New Methods of Celestial Mechanics

124 Pages·2010·2.1 MB·English

by  Jan Vrbik

New Methods of Celestial Mechanics
Jan Vrbik
Department of Mathematics 
Brock University 
Can ada
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To my family.
Contents
Foreword i
Preface iii
Chapter1. Linearized Kepler problem 1
Chapter2. Iterative solution of perturbed problem 18
Chapter3. Perturbing forces 34
Chapter4. Solar system 45
Chapter5. Oblateness perturbations 57
Chapter6. Lunarproblem 67
Chapter7. Resonances 79
Chapter8. OtherPerturbations 103
References 112
Index 115
i

Foreword
This book will be valuable to students and experts alike who are interested in elegant and efficient math-
ematical 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 tech-
niques 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 fieldof astromechanics to control spaceflight andartificial satellites. With its narrow mission-oriented
focus, NASA incorporated 19th century mathematics into the space program without recognizing the im-
portance 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 andlinearized theKeplerproblembytransformingittotheformofaharmonicoscillator. How-
ever, his technique employed complex variables in way that limited the result to planar motion, 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]. Unsat-
isfied 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].
AfterIheardthisstoryaboutKustaanheimofromaFinnishcolleague, IreformulatedtheKStransforma-
tion 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 had 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 me-
chanics. 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 al-
ii
 gebra 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
iii

Preface
Thisbookhas beeninspiredby the‘Celestial Mechanics’ chapter of [21]. Its purposeis to demonstratehow
theperturbedKeplerproblemcanbeelegantly formulatedandsolvedusingthepowerofquaternionalgebra
(to be seen as a special case of a more general, and universally applicable, Geometric Algebra). The solu-
tion covers, in a naturalandrather routinemanner, even the mostdifficultcases 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 tradi-
tional ‘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 can one 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
arediscussedinChapter5. For theMoon, themainperturbingforceistheSun;duetoitsrelative 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.
New Methods of Celestial Mechanics, 2010, 01-17 1
CHAPTER 1

Linearized Kepler problem
Abstract
Ourfirstobjectiveistoreformulatetheage-oldproblemofasatelliteorbitingaprimary(ofasubstantially
biggermass)bymeansofquaternionalgebra.1 Insubsequentchapters,thiswillfacilitate arelatively simple
construction of the problem’s iterative solution.
Here we also review some details of constructing the unperturbed solution.
1.1 QUATERNION ALGEBRA
We first briefly review the algebra’s basic rules and definitions [19], [16].
1.1.1 Basics
Quaternion algebra has three imaginary units usually called i, j and k, each of them squaring to 1. Under
−
the algebra’s multiplication (denoted ), any two of these anticommute (i.e. i j = j i), furthermore,
◦ ◦ − ◦
i j = k, j k= i and k i= j. Together with the ordinary (real) unit 1 which commutes with each of them,
◦ ◦ ◦
they constitute the algebra’s basis. An element of this algebra can thus be written as
A+a i+a j+a k A+a (1.1)
z y x
≡
and interpreted geometrically as a vector a (a , a and a representing its x, y and z coordinates,
x y z
respectively), appended to a scalar A (note the deliberate reversal of the components of a).
Quaternion multiplication thus amounts to
(A+a) (B +b) = AB a b+Ab+Ba a b (1.2)
◦ − · − ×
which can be easily verified by expanding (A + a i + a j + a k) (B + b i + b j + b k). Note that this
z y x z y x
◦
multiplication is noncommutative, but remains associative. The latter property is easily confirmed for any
selection of the algebra’s basis elements, e.g. (i j) k = i (j k), (i j) i = i (j i), etc., implying the
◦ ◦ ◦ ◦ ◦ ◦ ◦ ◦
general result.
From now on, all our vectors will be understood to be quaternions with zero real component, e.g.
f f i + f j + f k. A product of two vectors (such as, for example, g f) will always imply their
z y x
≡ ◦
quaternion multiplication. Note that, in this notation, f2 f f = f2, where f = f2+f2+f2 is the
≡ ◦ − x y z
usual length of f (alternately denoted f ). Also note that, for the usual dot producqt, we get
| |
g f +f g
g f = ◦ ◦ (1.3)
· − 2
and, correspondingly
g f f g
g f = ◦ − ◦ (1.4)
× − 2
1Formore details see [20], [21], [23], [24], [32]–[39].
Jan Vrbik
All rights reserved - © 2010 Bentham Science Publishers Ltd.
2 New Methods of Celestial Mechanics Jan Vrbik

for the cross product.
For quaternion quantities, we use the ‘blackboard’ type (e.g. A A+a i+a j+a k); A denotes the
z y x
≡
correspondingquaternion conjugate(changing the sign of thei, j,and k-components, i.e. thevector part,
of A). One can easily verify that
A B = B A (1.5)
◦ ◦
In general, the magnitude A of a quaternion A is computed by √A A √A A (a square root of the
| | ◦ ≡ ◦
sum of squares of its four components). Clearly, A B = √A B B A = A B . For a vector, this
| ◦ | ◦ ◦ ◦ | || |
implies f2 = f f f f.
◦ ≡ ◦
For complex quantities (with no j and k components), we use the calligraphic type (e.g. ), an asterisk
D
implying the complex conjugation, thus: . In subsequent chapters, we make an exception to this rule
∗
D
by employing the symbols z and q for complex variables.
1.1.2 Rotations
The process of rotation in three dimensions is greatly simplified in the quaternion-algebra representation.
First we note that any quaternion A can be written as A+aa where a and a are the length and unit
direction, respectively, of a. Since a2 = 1, a behaves like a single imaginary unit; the whole quaternion
−
thus has attributes of a complex number. Correspondingly, we can evaluate functions of A, e.g.
b b
b eA =b eA(cosa+asina) (1.6)
Now, let us assume that a rotation is defined by a vector w w i+w j+w k, with its unit direction w
z y x
b ≡
specifying the rotation’s axis (passing through the origin) and its length w representing the corresponding
angle.
b
One can easily verify that a point x (also represented by a pure quaternion) is moved, under such a
rotation, to a new point x by
†
x = e w/2 x ew/2 R x R (1.7)
† −
◦ ◦ ≡ ◦ ◦
Proof. Letusrewritexasx +x ,theparallelandperpendicularcomponentswithrespecttow.Obviously,
x commutes with w (in tekrms⊥of quaternion multiplication), whereas x and w anticommute (i.e. x
wk= w x = w x ), implying that ⊥ ⊥ ◦
b
− ◦ ⊥ × ⊥
x ew/2 = e w/2 x (1.8)
−
⊥◦ ◦ ⊥
since x wn = ( 1)nwn x , for each nonnegative integer n. One thus gets
⊥◦ − ◦ ⊥
x = e w/2 ew/2 x +e w/2 e w/2 x = x +e w x (1.9)
† − − − −
◦ ◦ k ◦ ◦ ⊥ k ◦ ⊥
w
Furthermore, since e = cosw wsinw, we finally get
−
−
x = x +x cosw+(w x )sinw (1.10)
†
b k ⊥ × ⊥
which is exactly what a right-handed rotation should do (w x is x , rotated by 90o). 2
b × ⊥ ⊥
In (1.7), R obviously meets
b
R R = R R = 1 (1.11)
◦ ◦
(i.e. R 1 R), since the powers of w commute with one another, and the law of exponents then (and only
−
then) imp≡lies that e w ew = ew w = e0 = 1.
− −
◦
The reverse is also true; any R which meets (1.11) represents a specific rotation and must have the form
of cos(w/2)+wsin(w/2) = ew/2.
b