A new set of integrals of motion to propagate the perturbed two-body problem

  • Giulio Baù
  • Claudio Bombardelli
  • Jesús Peláez
Original Article


A formulation of the perturbed two-body problem that relies on a new set of orbital elements is presented. The proposed method represents a generalization of the special perturbation method published by Peláez et al. (Celest Mech Dyn Astron 97(2):131–150, 2007) for the case of a perturbing force that is partially or totally derivable from a potential. We accomplish this result by employing a generalized Sundman time transformation in the framework of the projective decomposition, which is a known approach for transforming the two-body problem into a set of linear and regular differential equations of motion. Numerical tests, carried out with examples extensively used in the literature, show the remarkable improvement of the performance of the new method for different kinds of perturbations and eccentricities. In particular, one notable result is that the quadratic dependence of the position error on the time-like argument exhibited by Peláez’s method for near-circular motion under the \(J_{2}\) perturbation is transformed into linear. Moreover, the method reveals to be competitive with two very popular element methods derived from the Kustaanheimo-Stiefel and Sperling-Burdet regularizations.


Perturbed two-body problem Regularization Generalized orbital elements  Orbit propagation Linearization 



The study has been supported by the research project “Dynamic Simulation of Complex Space Systems” supported by the Dirección General de Investigación of the (former) Spanish Ministry of Innovation and Science through contract AYA2010-18796.

Supplementary material

10569_2013_9475_MOESM1_ESM.pdf (162 kb)
Supplementary material 1 (pdf 162 KB)


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Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Giulio Baù
    • 1
  • Claudio Bombardelli
    • 2
  • Jesús Peláez
    • 2
  1. 1.CISAS “G. Colombo”University of PadovaPadovaItaly
  2. 2.Space Dynamics GroupSchool of Aeronautics, Technical University of MadridMadridSpain

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