In astrophysics, the Lane-Emden equation, named after Jonathan Homer Lane and Robert Emden, describes the structure of a star with a polytropic equation of state P = Kρ(n+1)/n. This equation is a dimensionless form of Poisson’s equation for the gravitational potential of a Newtonian self-gravitating, spherically symmetric, polytropic fluid and therefore, it can be applied to different types of stars by varying n: the polytropic index.
In Section 2.1, we will show the case when n = 1, and the relevant physical solution applicaple to the system. Section 2.2 will consider some general n from where the n = 1 case can be retrieved. Lastly, in Section 2.3, we will demonstrate different polytropic indices that do not have an analytic solution to them, but instead we direct ourselves the numerical approach. Graphs are presented at the end of the section. In section 3 we present conclusions from the aftermath. Finally, two codes are presented in the appendix, we show the numerical output of one of them and give references.
For the system of a point charge q located at the point (ρ, φ, z) inside a grounded cylindrical box bounded by surfaces z=0, z=L, ρ = a, we have derived the Green’s function and thereby the potential inside the box using the modified Bessel functions of the first and second kinds: In(x) and Kn(x), respectively. Using this result, we find the potential inside a cylindrical box held at zero potential at all surfaces except for a disc in the upper end of radius b, where b < a, held at potential V. In the case that ρ = 0, z=L/2, and b=L/4=a/2, we find the ratio Φ/V to 10 significant figures to be Φ/V = 0.0715293729