Modification of Gravitational Instability Criteria for Astrophysical Disks within Nonadditive Thermodynamics

In contrast to several classical studies, in which gravitational instability criteria for astrophysical disks are derived within traditional hydrodynamics or kinetics, it is proposed to consider the set of loose gas-dust clusters of an accretion protoplanetary disk as a special type of continuous medium, i.e., a fractal medium whose phase velocity space contains points and areas not filled by its components. Within the Tsallis formalism of nonadditive statistics (thermodynamics), intended to describe the behavior of anomalous systems, i.e., systems with a strong gravitational interaction of its individual parts and the fractal nature of the phase space, linearized equations are obtained for oscillations of a solid-state rotating disk on the basis of modified Navier-Stokes hydrodynamic equations (the so-called q-hydrodynamics equations) and in view of dissipative effects, and a derivation is given of the dispersion equation in the WKB approximation. An analysis is conducted of axisymmetric oscillations of a differentially rotating astrophysical gas-dust space object to obtain modified Jeans and Toomre gravitational instability criteria for disks with a fractal structure.