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Research Collection Doctoral Thesis Regular orbital and rotational elements for artificial satellites Author(s): Vitins, Michael Publication Date: 1973 Permanent Link: https://doi.org/10.3929/ethz-a-000085444 Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection . For more information please consult the Terms of use . ETH Library
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Research Collection
Doctoral Thesis
Regular orbital and rotational elements for artificial satellites
Author(s): Vitins, Michael
Publication Date: 1973
Permanent Link: https://doi.org/10.3929/ethz-a-000085444
Rights / License: In Copyright - Non-Commercial Use Permitted
This page was generated automatically upon download from the ETH Zurich Research Collection. For moreinformation please consult the Terms of use.
ETH Library
Diss. Nr. 5174
REGULAR ORBITAL AND ROTATIONAL ELEMENTS
FOR ARTIFICIAL SATELLITES
ABHANDLUNG
zur Erlangung
des Titels eines Doktors der Mathematik der
EIDGENOESSISCHEN TECHNISCHEN
HOCHSCHULE ZUERICH
vorgelegt von
MICHAEL VITINS
Dipl. El.-lng. ETHZ
geboren am 27. April 1947
Kanadischer Staatsangehoriger
Angenommen auf Antrag von
Prof. Dr. E.Stiefel, Referent
Prof. Dr. Ch. Wehrli, Korreferent
aku-Fotodruck
Zurich
1973
To my wife Ursula
ACKNOWLEDGEMENTS
This work was carried out at the Seminar fur Angewandte Mathematik of the
Federal Institute of Technology under the guidance of Professor Dr. E.
Stiefel, to whom I would like to express my gratitude for his continued
interest and support.
I would like further to express my sincere thanks to Professor Dr. Ch.
Wehrli, who consented to examine this thesis.
ABSTRACT
Sets of regular elements describing Keplerian two-body motion and the
free motion of a rotating symmetrical rigid body are derived. Special
attention is paid to the desire that the elements are to be well adapted
for the application of the method of averages in solving certain classes
of perturbation problems.
A first order perturbation theory, based on these elements, for the orbital
position and for the attitude of a symmetrical satellite of finite dimensions,
orbiting an oblate central body, is developed.
INTRODUCTION
Analytical mechanics frequently makes use of general coordinates which
are well adapted to the problem at hand. Commonly used coordinates are,
for example, the three Euler angles of a rotating rigid body or the
classical orbital elements of Keplerian motion. The drawback of such
coordinates is that they often present singularities. If, for example,
the inclination of the equatorial plane of the rotating body with the
plane of reference vanishes, the angular distance of the node is unde¬
termined. For a similar reason the Keplerian elements break down when
the inclination of the orbit becomes zero. These singularities are pro¬
duced alone by the geometrical nature of the coordinates and are there¬
fore called topological singularities.
Analytical perturbation theories such as the averaging methods, which
were developed by N.M.Krylow, N.N.Bogoliobow and others, are based on
the use of elements, i.e. variables which vary linearly in the unperturb¬
ed motion. The set of Keplerian variables is an example of elements.
The elements derived in this study are required to satisfy the following
two criteria -
1) The elements are regular in the following sense: neither the differ¬
ential equations for the perturbed elements nor the formulae for
computing the coordinates from the elements should present topological
singularities.
2) The differential equations obtained by averaging the perturbation
equations have elementary solutions. Such variables are well adapted
to the problem at hand and are called suitable.
In Part I we derive a set of such elements for the free motion of a
rigid body with dynamical symmetry. The coordinates and the resulting
elements are essentially quaternions, however the actual construction
of the elements is based on the use of 2><2 complex matrices ( in the
complex spinor space ). The elements turn out to be suitable in many
oerturbation problems. In Part III, a theory of the rotational motion
of a symmetrical satellite is established which is based on these
elements.
Part II treats the unperturbed orbital motion, which is a pure Keplerian
motion. Based on a proposal of C.A.Burdet, we define the position of the
satellite by the unit vector directed from the center of the earth to
the vehicle together with the reciprocal distance. The true anomaly q>
is used as the independent variable. A set of regular and suitable ele¬
ments is then derived. This set contains a so-called time-element which
permits the computation of the physical time in terms of <p by explicit
formulae. A convenient property of the elements is that the dominant
terms of the oblateness potential of the earth are finite Fourier
series of (p, which simplifies the averaging procedure. The averaged
solution of the satellite problem is elementary, provided only oblate¬
ness terms are considered.
In Part III we present the simultaneous integration of the orbital and
rotational motion, using the physical time as the independent variable.
It is assumed that the rotational angular speed of the vehicle is large
in comparison with the orbital rate of the center of mass. Applying the
principle of averaging, we first eliminate the so-called short-periodic
terms and then proceed to solve the resulting average system. Due to
the suitability property of the elements, the solution is obtained in
a systematical way. The rotational motion requires elliptic integrals
and functions.
The exact definition of suitability and its consequences are presented
in Part IV. In contrast to canonical mechanics, in which the notion of
suitability originally arised, the requirements that an element system
is to be suitable and regular do not conflict with each other.
Since the degrees of freedom of a given mechanical system often has to
be artificially augmented in order to obtain regular variables, regular
elements generally satisfy a redundancy relation. It is shown, in Part
IV, that these constraint must have a special form for the set to be
suitable. This necessary condition is employed to eliminate several
sets of regular elements a priori, since they have no chance of being
suitable.
ZUSAMMENFAS SUNG
Storungstheorien in der analytischen Mechanik, wie etwa die Mittelungs-
methoden von Krylow und Bogoliubow, benutzen Elemente, d.h. Variable,
die im ungestorten Problem linear mit der unabhangigen Variablen varneren.
Als Beispiel seien die klassischen Keplerelemente genannt. In der vor-
gelegten Arbeit werden folgende zwei Forderungen an die Elemente gestellt:
1) Nach Anwendung der Mittellunsmethode sollen die resultierenden Storungs-
gleichungen elementar losbar sein. Solche Variablen sind dem vorgelegten
Problem gut angepasst und werden "suitable" genannt.
2) Die Differentialgleichungen fur die gestorten Elemente sollen frei von
topologischen Singularitaten sein.
Topologische Singularitaten sind Singularitaten, die nicht durch das
physikalische Problem gegeben sind, sondern solche, die durch die Wahl
von angepassten allgemeinen Lagekoordmaten eingeschleppt werden. Zum
Beispiel enthalten die Euler'schen Winkel emer Kreiselbewegung ernes
starren Korpers eine topologische Singularitat: 1st die Neigung der korper-
festen Aequatorebene zur Standebene gleich Null, so 1st der Polarwinkel
des Knotens unbestimmt. Diese Singularitat kommt dadurch zustande, dass
die Lagen des Kreisels nicht em System von 3 Winkeln ( Eulerwmkel ) ,
d.h. einen 3-dimensionalen Torus, sondern emen 3-dimensionalen pronektiven
Raum bilden.
Im ersten Teil der Dissertation werden nun Elemente, die die oben erwahnten
Forderungen erfullen, fur die freie Bewegung ernes symmetrxschen starren
Korpers bestimmt. Die Lagekoordmaten und die Elemente sind im wesentlichen
Quaternionen. Als Storung kommt das Drehmoment des Gradienten der Newton1
schen Zentralkraft hmzu.
Im zweiten Tell werden Elemente fur die Bahnbewegung des Massenmittelpunktes
des Satelliten hergeleitet. Es wird eine verallgemeinerte wahre Anomalie q>
als neue unabhangige Variable benutzt. Dies bringt den Vorteil mit sich, dass
die Mittelungsmethode nur auf endliche Founerreihen, bezuglich der unab-
hangigen Variablen <p, angewendet werden muss.
Im dritten Tell wird die simultane Integration der orbitalen und der
Rotationsbewegung durchgefuhrt. Das resultierende approximierende Problem
ist im Falle der Bahnbewegung vollig trivial losbar, wahrend die Rotations¬
bewegung des Satelliten mittels elliptischer Funktionen und Integralen ge-
geben ist.
Der Begriff der Suitabilitat, wie er ursprunglich fur kanomsche Systeme
definiert worden ist, ist mit der Forderung nach Regularitat nicht immer
verembar. Deshalb wird im letzten Teil erne neuartige Definition von
Suitabilitat vorgeschlagen, und deren weitere Konsequenzen werden bewiesen.
Schliesslich wird erne notwendige Bedmgung fur die Suitabilitat angegeben,
irit der verschiedene Elementensatze, die nicht suitable sind, a priori
elimimert werden konnen.
CONTENTS
Introduction Page
Part I. Regular Theory of Rotational Motion 2
1. The Euler Parameters 2
2. Free Motion of Symmetrical Rigid Body, Elements 9
3. Perturbation Equations 16
4. Connection with Classical Elements 17
5. Vector of Angular Momentum 19
Part II. Regular Orbital Elements Based on the True Anomaly 20
1. Equations of Motion 20
2. Elements22
3. Time-Element 27
4. Connection with Classical Elements 28
5. The Oblateness Problem 29
Part III. Gravity Effects on the Motion of a Rotating Satellite with
Dynamical Symmetry 32
1. Equations of Motion 3 3
2. Perturbation Procedure 36
3. Secular System41
4. Vector of Rotational Angular Momentum 44
5. Attitude of Satellite 47
6. Numerical Experiment51
Part IV. Redundancy and Suitability of Elements 55
1. Preliminaries55
2. Suitability58
3. Redundant Elements 64
Appendix68
2 -
PART I REGULAR THEORY OF ROTATIONAL MOTION
Euler Parameters
In this section, we state some well-known results on the analytical des¬
cription of a rotating rigid body. In particular, we introduce the
mathematical tools with which a regular theory of rotational motion can
be established and which will also be useful in our discussion of Kepler-
lan motion in Part II.
The proposed theory is based on the use of four real parameters, the so
-called Euler parameters. The three more commonly known Euler angles can¬
not be employed, since they contain a singularity when the inclination of
the body approaches zero. As a matter of fact, no set of three angles
can be regular, since they form a 3-dimensional torus, whereas the con¬
figuration space of a rotation is the real 3-dimensional projective
space.
1.1 Geometrical Properties
Consider two systems of rectangular axes both having their origin at a
fixed point as is shown schematically in
Fig.(1). The one system x ,x ,x remains
fixed in space whereas the other system
y.»y_»y, is fixed to a rigid body, which
is free to rotate about the origin.
It is well known that every displacement
of the body can be represented by a rota¬
tion about a fixed axis. Let ft be the angle
of rotation and let L ,L ,L be the com¬
ponents of a unit vector, directed alongFig. (1)
the axis, with respect ho the x ,x,x -frame. Obviously, the same three
components define the position of the axis in the rotated frame as well.
