On the evolution of gravitationally unstable protostellar disks

2D, N-body simulations of the nonlinear development of long-wavelength instabilities predicted to occur early in the formation stage of protostellar disks forming single stars are presented under conditions of controlled energy loss. It is found that the thermodynamics and long-term evolution of gravitationally unstable disks are intimately coupled: the global evolution rate is linearly proportional to the rate at which dissipated energy reaches and escapes from the surfaces of the disk. An upper limit is obtained to the rate at which energy is lost if it is assumed that the disk maintains vertical isothermality.