On an analytical method for the satellite problem revisited

TL;DR

This paper revisits and refines an analytical method for calculating satellite orbits around an oblate Earth, simplifying the core technique and extending its applicability with new formulas and computational tests.

Contribution

The paper reformulates Brouwer's classical method using power series compositions, deriving general formulas that unify and extend previous approaches.

Findings

Derived new formulas covering all cases in Brouwer's method

Simplified the core of the analytical approach

Performed computational tests validating the formulas

Abstract

An analytical method for solving the problem of the satellite around the oblated Earth is discussed. Such a method goes back to Dirk Brouwer (1959) and seems to be still widely used in real applications. We reformulate in a more concise way the core of the method which is essentially based on a technique of compositions of power series. In this way we derive some general formulas which cover, all at once, all the cases treated in the original paper and allow to compute some terms which were omitted there. We also show some simple computation we performed to test the formulas.