The Euler parameters are defined by
These parameters are not independent of each other : it is readily seen
- 3 -
that
fi * < * fi * £ = 1 (2)
The position of an arbitrary point with respect to the fixed triad will
T
be denoted by the vector x_= ( x ,x ,x ),where the superscript de-
T
notes the transpose. Let ^= ( y ,y ,y ) characterize the same
point in the body system. The transformation from one system to the
other is given by the well-known orthogonal transformation ( c.f.[l] )
l ' n y.' X~ Rl'£> (3-1)
where
'"ti «1x <*,}
R = \ <ki <?„. <7„ / (3.2)
is an orthogonal matrix with the elements
*"' $-£-$+ 11 , ai*r 2( **- fj $H ), V Z ( fr& + $z<Jr),
«» " 2 ( fa ft -
<?> 1> ), Qn- 2 ( Mr * f» It), «n = - jl- H " £ +$
In order to throw more light on the properties of the Euler parameters
and at the same time to obtain a convenient notation for the application
to mechanics, we introduce the two complex parameters
* * 1* + < f*. y -1-* 'ft .<4)
Relation (2) now reads
«/* + rr - 1, (5)
where a and Y a^e the complex conjugates of a and y respectively.
For the sake of latter use, we note that the matrix elements a, in
kD
Eq.(3.3) are given by
- 4 -
Furthermore, it will be convenient to introduce the spinor A,
A
J* = X.- < *», <J» = - Ck, * aO, -U,= -h>
which represents the general parametrization of an isotropic vector,2 2 2
for which x + x + x =0. Likewise, let A,A be the spinor
(7.1)
of an isotropic vector in the body system.
(7.2)
Corresponding to the orthogonal transformation (3), we have in terms of
our new parameters
Hence, since
/I, = I ( rf^ - f ^)_,we conclude that
f K + * O.
Thus we have obtained the representation of rotations by 2><2 matrices in
where
spinor space:
A - * Q L
L-4
,2-
,<?<?<v,r)
(8.1)
(8.2)
T-Due to relation (5), the matrix Q is unitary ( i.e. Q 2 = E> where E is
the unit 2X2 matrix ) and has unit determinant.
The signs appearing on the right-hand side of Eq.(8.1) are immaterial
since both Q and -Q give rise to the same A-matrix, due to the squaring
process which is involved. Thus to each rotation correspond two sets of
a,Y- or q- parameters which differ m sign. We may, of course, verify
this property by direct use of definition (1): due to the appearence of
"half-angles", an increase of % by the amount 277 will change the sign of
all the q ( k= 1,2,3,4 ).
- 5 -
Since the matrix Q is linear in a,Y,Oi,Y ( an<3 thus in the Euler para¬
meters q ), it may be used for determining the parameters of two succes-
kj I I -v/ I 1
sive rotations. Let a',Y' charaterize the first rotation and let a'',Y'
define the second one. Successive applications of Eq.(8.1) lead to a
resulting rotation, given by 0C,Y >
Q(<r)' QL<r')-Q<-<r').
This equation is equivalent to
-?"
*"J [r'\
(9.1)
(9.2)
Comments
1) Mobius-Transformation
The, equations
' - *a ' rr . a* - < *n = *?,
follow immediately from Eqs.(5),(6). Thus we have
r 1 - «n (10)
* au- t <?,!
This relation, which will be useful further on, connects the quotient
Y/0t vith the body components ( a ,a ,a ) of a unit vector direc¬
ted along the x -axis of the frame of reference.
This equation is a special case of a Mobius transformation. Introduce
the stereographic projection of an arbitrary vector onto the extended
Gaussian x ,x -plane of reference
Ar -
K,tt- -/ti + ti+x}')
and onto the y ,y -equator plane of the rotating body
M =A * <'Y*
r - Yi
We state without proof that the transformations(3),(8) are equivalent
to the Mobius-transformation given by
M (id
- 6 -
where a and y are the previously defined complex parameters.
Eq.(11) is easily motivated by considering isotropic vectors, i.e.
r= 0. In terms of spmors we have
A .x, - it*
m _J.
m -* + <y>-
. di-y,
'
i.'
The linear fraction (11) follows immediately from the spmor transforma¬
tion (8) .
The case r* 0 may be verified by using the method described in [2J.
TThe special case (10) is recovered by setting x.= ( 0, 0, 1 ) •
Tj£= ( a ,a ,a ) and bearing in mind that u= °°.
2) Quaternions
In the theory of rotation it is common to introduce the unitary
quaternion
where e ,e_,§- are three orthogonal unit vectors. In applications to
mechanics it seems to be advantageous to consider the complex pair
(a,Y) rather than the more symmetrical quaternion <r.
3) Brief Historical Account
The representation of a general orthogonal 3X3 natrix in terms of four
real parameters was probably first achieved by L.Euler [3]. A.Caley
[4J developed a general method of constructing such matrices.
A substantial progress was made by O.Rodrigues [5J ,who determined
the parameters of a product of two rotations. The quaternion formula¬
tion was found by W.R.Hamilton [6] and A.Caley [7]. Apparently C.F
Gauss was also familiar with quaternion multiplication,[8J.
The representation of a rotation by a Mobius-transformation was known
to C.F.Gauss |8J , introduced into literature by B.Riemann [9] and
elaborated by F.Klein [lOJ .
- 7 -
Finally we note that the complex representations (8),(11) have been
applied to classical rotational problems by G.Darboux [ll] ard main¬
ly by F.Klein [l2].
4) Derivation of Eg.(3.3)
Eq.(3.3) is elegantly derived by A.Schoenflies [l] and E.Cartan [l3],
whereby the second author makes use of the complex variables a and Y-
Again it was L.Euler [l4J who first found this result.
1.2. Equations of Motion
The position and rotational velocity of a rigid body are uniquely defined
by the complex parameters a,Y and the body components 0^,0)2,103 of the
instantaneous vector of rotational velocity.
If the body system y ,y ,y coincides with the principle axes system of
the spinning body, the components U) satisfy the well-known Euler Equations
>T "77" ( B- c) eSt£fj ~ M,
c -£-' - c/f - i) a, £> - /?,
(12)
where A, B, C are the moments of inertia and M ,M ,M are the components
of the exterior torques, both with respect to the body system, and where
t is the physical time.
The time variation of a and Y may be obtained by considering a point
fixed in space, given by the vector x_. As seen from the body frame, this
point moves with the velocity
(13)
where jj) = ( C1,!S'2,a)3 ) and where the vector y defines the body com¬
ponents of the mentioned point.
Differentiating the spmor Ai,A2 of Eq. (7.2) and inserting Eq. (13) we
straightforwardly obtain
3,
u>.A =
W, *• / 10, <*>,
On the other hand, differentiation of the linear transformation (8) gives
JQ7T '
cH A<
since X. is given in terms of x_ ( c.f. Eq.(7.1) ) and therefore remains
constant.
Hence, by comparing the last two equations , using A= Q-X ,we find
Q.
11 - M(14)
JOIf we drop the second column of Q and 5£-, we obtain the desired dif¬
ferential equations
AtSi
dr\.jt j
Lt>1-H C4)t
(15)
In the applications it is convenient to use the dimensionless quantities
A,B,C,(Ji ,0>2 ,0)3
A .± B'4- c
To '
a.
where I and OOo are appropriately chosen constants, e.g.
&»„ = / ZMl Ib = ( Aco,(o) * ?*£(') + cZ*(o))/cj?.
Using the dimensionless time T
as independent variable, Eqs.(12) , (15) become
B-C
A
C-A
3
A-t
<*i H"'wi <L
OJ, +,U)% ~^i .r
OJ,U>i -" f«, f,
a>3 to, £, f.
"1 v, - f,, £
f •-&) (16.1)
(16.?)
- 9 -
where f. fk(a,afY,Y,Ui,(o2,(i)3,T).
The total order of this simultaneous system is seven. Provided that the
torques are free of singularities, this system is regular, in contrast
to classical sixth order systems.
Comment
In real notation, Eq.(16.1) takes on the form
*0 u>t •1 «r 1i
n 1~
i.
Uj 0
4 ""-I 0 oj}
h
h -*!, -U>% -u>j 0 1*
The matrix on appearing on the right-hand side is skew-symmetric.
2. Free Motion of a Symmetrical Rigid Body, Elements
In free motion, that is when the external torque in Eq.(16.2) vanishes,
the motion of the rotational vector iO. no longer depends on the position
coordinates a and Y- In the symmetrical case, A= B, we readily find the
solution
^1 * ' ^a
CJ, « con
C /77<Vj 2* C-A
si
(17)
where a 3 is a complex constant. It is always possible to define the body
frame in such a way that (1)3 is non-negative. For convenience, we may
therefore assume that
(18)
The quantities 013 and (1)3 are elements since they are constants during
free motion.
Returning to the general case, A* B, and assuming that OJ is a known
function of T, we list some properties of the solution a,Y of Eq.(16.1)
- 10 -
which is now a linear differential system with time-dependent coefficients:
(a) The general solution is a linear combination of two complex linearly
independent fundamental solutions y (T), y (T) ,where y ,y are
complex 2-dimensional vector functions.
ret< v/r) ^ y2(v) = [ V,(t), V,Cr)] (19)
The quantities aj and M2 are complex constants, i.e. they are elements.
Since the determinant of the fundamental matrix [ v ,y J cannot
vanish, we can guarantee at this stage that aj and a2 will be well
defined for all initial conditions.
(b) If
&(r) - ( 4,(t), yP(r) )T
is a particular fundamental solution, then a second linearly inde¬
pendent solution is given by
This result follows immediately upon examination of the second
column of Q in Eq.(14).
(c) Using (a) and (b), we obtain
°LCt)
aif(t)(20)
where the matrix on the right-hand side is the fundamental matrix.
Since the matrix which appears on the right-hand side of the Equations
of motion (16.1) has a vanishing trace, the determinant of the fundamen¬
tal matrix
must remain constant during motion ( perturbed or unperturbed ).
Due to the linearity of the problem we may set det= 1. In this case
- according to Eq.(9.2) - the solution (20) is a product of two rota¬
tions. The first rotation is given by the initial conditions and it is
- 11 -
characterized by the elements (Xi and a2- These elements are the complex
parameters of a general rotation in space. The second rotation describes
the dynamical aspect of the motion.
Clearly, the choice det= 1 is not essential: if det* 1, then the
results of the two rotations are to be multiplied by det or 1/det
respectively.
The general solution (20) is given as soon as one fundamental solution
is known.
Construction of a Fundamental Solution
The following method is based on the fact that we may determine the fun¬
damental solution with respect to any fixed frame of reference. The
general solution (20) is then obtained simply by applying a preliminary
rotation given by the elements oti and aj.
Since the angular momentum vector remains fixed in space during free
motion, we may- for the time being - place the frame of reference in
such a way that its x -axis is directed along the angular momentum vector.
