Semi-Equatorial Orbits Around an Oblate Body

Abstract

In this equation, is the gravitational constant, is the radius vector, j j, and is the colatitude angle (See Fig. 1). R is the Earth radius, and P2 is the second order Legendre polynomial. The forces due to the J2 term as well as drag with the upper atmosphere have important impact on the motion of clusters of satellites [5] and are currently the subject of intense research efforts [4–13]. Starting with the seminal papers of Burns [3] and King-Hele andMerson [4], the impact of the J2 term on the orbit of a particle has been treated through various forms of perturbation expansions [6–8]. Using this approach and the Lagrange planetary equations in Eq. (6), it was possible to obtain simple and useful expressions for the average rate of change of some orbital elements. In previous papers byHumi [11] andHumi andCarter [12,13], this additional termwas treated analytically without any approximations, and exact equations of motion for the orbit were derived. The key element in this treatment was the use of two cylindrical coordinates systems. One (inertial) system is attached to the Earth’s center, and the other is comoving with the particle. It was shown that in these coordinates the equations for the orbit can be reduced to a system of two coupled harmonic oscillators. For semi-equatorial orbits, this paper develops an approximation scheme to the exact equations of motion in cylindrical coordinates. These equations are then solved analytically in some special cases. Approximate equations are derived also for the general case. The orbits described by these approximations are compared numerically to those obtained from the exact equations of motion.