Long-term dynamical evolution of Tundra-type orbits

Tundra-type orbits are elliptical geosynchronous orbits located at the critical inclination. The long-term dynamical evolution of this type of special orbit is investigated in this paper. First, the effect of Earth's gravitational potential is examined. A simplified Hamiltonian of Tundra-type orbits subjected to Earth's gravitational potential is presented through a strict magnitude comparison of the involved terms. Based on this simplified Hamiltonian with two degrees of freedom, the equilibrium points of the orbits subjected to Earth's gravitational potential and their stabilities are discussed. This simplified Hamiltonian is then reduced to a one-degree-of-freedom system dominating the intermediate-period motion of the orbits approximately. In particular, the main characteristic parameters of the intermediate-period motion for nominal Tundra-type orbits and the corresponding specific results for three Sirius satellites in such orbits are presented. Second, the effect of lunisolar perturbations is examined. A magnitude comparison elementarily illustrates that the effect of tesseral harmonics of the Earth's gravitational potential on the long timespan evolution of Tundra-type orbits is negligible compared to that of lunisolar perturbations. A simplified dynamical model including lunisolar perturbations is then presented. Based on this simplified dynamical model, the influences of lunar precession, the initial longitude of the ascending node, the initial argument of perigee, and the initial epoch on the long-term dynamical evolution of the orbits are comparatively analyzed. Finally, numerical calculations with exact perturbation models are conducted to verify the theoretical analysis and to provide more information about the dynamical evolution of Tundra-type orbits.