The body components of the vector of angular momentum are given by AUj,
B0)2 and COJ3. The body components of the unit vector along the x -axis
are therefore determined by
where d is the magnitude of the angular momentum,
Eq.(10) yields
(21)
*VM A^i?) -c 6^(f)
According to Eq.(16.1) we may write
(22)
Due to Eq.(22) the terms in the curly brackets are known functions of T.
- 12 -
In the unsymmetrical case A£ B, the solution a (T) of the aboveP
equation is given in terms of elliptic functions, as is derived by
F.Klein and A.Sommerfeld [l2j. These authors do not consider the
general solution and hence they do not introduce a complete set of
elements. In the following, we restrict ourselves to the more elementary
symmetrical case A= B, which is not treated in [l2].
Setting A= B, we obtain
hence
where ao is an arbitrary complex constant.
In a like manner, we find
i.( sL.+ MUi ) v
rPcr) = r. <?*A
where Yo is a complex constant.
The constants ao, Yo are not entirely independent of each other ;
T
inserting the solution y = ( a, Y ) into Eq.(16.1), we obtain
1 p p
*'-£''" (23>
where
L = {f ( J + ^).
Since d, C and - by virtue of assumption (18) - also (1)3 are all
positive constants, P2 cannot vanish.
Since no further condition necessarily needs to be imposed on ao, we
may choose Clo= 1, for convenience.
Thus
- 13 -
Uf(r)= e
it d.xt 4 Of
fr<-d«i(f * m^)V
?f»(24)
The free motion of the rigid body is completely defined by the elements
a.i, 0.2.1 a3 an<3 w3 ithat is to say by seven real quantities. They are
not independent of each other: it is readily seen that
-
1 A?*<t* «,
dei J(25)
Final Set of Elements
As has already been mentioned, the quantities a.\, 0.1, 03 and U3 are
constant elements.
At this stage we mention that the differential equations for the perturbed
elements are obtained by differentiating the solution (17),(19),(24), in¬
serting the original equations of motion (16) and then solving for the
derivatives of the elements. Clearly the differentiation of Eq.(24)
gives rise to terms of the type
( to C4J3T )' = mui} t mcjj f
i.e. terms which may grow indefinitely in the independent variable X. In
order to avoid such terms - usually called mixed-secular terms - we
propose to introduce two auxiliary elements <ji and q>2 defined by the
differential relations
elrn c*J7
The final set of elements ax, a2, a3, co3, <p l and q>2 is defined by
'9"»u>j w Element't
4.
r
s
4 itf. *%>
Sz e-i(**M'
<1
% -
A ' % - ^^J.
r ol .* V'Pui+A%4,~< . "''A'*- *-iV
4(J + <*i)J
:26>
- 14 -
The expressions in Eq. (26) are presented in real notation in [l5] .'
Initial Conditions
The values of the elements corresponding to the initial conditions
(1)1 (0), 0)2 (0) , U)3(0), a(0) and Y(0) are given by
%(*) ~
°, %w = o,
">M = q/re/i «
cJ .• V^ < A' c<3 */ , f
<. Afi."
cl
\ 1 -l•
(27)
Since p2 cannot vanish, the elements al and a2 are always well defined.
Comments
1) The complex elements 0^ and a2 are, up to a scaling factor, the
parameters of a general rotation, and it is therefore not surprising
that they satisfy the "quaternion"-relation (25)
<i 3 <%Ah4
A direct consequence of this property is that it is not possible to
replace <3.\ and 0l2 by three angles ( such as Euler angles ) which
remain well defined for all initial conditions. Thus the use of
redundant elements is necessary.
2) An alternative set of regular elements, which does not share the
difficulty of the denominator p2 as in the a-set (26), may be
obtained as follows. Differentiate Eq.(16.1), insert the Euler
Equations (16.2) of free motion and carry out several further
manipulations to obtain
f - crr,^ y- * lUi+2m- ft)**** J*? r- °
This is a linear differential system with constant coefficients.
- 15 -
Its general solution is readily found in terms of four complex elements
01, 02, 03 and 0„
«: ri= /}. ey,
e,
h - i ( t - ****),
fit) = ^e'v*
^ &"**, vx = { t J-
+ n>^).
The connection between the a- and 0- elements IS given by
a - <<, /.'-it**.
(30)
h'-it <>. A» <
Despite the simplicity of these relations, the 0 - set is not re¬
commended for use in analytical perturbation methods since they are
not suitable, that is, their averaged differential equations tend to
be untractable to analytical integration ( proof in Part IV ). We
therefore discard the 0- elements.
Applications
Let us list a number of applications in which the regular elements CXi,
0(2, C*3, (1)3, q>i, 92 turn out to be suitable, that is to say their
corresponding averaged system possesses elementary solutions:
(a)Free motion of a symmetrical top perturbed by a small asymmetry of
the transversal moments of inertia.
(b)Symmetrical spinning top perturbed by its own weight.
(c)Rapidly rotating satellite ( with finite dimensions ) orbiting the
earth on a Keplerian orbit and perturbed by the non-homogeneity of
the central Newtonian force.
(d)Same as in (c) restricting the orbit to have zero inclination but
allowing the orbit to precess under the influence of the oblateness
of the earth.
Problem (b) will be discussed in more detail in Part IV, Section 2, while
problems (c) and (d) are special cases of the more general problem solved
in Part III. The discussion of problem (a) is left to the reader.
- 16 -
3 Perturbation Equations
If we insert the solution (26) of the unperturbed problem into the
original perturbed differential system (16), we obtain by straight¬
forward computation
% -
A£ - OTOJj ,
«,- f.,
*> = a. *<•f> ) «"'"
'<,' -&{ Z3< e'*>.
% + f2fW," 5;/, p~'^j AA A
'< = iri Xi<« '*.2 ft.
*, 4- ( if* <i * «U i? ''5?) A I
where
and where f,
f, f are defined m Eq.(16.2) ,
A ( = B ) is the
normalized transversal moment of inertia and C is the normalized
longitudinal moment of inertia.
System (31) consists of 3 real differential equations for tpi, $2 and
(1)3 and of 3 complex ones for c*i , a2 and a3. Thus the total order
of the system ,considered as a real system ,
is 9.
Note that no singularities, except the rather unimportant case P2= 0,
are introduced in the transformation of the original problem (16) to the
perturbation equations (31) . However, singularities in the external
torque ( f ,f ,f ) ,i.e. physical singularities, continue to appear
in the above system. Elements whose differential equations of perturbed
motion are free of mathematical singularities, but not necessarily of
physical singularities, are said to be topologically regular.
- 17 -
Remark
The quantity P2 ,which appears in the denominator of the expressions
m Eqs.(31),(26) ,no longer remains constant during perturbed motion.
In rare cases it may occur, after some time, that the value of P2 be¬
comes zero, i.e. CU3 = -d, tii1 = 0, to2 = 0. This difficulty of vanishing
denominators can be circumvented, if P2 approaches zero, by replacing
Eq.(23) by the equivalent relation
<Z?*
r. (32)
and setting Yo = 1 Of course, we have thereby introduced a new singular¬
ity at Pi = 0, i.e. CU3 = d. Eqs.(26),(31) must be modified in a straight¬
forward manner.
4. Connection with Classical Elements
In the theory of rotation, a plane perpendicular to the angular momentum
vector is often introduced. The position of this plane with respect to the
reference plane may be ob¬
tained by two successive ro¬
tations through the angles
8 and £ as depicted in Fig.
(2). Three further angles
\i, O and X determine the
position of the body system
( y1.y2.y3 >•
The five angles 9, £, p, O,
X together with the length
of the vector of angular
momentum, d, uniquely
define the position and the
angular velocity of the ro¬
tating rigid body.Fig. (2)
These variables, which are closely related to the so-called Andoyer varia¬
bles, are elements m free motion ( with A= B ):
- 18 -
A< Z = - m caJ,
In general, the differential equations for the perturbed elements contain
singularities at a = 0 and e = 0 which are absent in the regular ele¬
ment equations (31).
Let Q„, Q ,0 ,0 and Q be the rotation matrices of the rotations9 e u *a x
through the five angles 9, E, \i, 0 and X • Tne first matrix, for example,
is defined by
( 0, 0, 1)\
cc = e'T
hence
n.e *
o
o e~'*
Consulting Fig. (2) , we see that the body components of the vector of angular
momentum is given by
A uj, at find SinX.
d sii ir' coix
tf/ cos. J.
(33)
Inserting Eq.(26) into Eq.(33) we obtain
ci3 = / -|- s-tntl' e-/ex* %;
Since the total rotation is given by
we may apply Eq.(20), which yields
(34)
(35)
< V. (w * r?yJ
/
#
r,
vp nseti tic 'xor-'oSior^ M) , (33) , (3 4 i ,
' ^5) an :[uate the mitru
- 19 -
C COj = ol cosS >
*C = oosi cot* e ,
*L ' I 4 tinS e
It is well known that the singularities in the six ( canonical ) Andoyer
variables can be eliminated by switching to a set of six variables which
are similar to the regular Poincare variables used in celestial mechanics.
However, it must be pointed out that these variables are not elements in
the free rotational motion of a symmetrical rigid body ( cf. [l6j ) .
The key to obtaining regular elements lies in the increase of dimensions.
5. Vector of Angular Momentum
Let us denote the dimensionless components of the angular momentum vector
with respect to the non-rotating reference system x ,x ,x by
a= (HlfH2fH3 )T.
T
If we insert _y_= ( Ad)1# AU2, CU3 ) into Eq.(3) or (8) and express
u ,a and y in terms of the elements of Eq.(26), we obtain the
constant vector H. = js. :
H,+ < «i = - z -£ <. «-,
(37)
. ( J. 'J. - J 1/ )"z =
rr- ( ^%- <%).
These equations will be referred to in Part III.
- 20 -
PART II REGULAR ORBITAL ELEMENTS BASED ON
THE TRUE ANOMALY
1. Equations of Motion
Consider an inertial frame with origin at the center of a rigid body of
mass fl . The Newtonian Equation of motion of a satellite of negligible
mass is then
-£a -r. f-i-%. (•••**) (i)
where f is the physical time, J£ the position vector1,
r its magnitude,
K * If" ftj A* is the gravitational constant, jg is the gradient of a pertur¬
bing potential V(]£,i), and ^$t,£,<7 contains the remaining perturbing forces.
, 2)The total negative energy n, which is defined by
A - a, - /. A* - -f - [ d,k). <2)
where h^ is the so-called Kepler energy, obeys the equation
f> ' ~ jf ~ <£,*.) (3)
In conservative problems, i.e. _£sgt Vfe-tl * V(x,~), the total negative energy
h remains constant and is an integral of system (1).
It is well-known that, in Keplerian motion, the unit vector directed from
the origin to the particle and the reciprocal radius are harmonic oscilla¬
tions with respect to the true anomaly. This fact can be utilized to trans¬
form the Newtonian Equation to a set of oscillator equations.
