The effects of thermal energetics on three-dimensional hydrodynamic instabilities in massive protostellar disks

We use numerical three-dimensional hydrodynamics to investigate how assumpt ions about local thermal conditions affect the strength and outcome of nona xisymmetric instabilities in massive protostellar disks. Building on work p resented in earlier papers, we generate two protostellar core models that r epresent equilibrium states that could form from the axisymmetric collapse of uniformly rotating, singular isothermal spheres. Both models are continu ous 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 mod els 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 a s the central isentropic star, giving a Toomre Q-parameter similar to 2.5 o ver the disk region. In the "cool" model, the entropy per gram decreases ra dially outward in the disk, resulting in more realistic, cooler disk temper atures and Q approximate to 1.5. Each of these protostellar star/disk syste ms is evolved in our three-dimensional hydrodynamics code under two differe nt assumptions about thermal equilibrium in the disk, namely that either th e entropy per gram or the temperature remains constant with position in the disk. We refer to these two cases as locally isentropic evolution and loca lly isothermal evolution, respectively. All four calculations have been run for at least two outer rotation periods of the disk. With either assumptio n about the thermal equilibrium, one- and two-armed spiral disturbances, wh ich grow in the hot models, saturate at low amplitude (similar to 1%) and d o not alter the protostellar core significantly. On the other hand, the coo l model is highly unstable to multiple low-order spirals, which induce sign ificant 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 un stable to a two-armed spiral, but all these modes saturate at moderate nonl inear (a few tens percent) amplitudes after about 1.5 outer rotation period s. The same instabilities occur under locally isothermal evolution; however , the two-armed spiral in the disk grows more vigorously and does not satur ate, ultimately disrupting the disk and concentrating material into thin, d ense arcs and arclets that approach stellar densities. In both cool model c alculations, there is substantial inward transport of mass and outward tran sport 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 isen tropic evolution when the two-armed spiral saturates. It is clear from thes e calculations that thermal energetics play a critical role in the developm ent of self-gravitating instabilities and that, under conditions of strong cooling, such instabilities can disrupt a disk very early in its developmen t. We compare these calculations with previous work on gravitational instab ilities in disks and discuss implications for star and planet formation.