#### Foreword

This book will be valuable both to students and experts who are interested in elegant and efficient mathematical methods for achieving quantitative understanding of satellite orbit perturbations. Until the 20th century, celestial mechanics was the primary domain for development of analytic and computational techniques in mathematics, from treatment of complex singularities by Cauchy to recognition of chaos by Poincare. Classical mechanics was then pushed to the background by the rise of quantum mechanics until the beginning of the space age in the 1950s, when celestial mechanics was revived as a foundation for the new field of astromechanics to control space flight and artificial satellites. With its narrow mission-oriented focus, NASA incorporated 19th century mathematics into the space program without recognizing the importance of sponsoring further development of mathematical and computational techniques. Nevertheless, development continued to be driven by intellectual forces. This book is a good example of one outcome. Let me place it in a historical context.

The singularity of the gravitational potential presented a persistent problem in orbital mechanics until 1920, when Italian mathematician Tullio Levi-Civita discovered a change of variables that regularized the singularity and linearized the Kepler problem by transforming it to the form of a harmonic oscillator. However, his technique employed complex variables in way that limited the result toplanarmotion, and he was at a loss to generalize it to 3D. Thus, it remained as a mathematical curiosity until the 1964 Oberwolfach conference on Mathematical Methods in Celestial Mechanics when, in an opening address in the grand tradition of David Hilbert, Swiss mathematician Eduard Stiefel posed the generalization of Levi-Civita regularization to 3D as one of the great unsolved problems of mathematics. Then in dramatic fashion at the afternoon session, an obscure young Finnish astronomer Paul Kustaanheimo clamored for the podium to announce that he had solved the problem.

Kustaanheimo employed idiosyncratic spinor methods to generalize Levi-Civita’s approach [23]. Unsatisfied with that, Stiefel transformed it to a more conventional matrix formulation now generally known as the Kustaanheimo-Stiefel (KS) transformation. Recognizing its advantages for perturbation theory in celestial mechanics, Stiefel promoted it in research and publicized it in a book [32].

After I heard this story about Kustaanheimo from a Finnish colleague, I reformulated the KS transformation in terms of geometric algebra, which I found clarifies its geometric structure and physical significance [20]. Pleased with the result, I called the method Spinor Perturbation Theory and incorporated it in my mechanics book [21]. Soon thereafter, a researcher in the space program informed me that he applied it with considerable success in the design of software to control artificial satellites, but that was never published or followed up by others.

As abundantly demonstrated in this book, Vrbik has adopted the method and pushed it to a new level of perfection with detailed applications, including some of the trickier problems in planetary celestial mechanics. The reception to this book may well determine if Spinor Perturbation Theory is ultimately placed among the celebrated achievements of celestial mechanics.

Vrbik tells me he chose to formulate his theory in terms of quaternions rather than geometric algebra because the former has an established tradition in celestial mechanics. On the other hand, geometric algebra enhances the value of quaternions by clearly integrating them with vector algebra and spinors, thus resolving a history of confusion and controversy and consolidating their place in the broader context of mathematical physics [16]. It is an easy and instructive exercise to relate the quaternion formulation in this book to the spinor formulation in mine. The serious student will be in for some surprises.

**
***Dr. David Hestenes*

Professor Emeritus of Physics

University of Arizona, Tempe

2002 Oersted Medal recipient

#### Preface

This book has been inspired by the ‘Celestial Mechanics’ chapter of [21]. Its purpose is to demonstrate how the perturbed Kepler problem can be elegantly formulated and solved using the power of quaternion algebra (to be seen as a special case of a more general, and universally applicable, Geometric Algebra). The solution covers, in a natural and rather routine manner, even the most difficult cases of resonant perturbations.

There are two technical reasons which enabled me to construct such a universal solution. Namely, I had to (i) modify the usual definition of auxiliary time s (making the clock ‘tick’ faster, as the satellite approaches the primary, in a way which reflects the third Kepler law), and (ii) dispense with the traditional ‘gauge’ (this came at the expense of complicating the resulting equation — yet, without it, one is bound to fail, as the history of this problem clearly demonstrates). Only then one can proceed to build a unique solution in a rather algebraic manner of matching coefficients of two Fourier-series expansions (see Chapter 2). The rest of the book demonstrates how relatively easily can the resulting formulas be applied to practically all perturbations encountered within our Solar system.

The book is organized as follows: Chapter 1 reformulates the Kepler problem in terms of the quaternion algebra, and solves its unperturbed version. Chapter 2 constructs a detailed solution of the perturbed version; this is, in a sense, the key development of the book — however, it can be skipped by readers who want to learn only how to apply the resulting formulas. Chapter 3 derives and discusses most of the common perturbing forces, and explains their origin. Each of the remaining chapters concentrates on one such perturbing force, starting with perturbations of planetary motion caused by the gravitational pull of other planets (Chapter 4). For artificial satellites, the most important perturbing force is due to the Earth’s slightly flattened shape (and similar deviations from a perfect sphere), the consequences of which are discussed in Chapter 5. For the Moon, the main perturbing force is the Sun; due to its relative strength, the Moon’s motion becomes quite complicated and difficult to compute with high accuracy (Chapter 6). Chapter 7 deals with an interesting (and also rather difficult) phenomenon of a resonant perturbing force (meaning that is oscillates with a frequency which is either an exact multiple, or an exact fraction, of the orbital frequency) — this is important for proper understanding of asteroid motion, and it also explains the formation of Kirkwood gaps (narrow regions in the asteroid belt, which are almost void of asteroids). The final chapter then deals, rather more quickly, with the remaining common perturbations, such as the relativistic correction and radiation pressure.

Finally I would like to acknowledge that, as with any significant project, things are rarely done entirely by oneself. To this end I would like to thank: Dr. Hestenes without whom this book would not have been possible and my son Paul Vrbik for his help and technical expertise in preparing the LATEX version of this book.

#### List of Contributors

##### Author(s):

**Jan Vrbik **

Department of Mathematics

Brock University

Canada