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An efficient code to solve the Kepler equation for elliptic and hyperbolic orbits
Research Areas
  • Aeronautical engineering
The Kepler equation for the elliptical motion, y ? e sin y ? x = 0, involves a nonlinear function depending on three parameters: the eccentric anomaly y = E, the eccentricity e and the mean anomaly x = M. For given e and x values the numerical solution of the Kepler equation becomes one of the goals of orbit propagation to provide the position of the object orbiting around a body for some specific time. In this paper, a new approach for solving Kepler equation for elliptical and hyperbolic orbits is developed. This new approach takes advantage of the very good behavior of the Laguerre method when the initial seed is close to the looked for solution and also of the existence of symbolic manipulators which facilitates the obtention of polynomial approximations. The central idea is to provide an initial seed as good as we can to the modified Newton-Raphson method, because when the initial guess is close to the solution, the algorithm is fast, reliable and very stable. To determine a good initial seed the domain of the equation is discretized in several intervals and for each one of these intervals a fifth degree interpolating polynomial is introduced. The six coefficients of the polynomial are obtained by requiring six conditions at both ends of the corresponding interval. Thus the real function and the polynomial have equal values at both ends of the interval. Similarly relations are imposed for the two first derivatives. Consequently, given e and x = M, selecting the interval [xi, xi+1] in such a way that M belongs to [xi, xi+1] and using the corresponding polynomial pi(x), we determine the starter value y0 = E0. However, the Kepler equation has a singular behavior when M is small and e close to unity (singular corner). In this case, the exact solution of the equation has to be described in a different way to guarantee the enough accuracy to be part of the seed used to start the numerical method. In order to do that, an asymptotic expansion in power of the small parameter ? = 1?e is developed. In most of the cases, the seed generated by the Space Dynamics Group at UPM (SDG-code) leads to reach machine error accuracy with the modified Newton-Raphson methods with no iterations or just one iteration. The final algorithm is very stable and reliable. This approach improves the computational time compared with other methods currently in use. The advantage of our approach is its applicability to other problems as for example the Lambert problem for low thrust trajectories. A method solving Kepler?s equation without transcendental function evaluations.
ICATT2016: 6th International Conference on Astrodynamics Tools and Techniques
Darmstadt (Germany)
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  • Autor: Virginia Raposo Pulido (UPM)
  • Autor: Jesus Pelaez Alvarez (UPM)
Research Group, Departaments and Institutes related
  • Creador: Grupo de Investigación: Dinámica Espacial (SDG-UPM)
  • Departamento: Física Aplicada a Las Ingenierías Aeronáutica y Naval
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Cofinanciación del MINECO en el marco del Programa INNCIDE 2011 (OTR-2011-0236)
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