Solution of the Kepler’s Equation with Four Iterations or Less

This paper presents an easy and efficient algorithm for solving the Kepler's equation. The main body of the algorithm uses the Newton-Raphson method to iteratively finding the solution. But, the contribution herein is the introduction of an initial condition so close to the solution that will result in four iterations or much less. This initial condition enables solving the equation for any eccentricity and for any anomaly regardless of its value. This is done by selecting two points close to the solution of the Kepler 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 true 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 cases as shown herein.

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The authors declare no conflict of interest.

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The datasets used in this study are openly available at [repository link] and the source code is available on GitHub at [GitHub link].

This work did not receive any external funding.