Gravitational Instabilities in Gaseous Protoplanetary Disks andImplications for Giant Planet FormationRichard H. DurisenIndiana UniversityAlan P. BossCarnegie Institution of WashingtonLucio MayerEidgen¨ossische Technische Hochschule Z¨urichAndrew F. NelsonLos Alamos National LaboratoryThomas QuinnUniversity of WashingtonW. K. M. RiceUniversity of EdinburghProtoplanetary gas disks are likely to experience gravitational instabilites (GI’s) duringsome phase of their evolution. Density perturbations in an unstable disk grow on a dynamictime scale into spiral arms that produce efficient outward transfer of angular momentum andinward transfer of mass through gravitational torques. In a cool disk with rapid enough cooling,the spiral arms in an unstable disk form self-gravitating clumps. Whether gas giant protoplanetscan form by such a disk instability process is the primary question addressed by this review.We discuss the wide range of calculations undertaken by ourselves and others using variousnumerical techniques, and we report preliminary results from a large multi-code collaboration.Additional topics include – triggering mechanisms for GI’s, disk heating and cooling, orbitalsurvival of dense clumps, interactions of solids with GI-driven waves and shocks, and hybridscenarios where GI’s facilitate core accretion. The review ends with a discussion of how welldisk instability and core accretion fare in meeting observational constraints.1. INTRODUCTIONGravitational instabilities (GI’s) can occur in any regionof a gas disk that becomes sufficiently cool or develops ahigh enough surface density. In the nonlinear regime, GI’scan produce local and global spiral waves, self-gravitatingturbulence, mass and angular momentum transport, anddisk fragmentation into dense clumps and substructure. Theparticular emphasis of this review article is the possibility(Kuiper, 1951; Cameron, 1978), recently revived by Boss(1997, 1998a), that the dense clumps in a disk fragmentedby GI’s may become self-gravitating precursors to gas giantplanets. This particular idea for gas giant planet formationhas come to be known as thedisk instabilitytheory. Weprovide here a thorough review of the physics of GI’s ascurrently understood through a wide variety of techniquesand offer tutorials on key issues of physics and methodol-ogy. The authors assembled for this paper were deliberatelychosen to represent the full range of views on the subject.Although we disagree about some aspects of GI’s and aboutsome interpretations of available results, we have laboredhard to present a fair and balanced picture. Other recent re-views of this subject include Boss (2002c), Durisen et al.(2003), and Durisen (2006).2. PHYSICS OF GI’s2.1 Linear RegimeThe parameter that determines whether GI’s occur in thingas disks isQ = csκ/πGΣ, (1)where csis the sound speed, κ is the epicyclic frequency atwhich a fluid element oscillates when perturbed from circu-lar motion, G is the gravitational constant, and Σ is the sur-face density. In a nearly Keplerian disk, κ ≈ the rotational1angular speed Ω. For axisymmetric (ring-like) disturbances,disks are stable when Q > 1 (Toomre, 1964). At high Q-values, pressure, represented by csin (1), stabilizes shortwavelengths, and rotation, represented by κ, stabilizes longwavelengths. The most unstable wavelength when Q < 1is given by λm≈ 2π2GΣ/κ2.Modern numerical simulations, beginning with Pa-paloizou and Savonije (1991), show that nonaxisymmetricdisturbances, which grow as multi-armed spirals, becomeunstable for Q . 1.5. Because the instability is both linearand dynamic, small perturbations grow exponentially on thetime scale of a rotation period Prot= 2π/Ω. The multi-armspiral waves that grow have a predominantly trailing pat-tern, and several modes can appear simultaneously (Boss,1998a; Laughlin et al., 1998; Nelson et al., 1998; Pickett etal., 1998). Although the star does become displaced fromthe system center of mass (Rice et al., 2003a) and one-armed structures can occur (see Fig. 1 of Cai et al., 2006),one-armed modes do not play the dominant role predictedby Adams et al. (1989) and Shu et al. (1990).2.2 Nonlinear RegimeNumerical simulations (see also Sections 3 and 4) showthat, as GI’s emerge from the linear regime, they may eithersaturate at nonlinear amplitude or fragment the disk. Twomajor effects control or limit the outcome – disk thermo-dynamics and nonlinear mode coupling. At this point, thedisks also develop large surface distortions.Disk Thermodynamics. As the spiral waves grow, theycan steepen into shocks that produce strong localized heat-ing (Pickett et al., 1998, 2000a; Nelson et al., 2000). Gasis also heated by compression and through net mass trans-port due to gravitational torques. The ultimate source ofGI heating is work done by gravity. What happens next de-pends on whether a balance can be reached between heatingand the loss of disk thermal energy by radiative or convec-tive cooling. The notion of a balance of heating and coolingin the nonlinear regime was described as early as 1965 byGoldreich and Lynden-Bell and has been used as a basisfor proposing α-treatments for GI-active disks (Paczy´nski,1978; Lin and Pringle, 1987). For slow to moderate cool-ing rates, numerical experiments, such as in Fig. 1, verifythat thermal self-regulation of GI’s can be achieved (Tom-ley et al., 1991; Pickett et al., 1998, 2000a, 2003; Nelsonet al., 2000; Gammie, 2001; Boss, 2003; Rice et al., 2003b;Lodato and Rice, 2004, 2005; Mej´ıa et al. 2005; Cai etal., 2006). Q then hovers near the instability limit, and thenonlinear amplitude is controlled by the cooling rate.Nonlinear Mode Coupling. Using second and third-order governing equations for spiral modes and comparingtheir results with a full nonlinear hydrodynamics treatment,Laughlin et al. (1997, 1998) studied nonlinear mode cou-pling in the most detail. Even if only a single mode ini-tially emerges from the linear regime, power is quickly dis-tributed over modes with a wide variety of wavelengths andnumber of arms, resulting in a self-gravitating turbulenceFig. 1.— Greyscale of effective temperature Tef fin degreesKelvin for a face-on GI-active disk in an asymptotic state of ther-mal self-regulation. This figure is for the Mej´ıa et al. (2005) evo-lution of a 0.07 M⊙disk around a 0.5 M⊙star with tcool= 1outer rotation period at 4,500 yr. The frame is 120 AU on a side.that permeates the
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