Polytropic gas spheres: An approximate analytic solution of the Lane-Emden equation

Abstract

Polytropic models play a very important role in galactic dynamics and in the theory of stellar structure and evolution. However, in general, the solution of the Lane-Emden equation can not be given analytically but only numerically. In the Lane-Emden equation can not be given analytically but only numerically. In this paper we give a very good and simple analytic approximate solution of the Lane-Emden equation. This approximation is very good for any finite polytropic index $n$ and for the isothermal case (n equals infinity) at a level $<1\%$. We also give analytic expressions of the mass, pressure, temperature, and potential energy as a function of radius.