Modification in the framework of nonadditive Tsallis statistics of the gravitational instability criterions of astrophysical disks

Unlike classical studies in which the gravitational instability criterions for astrophysical disks are derived in the framework of traditional kinetics or hydrodynamics, we propose to totality of fluffy dust clusters of various astrophysical objects, in particular, protoplanetary subdisks, as a special type of continuous medium, i.e., fractal medium for which there are points and areas not filled with its components. Within the deformed Tsallis statistics formalism, which is intended to describe the behavior of anomalous systems with strong gravitational interaction and fractal nature of phase space, we derive, on the basis of the modified hydrodynamic equations of Navier–Stokes (the so-called equations $q$-hydrodynamics) the linearized equations of oscillations a solid-state rotating disk and the conclusion of the dispersing equation in VKB-approach is given. Considering the linearization of the $q$-hydrodynamics equations for viscosity solid-state rotating clouds we investigate the instability of an infinitely homogeneous medium to obtain a simplified version of the modified gravitational instability Jeans and Toomre criterions for an astrophysical disk with fractal structure.