Richard H. Durisen

Clumps in the Outer Disk by Disk Instability: Why They are Initially Gas Giants and the Legacy of Disruption

Abstract We explore the initial conditions for fragments in the extended regions ( r ≳ 50 AU ) of gravitationally unstable disks. We combine analytic estimates for the fragmentation of spiral arms with 3D SPH simulations to show that initial fragment masses are in the gas giant regime. These initial fragments will have substantial angular momentum, and should form disks with radii of a few AU. We show that clumps will survive for multiple orbits before they undergo a second, rapid collapse due to H 2 dissociation and that it is possible to destroy bound clumps by transporting them into the inner disk. The consequences of disrupted clumps for planet formation, dust processing, and disk evolution are discussed. We argue that it is possible to produce Earth-mass cores in the outer disk during the earliest phases of disk evolution.

THE THERMAL REGULATION OF GRAVITATIONAL INSTABILITIES IN PROTOPLANETARY DISKS. III. SIMULATIONS WITH RADIATIVE COOLING AND REALISTIC OPACITIES

This paper presents a fully three-dimensional radiative hydrodymanics simulation with realistic opacities for a gravitationally unstable 0.07 Mdisk around a 0.5 Mstar. We address the following aspects of disk evolution: the strengthofgravitationalinstabilitiesunderrealisticcooling,masstransportinthediskthatarisesfromGIs,comparisons betweenthegravitationalandReynoldsstressesmeasuredinthediskandthoseexpectedinan � -disk,andcomparisons between the SED derived for the disk and SEDs derived from observationally determined parameters. The mass transport in this disk is dominated by global modes, and the cooling times are too long to permit fragmentation for all radii. Moreover, our results suggest a plausible explanation for the FU Ori outburst phenomenon. Subject headingg accretion, accretion disks — convection — hydrodynamics — instabilities — solar system: formation

The Thermal Regulation of Gravitational Instabilities in Protoplanetary Disks. II. Extended Simulations with Varied Cooling Rates

In order to investigate mass transport and planet formation through gravitational instabilities (GIs), we have extended our three-dimensional hydrodynamic simulations of protoplanetary disks from a previous paper. Our goal is to determine the asymptotic behavior of GIs and how it is affected by different constant cooling times. Initially, Rdisk = 40 AU, Mdisk = 0.07 M☉, M* = 0.5 M☉, and Qmin = 1.5. Sustained cooling, with tcool = 2 ORPs (outer rotation periods; 1 ORP ≈ 250 yr), drives the disk to instability in about 4 ORPs. This calculation is followed for 23.5 ORPs. After 12 ORPs, the disk settles into a quasi-steady state with sustained nonlinear instabilities, an average Q = 1.44 over the outer disk, a well-defined power law Σ(r), and a roughly steady ≈ 5 × 10-7 M☉ yr-1. The transport is driven by global low-order spiral modes. We restart the calculation at 11.2 ORPs with tcool = 1 and ORPs. The latter case is also run at high azimuthal resolution. We find that shorter cooling times lead to increased -values, denser and thinner spiral structures, and more violent dynamic behavior. The asymptotic total internal energy and the azimuthally averaged Q(r) are insensitive to tcool. Fragmentation occurs only in the high-resolution tcool = ORP case; however, none of the fragments survive for even a quarter of an orbit. Ringlike density enhancements appear and grow near the boundary between GI-active and GI-inactive regions. We discuss the possible implications of these rings for gas giant planet formation.

Dynamical and luminosity evolution of active galactic nuclei : models with a mass spectrum

A multimass energy-space Fokker-Planck code is used to follow the dynamical and luminosity evolution of an AGN model that consists of a dense stellar system surrounding a massive black hole. It is found that stellar evolution and tidal disruption are the predominant mass-loss mechanisms for low-density nuclei, whereas physical collisions dominate in high-density nuclei. For initial central densities greater than 10 million solar masses/cu pc the core of the stellar system contacts due to the removal of kinetic energy by collisions, whereas for densities less than this the core of the stellar system expands due to heating that results from the settling of a small population of stars into orbits tightly bound to the black hole. These mechanisms produce differing power-law slopes in the resulting stellar density cusp surrounding the black hole, -7/4 and -1/2 for low- and high-density nuclei, respectively. 60 refs.

