Exact analytic solution of an unsolvable class of first Lane–Emden equation for polytropic gas sphere

Abstract

This article provides for the first time a general analytical solution to the Lane-Emden equation of the first kind. So far only three known analytical solutions are found in the literature, for the following values of n: 0, 1 and 5. A common feature these three solutions share is their boundary conditions: θ(ξ)∣ξ=0=1 and dθ(ξ)dξ∣ξ=0=0. If a third boundary condition d2θ(ξ)dξ2∣ξ=0,= −1 is used, only the solution for n=1 is able to meet all three. In order to address this difference, our solution aims to be more inclusive and takes into account θ(ξ)=1ξ and the constant solution. By keeping τ in parametric form, we found out that θ(ξ(τ))=1ξ(τ)→1 when ξ → 0. Thus proving that 1ξ→1 in the origin. It is worth noting that upon integrating the Lane-Emden equation, we came across five parameters. Three of them depend on the three boundary conditions used and two can be adjusted numerically.In order to demonstrate the validity of our solution, we tested it on six cases of interest to the scientific community related to studies on real stars and exoplanets. The adiabatic exponents are n=1.5,n=2,n=2.592, n=3,n=3.23 and n ≃ 5 contained in the intervals 1 < n < 5 and 5 ≲ n < 9. It is worth noting that four of these cases are of particular importance; n=1.5, which corresponds to an adiabatic star supported by the pressure of non-relativistic gas; n=3, which corresponds to an adiabatic star supported by the pressure of an ultra-relativistic gas. Finally, n=2.592 and n=3.23, which correspond to exoplanets. The obtained solution of the Lane–Emden equation of the first kind proves valid for any value of n.