Very-large-scale spectral solutions for spherical polytropes of index m > 5 and the isothermal sphere

In view of astrophysical applications, we obtain very large scale numerical solutions to the Lane-Emden equations for spherical polytropes of index m > 5 and for the isothermal sphere (m=∞), by considering ϕ−1, the inverse function of the gravitational potential. Since the domain of ϕ−1 is bounded, and the asymptotic behavior of its end-point singularities is known, highly accurate solutions can be obtained by spectral collocation methods. This leads to solutions for ϕ that extend accurately to extremely large radii, well beyond those achieved by traditional numerical schemes on [0, ∞). As a reference, we include a table of values for the isothermal sphere (Lane-Emden function of the 2nd kind) spanning r=10−4 through r=10150 with at least nine significant figures. The corresponding semi-analytical asymptotic solution as r → ∞ is ϕa(r)=lnr22+1.1779636±1×10−7r1/2cos(72lnr+1.0623271±1×10−7).