Modification of the Jeans and Toomre instability criteria for astrophysical fractal objects within nonextensive statistics

Within the formalism of Tsallis nonextensive statistics designed to describe the behavior of anomalous systems, systems with a strong gravitational interaction between their individual parts and the fractal nature of phase space, we have obtained linearized equations for the oscillations of a rigidly rotating disk by taking into account dissipative effects and give a derivation of the dispersion equation in the WKB approximation. Based on the previously derived modified Navier-Stokes hydrodynamic equations (the so-called equations of q-hydrodynamics), we have analyzed the axisymmetric oscillations of an astrophysical, differentially rotating gas-dust cosmic object and obtained modified Jeans and Toomre gravitational instability criteria for disks with a fractal phase-space structure.