An Integral Approach to the Theory of Classical Polytropes

Abstract

A nonlinear Volterra integral equation of the second kind is used instead of conventional Lane-Emden differential equation to represent an alternative approach to finding exact solutions and analytical approximations to solutions of the Lane-Emden equation for classical polytropic models. This approach enables us to reproduce the well - known Lane- Emden (or, just Emden) functions for polytropic indices n=0,1,5 directly or by making use of the Laplace transform, and, being combined with some heuristic reasonings, derive analytical approximations to exact solutions for n = 1.5, 2 and ∞ in closed forms. The proximity of all suggested analytical approximations to the exact solutions are evaluated with the use of the mean square error estimator. Standard deviations are found to be of 10-3 by the order of magnitude. The approximating function of the isothermal density distribution enables us to calculate a theoretical rotation curve that reproduces main features of rotation curves of a set of spiral galaxies. Detailed mathematical calculations will be introduced in an extended paper which is under preparation.