the disks decays as r and



each disk is mo deled with only particles Due to the low resolution of the disk mo del

Neptunes migration results highly sto chastic and unpredictable

Fig A selfconsistentsimulation of the Petit et al scenario for the excitation

and dynamical depletion of the Kuip er b elt Neptune is originally assumed at AUand

an Earthmass embryoat AU Both planets are emb edded in a M disk extending



from to AU The pair of blackcurves showtheevolution of Neptunes p erihelion and

aphelion distance while the grey curves refer to the embryo Notice that the embryo is never

scattered by Neptune It migrates through the disk faster than Neptune until the disks outer

edge Neptune interacts with most of the mass of the disk thanks to the dynamical excitation

of the latter due to the presence of the embryo Therefore it migrates much further that it

would if the embryowere not present and reaches a nal p osition well b eyond AU

Fig The lo cation of the secular resonance as a function of the disk mass assuming

Neptune at AU and the disk inner and outer b oundaries at and AU As the disk mass

decreases the nu secular resonance sweeps the disk In the collisional grinding scenario

this phenomenon should haveprovided new material to the Neptunecrossing region and

restart Neptunes migration



Fig Examples of Neptunes migration in disks with an outer edge at AU r

surface density proles and masses equal to and M from b ottom to



top Only in the case of a M disk a massiveannulus is left b etween Neptunes p osition



and the original outer edge of the disk In all other cases the disk is completely depleted

Fig The dynamics of a Marsmass embryo initially placed outside Neptunes orbit

and AU resp ectively The disk mass is M between and AU The black curve



at the b ottom of the panel shows the evolution of Neptunes semima jor axis The three

light grey curves show the evolution of the embryos p erihelion distance semima jor axis

and aphelion distance resp ectively from b ottom to top The twocurves with intermediate

grey color whichevolve parallel to Neptunes semima jor axis show the lo cation of the

and resonances resp ectivelyTheembryo is initially in the former resonance and then

is capture in the latter at t My The embryo quits the resonance at t My

Fig The evolution of an embryo with M initially placed outside Neptune and



AUs resp ectively in a disk with M truncated at AU The black curves showthe



evolution of the semima jor axes of Jupiter Saturn Uranus and Neptune from b ottom to

top while the light grey curves show the p erihelion and aphelion distance of the embryo

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure

Figure Figure

Figure