Analytical solution of a satellite orbit disturbed by lunar and solar gravitation

Abstract

In this paper, we derive to the second order (5 x 10- 6 ) the analytical solution of a satellite orbit disturbed by the lunar gravitational force. The force vector is first expanded to omit terms smaller than the third order (10 -9 ). Then, four terms of potential functions are derived from the expanded force vector and set into the Lagrangian equations of satellite motion to obtain the theoretical solutions. For the first term of the potential functions, the solutions are derived directly. For the second term, mathematical expansions and transformations are used to separate disturbances into three parts: short-periodic terms with triangular functions of M, long-periodic terms with triangular functions of (ω, i, Ω) and secular terms with non-periodic functions of (a, e). The integrations are then carried out with respect to M, (ω, i, Ω) and t, to obtain the analytical solutions of satellite orbits with a program using mathematical symbolic operation software. The third potential function differs from the second by a factor and the fourth is simpler than the second. Therefore, the solutions are derived similarly using slightly modified programs, respectively. The results show that only two Keplerian elements (ω, M) are linearly perturbed by lunar gravitation; that is, the lunar attracting force will cause a linear regression (delay) of the perigee (orientation of the ellipse) and a linear delay of the position (mean anomaly) on an Earth satellite. The Keplerian element a (semimajor axis of the ellipse) is not perturbed long periodically as the others. The derived solutions are also valid for solar and planetary gravitational disturbances. Because of the distance differences between the Moon, the Sun and the planets to the Earth or an Earth satellite, the solutions are of third and fourth orders for solar and planetary gravitational disturbances on an Earth satellite, respectively.