From Kepler to Bessel
How Kepler's equation led to the definition of Bessel functions
https://www.johndcook.com/blog/2026/06/06/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 Newton’s method. I write more about that here.
Still approach is to expand _E_ in a sine series and find the series coefficients. An advantage to this approach is that once we have the coefficients, we now have an expression for _E_ and a function of _M_ , and we can plug in more values of _M_ without having to solve Kepler’s equation for each value of _M_ separately.
## Sine series coefficients
Kepler’s equation is easy to solve at _E_ = 0 and at _E_ = π. In both cases, _E_ = _M_. So the function _E_ − _M_ is zero at both ends of [0, π], which suggests we try to expand _E_ − _M_ in a sine series
We then calculate the Fourier coefficients _a_ _n_ as usual.
The second line uses integration by parts. The third line uses Kepler’s equation. The last line uses the definition of the Bessel functions _J_ _n_ given in the previous post.