The Thermal Regulation of Gravitational Instabilities in Protoplanetary Disks

We present a series of high-resolution, three-dimensional hydrodynamics simulations of a gravitationally unstable solar nebula model. The influences of both azimuthal grid resolution and the treatment of thermal processes on the origin and evolution of gravitational instabilities are investigated. In the first set of simulations, we vary the azimuthal resolution for a locally isothermal simulation, doubling and quadrupling the resolution used in a previous study; the largest number of grid points is (256, 256, 64) in cylindrical coordinates (r, , z). At this resolution, the disk breaks apart into a dozen short-lived condensations. Although our previous calculations underresolved the number and growth rate of clumps in the disk, the overall qualitative, but fundamental, conclusion remains: fragmentation under the locally isothermal condition in numerical simulations does not in itself lead to the survival of clumps to become gaseous giant protoplanets. Since local isothermality represents an extreme assumption about thermal processes in the disk, we also present several extended simulations in which heating from an artificial viscosity scheme and cooling from a simple volumetric cooling function are applied to two different models of the solar nebula. The models are differentiated primarily by disk temperature: a high-Q model generated directly by our self-consistent field equilibrium code and a low-Q model generated by cooling the high-Q model in a two-dimensional version of our hydrodynamics code. Here, high-Q and low-Q refer to the minimum values of the Toomre stability parameter Q in each disk, Qmin = 1.8 and 0.9, respectively. Previous simulations, by ourselves as well as others, have focused on initial states that are already gravitationally unstable, i.e., models similar to the low-Q model. This paper presents for the first time the numerical evolution of an essentially stable initial equilibrium state (the high-Q model) to a severely unstable one by cooling. The additional heating and cooling are applied to each model over the outer half of the disk or the entire disk. The models are subject to the rapid growth of a four-armed spiral instability; the subsequent evolution of the models depends on the thermal behavior of the disk. The cooling function tends to overwhelm the heating included in our artificial viscosity prescription, and as a result the spiral structure strengthens. The spiral disturbances transport mass at prodigious rates during the early nonlinear stages of development and significantly alter the disks vertical surface. Although dense condensations of material can appear, their character depends on the extent of the volumetric cooling in the disk. In the simulation of the high-Q model with heating and cooling applied throughout the disk, thin, dense rings form at radii ranging from 1 to 3 AU and steadily increase in mass; later companion formation may occur in these rings as cooling drives them toward instability. When heating and cooling are applied only over the outer radial half of the disk, however, a succession of single condensations appears near 5 AU. Each clump has roughly the mass of Saturn, and some survive a complete orbit. Since the clumps form near the artificial boundary in the treatment of the disk gas physics, the production of a clump in this case is a numerical artifact. Nevertheless, radially abrupt transitions in disk gas characteristics, for example, in opacity, might mimic the artificial boundary effects in our simulations and favor the production of stable companions in actual protostellar and protoplanetary disks. The ultimate survival of condensations as eventual stellar or substellar companions to the central star is still largely an open question.

THE INTERNAL ENERGY FOR MOLECULAR HYDROGEN IN GRAVITATIONALLY UNSTABLE PROTOPLANETARY DISKS

The gas equation of state may be one of the critical factors for the disk instability theory of gas giant planet formation. This Letter addresses the treatment of H2 in hydrodynamic simulations of gravitationally unstable disks. In our discussion, we point out possible consequences of erroneous specific internal energy relations, approximate specific internal energy relations with discontinuities, and assumptions of constant Γ1. In addition, we consider whether the ortho/para ratio for H 2 in protoplanetary disks should be treated dynamically as if the species are in equilibrium. Preliminary simulations indicate that the correct treatment is particularly critical for the study of gravitational instability when T = 30-50 K.

THE EFFECTS OF METALLICITY AND GRAIN SIZE ON GRAVITATIONAL INSTABILITIES IN PROTOPLANETARY DISKS

Observational studies show that the probability of finding gas giant planets around a star increases with the stars metallicity. Our latest simulations of disks undergoing gravitational instabilities (GIs) with realistic radiative cooling indicate that protoplanetary disks with lower metallicity generally cool faster and thus show stronger overall GI activity. More importantly, the global cooling times in our simulations are too long for disk fragmentation to occur, and the disks do not fragment into dense protoplanetary clumps. Our results suggest that direct gas giant planet formation via disk instabilities is unlikely to be the mechanism that produced most observed planets. Nevertheless, GIs may still play an important role in a hybrid scenario, compatible with the observed metallicity trend, where structure created by GIs accelerates planet formation by core accretion.

