Symmetric orbits in the presence of J₂ perturbation

In this paper, for Earth-orbiting spacecraft, we present a numerical algorithm to determine initial conditions that yield periodic, symmetric equatorial and polar orbits in the presence of Earth's oblateness. By restricting consideration to orbits which contain trajectories crossing a plane of symmetry orthogonally, we develop a numerical method that efficiently converges to initial conditions yielding a periodic, symmetric orbit. Moreover, we provide two heuristic expressions to approximate numerical values of the initial conditions yielding periodic, symmetric orbits. Simulation results are presented to illustrate the efficacy of the numerical algorithm and the heuristic expressions in generating periodic, symmetric equatorial and polar orbits.