Planetary migration in a planetesimal disk why did Neptune 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 planetesimals In the case of our Solar System 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 planets 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 Earthmass 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 planet with the gaseous disk Ward Masset After the gas disk dissipates the energy and angular momentum 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 Pluto 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 Uranus and Neptune is unlikely suggesting that these planets formed much closer to Jupiter and Saturn 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 Sun 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