THE THERMAL REGULATION OF GRAVITATIONAL INSTABILITIES IN PROTOPLANETARY DISKS. IV. SIMULATIONS WITH ENVELOPE IRRADIATION

It is generally thought that protoplanetary disks embedded in envelopes are more massive and thus more suscep- tible to gravitational instabilities (GIs) than exposed disks. We present three-dimensional radiative hydrodynamic simulationsof protoplanetarydiskswiththepresenceof envelopeirradiation.Foradiskwitharadiusof 40AUanda mass of 0.07 Maround a young star of 0.5 M� , envelope irradiation tends to weaken and even suppress GIs as the irradiating flux is increased. The global mass transport induced by GIs is dominated by lower order modes, and irradiation preferentially suppresses higher order modes. As a result, gravitational torques and mass inflow rates are actually increased bymild irradiation. None of the simulations producedense clumps or rapid cooling byconvection, arguing against direct formation of giant planets by disk instability, at least in irradiated disks. However, dense gas rings andradial mass concentrationsare produced,and these mightbeconduciveto accelerated planetary coreforma- tion. Preliminary results from a simulation of a massive embedded disk with physical characteristics similar to one of thedisksintheembeddedsourceL1551IRS5indicatealongradiativecoolingtimeandnofragmentation.TheGIsin this disk are dominated by global two- and three-armed modes. Subject headingg accretion,accretiondisks — hydrodynamics — instabilities — planetarysystems:formation — planetary systems: protoplanetary disks

Three-Dimensional Radiative Hydrodynamics for Disk Stability Simulations: A Proposed Testing Standard and New Results

Recent three-dimensional radiative hydrodynamics simulations of protoplanetary disks report disparate disk behaviors, and these differences involve the importance of convection to disk cooling, the dependence of disk cooling on metallicity, and the stability of disks against fragmentation and clump formation. To guarantee trustworthy results, a radiative physics algorithm must demonstrate the capability to handle both the high and low optical depth regimes. We develop a test suite that can be used to demonstrate an algorithms ability to relax to known analytic flux and temperature distributions, to follow a contracting slab, and to inhibit or permit convection appropriately. We then show that the radiative algorithm employed by Mejia and Boley et al. and the algorithm employed by Cai et al. pass these tests with reasonable accuracy. In addition, we discuss a new algorithm that couples flux-limited diffusion with vertical rays, we apply the test suite, and we discuss the results of evolving the Boley et al. disk with this new routine. Although the outcome is significantly different in detail with the new algorithm, we obtain the same qualitative answers. Our disk does not cool fast due to convection, and it is stable to fragmentation. We find an effective α ≈ 10-2. In addition, transport is dominated by low-order modes.

Bombardment of planetary rings by meteoroids: General formulation and effects of Oort cloud projectiles

Abstract We present a general solution for the angular distribution of the intensity and velocity of interplanetary projectiles as they impinge on a planetary ring system, given the original distribution in inertial space. Our solution lends itself immediately to three results of importance. First, we demonstrate a variation with orbital longitude of an impact-velocity-weighted impact rate of cometary meteoroids in planetary rings. The largest rate of energetic impacts occurs at orbital longitudes near solar midnight, consistent in location and form with observed creation rates of “spokes” in Saturns rings. This provides support for spoke formation hypotheses which rely on meteroid impact. Second, we determine the angular distribution of the intensity of ejecta which result from the bombardment of a planetary ring composed of centimeter- to meter-sized particles by interplanetary meteroids in the subcentimeter-size range. We do this using a radiative transfer formalism. The angular distribution of ejecta resulting from the bombardment of a single target particle is first obtained as a function of angle from a single incident direction. In this step, we incorporate results from laboratory studies of nondisruptive hypervelocity impacts into granular and powdery surfaces. This single particle “phase function” is part of a standard expression for the net scattering function of the layer as a whole, which accounts for the escape probabilities of ejecta emerging from impacts occurring at different depths within the ring layer. We assume that multiple scattering terms resulting from impacts into nearby particles by the more slowly moving ejecta are negligible. Finally, the net scattering function of the layer is integrated over the angular distribution of the incoming meteoroids, which is determined in the frame of an orbiting ring particle by accounting for aberration effects arising from the orbital motion of the ring particle and its parent planet. Our calculated ejecta distributions are incorporated into evolutions of ring mass density and radial structure in a companion paper (Durisen et al. 1989, Icarus , 80 136–166). Third, we calculate the radial drift velocity of a planetary ring of arbitrary optical depth which results from two factors: simple mass loading, and aberration-induced asymmetry in the impact rate. Bombardment of ring systems at the currently accepted rate of interplanetary meteoroid flux leads to an inward radial drift of several cm yr −1 in regions of moderate optical depth. This rate is large enough for the Uranian α and β rings, and the entire C ring of Saturn, to fall into the atmosphere of the parent planet in about 10 8 years under only the known flux of projectiles on Oort cloud orbits. These effects can vary significantly with the orbital distribution of the projectile population. In this paper, we present results for projectiles with “Oort cloud” orbits. In a subsequent paper, we will present the results anticipated for projectiles in “local family” prograde, low inclination orbits.