Perturbation Theory

Abstract

In this section we will discuss the changes in the orbital elements produced by the action of a perturbing acceleration F, corresponding to the equations of motion in cartesian coordinates:

$$frac{{{d^2}r}} {{d{t^2}}} = - \frac{{Gmr}}{{{r^3}}} + F$$

(11.1)

Here the perturbing acceleration F is assumed to be a known function of r, r and t. The main idea underlying the perturbative equations is the osculating orbital elements: given an orbit r(t) solution of eq. (11.1), for every time t’ we consider the keplerian elements a, e, I,..., of a keplerian orbit with initial conditions r(t’), r(t’). These osculating elements give the orbit that the body would follow if at time t’ the perturbation F were suddenly turned off; of course, they are functions of the time t’. As a consequence, the osculating orbit changes with time, but does so slowly, provided the magnitude of F is small with respect to the monopole gravitational acceleration Gm/r² and its time scale is longer. We want the differential equations which describe the time dependence of the osculating elements. In this chapter we shall derive them in two different forms (the so called Gauss’ and Lagrange’s equations), and will show how they can be applied to solve two classical problems of artificial satellite dynamics, i.e. the orbital effects of drag and of the oblateness of the planet. The approximation methods commonly used to obtain the solutions are also discussed, as well as the problems and limitations of these methods.