1. Solution of Lame-Emden equation and 2.) Determining the potential of a cylindrical system using the modified bessel functions
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