Analytical modelling of the motion of a drag-free satellite

The evolution of a satellite orbit is described by a first-order vector differential equation which can be expressed linearly as a function of the zonal and tesseral coefficients of the earth potential. The use of a given model of potential requires the truncation of this development and a choice in the values of these coefficients. Linearization of the differential equation about an orbit close to the solution yields direct or indirect effects of very different significance. Analysis of these effects permits expansion of the linearized equation by means of the Lagrange perturbation equations. A complete solution of the problem leads to simple analytical relationships describing, for small orbital eccentricities, the motion of a satellite subjected to gravitational forces only. Two analytical models were developed, and the most accurate was compared with a high-performance numerical integration. This model, applied to a quasi-polar small-eccentricity (0.001) orbit, provides a precision of better than 200 m in the orbital parameters after 40 days, although less along the orbit. An extension of this model is proposed for orbits of larger eccentricities.