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 which allows to obtain approximate analytical solutions, $ψ_{n+ε}({\bf x})$, of the nonlinear Poisson equation with the polytropic index $n+ε$, given the solution $ψ_n({\bf x})$ with the polytropic index n, for any positive or negative $ε$ such that $|ε|\ll1$. Application of this method to the spherically symmetric stellar polytropes with $n\simeq5$ 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 $γ\simeq 6/5$. Stability of such systems and observational consequences for both stellar and gaseous systems are discussed.