The Structure of Self-gravitating Polytropic Systems with N around 5

We investigate the structure of self-gravitating polytropic stellar systems. We present a method which allows to obtain approximate analytical solutions, ψn+ǫ(x), of the nonlinear Poisson equation with the polytropic index n+ ǫ, given the solution ψn(x) with the polytropic index n, for any positive or negative ǫ such that |ǫ| ≪ 1. Application of this method to the spherically symmetric stellar polytropes with n ≃ 5 yields the solutions which describe spatially bound systems if n < 5 and the formation of a second core if n > 5. A heuristic approximate expressions for the radial profiles are also presented. Due to the duality between stellar and gas polytropes, our results are valid for gaseous, self-gravitating polytropic systems (e.g., molecular clouds) with the index γ ≃ 6/5. Observational consequences for both stellar and gaseous systems are discussed. Subject headings: galaxies: kinematics and dynamics — galaxies: star clusters — stars: kinematics — stars: formation — ISM: structure — ISM: clouds