Following C.A.Burdet [17J , we introduce the four coordinates U,, <ft, Uj, Uv,
«,*-£-, ^'-jK *,--£-. C/y.-f-. (4)
which uniquely determine the position of the vehicle ( <^ is a normaliza¬
tion constant, e.g. the initial value of the semi-major axis ).
The unit vector U * (u,, U,, <4})T, whereTdenotes the transpose, satisfies the
relation (H.ii) = 1.
1The notations employed throughout Part II are independent of those of
Part I.
The symbol f, ) denotes the scalar product of the two inserted vectors.
- 21 -
Let us define a new independent variable f by the differential relatic
dt Jf, I )'•
whereK-JF
jo
K/p( r (5)
(6)
In Keplerian motion, the parameter p is found to be the semi-latus rectum
of the orbit and f is the true anomaly up to a constant of integration.
In contrast to C.A. Burdet, we include the perturbing potential V(_x,i)
in the definition of p . Consequently, the total negative energy h rather
than the Kepler energy hK is related to p as follows
hence,
h -.->£
-j-r
Y 21
,/r* :% h
h ~ -£[ «> -i i(«;-«r> ] (7)
The discussion of this equation is postponed to the third section.
The new equations of motion are given by
where
u" + u_ - i> < + «» - b * 9*.
p = ?r -
/' &t
(8.1)
(8.2)
i-f,
(8.3)
( The components of g_ are %, o/x c/3 )
Provided that £, V are regular, this tenth order system is regular. The
collision singularity, p-0 ,is not discussed.
The solution of Eq.(8) satisfies the following integrals
(U,'u'}= Wl, where |A/1 1 - I
(9.1)
(9 2)
In Keplerian motion, the unit vector t£ moves on the surface of the 3-di-
mensional sphere with constant velocity of unit magnitude with respect to <f
- 22 -
The initial conditions at "f-f'O are well defined for all orbits except
collision orbits and they are given by
where ^ is given in Eq.(6).
Finally, the physical coordinates and velocities are obtained from
/ '
V, 1= r^; r'-^u** *_* /?/?«.' +rX (11)i/f s» *—
2. Elements
Elements are introduced by considering pure Keplerian motion; thus we are
entitled to put J, *o,
le-% ,t.
The parameter p is a first element since it remains constant.
An obvious set of regular elements oltdj attached to u,u' is given by
U, * Cj cssf + dj r/1f , "J * wf- C, CMf * cfj coif J, Cj *f,2,t)
The insertion of the factor w , which is equal to one in pure motion, into
the expression for Ut is motivated by Eq.(9.2).
It turns out that the above elements of Burdet's type are not suitable in
the oblateness problem (c.f. Section 5 and Part IV ). We therefore proceed
to derive a set of elements which will be suitable in the considered problem.
More compact formulae are obtained if we adopt the complex set #,,&, fj.f*
Vm = l ( c<- " < dm). (»-f.2.i.f).
d3)
If we insert Eq.(12) into Eq.(9) we obtain
b* - Si* * y," = °. Kb + a?x * y, » - I • <14>
The vector t f,, Ji ft) is an isotropic vector of length »A5*. The general
solution of the first relation in Eq.(14),
' r, * • h) (-r< * ' ?%) * ?*,
is given by the spmor d,, £,
£s(r<,*>r*). %* = (-r,*'K.) (15.D
and inversely
y, - \ u?- -cf,x), r, '•*(<''* if), r, = •, 3f . (i5.2)
This transformation Ifafi Yt)" (£4) reduces the number of complex elements by
- 23 -
Transformation (12) now reads, putting 3 * U< t t U% and J, » /> , e^ = p-^. y
(16.1)
tfi = p = Element,
* * oz e - d] e *'. ,»[ d-^'**^<*-'*),
w, = ^,e'f* %</ e"r, u;.,*/Cdtfe'r-%j; e-'?],
t/Y = /r e,9>
+ cTv e f*-bt *4 = ' f <^ *,f- st «"^-(16.2)
The second relation in F^.(14) yields the constraint
< ST, + </, 3j - 1.
(17)
The elements J+ Ixl J3, <JY will turn out to be suitable in the J -oblate-
ness problem ( see Pa' IV ).
Geometrical Interpretation
Consider a rotating coordinate system the axes of which are given by the
following three mutually perpendicular unit vectors
;.-if(/x W) ". (18)
( see Eq.(9) )
The first two vectors span the orbital plane, whereas the third one, which
is perpendicular to this plane, points in the direction of the angular mo¬
mentum vector of the satellite.
Let us denote the position of any point with respect to this rotating triad
by £= C/t, )i, ft I. The coordinates of the same point but with respect to the non
-rotating mertial frame are given by J£ = fXf.Xj.Xj) We then have
±= R x, jl-F?t£, *-[«.£•'.«], (19)
where B is an. orthogonal matrix with unit determinant. Using the abbrevi-
ationsu =^ efft Jrs^gTir (2Q)
and applying Eq. (16), the matrix
J(^x'-j-'-pV ii<x- **«• rK- r1) -(«y + *?)
<y + * r i f «cp - z r) j.*. -yy
is found. The variables aL,y are Euler parameters ( see Part I, Eq. (8) )
The symbol X denot s the cross product.
- 24 -
Eq.(17) yields the required condition
ecx * yp- - 1
The direction cosines U_ of the orbiting satellite are obtained by setting
,y - (t, i,o )T,
hence L '
'111- (22)
The matrix R is isomorphic to the 2x2 unitary matrix Q ( see Part I, Sec¬
tion 1 )
<?« f "
L r
where
r
2
det Q = ux. * YY » * 1
The solution matrix Q splits up into a product of two unitary matrices 9,, Q,
where
<?, • <?»•
ir
,-if (23.2)
Thus the solution is given by two successive rotations. The first one,Q,, is
defined by the elements all dt . They constitute a set of Euler parameters.
The second rotation <JZ represents a rotation about the constant unit vec¬
tor
y = (t,o, l )r
through the angle <p ( see Eqs. (1,11) , (1,9) ).
(24)
According to Eqs.(18),(19), the axis of rotation (24) coincides with the
normal to the orbital plane, 2. ,
»[;i1
Thus the transformation from the coordinate vector u, to the elements o£,°^
simply expresses the fact that the vector it, rotates with uniform rate in
f in the orbital plane.
* By isomorphic we mean that the multiplication of two I matrices is equiva¬
lent to the product of two Q matrices.
- 25 -
Perturbation Equations
Assuming that aq^fc in Eq.(8) are small, transformation (16) carries
the differential Equations (8) over into a system for the perturbed time
-varying elements J,, <fl, Jt, JH
<' = 9r,
<
4.
( '<
f*. /
< - i ( h$ - <% ) *".
(25)
where
w4 = «l (p* a)
This element system is entirely free of singularities ( except, of course,
in case of collision ), if they are absent in the original perturbing po¬
tential V and force E_ .
The above seventh order system must be supplemented by an equation for a
time element. This problem will be dealt with in Section 3.
Initial Conditions
Given: f-o, /=£> p, u, u*,Jl', »i, W ( See Eqs. (6) , (10) , (9.2) )
<t,lo) ' P,
dv(o)= X. { u,~b -< Vi /
(26.1)
<**vwhere
A
A
»w > '
•u**w-4-**-^A**ft*'.(26.2)
All square roots are to be taken with the positive sign. Since the Euler
parameters <fx, cf are only defined up to their sign, we may choose the up¬
per signs in Eq.(26.1).
26
Comments
1) If we put
then Eq.(22) yields the so-called KS-transformation
(27)
(28)
In the book [18J by E.Stiefel and G.Scheifele,this transformation is
employed as a coordinate transformation &.•*£ which augments the dimen¬
sions of the system from three to four. In the present application, how¬
ever, it appears in a transformation (t/,u')-* £ which reduces the order
of the system from six to four.
A condition similar to the so-called bilinear relation ( c.f. [18] ) also
persists in the present application. The third components of the relations
[si'**' [J]-^which result from Eq.(19), yield
(cr,u) » e>J (g, u_') = O
Inserting the second relation into the differentiated form of the first
one, the orthogonality relation
(g_'t U)zzO (29)
results. This relation expresses the fact that the rotating triad cannot
rotate about the y»-axis.
Upon application of Eq.(20), Eq.(29) takes on the form
0.
f.A Afi - fif* ft J", (30)
The use of Euler parameters in celestial mechanics has been proposed
recently by several authors ( e.g. R.Broucke,H.Lass,M.Ananda [20] ,S.P.
Altman [l9J ). These authors operate with the physical time as indepen¬
dent variable and do not introduce elements.
Using the true anomaly as independent variable, the Euler parameters are
found to satisfy the system
r'
w
-w
(31)
obtained from Eq. (25) . According to Eq. (1,16.1), the rotation vector m =
( w,, 0, w ) is attached to the rotating frame. As expected, the second
component vanishes.
- 27 -
3. Time Element
In the unperturbed case, the physical time t is obtained by integrating
expression (8.2)
r f~$- = f0*' (32)
where UY is a given function of 9 , Eq.(16.2). Following the classical
line of approach, we introduce the true anomaly f and the eccentricity
hence
f • y + or? (<f¥), t - i /<£|,0
"*" y* b ( i * £ cesf), uj = - e t> r,nf.
(33)
Furthermore, the total negative energy h of Eq.(7) expressed in terms of
the elements p > «£ ,«/»
^ = -£f«v - £^*<> 7
is needed.*
Introducing the eccentric anomaly £ by
£ =Z arc+anf-yjZTtonf jEq.(33) yields
«.An- **;
(f - e coi^-;
Since J<f r cff, Eq. (32) takes on the form
-£• /" f i - e c*e ]dF * (cn-
which leads to the result
(34.1)
where cf9 is a real constant of integration, the time element, and where
'= i£>*{J- *f + t£~s) ~£f'"^ }-
(£-f)= ia«^f-4 7(34.2)
The right-hand side of Eq. (34) is well defined for any elliptic orbit ( £ < 1 ).
- 28 -
Modifications are necessary in the parabolic and hyperbolic cases.
In conservative problems, f> *0, differentiation of Eq. (34) yields
«'. -1
(i* I) «v/ <hl ((L + e) +
4- (f-r i
I+i"<>[ Kt E) -
ii
CI - Vf-it)
(35)
where % and $. are given in Eq.(8.3).
Remark: In non-conservative problems it is convenient to replace Eq.(34) by
tS,nC]i (36)
where / is a new time element.
' - e + 7& !<*-'>
The final set of ( eight real ) elements is given by <f0l e£, ox, cf3, cfY .
Whereas c£, Jx, «j,e^ are valid for any type of orbit, the time element e£ is
only applicable for elliptic trajectories.
