We Ðnd a linear instability of nonaxisymmetric Rossby waves in a thin nonmagnetized Keplerian disk when there is a local maximum in the radial proÐle of a key function L(r) 4 F(r)S2@!(r), where F~1 \ is the potential vorticity, S \ P/&! is the entropy, & is the surface mass density, P is the zü AE ($ Â ¿)/& vertically integrated pressure, and ! is the adiabatic index. We consider in detail the special case where there is a local maximum in the disk entropy proÐle S(r). This maximum acts to trap the waves in its vicinity if its height-to-width ratio max(S)/*r is larger than a threshold value. The pressure gradient derived from this entropy variation provides the restoring force for the wave growth. We show that the trapped waves act to transport angular momentum outward. A plausible way to produce an entropy variation is when an accretion disk is starting from negligible mass and temperature, therefore, negligible entropy. As mass accumulates by either tidal torquing, magnetic torquing, or Roche-lobe overÑow, conÐnement of heat will lead to an entropy maximum at the outer boundary of the disk. Possible nonlinear developments from this instability include the formation of Rossby vortices and the formation of spiral shocks. What remains to be determined from hydrodynamic simulations is whether or not Rossby wave packets (or vortices) "" hold together ÏÏ as they propagate radially inward.

We show that planet formation via both gravitational collapse and core accretion is unlikely to occur in equal mass binary systems with moderate (∼ 50 AU) semimajor axes. Internal thermal energy generation in the disks is sufficient to heat the gas everywhere so that spiral structures quickly decay rather than grow or fragment. This same heating will inhibit dust coagulation because the temperatures rise above the vaporization temperatures of many volatile materials. We consider other processes not included in the model and conclude that our temperatures are conservatively estimated (low), i.e. planet formation is less likely in real systems than in the model.

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We define three requirements for accurate simulations that attempt to model circumstellar discs and the formation of collapsed objects (e.g. planets) within them. First, we define a resolution requirement based on the wavelength for neutral stability of self-gravitating waves in the disc, where a Jeans analysis does not apply. For particle-based or grid-based simulations, this criterion takes the form, respectively, of a minimum number of particles per critical ('Toomre') mass or maximum value of a 'Toomre number', T = δx/λ T , where the wavelength, λ T , is the wavelength for neutral stability for waves in discs. The requirements are analogues of the conditions for cloud collapse simulations as discussed in Bate & Burkert and Truelove et al., where the required minimum resolution was shown to be twice the number of neighbours per Jeans mass or four-five times the local Jeans wavelength, λ J , for particle or grid simulations, respectively.We apply our criterion to particle simulations of disc evolution and find that in order to prevent numerically induced fragmentation of the disc, the Toomre mass must be resolved by a minimum of six times the average number of neighbour particles used. We investigate the origin of the apparent discrepancy between the number of particles required by the cloud and disc fragmentation criteria and find that it is due largely to ambiguities in the definition of the Jeans mass, as used by different authors. We reconcile the various definitions, and when an identical definition of the Jeans mass is used, the condition that J 1/4 in the Truelove condition is equivalent to requiring about 10-12 times the average number of neighbour particles per Jeans mass in a smoothed particle hydrodynamics (SPH) simulation, reducing the difference between simulations of discs and clouds to about two. While the numbers of particles per critical mass are similar for both the Jeans and Toomre formalisms, the Toomre requirement is more restrictive than the Jeans requirement when the local value of the Toomre stability parameter Q falls below about one half.Second, we require that particle-based simulations with self-gravity use a variable gravitational softening, in order to avoid inducing fragmentation by an inappropriate choice of softening length. We show that using a fixed gravitational softening length for all particles can lead either to artificial suppression or enhancement of structure (including fragmentation) in a given disc, or both in different locations of the same disc, depending on the value chosen for the softening length. Unphysical behaviour can occur whether or not the system is properly resolved by the new Toomre criterion.Third, we require that three-dimensional SPH simulations resolve the vertical structure with at least ∼4 particle smoothing lengths per scaleheight at the disc mid-plane, a value which implies a substantially larger number per vertical column because the disc itself extends over many scaleheights. We suggest that a similar criterion applies to grid-based simulations...

We present a series of two-dimensional hydrodynamic simulations of massive disks around protostars. We simulate the same physical problem using both a Piecewise Parabolic Method (PPM) code and a Smoothed Particle Hydrodynamic (SPH) code and analyze their di †erences. The disks studied here range in mass from to and in initial minimum Toomre Q value from 1.1 to 3.0. We adopt 0.05M * 1.0M * simple power laws for the initial density and temperature in the disk with an isothermal (c \ 1) equation of state. The disks are locally isothermal. We allow the central star to move freely in response to growing perturbations. The simulations using each code are compared to discover di †erences due to error in the methods used. For this problem, the strengths of the codes overlap only in a limited fashion, but similarities exist in their predictions, including spiral arm pattern speeds and morphological features. Our results represent limiting cases (i.e., systems evolved isothermally) rather than true physical systems. Disks become active from the inner regions outward. From the earliest times, their evolution is a strongly dynamic process rather than a smooth progression toward eventual nonlinear behavior. Processes that occur in both the extreme inner and outer radial regions a †ect the growth of instabilities over the entire disk. E †ects important for the global morphology of the system can originate at quite small distances from the star. We calculate approximate growth rates for the spiral patterns ; the one-armed (m \ 1) spiral arm is not the fastest growing pattern of most disks. Nonetheless, it plays a signiÐcant role because of factors that can excite it more quickly than other patterns. A marked change in the character of spiral structure occurs with varying disk mass. Low-mass disks form Ðlamentary spiral structures with many arms while high-mass disks form grand design spiral structures with few arms. In our SPH simulations, disks with initial minimum Q \ 1.5 or lower break up into protobinary or protoplanetary clumps. However, these simulations cannot follow the physics important for the Ñow and must be terminated before the system has completely evolved. At their termination, PPM simulations with similar initial conditions show uneven mass distributions within spiral arms, suggesting that clumping behavior might result if they were carried further. Simulations of tori, for which SPH and PPM are directly comparable, do show clumping in both codes. Concerns that the pointlike nature of SPH exaggerates clumping, that our representation of the gravitational potential in PPM is too coarse, and that our physics assumptions are too simple suggest caution in interpretation of the clumping in both the disk and torus simulations.

