7 - Secular Perturbations

Summary

Past and to come seem best, things present worst.

William Shakespeare, Henry IV, (2), I, iii

Introduction

In the last chapter we saw how the disturbing function can be expanded in an infinite series where the individual terms can be classified as secular, resonant, or short period, according to the given physical problem. We have already stated in Sect. 3 that the N-body problem (for N ≥ 3) is nonintegrable. However, in this chapter we will show how, with suitable approximations, it is possible to find an analytical solution to a particular form of the N-body problem that can be applied to the motion of solar system bodies. We can do this by considering the effects of the purely secular terms in the disturbing function for a system of N masses orbiting a central body. The resulting theory can be applied to satellites orbiting a planet, or planets orbiting the Sun, and then used to study the motion of small objects orbiting in either of these systems. This is the subject of secular perturbation theory.

Secular Perturbations for Two Planets

Consider the motion of two planets of mass m₁ and m₂ moving under their mutual gravitational effects and the attraction of a point-mass central body of mass m_c where m₁ ≪ m_c and m₂ ≪ m_c. Let R₁ and R₂ be the disturbing functions describing the perturbations on the orbit of the masses m₁ and m₂ respectively, where R₁ and R₂ are functions of the standard osculating orbital elements of both bodies.