Analytic expansions of luni-solar gravity perturbations along rotating axes for trajectory optimization: Part 1: The dynamic system

An analytic form of the accelerations due to the luni-solar perturbations resolved along the rotating Euler-Hill frame is devised by using the expansion method. The addition of higher order terms to the main gravity gradient term linear in the spacecraft radial distance, carried out to the third order, provides a very high level of accuracy in accounting for the gravity perturbations experienced by a vehicle in orbit due to the sun and the moon. The nodal precession as well as the perigee advance of the lunar orbit is taken into account analytically by using the analytic lunar theory of de Pontécoulant. The analytic description of the apparent solar orbit and the motion of the moon remove the need to call an epherneris generator at each integration step during the numerical integration of the spacecraft trajectory, leading to the self-contained software for rapid and efficient optimal trajectory generation through iterations. Equinoctial elements are used to describe the spacecraft state and the luni-solar accelerations are given in terms of the apparent solar and lunar longitudes as well as Eulerian angles of the spacecraft orbit with respect to the inertial ecliptic system. The analysis is useful in optimal low-thrust orbit transfers complementing previous analyses carried out by this author, in which thrust and Earth zonal perturbations such as J 2, J 3 and J 4 in terms of the nonsingular equinoctial elements are included.