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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, ph, 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.

In this paper, the effects of thermal energetics on the evolution of gravitationally unstable protostellar disks are investigated by means of three-dimensional hydrodynamic calculations. The initial states for the simulations correspond to stars with equilibrium, self-gravitating disks that are formed early in the collapse of a uniformly rotating, singular isothermal sphere. In a previous paper (Pickett et al.), it was shown that the nonlinear development of locally isentropic disturbances can be radically different than that of locally isothermal disturbances, even though growth in the linear regime may be similar. When multiple low-order modes grew rapidly in the star and inner disk region and saturated at moderate nonlinear levels in the isentropic evolution, the same modes in the isothermal evolution led to shredding of the disk into dense arclets and ejection of material. In this paper, we (1) examine the fate of the shredded disk with calculations at higher spatial resolution than the previous simulations had and (2) follow the evolution of the same initial state using an internal energy equation rather than the assumption of locally isentropic or locally isothermal conditions. Despite the complex structure of the nonlinear features that developed in the violently unstable isothermal disk referred to above, our previous calculation produced no gravitationally independent, long-lived stellar or planetary companions. The higher resolution calculations presented here confirm this result. When the disk of this model is cooled further, prompting even more violent instabilities, the end result is qualitatively the same--a shredded disk. At least for the disks studied here, it is difficult to produce condensations of material that do not shear away into fragmented spirals. It is argued that the ultimate fate of such fragments depends on how readily local internal energy is lost. On the other hand, if a dynamically unstable disk is to survive for very long times without shredding, then some mechanism must mitigate and control any violent phenomena that do occur. The prior simulations demonstrated a marked difference in final outcome, depending upon the efficiency of disk cooling under two different, idealized thermal conditions. We have here incorporated an internal energy equation that allows for arbitrary heating and cooling. Simulations are presented for adiabatic models with and without artificial viscosity. The artificial viscosity accounts for dissipation and heating due to shocks in the code physics. The expected nonaxisymmetric instabilities occur and grow as before in these energy equation evolutions. When artificial viscosity is not present, the model protostar displays behavior between the locally isentropic and locally isothermal cases of the last paper; a strong two-armed spiral grows to nonlinear amplitudes and saturates at a level higher than in the locally isentropic case. Since the amplitude of the spiral disturbance is large, it is expected that continued transport of material and angular momentum will occur well after the end of the calculation at nearly four outer rotation periods. The spiral is not strong enough, however, to disrupt the disk as in the locally isothermal case. When artificial viscosity is present, the same disturbances reach moderate nonlinear amplitude, then heat the gas, which in turn greatly reduces their strength and effects on the disk. Additional heating in the low-density regions of the disk also leads to a gentle flow of material vertically off the computational grid. The energy equation and high-resolution isothermal calculations are used to discuss the importance and relevance of the different thermal regimes so far examined, with particular attention to applications to star and planet formation.

We present a simplified analysis of some effects of disk accretion on the early evolution of fully convective, low-mass pre-main-sequence stars. Our analysis builds on the previous seminal work of Stahler, but it differs in that the accretion of material occurs over a small area of the stellar surface, such as through a disk or magnetospheric accretion column, so that most of the stellar photosphere is free to radiate to space. This boundary condition is similar to the limiting case considered by Palla & Stahler for intermediate-mass stars. We argue that for a wide variety of disk mass accretion rates, material will be added to the star with relatively small amounts of thermal energy. Protostellar evolution calculated assuming this "low-temperature" limit of accretion generally follows the results of Stahler because of the thermostatic nature of deuterium fusion, which prevents protostars from contracting below a "birthline" in the H-R diagram. Our calculated protostellar radii tend to fall below Stahler's at higher masses; the additional energy loss from the stellar photosphere in the case of disk accretion tends to make the protostar contract. The low-temperature disk accretion evolutionary tracks never fall below the deuterium-fusion birthline until the internal deuterium is depleted, but protostellar tracks can lie above the birthline in the H-R diagram if the initial radius of the protostellar core is large enough or if rapid disk accretion (such as might occur during FU Ori outbursts) adds significant amounts of thermal energy to the star. These possibilities cannot be ruled out by either theoretical arguments or observational constraints at present, so that individual protostars might evolve along a multiplicity of birthlines with a modest range of luminosity at a given mass. Our results indicate that there are large uncertainties in assigning ages for the youngest stars from H-R diagram positions, given the uncertainty in birthline positions. Our calculations also suggest that the relatively low disk accretion rates characteristic of T Tauri stars below the birthline cause low-mass stars to contract only slightly faster than normal Hayashi track evolution, so that ages for older pre-main-sequence stars estimated from H-R diagram positions are relatively secure.

