Dynamics of Frozen Orbit and Its Critical Case in the Zonal Problem

Abstract

This paper investigates the dynamical behaviors of Earth frozen orbits including the critical inclination orbit in the zonal problem with terms of perturbations up to J15. From the mean element theory, an analytical expansion of the dynamics model is derived to obtain the frozen conditions. The global existence of Earth frozen orbits is numerically illustrated by exploration of equilibria in the equations of motion based on Lagrangian formulations. Frozen orbits are widely located at the inclination between 0° and 90°, which may be grouped into families: one for the elliptic orbit case, another for the quasicircular orbit case and the other for the critical inclination orbit case. For the elliptic and quasicircular orbit cases, orbits evolve periodically around the frozen ones, and elliptic orbits tend to be more sensitive than quasicircular ones with respect to variation of initial orbital elements. Frozen orbit bifurcations near the critical inclination are shown to illustrate the origins, evolutions, and stability of frozen orbits. For the last critical inclination case, it is observed that eccentricities keep evolving periodically instead of being frozen at the specified ones compared with that in the J2 problem, and then a relationship between the evolution period and terms of zonal perturbations is illustrated.