4. Connection with Classical Elements
The position of the satellite is characterized by three Euler angles (A,I, ***f(
where St is the longitude of the ascending node, J is the inclination, <+*
is the angular distance from the node to the pericenter, and f denotes the
true anomaly defined in Eq.(33).
The corresponding Euler parameters U.,y~ are given by the well-known formu¬
lae ( see e.g. [2l] )
ci m COS ± e2. '
From this it follows
.tf+t~- Si.)
Eq.(33) may be written as follows
(37.1)
(37.2)
The time element <f. is essentially the same time element employed m the
canonical Poincare-set in [22j found by N.Sigrist.
Pomcare variables ( cf. [22j , [23] ) present a singularity at I" "-
In the following discussion we will assume that Jl, I, as, l and f are defined
- 29 -
by Eqs.(37). These variables are not all "osculating" elements.
Let us introduce a triad (J£, j£,j!j/. The J^-axis is normal to the orbital
plane
£ - a. - («,,«,, «,y
The _|^ -axis is directed towards the ascending node
X, " c' ("*„ a, o)T.
where Co is a positive constant. Finally, the )j_-axis is given by
£ - I * £ •
It lies in the orbital plane.
The vectors X, X ar2 unit vectors if
Applying Eqs. (16.1) , (19) , (20) , (37), we have
I'
cos A. - tin A <*»! tin Jl si*r'
Sin Jl.
1- COSJl COif,
I,= - cosJl tint
O Sltil cosS
(38)
If we denote a vector in the inertial frame by _X_= (K,, Xv)f,) and the compo¬
nents of the same vector with respect to the ]ust mentioned orbital triad
by l^,, we have
This equation will be useful in Part III.
% (39)
5. The Oblateness Problem
v- 4t£(j-S -*)' (40)
Adopting the dominating part of the oblateness potential of the earth
where R. is the radius of the equator of the central body and ^ is a small
dimensionless constant, and neglecting all other perturbing effects, we ob¬
tain in Eq.(1)
1 KJ M '
- 30 -
The element Equations (25) , (35) then yield
<'-±±W-S) \ * * fu)J^L },
S/a-igLrt-Vu',,k'l
1w-1 \m, e~,cr k"
kJ z w,e'r i. - k/ /*. /
< *
-y- (Q-V{ "t* < - zib u'» } e~'*<where
Fit
w
) - 41* t
>'$, £• $-IJ*l. i-^^1,
= Vi - *(«i-V*.
wf - - a i/jU. (JA-sr^Vw,
(41.1)
(41.2)
and where Uj,U*,ui are given in Eq. (16) . The function /%) is regular
at CO. The parameter A is a small dimensionless quantity.
Kxpanding W with respect to A and neglecting higher orders of h, Eqs(41)
yield
K- 0,1,1,1,*, (42.1)
3where
< £- dyUAA,44,%.T(.r.) &(42.2)
Qt » 2 b-ff /c
a'.0 ' F(t)s,/t «N **/,
Q?> . FU)l/c <£"- •£*;/<
«?. -
<*•-< 4-i
«*. < x, .i
2 » *
r*,<
(42.3)
- 31 -
<» i fa «S - r¥ < j> 7 «*'. -' •, // J, - •!•„•, < r¥ ]
<<»**.', a^.JU ^-"h^
(42.3)
(42.4)
Note that the right-hand sides of the differential equations (42) are
finite Fourier polynomials in the true anomaly
- 32 -
PART III GRAVITY EFFECTS ON THE MOTION OF A ROTATING
SATELLITE WITH DYNAMICAL SYMMETRY
The gravitational field of the earth has a dominating influence on the
translational and rotational motion of earth satellites having a suffi¬
ciently high perigee so that atmospheric resistance may be neglected.
The present treatment may be characterized as follows:
(a) The differential equations for the perturbed elements are regular and
therefore well suited to numerical integration.
(b) The method of averages is applied to the regular elements system.This
method eliminates short periodic terms from the original differential
equations by means of near-identity transformations. The resulting
averaged system and the appropriate transformations are regular a priori,
hence no additional transformations are necessary for obtaining a regular
solution.
It is worthy of note that the averaging scheme, which must be modified
in order to meet the conditions of the present problem, is applied with
respect to finite Fourier polynomials.
(c) The translational and rotational motion are treated simultaneously.
Although the former may be considered to be independent of the latter,
this line of attack does offer a self-contained and adapted approach
to the whole problem.
(d) The averaged system is solved analytically. The orbital motion posses¬
ses trivial solutions whereas the rotational motion involves elliptic
integrals and functions except in certain special cases.
(e) We restrict ourselves to the J» -oblateness problem.
Since common perturbation theories operate with singular elements, addi¬
tional transformations of the solution are generally called for. This dif¬
ficulty is avoided in the current approach. The solution obtained has the
desirable property that the formulae remain valid in the singular cases and
are in addition simpler in these cases.
An orbital theory partially satisfying (a), (b), (d) was proposed by W.T.
Kyner [24]. His theory still presents a singularity for vanishing inclination,
and he does not introduce a time element.
*
Collision is not taken into account.
- 33 -
The motion of the rotational angular momentum vector has been studied by
G.Colombo [25], V.V.Beletskn [26], and R.L.Holland and H.J.Sperling [27].
The complete rotational motion of a spinning uniaxial rigid satellite, pre-
cessing uniformly on a circular orbit, is considered by J.W.Crenshaw and
P.M.Fitzpatrick [28J. F.L.Chernousko [29] investigates the complete rota¬
tional motion of a triaxial satellite, whereas J.Cochran [30] allows the
orbit ( with sufficiently small eccentricity ) to precess and spin ( along
the apsidal line ). The complete theories just mentioned suffer from the
presence of singularities.
1. Equations of Motion
Let the vector £= CXtl Xx, *» f denote the center of mass of the satellite with
respect to a non-rotating reference triad located at the center of the
earth such that the *, vplane coincides with the equator of the earth.
The vector y(Yt,A,/,? will represent the components of a point with res¬
pect to the principle axis frame attached to the vehicle and with origin
in the center of mass,X. Denoting the coordinates of the point vin the
reference frame by X^f ,we have
£*« • ! + RX'
where the matrix /7 is given in terms of complex Euler parameters *LtY~ in
Eq. (1,8)!
The differential equations for the orbital motion are given in Part II,
Section 5.
Due to the finite dimensions of the satellite, the non-homogeneity of the
gravitational force will induce a torque with respect to the center of
mass ( see e.g. [26] ). If the satellite is sufficiently small in com¬
parison with its distance to the earth, the body components hK of this
torque are given by
n, = 3K* (c - S) i„ b„/r'
Mx* 3 n" (A - c) btf h„/n (l.D
/V 3 K* (* -*) <*« b**/rl
The notations valid throughout Part III are taken from Part I or II as
is indicated. ( For example, (1,8) refers to Eq.(8) in Part I.)
- 34 -
where the vector Mt'^^n, ^n, *<j) denotes the body components of the unit vec¬
tor U » («,,<{, U, )r, Eq. (2,4),
R'u.(1.2)
To the present accuracy, the oblateness of the earth does not appear in
the exterior torque directly. It influences the rotational motion indirect¬
ly, namely through the vector u_ and the radius T.
Two basic types of rotational motion are to be distinguished: A librational
motion about some equilibrium position and a motion with large rotational
kinetic energy. The present study is concerned with the latter type.
The rotational rate <u« ,Eq. (1,15), is assumed to be much larger than the
orbital rate which is roughly given by the constant Jl»
_n
•^ " "l3?r . (2
( 9, is approximately the semi-ma]or axis of the orbit, see Eq.(2,4). )
Hence
is a small perturbation parameter.
In case of dynamical symmetry,A *B, Eqs.(1,16.2),(1.1) yield
(3)
A -j *•>• iff A,, *„, f* '°' (4)
and the rotational motion is given by Eq.(1,27).
The element bt} is an exact integral of the problem, since fs vanishes. We
therefore obtain the exact solution
<)>x(t)~ m<jj, r (5)
If we use the dimensionless time T of Eq. (1,15.2) as the independent va¬
riable, the remaining elements introduced in Parts I and II satisfy the
system
4&dt
"
A '
/+ ?(',.?%,*) £(4.*t.T, -Cr.'-ie. r->.4<fndt
= r\ *u„ ?f, r) sn(4. *i,T).
(6.1)
(6.2)
(6.3)
( K, I ' J, i, J. o,f> = o i 2,3, y f *.*.* '
- 35 -
where
* -1/£ 4<6."4, f), 4= V fV - -)**# (7)
and where (?„ is defined in Eq. (2,42). £ is a finite Fourier-series in
% ( note that fx is not present )
£~ a '
f-->
.iff
(8.1)
Since the coefficients ^ are rather lengthly, it will suffice to list the
ones which are needed later on
where
f)
c. */ < - ^ ^. <,
6f =- j«m*«v ** ft/cm),
(8.2)
(8.3)
Pi (*-z)
(8.4)
( for 4,4,VV see Eq. (2,16) ).
System (6) is a 15-th order system for the three real variables %, <£, e£ and
the six complex quantities <fXl <ft, tfM , •£,, a(x,*(.3 .
The instantaneous value of the generalized true anomaly y> is obtained by
solving the transcendental Eq.(2,34), which now takes on the form
/«r = e( <fP,-<ft,?),
e > [i(l-£')]',/x { <fQ * L£-f) -
e s-mf },
where (f-f) and ttixi are abbreviations defined in Eq. (2,34.2).
(9.1)
(9.2)
- 36 -
The following formulae are recorded for reference purposes
9(<t*.*f. o) * o, efo.*f. zv) - nr [ia-i-)]'^ (io)
$r -/r V(4.W, $*..£-.*. ,11)
The period of one orbital revolution, r? ,is given by
4L fid-**)]-7'* (12)
The small divisor /i indicates that Z> is much larger than the reference
rotational period, which is of the order 27T.
The two small parameters X and ft are usually of comparable smallness in
the applications: d is of the order /0~ whereas /*i varies from 10 to to.
The right-hand sides of Eq. (6) are -i^-periodic in <fi, and f.
2. Perturbation Procedure
The goal of this section is to eliminate the variables % and J» in the
perturbation Eq.(6) by applying appropriate near-identity transformations.
The procedure to be described in what follows leads to a first-order solu¬
tion in the following sense:
Since J and /* are of comparable magnitude, they may be considered to be
proportional to a single perturbation parameter rf where the proportion¬
ality factor is of the order 1. A first-order solution deviates from the
exact solution by an amount of the order tj% provided the independent
variable T is restricted to the interval f», ^ J ( c arbitrary but fixed )
and provided q is sufficiently small.
The disposable time interval corresponds to about one orbital revolution
( cf.Eq.(12) ). It is possible to extend this time interval to r orbital
periods. However, the order of the error then increases to 7 .
