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

Abstract

This article provides for thefirst time a general analytical solution to the Lane-Emden equation of thefirst kind.So far only three known analytical solutions are found in the literature, for the following values ofn: 0, 1 and 5.A common feature these three solutions share is their boundary conditions:∣==θξ()1ξ0and∣=dθ ξdξξ()0=0.Ifathird boundary condition∣=dθξdξξ2()20,=−1is used, only the solution for=n1is able to meet all three. In order toaddress this difference, our solution aims to be more inclusive and takes into account=θξ()ξ1and the constantsolution. By keepingτin parametric form, we found out that=→θξτ(())1ξτ1()whenξ→0. Thus proving that→1ξ1in the origin. It is worth noting that upon integrating the Lane-Emden equation, we came acrossfiveparameters. 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 scientificcommunity related to studies on real stars and exoplanets. The adiabatic exponents are=n1.5,=n2,=n2.592,=n3,=n3.23andn≃5 contained in the intervals 1 <n< 5 and 5≲n< 9. It is worth noting that four of thesecases are of particular importance;=n1.5,which corresponds to an adiabatic star supported by the pressure ofnon-relativistic gas;=n3,which corresponds to an adiabatic star supported by the pressure of an ultra-re-lativistic gas. Finally,=n2.592and=n3.23,which correspond to exoplanets. The obtained solution of theLane–Emden equation of thefirst kind proves valid for any value ofn