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, φ, 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 disk's 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.