From Kepler to Bessel

The previous post very briefly said that the integral representation for Bessel functions was motived by solving Kepler’s equation. This post will go into more detail. Kepler’s equation There are multiple ways to describe the position of a planet in an elliptical orbit around a star. For historical reasons, these descriptions have arcane names such as mean anomaly, true anomaly, and eccentric anomaly. This post explains how these three are related. For this post, it is enough to say that often you know mean anomaly M and want to know eccentric anomaly E. These are related via Kepler’s equation where e is the eccentricity of the orbit. You’d like to solve for E as a function of M and e, but there’s no elementary way to do that. One way to solve Kepler’s equation is to take a guess at E and plug it into the right hand side of to get a new E, and keep iterating until the two sides are closer together. I write more about this here. Another approach to solving Kepler’s equation is to use…