Secular evolution of viscous and self-gravitating circumstellar discs

We add the effect of turbulent viscosity via the alpha-prescription to models of the self-consistent formation and evolution of protostellar discs. Our models are non-axisymmetric and are carried out using the thin-disc approximation. Self-gravity plays an important role in the early evolution of a disc, and the later evolution is determined by the relative importance of gravitational and viscous torques. In the absence of viscous torques, a protostellar disc evolves into a self-regulated state with the Toomre parameter Q similar to 1.5-2.0, non-axisymmetric structure diminishing with time and maximum disc-to-star mass ratio xi = 0.14. We estimate an effective viscosity parameter alpha(eff) associated with gravitational torques at the inner boundary of our simulation to be in the range 10(-4)-10(-3) during the late evolution. The addition of viscous torques with a low value alpha = 10(-4) has little effect on the evolution, structure and accretion properties of the disc, and the self-regulated state is largely preserved. A sequence of increasing values of alpha results in the discs becoming more axisymmetric in structure, being more gravitationally stable, having greater accretion rates, larger sizes, shorter lifetimes and lower disc-to-star mass ratios. For alpha = 10(-2), the model is viscous-dominated, and the self-regulated state largely disappears by late times. The axisymmetry and low surface density of this model may contrast with observations and pose problems for planet formation models. The use of alpha = 0.1 leads to very high disc accretion rates and rapid (within 2 Myr) depletion of the disc, and seems even less viable observationally. Furthermore, only the non-viscous-dominated models with low values of alpha = 10(-4)-10(-3) can account for an early phase of quiescent low accretion rate (M)over dot similar to 10(-8) M-circle dot yr(-1) (interspersed with accretion bursts) that can explain the recently observed Very Low luminosity Objects (VeLLOs). We also find that a modest increase in disc temperature caused by a stiffer barotropic equation of state (gamma = 1.67) has little effect on the disc accretion properties averaged over many disc orbital periods (similar to 10(4) yr), but can substantially influence the instantaneous mass accretion rates, particularly in the early embedded phase of disc evolution.