RUNAWAY MIGRATION AND THE FORMATION OF HOT JUPITERSF. S. Masset1Service d’Astrophysique, Centre d’Etudes de Saclay, Batiment 709, Orme des Merisiers, F-91191 Gif-sur-Yvette Cedex, France;[email protected]. C. B. PapaloizouAstronomy Unit, School of Mathematical Sciences, Queen Mary University of London, Mile End Road, London E1 4NS, UK;[email protected] 2002 September 16; accepted 2003 January 10ABSTRACTWe evaluate the coorbit al corotation torque on a migrating protoplanet. The coorbital torque is assumedto come from orbit crossing fluid elements that exchange angular momentum with the planet when theyexecute a U-turn at the end of horseshoe streamlines. When the planet migrates inward, the fluid elements ofthe inner disk undergo one such exchange as they pass to the outer disk. The angular momentum they gain isremoved from the planet, and this corresponds to a negative contribution to the corotation torque, whichscales with the drift rate. In addition, the material trapped in the coorbital region drifts radially with theplanet, giving a positive contribution to the corotation torque, which also scales with the drift rate. These twocontributions do not cancel out if the coorbital region is depleted, in which case there is a net corotation tor-que that scales with the drift rate and the mass deficit in the coorbital region and has the same sign as the driftrate. This leads to a positive feedback on the migrating planet. In particular, if the coorbital mass deficit islarger than the planet mass, the migration rate undergoes a runaway that can vary the protoplanet semimajoraxis by 50% over a few tens of orbits. This can happen only if the planet mass is sufficient to create a dip orgap in its surrou nding region and if the surrounding disk mass is larger than the planet mass. This typicallycorresponds to planet masses in the sub-Saturnian to Jovian mass range embedded in massive protoplanetarydisks. Runaway migration is a good candidate to account for the orbital characteristics of close orbiting giantplanets, most of which have sub-Jovian masses. These are known to cluster at short periods, whereas planetsof greater than two Jovian masses are rare at short periods, indicating a different type of migration processoperated for the two classes of object. Further, we show that in the runaway regime, migration can bedirected outward, which makes this regime potentially rich in a varie ty of important effects in shaping aplanetary system during the last stages of its formation.Subject headings: accretion, accretion disks — hydrodynamics — methods: numerical —planetary systems: formation — planetary systems: protoplanetary disks1. INTRODUCTIONThe study of the tidally induced migration of proto-planets embedded in protoplanetary disks has receivedrenewed attention in the last few years following thediscovery of extrasolar giant planets (EGPs). It is in particu-lar the best candidate to explain the short-period EGPs (theso-called hot Jupiters) that are likely to have begun to formfarther out in the disk a nd migrated radially inward.When the planet mass is small (i.e., when its Hill radius ismuch smaller than the disk thickness), the migration ratecan be evaluated using linear analysis and is shown to beproportional to the planet mass and the disk surface densityand inversely proportional to the square of the disk aspectratio (Ward 1997). The linear regime is often called the typeI regime. It corresponds to a fast migration rate, althoughrecent estimates (Miyoshi et al. 1999; Tana ka, Takeuchi, &Ward 2002; Masset 2002) show that the linear analyticalestimate assuming a flat two-dimensional disk has to bereduced by a factor of 2–3 or more in a more realistic calcu-lation that accounts for the disk vertical structure and apossible nons aturation of the corotation torque if the disk isviscous enough. Migration in the type I regime is neverthe-less still too fast, in the sense that the migration timescale ityields is shorter than the buildup timescale of a giant proto-core (see, e.g., Papaloizou & Terquem 1999). We shall notaddress this issue here but rather assume that a giant planetembryo can form in the disk at a distance r 1 AU and witha mass M > Mcrit, wher e Mcrit 15 M is the critical massabove which rapid gas accretion begins.When this embryo mass is large enough, it entersanother well-studied migration regime, called type IImigration (Ward 1997). In this regime, the protoplanethas a mass sufficient to open a gap in the disk, which istherefore split into an inner disk and an outer disk. Theprotoplanet then finds itself locked into the disk viscousevolution drifting inward with it (Lin & Papaloizou 1986;Trilling et al. 1998). As the protoplanet undergoes type IImigration toward the central object, it may accrete thesurrounding nebula material. The accretion rate scaleswith the mass accretion rate onto the central object_MMp 3, where is the disk surfa ce density. Hereone assumes that the processes at work in the disk thatcontribute to the angular momentum exchange betweenneighboring rings can be adequately modeled by a1Corresponding author.The Astrophysical Journal, 588:494–508, 2003 May 1# 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.494phenomenological kinematic viscosity . On the otherhand, the type II migration timescale is of the order ofthe disk viscous timescale IImig r2=3. The maximummass that a giant protoplanet can accrete on its way tothe central object should be therefore of the order ofMp_MMp IImig r2, that is, of the order of the diskmass interior to the planet starting distance. Noticeably,this mass does not depend on the disk viscosity. If theplanet does not migrate all the way to the central objectbefore the disk is dispersed, then because more time isspent at larger radii, it is most likely to be left with asemimajor axis larger than the typical one for hotJupiters (0.05–0.2 AU; see Trilling, Lunine, & Benz2002). This is consistent with the observed paucity ofplanets with masses exceeding two Jovian masses at shortperiods (Zucker & Mazeh 2002). Note too that planetsundergoing type II migration should tend to have highermasses at shorter periods. This is contrary to theobserved trend. Furthermore, as the planet mass grows,it becomes eventually larger than the surrounding diskmass, and its migration rate tends to drop, as the diskcannot remove enough
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