The applicability of the viscous α-parameterization of gravitational instability in circumstellar disks

We study numerically the applicability of the effective-viscosity approach for simulating the effect of gravitational instability (GI) in disks of young stellar objects with different disk-to-star mass ratios ξ. We adopt two α-parameterizations for the effective viscosity based on Lin & Pringle (1990) and Kratter et al. (2008) and compare the resultant disk structure, disk and stellar masses, and mass accretion rates with those obtained directly from numerical simulations of self-gravitating disks around low-mass (M∗ ∼ 1.0 M⊙) protostars. We find that the effective viscosity can, in principle, simulate the effect of GI in stellar systems with ξ ∼< 0.2−0.3, thus corroborating a similar conclusion by Lodato & Rice (2004) that was based on a different α-parameterization. In particular, the Kratter et al’s α-parameterization has proven superior to that of Lin & Pringle’s, because the success of the latter depends crucially on the proper choice of the α-parameter. However, the α-parameterization generally fails in stellar systems with ξ ∼> 0.3, particularly in the Class 0 and Class I phases of stellar evolution, yielding too small stellar masses and too large disk-tostar mass ratios. In addition, the time-averaged mass accretion rates onto the star are underestimated in the early disk evolution and greatly overestimated in the late evolution. The failure of the α-parameterization in the case of large ξ is caused by a growing strength of low-order spiral modes in massive disks. Only in the late Class II phase, when the magnitude of spiral modes diminishes and the mode-to-mode interaction ensues, may the effective viscosity be used to simulate the effect of GI in stellar systems with ξ ∼> 0.3. A simple modification of the effective viscosity that takes into account disk fragmentation can somewhat improve the performance of αmodels in the case of large ξ and even approximately reproduce the mass accretion burst phenomenon, the latter being a signature of the early gravitationally unstable stage of stellar evolution (Vorobyov & Basu, 2006). However, further numerical experiments are needed to explore this issue.