An Alternative Solution to Kepler’s Equation

Abstract

This paper presents an easy and efficient algorithm for solving Kepler’s equation. The main body of the algorithm uses the Newton-Raphson method to iteratively find the solution. The contribution herein is the introduction of an initial condition so close to the solution that results in four iterations or less for an error of 10 − 10 rad. With little effort, the initial conditions could lead to a number of iterations of three or less. This initial condition enables solving the equation for any eccentricity or anomaly, regardless of their values. This is done by selecting two points close to the perceived solution to Kepler’s equation, from which we interpolate to get the initial condition. This method is called the linear method. Another method, called the quadratic, is one in which we select three points close to the perceived solution and interpolate to get a close initial condition. Both methods are tested and compared against all possible conditions and are found to perform favorably even for near-parabolic and parabolic cases, given in detail below.