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The Effects of Thermal Energetics on Three-dimensional Hydrodynamic Instabilities in Massive Protostellar Disks

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.