Numerical integration of periodic orbits in the main problem of artificial satellite theory

Abstract

We describe a collection of results obtained by numerical integration of orbits in the main problem of artificial satellite theory (theJ2 problem). The periodic orbits have been classified according to their stability and the Poincare surfaces of section computed for different values ofJ2 andH (whereH is thez-component of angular momentum). The problem was scaled down to a fixed value (−1/2) of the energy constant. It is found that the pseudo-circular periodic solution plays a fundamental role. They are the equivalent of the Poincare first-kind solutions in the three-body problem. The integration of the variational equations shows that these pseudo-circular solutions are stable, except in a very narrow band near the critical inclincation. This results in a sequence of bifurcations near the critical inclination, refining therefore some known results on the critical inclination, for instance by Izsak (1963), Jupp (1975, 1980) and Cushman (1983). We also verify that the double pitchfork bifurcation around the critical inclination exists for large values ofJ2, as large as |J2|=0.2. Other secondary (higher-order) bifurcations are also described. The equations of motion were integrated in rotating meridian coordinates.