2.1. Elimination of % . This can be achieved straightforwardly by applying
the well-known method of averaging, which was established by N.N.Bogol]u-
bow and J.A.Mitropolski [3l]. We seek a transformation, expanded in powers
Of /I* ,
% * £ + /»* «. ('*. *i-Y. ^ ,2,
. f,) *•
( the variables <f„ ,and therefore also f , are left unchanged because
- 37 -
their differential equations are already free of % ) such that the orig¬
inal Equations (6.1),(6.2) take on the form
Insertion of Eqs.(13), (14) into Eqs.(6.1),(6.2) yields an identity m pow¬
ers of /i* .Since the coefficient of /** must vanish, we have
v ZcWtXlZ)*^^'**-^-*) - ii^^* *%&**,}. <15-2>
As usual, we require the functions l4lo,Hitr to be If -periodic in the phase
variable f, . Averaging Eq.(15.1) and inserting the Fourier-polynomial (8.1),
we obtain
Hence the integration of Eq.(15.1) yields
<» 1^ £ *«+'**> .**fc. (16.2)
Ulil( is determined up to the arbitrary function J» of ^'^fif&Jii tne choice
of which does not effect the final solution. It is very convenient to
choose this integration constant m such a way that uKk has zero mean with
respect to <f>, ,i.e. $?». The average of Eq.(15.2) then gives the simple
result «.
£ "°- (17)
Since i, is real (cf. Eq. (8.2)), system (14) possesses the trivial inte¬
gral y, 7, - aiu/. Thus, to the present order, fyc) is given by
ftCv) =-%-*. (18.1)
where D * / ^' + A* V,l>-) V> '. <18-2>
The initial condition of the constant frequency -^- must be correct to the
order /*»' in order to guarantee an error not larger than of the order /i* in
the solution f>jti throughout the time-interval o i X < fa ,
Thus the initial condition of %j in Eq.(18.2) is given by
»/«) - ^M - /»* U,it ( <f,''l71">,f-'.«',<i>),--i-<''>,fi">l (19)
The functions IL^ and u may be omitted in the transformation (13), since
they are of the same order a the tolerated error.
- 38 -
(20)
(21)
(22)
2.2 Elimination of y .
The remaining equations of motion ( t = ft 2" /
are now subjected to the transformation
*a - «i* * /* u*(4**t*r. A. A) *..
^ • <* * J v, (A, fir) *..
such that the resulting system
is free of the generalized true anomaly 9* •
The present situation differs from the previous one inasmuch as the varia¬
ble f, which is to be eliminated,is defined by a transcendental equation
rather than a differential equation.
Inserting Eqs.(21),(22),(11) into Eq.(20) and equating powers of /» and A,
we obtain
The functions U^ t Vn are again uniquely determined by requiring them to
be 2^-periodic in f and to have zero mean with respect to <f ( ^j,«i then
represent average motion ).
Averaging Eq.(23) with respect to f, we are left with
f,'( A. 4 A. -A) = -^ '(k tA.A.% A.-A) </r >
ti(4.%) -
~r T&*(A,4r) Jr •
where, y r
9* • 9(<f,. <ft,zr). (25)
The integration of Eq.(23) then gives
(24)
(26)
where the integration constants are to be adjusted to obtain zero mean in if.
- 39 -
Evaluation of Formulae
In the present problem Eq.(16.l) yields
ft,'*' if%,
R?- X,K% +»-,</*. .
"'"' i l,Zi,
f>f = *,a, </,,
where
*?'*»*.
/% = x, x, *x, xx, n% - K - i x, k7 .
n* = ix*k - {( *.*» * *,x,),
and where *., &, Xt are defined by (see Eq. (8.3))
if — -.9 . '? -if
h, * X1 e r+ x, e r, hx = Xt e \ x, &.
Hence
X, = <C'<^ - ^V„< - <*3 ( 4H-<k)
C,, C„(i,i are given in Eq. (8.4).
Eqs.(19)-(25) yield
<f • '"£ .*»" ~%r, (28)
Iff
(29)it *•>>
Use was made of relation (11) to obtain ^C*i,.'<f<i,f). This function was then
replaced by 8<VS Jx,,f)-^" since these two expressions only differ by an amount
of the order /4
- 40 -
The First-Order Solution
(a) Average (secular) System:
*rt • -%r aFd.-ti), (j>,fo.i.*,x*. 1-v.i) (30.2)
zrr~ £ c?(J,?l4,£),(Ke.tii3) (3o.i)
(30.3)
where £*%'* and e* are <3lven ln Eqs. (27.2) , (2 ,42.3) , (25) .
(b) Initial Conditions:
Given at r » J» - » > <, <•>. <& W
where Uk, v„ are defined in Eq. (29) .
(c) By adding short periodic terms to the average solution, we obtain the
approximate solution:
ut iz) = utOit J» it) '
j Z, ft (z) « m u, z_,
<klz) = <lCt) * ffU„( <fplt), <%Cz), fCt), S.lc), S.U)), (304)
<fnlz) . J'tz) »JV,( •f'n), 7^cc), ?<c) ),
where D is given in Eq.(18.2).
The generalized true anomaly f is the solution of the transcendental
Eq'(9'1)r» [.^i). e(SPct), <r%cv>, fn). O0.5)
Comments
System (30.1-2) is regular. Its solution depends on the slow variables/""?
and /fJZ"
The approximating transformation (30.4) is free of the difficulties which
are common to most general perturbation theories: singularities are absent.
For machine computation, compact programms are obtained if complex arith¬
metic is employed. This approach may even reduce the execution time.
The formulae are not much more lengthly than those in known solutions,e.g.
[22]. The advantage of having one algorithm for all cases outweighs this
small disadvantage.
Transformation (30.4) has the desirable property that it becomes simpler
in the singular cases. These cases are characterized by the fact that one
of the elements vanishe* e.g. if the eccentricity is zero, we have t^ « O.
- 41 -
3. Secular System
The secular system (30.1-2) is completely free of singularities ( except
collision singularity ). It can be integrated numerically with a much
larger step size than may be used in the original system (6).
Orbital Motion
Eq.(30.2) can be written in the form
where
Jtft -ftf, o.
,*dtf,It
f
{fib Jlf
• •J, I *
<C
Jl -
re- { 1 - 3 c,s*I* } ,
<**}.
e0
6* r>nl** /<£%>/,
(31.1)
(31.2)
are real quantities.
Clearly, the expressions <fx 3J ,<^3f, <f¥ 3J, and thus Jl,, Jlj.Jl*.
constant. Hence we obtain a solution which is valid for all orbits
AsftlZ *<£'(»),
fa? • A * c)f= *.h*
(32)
The average eccentricity *,inclination J-
,and semi-latus rectum a, ,
and therefore also the semi-major axis <f?l/l ~ £** ', remain constant (cf.
Eq.(2,37) ). The classical Euler angles Jl, c, f,see Part II, Section 4,
vary linearly in the independent variable r in averaged motion
JT •j e, ceil* faz + Jl*Co).
f- Jl¥ \/*T fto).
(33)
Eq.(33) is consistent with known theories ( see e.g. [22] ).
Without loss of generality, we may assume that
0 & I* < ff, hence c£ * 0, ( See Eqs.(2,37) (34)
- 42 -
Rotational Motion
In order to solve Eq.OO.l) analytically, we must rewrite them in the form
(35.1)
<'- '/"'» & + '« *. } *:
<' • ' { c>i -& " r« ** * <
< - - < 0, *, •,*( '-*
are real quantities,
and where
/"A* . _5t_,see Eq.(8.4) \
-<**
(35.2)
(35.3)
In spite of the (removable) singularities h^ = <?, n/z = ^, the solution
will be obtained with no additional effort, even when tvf or M4 vanishes.
These critical cases cause difficulties when the Andoyer set is employed.
They correspond to the angles t' I and E'tT ( cf. Part I, Section 4, Eq. (36) )
Since i? is real, the aggregate </,Js is an integral of secular motion.
Thus the magnitude of the average rotational angular momentum
d(^7,') - d* = V CW * Ax «V st,' (36)
and the coefficients Cff/ cM , Ct, remain constant in turn.
Special Case:
If the orbital and the equator planes coincide, we have I =0„rT, ?,»22=0
- 43 -
one
-f,V> = «e>; e- <i.
#, 7-r
<w • <%) if •H/
«£w s <W p"-< fn "3 J'
( s - rx r)
Since ^r has become real, we have the further integrals
W, » To mir Wx = fo o,r/
hence #, and ^j are constants. The solution of Eq.(35.1) is thus the
(37)
It is valid for all initial conditions.
The above formulae also hold true when the central body is assumed to be
spherical rather than oblate, i.e. h -0. in this case, the orbit is Kep-
lerian. The equator plane of the reference triad may be chosen to coincide
with the fixed orbital plane. Hence I -0 and Eq(37) follows immediately.
General Case:
The expressions in the curly brackets of Eq.(35.4) depend only on the quan¬
tities w,, w», 2t .These abbreviations are closely related with the com¬
ponents Hk of the secular rotational angular momentum with respect to the
inertial frame (X,,XXlXa)
Wt - *r. ( cr - ul),
2^ = - X. C «,* * ' *?) *i .
obtained by inverting Eqs.(l,37) and employing Eq.(1,25).
(38)
It is convenient to introduce the orbital components LK of the average
angular momentum vector ( cf. Eq.(2,39) ),
H* = I* cojJI* - L* c*>rf* c/oA* + L* s„I*nnJl*,
H* = £ tinsf + L\ c<x2*cosJ? - /,* tinl' cotj?_
(39)
«; - l; s,»i* + L, casJ
Eq.(38) then yields
Wi = ft. { J* - if «*/* - < -J* }, (40)
- 44 -
These expressions may be inserted into if,, X^ , Eq. (35.4), which yields
the simple result
(41)
,+ ,tWe now proceed to determine the components <,, /,, lj of the average angular
momentum vector as functions of the independent variable f. Thereupon,
Eq. (35.1) will render the attitude elements «V, V,,tC^ by quadratures.
4. The Vector of Rotational Angular Momentum
The mentioned orbital components lk of the angular momentum satisfy the
third order sub-system
(42.1)
c- *.t: - i.?, * * f.t.;.*r--*/: - c. it I*,
',*'- iC »»/.lr)
where
«. J4* n/>2"*z /"
ft -- ¥ c/**; c„ . (42.2)
These equations are determined by inverting Eq.(40), differentiating the
result and inserting the secular system (35).
This sub-system has been obtained by R.L.Holland and H.J.Sperling [27].
In the following we will not consider the cases
(a) C,*0,<e VTCcj, - ef*t ( see Eq. (8.4) )
( cf. Eq.(37) )(b) V= o, -rr )
(C) i =-o J
The system then becomes linear and its solution is readily found.
