Disk-Planet Interactions During Planet FormationJ. C. B. PapaloizouUniversity of CambridgeR. P. NelsonQueen Mary, University of LondonW. KleyUniversit¨at T¨ubingenF. S. MassetSaclay, France and UNAM MexicoP. ArtymowiczUniversity of Toronto at Scarborough and University of StockholmThe discovery of close orbiting extrasolar giant planets led to extensive studies of disk planet interactions andthe forms of migration that can result as a means of accounting for their location. Early work establishedthe type I and type II migration regimes for low mass embedded planets and high mass gap forming planetsrespectively. While providing an attractive means of accounting for close orbiting planets intially formed atseveral AU, inward migration times for objects in the earth mass range were found to be disturbingly short,making the survival of giant planet cores an issue. Recent progress in this area has come from the applicationof modern numerical techniques wich make use of up to date supercomputer resources. These have enabledhigher resolution studies of the regions close to the planet and the initiation of studies of planets interactingwith disks undergoing MHD turbulence. This work has led to indications of how the inward migration of lowto intermediate mass planets could be slowed down or reversed. In addition, the possibility of a new very fasttype III migration regime, that can be directed inwards or outwards, that is relevant to partial gap formingplanets in massive disks has been investigated.1. INTRODUCTIONThe discovery of extrasolar planets around sun–like stars(51 Pegasi b) (Mayor and Queloz, 1995; Marcy and Butler,1995; Marcy and Butler, 1998) has revealed a population ofclose orbiting giant planets with periods of typically a fewdays, the so called ’hot Jupiters’. The difficulties associatedwith forming such planets in situ, either in the critical coremass accumulation followed by gas accretion scenario, orthe gravitational instability scenario for giant planet forma-tion has led to the realisation of the potential importanceof large scale migration in forming or young planetary sys-tems.This in turn led to more intensive theoretical develop-ment of disk protoplanet interaction theory that had alreadyled to predictions of orbital migration (see Lin and Pa-paloizou, 1993; Lin et al., 2000 and references therein). Atthe time of PPIV, the type I and type II migration regimes,the former applying to small mass embedded protoplanetsand the latter to gap forming massive protoplanets, had be-come apparent. Both these regimes predicted disturbinglyshort radial infall times that in the type I case threatened thesurvival of embryo cores in the 1 − 15M⊕regime beforethey could accrete gas to become giant planets. The mainquestions to be addressed were how to resolve the type Imigration issue and to confirm that type II migration appli-cable to giant planets could indeed account for the observedradial distribution and the hot Jupiters.Here, we review recent progress in the field of diskplanet interactions in the context of orbital migration. Forreasons of space constraint we shall not consider the prob-lem of excitation or damping of orbital eccentricity. Themost recent progress in this area has come from carryingout large scale two and three dimension simulations that re-quire the most up to date supercomputer resources. Thishas enabled the study of disk planet interactions in disksundergoing MHD turbulence, the study of the regions closeto the planet using high resolution multigrid techniques, ledto suggestions for the possible resolution of the type I is-sue and revealed another possible type III migration regime.However, the complexnature of these problems makes themchallenging numerically and as a consequence numericalconvergence has not been attained in some cases.In sections 2, 3, and 4 we review type I migration, type II1migration and type III migration respectively. In section 5we review recent work on disk planet interactions in diskswith MHD turbulence and in section 6 we give a summary.2. TYPE I MIGRATIONWhen the mass of the protoplanet is small the responseit induces in the disk can be calculated using linear theory.When the disk flow is non magnetic and laminar, densitywaves propagate both outwards and inwards away from theprotoplanet. These waves carry positive and negative an-gular momentum respectively and accordingly a compen-sating tidal torque is applied to the orbit resulting in type Imigration.2.1. The tidal torqueThe problem of determining the evolution of the planetorbit amounts to an evaluation of tidal torques. For a suf-ficiently small planet mass (an upper limit of which willbe specified below) one supposes that the gravitational po-tential of the protoplanet forces small perturbations. Thehydrodynamic equations are then linearized about a basicstate consisting of an unperturbed axisymmetric accretiondisk and the response calculated. The gravitational potentialψ of a proplanet in circular orbit is expressed as a Fourierseries in the formψ(r, ϕ, t) =∞Xm=0ψm(r) cos{m[ϕ −ωpt]}, (1)where ϕ is the azimuthal angle and 2π/(ωp) is the or-bital period of the planet of mass Mpat orbital semi-majoraxis a. The total torque acting on the disk is given byΓ = −RDiskΣ~r × ∇ψd2r where Σ is the surface densityof the disk.An external forcing potential ψm(r, ϕ) with azimuthalmode number m that rotates with a pattern frequency ωpina disk with angular velocity Ω(r) triggers a response thatexchanges angular momentum with the orbit whenever, ne-glecting effects due to pressure, m(Ω − ωp) is equal either0 or ±κ, with, for a Keplerian disk to adequate acuracy,κ ≡ Ω being the epicyclic frequency. The first possibil-ity occurs when Ω = ωpand thus corresponds to a co-rotation resonance. The second possibility corresponds toan inner Lindblad resonance located inside the orbit forΩ = ωp+ κ/m and an outer Lindblad resonance outsidethe orbit for Ω = ωp− κ/m.2.1.1. Torques at Lindblad resonances. Density waves arelaunched at Lindblad resonances and as a consequence ofthis a torque acts on the planet. It is possible to solve thewave excitation problem using the WKB method. In thatapproximation an analytic expression for the torque can befound. The torque arising from the component of the poten-tial with azimuthal mode number m is found, for a Keple-rian disk, to be given byΓLRm=sign(ωp− Ω)π2Σ3ΩωpΨ2, (2)withΨ =
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