On quasi-integrable problems. The example of the artificial satellites perturbed by the Earth's zonal harmonics

Abstract

With the Hamiltonian parameters developed for the two-fixed-centers problem a simple and very accurate expression of the ‘quasi-integral’ can be given for the motion of artificial satellites perturbed by the Earth's zonal harmonics. This motion can be considered as integrable.

A theoretical analysis shows that Hénon's ‘semi-ergodic regions’ or ‘chaotic regions’ are extremely small in this problem, and almost all orbits are of the ‘regular’ or ‘quasi-periodic’ type. Furthermore, the relative difference between the true motion and the corresponding integrable motion remains forever less than 10⁻¹⁴ for all regular orbits even in the vicinity of critical inclinations.

For chaotic orbits this very small difference remains verified at least for centuries, nevertheless there are some exceptional orbits that finally diverge from the integrable model.