Hence, let
e. * o, k * o
Eq.(42.1) possesses the integral
where
Q ' - ~- b p— •- /an I*, (t. +o)
(43)
(44.1)
(44.2)
- 45 -
and where P is a constant of integration. This relation represents a fam¬
ily of parabolic cylinders which open in the (positive or negative)^-direc¬
tion.
A second integral is given by ( cf. Eq.(36) )
d*x= C" - C l;\ (45)
The tip of the vector of rotational angular momentum thus lies on the in¬
tersection of a sphere with a parabolic cylinder.
Eqs.(45;,(44) may be combined to express l1 in terms of L}
where
w - -ir {«/**- f* if* n: * s r - C7 <46.2>
(a +o)
is a polynomial of fourth degree in L{.
The last equation in Eq.(42.1) then yields
! "'(it;) -£=; = l«U(!-f.), (47)
where fjlStO will be conveniently chosen later on. The f'j - function
switches sign whenever the polynomial fy becomes zero.
The integration constant r. is obtained by setting f = °
J. sfc- X" **v#- •
The solution of Eq.(47), which is expressable in terms of Jacobian ellip¬
tic functions of J",has been presented by J.Cochran [30] .
The roots of the quartic equation %lt.*)=0 , L,'0 ,are the values of Lz
at which the circle iT'i,*^ and the parabola S " °r»
* H-t*e intersect.
Multiple roots occur if the circle and parabola are tangent at a point- the
solution l*ts) will then remain constant (in a stable or unstable manner) .
Otherwise, there can only be two or four real distinct roots.
- 46 -
Case 1. Four Distinct Roots
Denote the four real roots by rk , where H * 1 * 1 * rv . If i, librates be-
*
tween the larger two roots f, and *i , we may write
Now, consider the function
/ - x?J»'6r,tf1 x
* - *^/7>V«,*) (49.1)
where fn(u,k)J or 0>« for short, is the sine-amplitude function with modulus It,
**C- r.) (r,.r>)
and where
(49.2)
(49.3)
This function satisfies the relation (cf. P.F.Byrd and M.D.Friedman [32] )
j ra <f* = <j u. u~ F(v,k). (50)
S -'
s/ny- - tn(u.h).
The function 19 is either *1 or -J and switches sign whenever the radicand
\(x) becomes zero. We may choose fj - ftf (b, i*)
F(f,k) is the normal elliptic integral of first kind.
If we put i*COr>rz in Eq. (47) , comparison of Eq. (47) with Eq. (50) yields
(51)U = lai.1 ( t - r.lq
Eq.(48) gives
, _
9 Ft*.,*) tufa)Iff U
''
(52)
where
T/ni
*
Thus we have determined the motion of L,(s).
See [30] or [32] for the case when *j librates between /J and 'J.
- 47 -
(53)
Whereas Z, is straightforwardly given in Eq. (44) , L, is obtained by dif¬
ferentiating I; , Eq.(42.1),
^ z l'4. fi -»*/„»«)«
'
One period of motion Z"» T, is given by
r~
/»» H.S1
This interval covers approximately — orbital revolutions.
Case 2. Two real and two complex distinct roots: See [30j .
5. Attitude of Satellite
Elimination of ix in Eqs.(41) and (40) by means of the integral (44)
yields
«,«;, i*> i*{- tf%ut* s,„f /; [ 4%P. ] * /Site)s,al' jt
v#- *? {J**- 34*}.(54a)
( p. - 4r)
Wt (l*,) - -*. S t.nf f ^- rf }, A - [- 4- (t- -£-.)]"\ (54. 2)
where l^(l*) is given by Eq. (46) . The choice of the branches in the square
root defining />,,/, is irrelevant.
Since l7(s) is a known function of S,the secular system (35.1) leads to
the solution
•(,(') * < ^ e' (55)
where
o WtCl'M)'
The integrals /,, J, need not be evaluated explicitly. Inserting the
above solution (55) into Wk > t<\ t(l (cf- Eq.(35.3P, we have
- 48 -
&,
(57)
where the right-hand sides are known functions of /, and thus of -S".
The integrand of T, is rational in i}
a /« J* i?+ i
1 *» * *» ?
Partial fraction decomposition yields
<? * P.
^-^-" *-<»+»'>("**(£» - -^rJl
(58)
(59.1)
(59.2)
Introducing the abbreviation
where
and
(60.1)
(60.2)
we cJbtain
z,V; - JY(s,/>J. V> = *: {**** -- 3 Us) }.
It is not difficult to show that
I* Hi •-lHUfx).
(61)
(62)
Since ^j is defined through the elliptic integral (47) , ^ and £ are
standard elliptic integrals. It is known that they may be expressed by
means of elliptic integrals of 1-st, 2-nd and 3-rd kind and Jacobian ellip¬
tic functions.
Assume, again, the case of four real distinct roots. l*(') is then given by
Eq.(49.1). Inserting this expression into Eq.(60.2), we obtain, provided
/> -t K (It will be shown latter on how to treat this case. )rf> *
- 49 -
where
-m.i(r,-P„) i t -*>*?*
(63i)
itin
T*(r> = ikr f( ^"ifJQ.ISk.i J ( j -xsnli I
**0
«.*- I8i.lt., ^*>^f x~.
(63.2)
Consulting P.F.Byrd and M.D.Friedman [32] ,we find, provided that X^-4*o,
^>l(T^Su)dt>'k{(^-^n^] * *%"l (64-1)
* in--*;)*- y(«) },(64.2)
where 7T(H??) is the normal elliptic integral of 3-rd kind
„J l -xxtfrtfr,*) .
(65)
and where
* (i ve'k*- * Z x? - M* -JIT) TC".*'-) (66)
1 - XxttfU J .
£(u) is the normal elliptic integral of 2-nd kind
u
JPlu) ' J" Jf)x(Z,k) du.
<67>
Hence
Comment: The above formulae are also valid when fn is imaginary. However,
the evaluation of WUi^M) then becomes rather involved (cf. [32] ).
Fortunately, in many cases, we may dispense with this complex expres¬
sion as will be shown shortly.
If ^4«o we have, instead of Eq. (b4.1),
2,1", f„) . (i - -£)u * 4r £lu)- <69>
- 50 -
Insertion of Eqs.(64),(68) into Eqs.(61),(60),(62) yields the desired ex¬
pressions for L,(s),Ithl I}f'). Thus the solution of the secular system, given
in Eq.(55), results in :
4?(S) = e'{c*' l?M * f„ I,M}
<« "^ V *4<cf«)
e'fc" J^fi) - CUI,«)}
•$(!)' C> e"***">
W,(0 :: M>.$Si»l*{ l**- »:}.H",*» X. a s«.X'f fx- **i
(70.1)
(70.2)
Comments
(a) In average motion, the angular momentum d,cf. Eq.(36), and the angle
between the body /,-axis and the vector of angular momentum, 4", re¬
main constant (cf. Part I, Section 4). The angle between the latter
vector and the inertial JT,- axis, e, may vary by large amounts over a
long time interval in a periodic manner. The remaining Andoyer varia¬
bles &,ft,X grow indefinitely and suffer from large long-periodic
fluctuations.
(b) If 4,ts) does not vanish - in the Andoyer set this means that £*Ji"-,
we may compute t(x by employing Eq. (35.3),
where
<<y =
?, = *>»{ e
,JL(I)
?i= *. *.jy-,j/;*2w £.. i#$)}.
(71)
This formula helps reduce labor and is especially recommended when pn
( cf. Eq.(54.2) ) is complex.
If ^J'i does not vanish, we may determine •£,&) in like manner.
«. "
«2. *x
-Si¬
te) The solution PJ.1) breaks down when K (L*/o)) =0 or vj±(l"M) *0
Due to the relation
iv, •< W., = 2 "/**,, v °'
which follows from Eq. (38 ) , V*» and W, cannot vanish simultaneously.
If tv, -o, say, then the solution <tlV can be obtained by ap¬
plying Eq. (71) since ^/s) is well defined in Eq.(70.1).
It is not difficult to show that <' cannot vanish if <x U)~o at some
instant.
(d) We now return to the case fn-"i, which we left out in Eq.(63.1). In
this case, £.(s) takes on the value P„ at some moment. At this instant,
Eq. (70.2) leads to the result that IV, or k£ and thus V, or <tx must va¬
nish. Since «:, and ^ cannot both vanish, we may employ Eq. (71) in
order to avoid the expression under consideration.
In the present theory, solutions which pass through the classical sin¬
gular cases t*o,ir are thus easily obtained.
(e) Similar formulae are obtained when lt liberates between the two smaller
roots rt and K, . The expressions for I,,IX and I} become slightly more
complicated when two of the roots are complex.
6. Numerical Experiment
The ESRO-II satellite, which was launched in May 1968, is a typical exam¬
ple of a spin stabilized near-earth satellite. This satellite spinned at
a rate of 40 rpm about the spin-axis, which was required to be perpen¬
dicular to the satellite-sun line, for reasons of solar power. Once in
the selected position, the dynamic motion was a free-drift over several
days.
The theory of tiie previous sec^i }ns was put to test by roughly simulating
the situation of tie ESRO-II satellite. The initial conditions were
cho~^n \s t
- 52 -
Orbital motion X, » -iJfrSf tit m, K, • * " /n *"
,
XT= -iitfltO «'? rn, X3= ft mf'.
Rotational motion ( cf. Part I, Section 2 )
1i= e o
t f¥*o ?9Jo*s-¥?z,
V o» c', £«, - <?.<? /-', w, = f r"'
The orbital motion is characterized by the data
Pericenter height
Eccentricity
Inclination
ctol km ( from the center of the earth )
0 077
?? 2 ale?
Other pertinant data is given by
Moment of inertia about spin-axis
Average equatorial moment of inertia
Radius of the earth ( at equator )
Constant of gravitation
The perturbation parameters are given by
Jx • 0 0070 PZir,
C ' f 00 I0f J an\
X. • till km.
tC* ? "itC01 /cf km' r*
ft * o oorct6,
The satellite spins approximately 600 times per orbital revolution.
The solution of the rotational motion which was derived explicitly in
the previous sections is directly applicable, since the roots of the
quartic equation (46) are real and distinct for the given initial con¬
ditions.
The accuracy of the orbital and rotational position as computed by the
first-order analytical theory, see Eqs.(30),(32),(70), was checked by
comparing it with an 8-th order Runge-Kutta-Fehlberg integration of
the elements system (6). The numerical solution is assumed to be exact.
The results are contained m the following table for different values of
the physical time t. The first column presents the approximate number n
of rotations of the vehicle about the spin-axis, and the second column
- 53 -
contains the corresponding number N of orbital revolutions. The remain¬
ing columns list the values of the independent variable t and the depen¬
dent variables x,x
,x and q , q , q , q :
first and second lines : "exact" solution of xk<1k respectively,
second and fourth lines: analytical solution of xv'1v respectively.
