Family of spherical models with special gravitational properties

A new method for studying the structural and gravitational properties of spherical systems based on an analysis of the ratio of the potentials for their subsystems and shells has been developed. It has been proven for the first time that the gravitational virial Z(r) of the subsystem without allowance for the influence of the outer shell is equal to twice the work done to disperce the subsystem’s matter to infinity. A new class of spherical models has been constructed in which: (1) the ratio of the contribution to the potential at point r from the spherical subsystem to the contribution from the outer shell does not depend on radius and is equal to a constant γ; (2) the ratio of the gravitational energy W(r) to Z(r) for the spherical subsystem does not depend on r; and (3) the models are described by a power law of the density ρ = cr−κ and potential \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi (r) = \frac{{4\pi Gc}} {{(\kappa - 2)(3 - \kappa )}}r^{2 - \kappa } \left( {2 \leqslant \kappa \leqslant \frac{5} {2}} \right)$$\end{document}. Expressions for the gravitational energy W(r) and virial Z(r) have been found for the subsystem. The limiting case of ρ(r) ∝ r−5/2, where the subsystem’s potential at any sampling point is exactly equal to the potential from the outer shell and Z(r) is equivalent to its gravitational energy W(r), is considered in detail. The results supplement the classical potential theory. The question about the application of the models to the superdense nuclear star cluster in the center of the Milky Way is discussed.