An Analytical Theory for the SPOT Satellite

Resumen

The case of the SPOT satellite is studied. The perturbing force considered corresponds to a model of gravity field where some zonal and tesseral harmonics are included. Three adequate canonical simplifications of Lie type are performed up to order eight for the short period transformations and, also to order eight for the Hamiltonian and to order six for the generator of the long period transformation. By using these canonical transformations, the initial perturbed Hamiltonian is reduced in such a way that the final Hamiltonian is a system of zero degrees of freedom. Applying the technique of "the elimination of the parallax" transformation followed by the Delaunay normalization, the dynamical system associated with the Hamiltonian is independent of short period terms. These transformations are developed in close form of the eccentricity, avoiding thus an enormous amount of terms in the intermediate expressions. Once the mean anomaly has been removed, the averaged Hamiltonian depends on the ascending node, the argument of the perigee and the conjugate momenta. Hence, in order to obtain a secular Hamiltonian, a double elimination of the long period angles is performed. This transformation is analogous to the elimination of the two mean anomalies in the stellar case of the three body problem. Numerical comparisons are done in order to check the analytical results, showing the benefit of our approach.