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 \xi .
We adopt two \alpha -parameterizations for the effective viscosity based on ? ) and ? ) 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 _ { \ast } \sim 1.0 ~ { } M _ { \odot } ) protostars .
We find that the effective viscosity can , in principle , simulate the effect of GI in stellar systems with \xi \mathrel { \hbox { \hbox to 0.0 pt { \hbox { \lower 4.0 pt \hbox { $ \sim$ } } } \hbox { $ < $ } } } % 0.2 - 0.3 , thus corroborating a similar conclusion by ? ) that was based on a different \alpha -parameterization .
In particular , the Kratter et al ’ s \alpha -parameterization has proven superior to that of Lin & Pringle ’ s , because the success of the latter depends crucially on the proper choice of the \alpha -parameter .
However , the \alpha -parameterization generally fails in stellar systems with \xi \mathrel { \hbox { \hbox to 0.0 pt { \hbox { \lower 4.0 pt \hbox { $ \sim$ } } } \hbox { $ > $ } } } % 0.3 , particularly in the Class 0 and Class I phases of stellar evolution , yielding too small stellar masses and too large disk-to-star 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 \alpha -parameterization in the case of large \xi 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 \xi \mathrel { \hbox { \hbox to 0.0 pt { \hbox { \lower 4.0 pt \hbox { $ \sim$ } } } \hbox { $ > $ } } } % 0.3 .
A simple modification of the effective viscosity that takes into account disk fragmentation can somewhat improve the performance of \alpha -models in the case of large \xi and even approximately reproduce the mass accretion burst phenomenon , the latter being a signature of the early gravitationally unstable stage of stellar evolution ( ? ) .
However , further numerical experiments are needed to explore this issue .