We use numerical three-dimensional hydrodynamics to investigate how assumptions about local thermal conditions affect the strength and outcome of nonaxisymmetric instabilities in massive protostellar disks. Building on work presented in earlier papers, we generate two protostellar core models that represent equilibrium states that could form from the axisymmetric collapse of uniformly rotating, singular isothermal spheres. Both models are continuous star/disk systems, in which the star, the disk, the star/disk boundary, and the free disk outer boundary are resolved in three dimensions. The models are distinguished primarily by the temperature distribution in the disk, and both can be considered to represent the same early evolutionary stage of disk development, when the disk is massive but small in radial extent. In the "hot" model, the disk is assumed to have the same entropy per gram as the central isentropic star, giving a Toomre Q-parameter ~2.5 over the disk region. In the "cool" model, the entropy per gram decreases radially outward in the disk, resulting in more realistic, cooler disk temperatures and Q [?] 1.5. Each of these protostellar star/disk systems is evolved in our three-dimensional hydrodynamics code under two different assumptions about thermal equilibrium in the disk, namely that either the entropy per gram or the temperature remains constant with position in the disk. We refer to these two cases as locally isentropic evolution and locally isothermal evolution, respectively. All four calculations have been run for at least two outer rotation periods of the disk. With either assumption about the thermal equilibrium, one- and two-armed spiral disturbances, which grow in the hot models, saturate at low amplitude (~1%) and do not alter the protostellar core significantly. On the other hand, the cool model is highly unstable to multiple low-order spirals, which induce significant mass and angular momentum transport in a few dynamical times. Under locally isentropic evolution, the star and star/disk boundary in the cool model are unstable to three- and four-armed disturbances and the disk is unstable to a two-armed spiral, but all these modes saturate at moderate nonlinear (a few tens percent) amplitudes after about 1.5 outer rotation periods. The same instabilities occur under locally isothermal evolution; however, the two-armed spiral in the disk grows more vigorously and does not saturate, ultimately disrupting the disk and concentrating material into thin, dense arcs and arclets that approach stellar densities. In both cool model calculations, there is substantial inward transport of mass and outward transport of angular momentum during the growth phase of the two-armed spiral, but the transport rate drops by over an order of magnitude for locally isentropic evolution when the two-armed spiral saturates. It is clear from these calculations that thermal energetics play a critical role in the development of self-gravitating instabilities and that, under conditions of strong cooling, such instabilities can disrupt a disk very early in its development. We compare these calculations with previous work on gravitational instabilities in disks and discuss implications for star and planet formation.

The heating and vaporization of dust grains in the protosolar environment is modeled in order to assess the survivability of interstellar solids during the formation of the solar system. A multidimensional, discrete ordinate radiative transfer code is used to compute thermal transport in the collapsing protosolar cloud. The results are combined with estimates of heating at the shock where infalling material arrives at the surface of the solar nebula/accretion disk, and in the interior of the disk, to determine the distances at which various solid phases are vaporized. The thermal coupling between the envelope and the accretion disk (backheating) is treated self-consistently, so its effect on the disk's radial temperature profile is included. This treatment also permits evaluation of the effect of backheating on the observational inference of disk properties.