Power series solutions of the polytrope equations

We derive recurrence relations for the coefficients ak in the power series expansion θ(ξ)=∑ akξ^2k of the solution of the Lane—Emden equation, and examine the convergence of these series. For values of the polytropic index n1≈1.9 the series appear to converge everywhere inside the star. For n>n1 the series converge in the inner part of the star but then diverge. We also derive the series expansions for θ, ξ in powers of m=q^2/3, where q=-ξ²dθ/dξ is the polytropic mass. These series appear to converge everywhere within the star for all n ≤ 5. Finally we show that θ(ξ) can be satisfactorily approximated (∼ 1 per cent) by (1-cξ²)/(1+eξ²)^m, and give the values of the constants determined by a Padé approximation to the series, and by a two-parameter fit to the numerical solutions.