Families of frozen orbits of lunar artificial satellites

In this study, we investigate special families of frozen orbits for lunar artificial satellites. First, the Hamiltonian of a lunar artificial satellite is constructed. Next, the Hamiltonian is singly averaged by removing the short period coordinate while retaining the secular terms up to the first order using Lie transforms. A family of orbits with a frozen line of apses is obtained, which are then plotted. The solution for the argument of periapsis of Eq. (14) in the text and its corresponding plots show that the argument of periapsis splits the regions into two remarkable half planes. This actually restricts the selection of the inclination that can satisfy the argument of periapsis for the frozen orbits. A number of families of frozen eccentricity orbits are derived and graphically represented. All of the families are merged at the maximum critical inclination, which is around IC ≈ 64° The sensitivity to the variation in the semimajor axis is demonstrated. For very low nearly circular lunar orbits, different families of critical inclinations are obtained but they disappear very rapidly when either the semimajor axis or the eccentricity changes slightly. The complementaries of these values are also obtained. Three-dimensional figures of the critical inclination versus the argument of periapsis and the eccentricity of the different roots given by Eq. (22) are plotted. The flow of the eccentricity vector expressed in terms of nonsingular variables is visualized for different singly-averaged lunar orbiters at different orbital elements.