The motion of a close earth satellite

TLDR

The paper derives perturbations of six orbital elements of a close Earth satellite in the gravitational field without air resistance, expressed as functions of mean orbital elements and time. The analysis assumes a symmetric Earth density distribution, treats the second harmonic coefficient as first‑order small, and the third and fourth harmonics as second‑order, without restricting eccentricity or inclination magnitudes. The study finds first‑order periodic and up to second‑order secular perturbations, notes singularities for very small eccentricity or inclination orbits, and proves that the semi‑major axis contains no first‑order long‑periodic terms using Delaunay elements. Abstract.

Abstract

A bstract. In the present paper perturbations of six orbital elements of a close earth satellite moving in the gravita- tional field of the earth without air-resistance are derived as functions of mean orbital elements and time. No assumptions are made about the order of magnitude of eccentricity and inclination. However, it is assumed that the density distribu- tion of the earth is symmetrical with respect to the axis of rotation, that the coefficient of the second harmonic of the potential is a small quantity of the first order and that those of the third and the fourth harmonics are of the second order. The results include periodic perturbations of the first order and secular perturbations up to the second order. However, the solutions have some singularities for an orbit whose eccentricity or inclination is smaller than a quantity of the first order, and this case is treated in a different way. By using Delaunay's canonical elements a theorem is proved that there are no long-periodic terms of the first order in the expression of the semi-major axis.