Elementary derivation of the perturbation equations of celestial mechanics

Abstract

The equations of celestial mechanics that govern the time rates of change of the orbital elements are completely derived using elementary dynamics, starting from only Newton’s equation and its solution. Two orbital equations and the four most meaningful orbital elements—semimajor axis a, eccentricity e, inclination i, and longitude of pericenter Ω—are written in terms of the orbital energy E and angular momentum H per unit mass. The six resulting equations are differentiated with respect to time to see the effect on the orbital elements of small changes in E and H. The usual perturbation equations in terms of disturbing force components are then derived by computing the manner in which perturbing forces change E and H. The results are applied in a qualitative discussion of the orbital evolution of particles in nonspherical gravitational fields, through atmospheres, and under the action of tides.