Order and Chaos in Satellite Encounters

In order to describe the motion of two weakly interacting satellites of a central body we suggest to use orbital elements based on the the linear theory of Kepler motion in Levi-Civita's regularizing coordinates. The basic model is the planar three-body problem with two small masses, a model in which both regular (e.g. quasi-periodic) as well as chaotic motion can occur.
This paper discusses the basics of this approach and illustrates it with a typical example. First, we will revisit Levi-Civita's regularization of the two-dimensional Kepler motion and introduce sets of orbital elements based on the differential equations of the harmonic oscillator. Then, the corresponding theory for the three-dimensional motion will be developed using a quaternion representation of Kustaanheimo-Stiefel (KS) regularization; we present it by means of an elegant new notation.