No abstract

We present a series of two-dimensional (r, /) hydrodynamic simulations of marginally self-gravitating disks with and with disk radius and 100 AU) around protostars (M D /M * \ 0.2, M * \ 0.5 M _ R D \ 50 using a Smoothed Particle Hydrodynamic (SPH) code. We implement simple and approximate prescriptions for heating via dynamical processes in the disk. Cooling is implemented with a simple radiative cooling prescription, which does not assume that local heat dissipation exactly balances local heat generation. Instead, we compute the local vertical (z) temperature and density structure of the disk and obtain a "" photosphere temperature,ÏÏ which is then used to cool that location as a blackbody. We synthesize spectral energy distributions (SEDs) for our simulations and compare them to Ðducial SEDs derived from observed systems, in order to understand the contribution of dynamical evolution to the observable character of a system. We Ðnd that these simulations produce less distinct spiral structure than isothermally evolved systems, especially in approximately the inner radial third of the disk. Pattern amplitudes are similar to isothermally evolved systems farther from the star, but patterns do not collapse into condensed objects. We attribute the di †erences in morphology to increased efficiency for converting kinetic energy into thermal energy in our current simulations. Our simulations produce temperatures in the outer part of the disk that are very low (D10 K). The radial temperature distribution of the disk photosphere is well Ðtted to a power law with index q D 1.1. Far from the star, corresponding to colder parts of the disk and long-wavelength radiation, known internal heating processes (P dV work and shocks) are not responsible for generating a large fraction of the thermal energy contained in the disk matter. Therefore gravitational torques responsible for such shocks cannot transport mass and angular momentum efficiently in the outer disk. Within D5È10 AU of the star, rapid breakup and reformation of spiral structure causes shocks, which provide sufficient dissipation to power a larger fraction of the nearinfrared radiated energy output. In this region, the spatial and size distributions of grains can have marked consequences on the observed near-infrared SED of a given disk and can lead to increased emission and variability on yr timescales. The inner disk heats to the destruction temperature of dust [10 grains. Further temperature increases are prevented by efficient cooling when the hot disk midplane is exposed. When grains are vaporized in the midplane of a hot region of the disk, we show that they do not reform into a size distribution similar to that on which most opacity calculations are based. With rapid grain reformation into the original size distribution, the disk does not emit near-infrared photons. With a plausible modiÐcation of the opacity, it contributes much more.

We present a numerical code for simulating the evolution of astrophysical systems using particles to represent the underlying fluid flow. The code is written in Fortran 95 and is designed to be versatile, flexible, and extensible, with modular options that can be selected either at the time the code is compiled or at run time through a text input file. We include a number of general purpose modules describing a variety of physical processes commonly required in the astrophysical community and we expect that the effort required to integrate additional or alternate modules into the code will be small. In its simplest form the code can evolve the dynamical trajectories of a set of particles in two or three dimensions using a module which implements either a Leapfrog or Runge-Kutta-Fehlberg integrator, selected by the user at compile time. The user may choose to allow the integrator to evolve the system using individual time steps for each particle or with a single, global time step for all. Particles may interact gravitationally as N-body particles, and all or any subset may also interact hydrodynamically, using the smoothed particle hydrodynamic (SPH) method by selecting the SPH module. A third particle species can be included with a module to model massive point particles which may accrete nearby SPH or N-body particles. Such particles may be used to model, e.g., stars in a molecular cloud. Free boundary conditions are implemented by default, and a module may be selected to include periodic boundary conditions. We use a binary "Press" tree to organize particles for rapid access in gravity and SPH calculations. Modules implementing an interface with special purpose "GRAPE" hardware may also be selected to accelerate the gravity calculations. If available, forces obtained from the GRAPE coprocessors may be transparently substituted for those obtained from the tree, or both tree and GRAPE may be used as a combination GRAPE/tree code. The code may be run without modification on single processors or in parallel using OpenMP compiler directives on large-scale, shared memory parallel machines. We present simulations of several test problems, including a merger simulation of two elliptical galaxies with 800,000 particles. In comparison to the Gadget-2 code of Springel, the gravitational force calculation, which is the most costly part of any simulation including self-gravity, is ∼4.6-4.9 times faster with VINE when tested on different snapshots of the elliptical galaxy merger simulation when run on an Itanium 2 processor in an SGI Altix. A full simulation of the same setup with eight processors is a factor of 2.91 faster with VINE. The code is available to the public under the terms of the Gnu General Public License.