Analytic Expressions for Derivatives from Series Solutions to the Three Body Problem

Abstract

This paper presents a notation system to facilitate to solution of differential equations via Taylor series expansions and applies it to solve the circular restricted three body problem. Unlike previous Taylor series methods in the astrodynamics literature, computer algebra solvers are not used. Instead the notation system allows one to solve a system of differential equations analytically “by hand” without resorting to computer algebra software. This method produces recurrence relations explicitly in terms of a sequence of derivatives of the state with respect to time for the coefficients of Taylor Series solutions that can be evaluated numerically or manipulated further to investigate properties of the solution. For example, additional derivatives with respect to other parameters may also be found, including those that describe the dependence of the solution on initial conditions.