Observatorio de I+D+i UPM

Memorias de investigación
Communications at congresses:
The new SDG-code to solve the Kepler equation
Research Areas
  • Aeronautical engineering
The Kepler equation for the elliptical and hyperbolic motion involves a nonlinear function depending on three parameters: the eccentric/hyperbolic 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. The main 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. This new approach takes advantage of the very good behavior of the Laguerre method and also of the existence of symbolic manipulators which facilitates the obtaining of polynomial approximations. 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 according to which M belongs to and using the corresponding polynomial pi(x), we determine the starter value yo = Eo used to start the numerical method. When M is small and e close to unity (singular corner), the Kepler equation has a singular behavior. In this case, an asymptotic expansion in power of the small parameter 1- e (for elliptic case) and e -1 (for hyperbolic case) is obtained to describe the exact solution of the equation. Two different calculations have been carried out, using standard double precision and quadruple precision. Besides, a complete analysis of the code has been performed to assess the speed of the algorithm and the accuracy. 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 method with no iterations or just one iteration. Comparing with other methods, the SDG-code provides a greater accuracy when the quadruple precision is applied. The final algorithm is very stable and reliable. The advantage of our approach is its applicability to other problems as for example the Lambert problem for low thrust trajectories.
Stardust Final Conference on Asteroids and Space Debris
Noordwijk (Netherlands)
Start Date
End Date
From page
To page
  • 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
S2i 2021 Observatorio de investigación @ UPM con la colaboración del Consejo Social UPM
Cofinanciación del MINECO en el marco del Programa INNCIDE 2011 (OTR-2011-0236)
Cofinanciación del MINECO en el marco del Programa INNPACTO (IPT-020000-2010-22)