On the definition of the power-law of particle distribution and of the criterion of thermal stability for self-gravitating astrophysical systems within the limits of non-extensive kinetics

We propose a procedure of introduction of the newtonian potential of self-gravitation and the centrifugal potential in quasi-equilibrium of particle distribution in the phase space, received within the non-extensive statistics on the basis of the modified kinetic equation of Boltzmann when averaged on non-normalized distribution. It is shown that if power distribution satisfies to the kinetic stationary equation this the latter imposes clearly defined restrictions on the character of a long-range force field, and on possible dependence of hydrodynamic parameters on coordinates, thereby defining these parameters uniquely. The thermodynamic stability criterion of equilibrium of a non-extensive system is given. The obtained results allow to model more adequately the evolution of gaseous astrophysical systems, and, in particular, the gravitational stability of twirled protoplanetary accretion disks.