Today, the motion of spacecrafts is still described according to the
classical Newtonian equations plus the so-called "relativistic corrections",
computed with the required precision using the Post-(Post-)Newtonian formalism.
The current approach, with the increase of tracking precision (Ka-Band Doppler,
interplanetary lasers) and clock stabilities (atomic fountains) is reaching its
limits in terms of complexity, and is furthermore error prone. In the
appropriate framework of General Relativity, we study a method to numerically
integrate the native relativistic equations of motion for a weak gravitational
field, also taking into account small non-gravitational forces. The latter are
treated as perturbations, in the sense that we assume that both the local
structure of space-time is not modified by these forces, and that the
unperturbed satellite motion follows the geodesics of the local space-time. The
use of a symplectic integrator to compute the unperturbed geodesic motion
insures the constancy of the norm of the proper velocity quadrivector. We
further show how this general relativistic framework relates to the classical
one.Comment: 13 pages, 5 eps figures, 1 table, accepted in Acta Astronautica,
presented at the International Astronautical Congress, Vancouver 2004,
reference IAC-04-A.7.0