A Simple Explicit Solution for the Kepler Problem

摘要

In order to determine the position of a planet in its orbit about the Sun we have to solve Kepler's equation. This is a transcen-dental equation and no completely satisfactory mathematical-solution has ever been determined. Numerical methods of varying degrees of sophistication have been used. They will deliver an answer but can never reveal the underlying structure of the solution. Lagrange determined a complicated solution in the form of an infinite series of Bessel functions, but this solution is only par-tially successful for reasons that are given subsequently. What is required is a closed mathematical formula that gives the solution for all values of the parameters involved. In this paper we deter-mine such a solution and then compare it with the answer given by Lagrange. We show, in effect, that the Lagrange series solution has a sum-function of a simple type. To fully illustrate the procedure we also calculate the orbit of Halley's Comet since perihelion on 9 February 1986 and add a few final comments.