Expansion of the third-body disturbing function

Abstract

HE expansion of the disturbing function still remains one of the most important problems in celestial mechanics, both in planetary applications and artificial satellite problems. The expansion of the gravitational disturbing function in zonal and tesseral harmonics based on the Legendre polynomials is now relatively easy to perform with the help of modern digital computers. On the other hand, the disturbing function for the thirdbody effects (solar and lunar perturbations on artificial satellites) is more difficult to expand, even with the availability of fast digital computers. The classically known expansions come in two categories: with the Laplace coefficients l and with the Legendre polynomials. 27 In the present Note we develop a new form of the thirdbody disturbing function which is in the category of the Legendre expansions. The purpose of this work is to find an expansion which is especially efficient and easy to carry out to a high order with the use of a package of Poisson series programs for algebraic operations on a computer. The reader should be aware of the fact that the most efficient expansion for hand calculations is usually not the most efficient expansion with a computerized algebraic processor. Simplicity of programming is another factor that needs to be considered on a computer. In general, the computer will prefer simple recurrence relations. The Poisson series processor will usually have a certain number of standard operations available: repeated partial differentiation, substitution of a series for a polynomial variable, the bionomial theorem, and the Bessel series of the Kepler problem are all standard one-line operation with our processor. It is clear that many of the classical celestial mechanics problems must be completely reformulated in the light of these modern tools. The expansion of the disturbing function, which is the object of the present text, is a literal expansion of the following general form: with the remarkable property that it is separated in factors each depending only on one body. In other words, Xn depends only on the elements (e, /, M, co, ft) while X^3) depends only on the elements (e3,i3,M3,u3,Q3) of the third (perturbing) body. These factors are therefore Poisson series in five variables only. They are relatively short series, in comparison with the resulting series R which is an expansion