Self-Similar Evolutionary Solutions of Self-Gravitating, Polytropic $\beta$-Viscous Disks

We carry out the effect of $\beta$-prescription for viscosity which introduced by Duschel et al. 2000 & Hure, Richard & Zhan 2001, in a standard self-gravitating thin disks. We were predicted in a self-gravitating thin disk the $\beta$-model will have different dynamical behavior compare the well known $\alpha$-prescriptions. We used self-similar methods for solving the integrated equations which govern the dynamical behavior of the thin disks. We present the results of self-similar solutions of the time evolution of axisymmetric, polytropic, self-gravitating viscous disks around a new born central object. We apply a $\beta$-viscosity prescription which has been derived from rotating shear flow experiments ($\nu=\beta r^2\Omega$). Using reduced equations in a slow accretion limit, we demonstrate inside-out self-similar solutions after core formation in the center. Some physical quantities for $\beta$-disks are determined numerically.We have compared our results with $\alpha$-disks under the same initial conditions. It has been found that the accretion rate onto the central object for $\beta$-disks more than $\alpha$-disks at least in the outer regions where $\beta$-disks are more efficient. Our results show that Toomre instability parameter is less than one everywhere on the $\beta$-disk which means that in such disks gravitational instabilities can be occurred, so the $\beta$-disk model can be a good candidate for the origin of planetary systems. Our results show that the $\beta$-disks will decouple in the outer part of the disk where the self-gravity plays an important role which is in agreement with Duschl predictions.