Periodic and quasi-periodic satellite relative orbits at critical inclination

Orbit at the critical inclination is a practical option for long-term satellite formations flying around an oblate Earth. This paper establishes that passive and periodic relative orbits with cross-track motion are only possible at the critical inclination under conditions that all satellites share the same inclination, the same semi-major axis and the same eccentricity. Furthermore, a quasi-periodic relative orbit is achieved near the critical inclination by matching the orbital energy and the polar component of the orbital angular momentum. These matching conditions can be expressed in terms of the actual orbit variables. Our theoretical result shows that these conditions are equivalent to matching the orbital period and the rate of the mean anomaly.