Approximate analytic solutions to the isothermal Lane–Emden equation

We derive accurate analytic approximations to the solution of the isothermal Lane–Emden equation, a basic equation in Astrophysics that describes the Newtonian equilibrium structure of self-gravitating, isothermal fluid spheres. The solutions we obtain, using analytic arguments and rational approximations, have simple forms, and are accurate over a radial extent that is much larger than that covered by conventional series expansions around the origin. Our best approximation has a maximum error on density of 0.04 % at 10 core radii, and is still within 1 % from an accurate numerical solution at a radius three times larger.