Theory of Classical Gaseous Polytropes in an Integral Representation. II. Analytic Approximations to the Emden Functions and Density Profiles in a Closed Form

Analytic approximations are presented for the exact solutions of the Volterra type nonlinear integral equation of the second kind for classical gaseous polytropes in closed form. This equation is considered as the integral equivalent of the Lane-Emden differential equation with boundary conditions which describes the standard polytropic models in terms of a Cauchy problem. With the aid of a linear approximation of this equation and general heuristic considerations of a physical character, as well as with the aid of a graphical model and variation of the parameters of the approximating functions, approximate expressions for the Emden functions and the dimensionless density are obtained in closed form with a mean square accuracy from ~10^-4 to a few percent for a series of values of the polytropic index n of practical interest (n = 0.5, 3, 4, 6, ∞). Our previous approximation for the spatial density of the isothermal model is compared with a pseudo-isothermal law describing the distribution of the density of dark matter surrounding spiral galaxies and used by various authors for studying their rotation curves.