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

We carry out the effect of $β$-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 $β$-model will have different dynamical behavior compare the well known $α$-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 $β$-viscosity prescription which has been derived from rotating shear flow experiments ($ν=βr^2Ω$). 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 $β$-disks are determined numerically.We have compared our results with $α$-disks under the same initial conditions. It has been found that the accretion rate onto the central object for $β$-disks more than $α$-disks at least in the outer regions where $β$-disks are more efficient. Our results show that Toomre instability parameter is less than one everywhere on the $β$-disk which means that in such disks gravitational instabilities can be occurred, so the $β$-disk model can be a good candidate for the origin of planetary systems. Our results show that the $β$-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.