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This work is part of the project entitled Dynamical Analysis of Complex Interplanetary Missions with reference ESP2017-87271-P, supported by Spanish Agencia Estatal de Investigacion (AEI) (Spanish Research Agency) of Ministerio de Economia, Industria y Competitividad (MINECO) (Ministry of Economy, Industry and Competitiveness) and by European Found of Regional Development (FEDER). Authors thank to the referee for the constructive comments and suggestions which helped to improve the paper.

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An efficient code to solve the Kepler equation Hyperbolic case

Publicado en:Astronomy & Astrophysics. 619 (A129): - 2018-11-01 619(A129), DOI: 10.1051/0004-6361/201833563

Autores: Raposo-Pulido, V.; Pelaez, J.;

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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 - vertical bar M vertical bar < 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 (similar to 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 (similar to 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.

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