The Effects of Thermal Energetics on Three‐dimensional Hydrodynamic Instabilities in Massive Protostellar Disks. II. High‐Resolution and Adiabatic Evolutions

Abstract

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.