Introduction

We saw earlier, in Section 1.9, that an isolated dynamical system consisting of two freely moving point masses exerting forces on one another—which is usually referred to as a two-body problem—can always be converted into an equivalent one-body problem. In particular, this implies that we can exactly solve a dynamical system containing two gravitationally interacting point masses, as the equivalent one-body problem is exactly soluble. (See Sections 1.9 and 3.16.) What about a system containing three gravitationally interacting point masses? Despite hundreds of years of research, no useful general solution of this famous problem—which is usually called the three-body problem—has ever been found. It is, however, possible to make some progress by severely restricting the problem's scope.

Circular restricted three-body problem

Consider an isolated dynamical system consisting of three gravitationally interacting point masses, m1, m2, and m3. Suppose, however, that the third mass, m3, is so much smaller than the other two that it has a negligible effect on their motion. Suppose, further, that the first two masses, m1 and m2, execute circular orbits about their common center of mass. In the following, we shall examine this simplified problem, which is usually referred to as the circular restricted three-body problem. The problem under investigation has obvious applications to the solar system. For instance, the first two masses might represent the Sun and a planet (recall that a given planet and the Sun do indeed execute almost circular orbits about their common center of mass), whereas the third mass might represent an asteroid or a comet (asteroids and comets do indeed have much smaller masses than the Sun or any of the planets).

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.