Semianalytic Theory of Long-Term Behavior of Earth and Lunar Orbiters

Abstract

A semianalytical method capable of analyzing both lunar and Earth orbiters is presented. Primary attention is focused on predicting the evolution of the orbit as affected by third body perturbations together with those of the rotating primary. The singly averaged (literal) equations of motion are expanded by machine to high order in the parallax factor and the mean motion ratio. The equations are numerically integrated to yield the orbital evolution for a wide range of initial conditions. In addition, a purely analytical method is introduced to yield the orbital lifetimes for a special class of orbits. this assumption is clearly violated in the case of the Earth-moon system. Thus in carrying out the expansion, a time rate of change for all terms containing the third body position must be included. This yields a further expansion of the disturbing function in terms of the mean motion ratio ri\n the ratio of the mean motion of the disturbing body to that of the satellite. To carry out the expansions in terms of the parallax factor and the mean motion ratio and then to average the equations of motion over one orbit requires an excessive amount of algebra for the higher order terms. To aid in the algebraic computations, a general algebraic manipulation routine was developed and was used to compute the average (literal) equations of motion to eighth order in the parallax factor and to second order in the mean motion ratio with corresponding cross terms up to and including fifth order in parallax. In addition to the third body effects, the gravity harmonics of the rotating primary must be considered. For the moon, these equations may be averaged over the orbital period since the moon rotates slowly. However, for the more rapidly rotating Earth, this analysis is invalid as the orbital mean motion may be nearly commensurate with the rotation of the primary. To avoid this problem, the equations of motion are numerically averaged from one-half orbit behind to one- half orbit ahead of the present position of the satellite. These averaged rates are then used in the total variations of the ele- ments. At present a full 7x7 and 4x4 field is used for the Earth and moon, respectively. When the tesseral harmonics are not required, only terms containing 72, /22, /3 and /4 are used and the variational equations are calculated explicitly.