The Structure of Self‐gravitating Polytropic Systems with n around 5

Abstract

We investigate the structure of self-gravitating polytropic stellar systems. We present a method that allows us to obtain approximate analytical solutions, of the nonlinear Poisson equation with the t n`v (x) polytropic index n ] v, given the solution with the polytropic index n, for any positive or negative t n (x) v, such that o v o > 1. A similar technique has been developed independently by Seidov & Kuzakhmedov. Application of this method to the spherically symmetric stellar polytropes with n ^ 5 yields the solutions that describe spatially bound systems, if n \ 5, and the formation of a second core, if n [ 5. A heuristic approximate expression for the radial proÐle is also presented. Because of the duality between stellar and gas polytropes, our results are valid for gaseous, self-gravitating, polytropic systems (e.g., molecular clouds) with index c ^ 6/5. The stability of such systems and observational consequences for both stellar and gaseous systems are discussed.