Satellites in the gravitational field of the moon

Expressions are derived correct to the first order in the ellipticities of the Moon for the radial shift and the rotation of the unperturbed orbit of a close satellite in the plane of its instantaneous motion. Criteria for the rotation to be a libration lead to the existence of a critical range of values of orbital inclination as against a single value cos^?1 (·2)^1/2=63·4° quoted for the Earth satellites. It appears that mean radial shift and the perturbation in the period of the satellite, also possess similar ranges. The perturbation in the velocity vector is then analysed and it is shown that the angle between the unperturbed and real trajectories vanishes only at points where the radial distance of the satellite from the selenocentre attains extreme values. The interesting case of nearly circular trajectories is considered next in some detail. It is deduced that the critical inclination for the orbital libration is given byα=60° for an Earth satellite in this case. The paper concludes with numerical results based on Jeffreys’ estimates.