An efficient code to solve the Kepler equation

Hyperbolic case^⋆

Space Dynamics Group-Technical University of Madrid, 28040 Madrid, Spain
e-mail: v.raposo.pulido@upm.es, j.pelaez@upm.es

Received: 4 June 2018
Accepted: 6 September 2018

Abstract

Context. This paper introduces a new approach for solving the Kepler equation for hyperbolic orbits. We provide here the Hyperbolic Kepler Equation–Space Dynamics Group (HKE–SDG), a code to solve the equation.

Methods. Instead of looking for new algorithms, in this paper we have tried to substantially improve well-known classic schemes based on the excellent properties of the Newton–Raphson iterative methods. The key point is the seed from which the iteration of the Newton–Raphson methods begin. If this initial seed is close to the solution sought, the Newton–Raphson methods exhibit an excellent behavior. For each one of the resulting intervals of the discretized domain of the hyperbolic anomaly a fifth degree interpolating polynomial is introduced, with the exception of the last one where an asymptotic expansion is defined. This way the accuracy of initial seed is optimized. The polynomials have six coefficients which are obtained by imposing six conditions at both ends of the corresponding interval: the polynomial and the real function to be approximated have equal values at each of the two ends of the interval and identical relations are imposed for the two first derivatives. A different approach is used in the singular corner of the Kepler equation – |M| < 0.15 and 1 < e <  1.25 – where an asymptotic expansion is developed.

Results. In all simulations carried out to check the algorithm, the seed generated leads to reach machine error accuracy with a maximum of three iterations (∼99.8% of cases with one or two iterations) when using different Newton–Raphson methods in double and quadruple precision. The final algorithm is very reliable and slightly faster in double precision (∼0.3 s). The numerical results confirm the use of only one asymptotic expansion in the whole domain of the singular corner as well as the reliability and stability of the HKE–SDG. In double and quadruple precision it provides the most precise solution compared with other methods.