Kepler's equation and some of its pearls

Kepler's equation, rarely discussed in undergraduate textbooks, was enunciated by Johannes Kepler in his Astonomia Nova, published in 1609, much before the advent of the integral and differential calculus. The search for its solutions challenged the minds of brilliant researchers like Newton, Lagrange, Cauchy, and Bessel, among others. In this work, we start with a standard derivation of Kepler's equation and emphasize how it gave rise to new mathematics, like approximation methods, Bessel functions, and complex analysis. Then we apply it in two non-trivial examples. In the first one, we compute the distance reached by a projectile launched from a point at the equator of the rotating Earth. This result could be used to prove the rotation of the Earth without the need of a Foucault pendulum. In the second example, we show how two astronauts moving around the Earth along the same circular orbit could exchange a sandwich. These two apparently innocent problems are quite involved because their solutions demand the calculations of the time of flight.