Introduction

The two-body orbit theory described in Chapter 3 neglects the direct gravitational interactions between the planets, while retaining those between each individual planet and the Sun. This is an excellent first approximation, as the former interactions are much weaker than the latter, as a consequence of the small masses of the planets relative to the Sun. (See Table 3.1.) Nevertheless, interplanetary gravitational interactions do have a profound influence on planetary orbits when integrated over long periods of time. In this chapter, a branch of celestial mechanics known as orbital perturbation theory is used to examine the secular (i.e., long-term) influence of interplanetary gravitational perturbations on planetary orbits. Orbital perturbation theory is also used to investigate the secular influence of planetary perturbations on the orbits of asteroids, as well as the secular influence of the Earth's oblateness on the orbits of artificial satellites.

Evolution equations for a two-planet solar system

For the moment, let us consider a simplified solar system that consists of the Sun and two planets. (See Figure 9.1.) Let the Sun be of mass M and position vector Rs. Likewise, let the two planets have masses m and m′ and position vectors R and R′, respectively. Here, we are assuming that m, m′ ≪ M. Finally, let r = R − Rs and r′ = R′ − Rs be the position vectors of each planet relative to the Sun. Without loss of generality, we can assume that r′ > r.