Series solutions for polytropes and the isothermal sphere

Christopher Hunter

The Lane--Emden equation for polytropic index $n>1$ and its $n \to \infty$ limit of the isothermal sphere equation are singular at some {\it negative} value of the radius squared. This singularity prevents the real power series solutions about the centre from converging all the way to the outer surface when $n>1.9121$. However, a simple Euler transformation gives series that do converge all the way to the outer radius. These Euler--transformed series converge significantly faster than the series in the contained mass derived by Roxburgh \& Stockman (1999), which are limited to finite radii whenever $n>5$ by a complex conjugate pair of singularities. We construct some compact analytical approximations to the isothermal sphere, and give one for which the density profile is accurate to 0.001 percent out to the limit of stability against gravothermal collapse.