For a given value of the polytropic index n , denote the solution to the Lane–Emden equation as θ θ n ( ξ ξ ) . In general, the Lane–Emden equation must be solved numerically to find θ θ n . There are exact, analytic solutions for certain values of n , in particular: n = 0 , 1 , 5 . For n between 0 and 5, the solutions are continuous and finite in extent, with the radius of the star given by R = α α ξ ξ 1 , where θ θ n ( ξ ξ 1 ) = 0 .