Analytical and Semi‐Analytical Treatment of the Satellite Motion in a Resisting Medium

Abstract

The orbital dynamics of an artificial satellite in the Earth's atmosphere is considered. An analytic first-order atmospheric drag theory is developed using Lagrange's planetary equations. The short periodic perturbations due to the geopotential of all orbital elements are evaluated. And to construct a second-order analytical theory, the equations of motion become very complicated to be integrated analytically; thus we are forced to integrate them numerically using the method of Runge-Kutta of fourth order. The validity of the theory is checked on the already decayed Indian satellite ROHINI where its data are available. 2Journal of Applied Mathematics unforeseen complexity 3-8 . In order to be able to solve the equations of motion including the drag force analytically, one is forced to do many approximations. On the other hand, from the computational point of view, having an analytical solution at hand, we can jump from the initial conditions to the new state, which makes the computation extremely fast.The order of magnitude of the atmospheric drag acting on a satellite depends on the altitude of the satellite. The orbital behavior of an LEO satellite could be strongly influenced by atmospheric drag 4, 9, 10 , and therefore affects the quality of the remote sensing of the satellite 11, 12 . Traditionally, the problem of orbit disturbance by atmospheric drag is solved by numerical integration 13-18 . An analytical solution gives the theoretical integrals and shows the physical effects with very clear spectral properties 19-24 . This may give a direct insight into the physical phenomenon of the disturbance 25-27 . Bezdeka and Vokrouhlicky 28 developed a semianalytic theory of motion for close-Earth spherical satellites including drag and gravitational perturbations. Xu et al. 29 derived an analytical solution of a satellite orbit disturbed by atmospheric drag. They first transformed the disturbance force vector and rotated it to the orbital frame so that it can be used in the simplified Gaussian equations of satellite motion. The disturbances are separated into three parts: shortperiodic terms with triangular functions of the mean anomaly M, longperiodic terms with triangular functions of the argument of perigee and inclination ω, I , and secular terms nonperiodic functions of semimajor axis and eccentricity a, e .The main effects of the force are to bring out large secular decreases in the elements a, the semimajor axis, and e, the eccentricity, that cause the satellite plunges toward Earth. This perturbing influence is one of the most important perturbing forces in the altitude regime from 150 to 600 km which is called the drag force. The drag force, F D , per unit mass of the satellite can be represented by Submit your manuscripts at