The residual Ar is defined as the distance between the "exact" solution
of the orbital motion and the analytical one, and it is listed in the
second line of the last column.
Note that the quaternions q ,and thus also the elements of the ortho¬
gonal matrix ( Eq.(2), Part I ), are correct to 7 digits.
- 54 -
1- ^-0.
(N» 0/\
U to
«•* s>
»> \^CN,*^ -J- J-*-
V V-
Oo Oo
O* !*-»
NO N©
•-* or»
O <N-1 J.
NO No
Dado
4- *•
*>-
<r* No NO
UU
6s 6.
o^»n
rv-i »»<*.*"-
*.*>
NiO
111)
••So
0. Do
^<5k »o O
*-»> frN<i
'*M CM <**>
*. J- vfiv.
No'v* Q^
1 1
UNO l^
UN ***"
coo «v r<
v v * *-
1 ' 1 1
•o 0, ^ «•
0-. «-. N SNl
f, t-> nN) c<
W Vo u
»o*» U U
v- v- (ni n
i i t i
J-0» t> 0-.
"no "no <- ^
1 I
N-.IS WN
*• * <vj o,
(VJ'N, vo vo
*>»n ©oo,uu ^,»-
NO NO "nV sV-
> I , ,
5
io NO 0« Oo
CM *v d* N.
t- V- Q o
1 >
OoOa(^ N
o->o> O d
VONO CN) w
On, cy «*-NO
{^. No •"» On
n<V fNj o o
O0 Oo o-xw
*5> Ci No NO
**"«. B-nO-n
^»"no n» V
1 '
no* <vN
0» <\, M (Nj
V »~ "N^Oo
Co "K, &o Oo
no U Q Q
» J- a^V.
n "o o»^
> !? <V|V* *• IN. K,
Oo Oo o. oo
Oo D» ts, t--
7
H (V, *.0o
*. *. *.»-
"•,00 e.°»
i l
NO Nfi ««
II '1
*• V> (TnnoO ^ Ooo,
VO N. ^.J.
1- NO (N, ft
U Inf*.„.
NfiNI <V» fNi
W CN u u
>i il
t? ^"^
^^ •« 6o
& O »v o->
OO O 0
n« tv. /v.
Oo 6o «a Q
i" 1
^.P-. <*-<-/
*»^ »-o.
*» *" 0-.0-.
**<> NOV,,
(V> M oo o.
o« oo o <a
ii it
e
NO
•NO
O Q
•o
o
0-
*--
sr «-.|*»,-iu rl|l>
OO
c
CNl
Oo
4-
NO
NO
**-
v.
U
N>-
- 55 -
PART IV REDUNDANCY AND SUITABILITY OF ELEMENTS
1. Preliminaries
Consider a perturbed mechanical system given by N complex variables
x ( n= 1,2,... ,N ) and by M real variables <p { m= 1,...,M )
which are assumed to be elements, that is to say that x and <pn m
vary linearly during the unperturbed motion ( e = 0 ). We assume, in
particular, that the x remain constant in unperturbed motion whereasn
the a> - variables grow linearly. The <p- elements will be called
m m
phase-variables.
Let the differential equations of the problem at hand be given by
L = e Fn (x,j>,%) ,
(1)
(. e, m = 1, ,
M,
n p = 1.,n )
where the dot denotes differentiation with respect to the independent
variable t, e is a small real perturbation parameter, and where oj,
m
G are real and F are complex functions of the x,
<p„ and of them n p £
complex conjugates x of x . It is assumed that the "frequencies" 0)
of the unperturbed problem are functions of the magnitude of the complex
elements x . The functions G,F are assumed to be 2ir-periodic
p m n————-—
with respect to all of the cp .
m
System (1) is fit for applying analytical perturbation theories, such
as the method of averages. In order to motivate the definitions in the
following sections, we briefly outline this method, which is described
in detail in the book by N.N.Bogoliubow and J.A.Mitropolski [3l].
Let us seek a solution of (1) in the form
(2)
such that the functions v ,u are all 2TT-penodic m the new phase
m n
- 56
variables (p .It is convenient to require that these functions have
m
zero mean with respect to the phase variables <p . This will determine
^ c(k) (k)
m
the functions v, u ma unique way.
m n
Eq. (2) defines a near-identity transformation, expanded in powers of the
* *
perturbation parameter e, from the new set x , q> to the original setn m
x, <p . It is the aim of the averaging method to construct the transfor-
n m
mation (2) in such a way that the resulting differential equations for
* * *
the new set x , cp are free of the phase variables q> . Thus wen m
c
Tm
require that
fm - "*(&)* e G(*(xt.x;) e' cgtfX) + ^
i; - e Ffbe.V) + *x t"WX) + 0.2)
This differential system is called the secular system.
If we insert transformation (2) into the original system (1) and apply
Eq. (3), we obtain a first condition of compatibility by equating the
coefficients of powers of £ :
^ *
h "< *>?- **
fr l >*'>" ~*V > J,(4-1)
where the u,G
,F are to be considered to be functions of the new
m m n
variables. We thus obtain linear partial differential equations for v,
u, in which G
,F are still open for choice,
n m n
By virtue of the assumption of periodicity with respect to the <p we
m
may assume that F is formally given by a Fourier-seriesn
,IX2.X2) eFn(x;X,f;)-T. fl
rjxffl
e
""". (5)'It- t rl
L!
of Eq.(4.2)
Adopting this representation of F,we readily obtain the solution u
,«)_
x / fr„ r„ J(*<**+ -'mYm)
( f _ f*) f-£ -£ U * 9
<6>+
- 57 -
where g is an arbitrary constant of integration, which may depend on
* _*n
x,x and where the dash indicates the absence of the summation vector
P P
( r1# r2,..., rM ) = (0, 0,..., 0 ).
Since we require that the functions u are periodic in the phase
*
"
(1)variables <p. ,
we are obliged to choose F in Eq. (6) as follows
or iv
s"to*) - C = £>»[•1Fn {xp;' r?] J%*~ Jf» • (7)
Thus the first term of the secular system (3.2) ,F
,is obtained by
averaging the original differential equations with respect to the phase
variables.
The additional requirement that the average of u must vanish leads
to the choice g = 0 in Eq.(6). Thus the functions F ,u have
nn n
been uniquely defined.
Obviously Eq.(4.1) may be treated in a similar way, as well as all higher
order equations for u ,v ,
F,
G ( k- z,3 ,....).
n m n m
The procedure outlined above for eliminating the phase variables in F
(1)"and G breaks down, if the denominator appearing in the function u of
m n
Eq.(6) vanishes. We therefore adopt the following necessary condition
for the boundedness of u :
BASIC ASSUMPTION:
The"
frequencies'
to are linearm
ly independent in the foilowing
sense : in the r ange of interest ,the relation
r, «><(x. *•,) +...
* ^M U>,/X,-X,) = 0'
where r,
1• • ,r
m
are integers implies r =...
= r = 0.m
In the cases of finite Fourierseries, this condition is sufficient and
necessary for the existence of u , while, for infinite Fourierseries,n
it is no longer sufficient.
Due to the decoupling of tne -.ocular system (3) ,we may always obtain the
* *
pr.ase-vai j ables a by quadratures, once the x(t) are known. In the following
sections, we restrict ourselves to the discussion of the solution v_ 't).
Adopting the new independent variable T= z t, we obtain from Eq.(3.2)
- 58 -
This is a general perturbation problem. The corresponding unperturbed
problem, obtained by setting e = 0,
JK» ci * * 7* ^
"7P=
1 lV*P/ , (8.2)
is by no means trivially solvable. An important special case of Eq.(8.2)
is treated in the next section.
2. Suitability
a system of differential equations of the type (1) for NConsider
complex elements x and M real phase-variables <p This
problem is said to be suitable if
1) the average of the differential equations for the x-vanables
with respect to the phase-variables q> is given bym
where Q is a real function of the magnitudes x xn P P
(n,p=l,..,N).
2) the quantities Q. are linear independent in the sense of the
basic assumption of the previous section, thus
where r ,...,r are integers, implies that r = r =••
= r = 0.N
The significance of the first requirement is borne out by the fact that
*
the solution x (t) of the resulting unpertubed problem (8.2) is a
n
basic elementary periodic motion of the form
*Ur) = yn e'%(r),
o)
- 59 -
where
%(r) = Jl'n1>(yfyP) r, r = *t
and where y is a complex constant,n
Thus y and t are elements of the unperturbed problem (8.2). These
n n
quantities will be called intermediate elements.
The perturbation problem (8.1), which remains to be solved, can be
written in terms of the perturbed intermediate elements y ,f
. Settingn n
-^r-= Jln(yPyf) (lo.i)
and inserting Eq.(9) into Eq.(8.1), we obtain
{e*<n*'». I*'*) . }ef'
System (10) is again fit for applying the averaging scheme, provided
the second requirement mentioned above is valid ( cf. foregoing section ).
Suitability as defined in the above sense thus not only permits a very
elementary integration of the averaged system, in the first approximation,
but it also allows us to carry out a second averaging "cycle" to the inter¬
mediate elements.
For the sake of later reference, we denote the resulting secular system
obtained by applying the averaging method to the equations for the inter¬
mediate elements, (10), by
# - jl»iy;.%\ «\
dfHn (y; ,*',«>
A definition of suitability for canonical systems is offered in [22j
That definition corresponds roughly to the first requirement in the
definition presented above. The setond requirement nu't also be in¬
cluded in the canonical version if a further iveragmq cycle is re¬
quired.
- 60 -
The disadvantage of the canonical formulation in [22J is that it
only applies to variables with "amplitude" and "phase" character,
i.e. variables such as the radius and polar angle of the complex
element x . Such variables are, in general, singular, that is ton
say that their differential equations present singularities.
Definition
A problem of the type (1) is said to be strictly suitable if
1) it is suitable
2) the right-hand sides of the original differential equations are
of the type
<*.
, (m) r, „*,«; zu. *<T.T)U1)
Fn -/n t;%(*,w f; car C *
(m) (n)i,.,. , i „*where a ,
b are real quantities, r ,s (p= 1,...,N)
q,r,s q,r,s p p
are non-negative integers, and where q (m= 1,...M) are integers.
Theorem 1
Assumption: The following problem is strictly suitable
fm ' um + e am ,
L = e Fn.
Conclusion: The application of the averaging method to the inter¬
mediate system yields a secular system of the form
where Q and V are real quantities.
n n^
- 61 -
The proof of this theorem is given in the appendix.
Corollary 1
The solution of Eq.(12) is given by
t„'n) = Jinr * v>>,
( Jln, Vn = constants )
After two averaging cycles, the resulting secular system of strictly
suitable problems thus yields, elementary solutions for any arbitrary
order of £.
Corollary 2
Provided that all of the required E-series converge, the solution
of a strictly suitable problem, obtained by averaging, is quasi-periodic
( The proof consists in verifying that the solution satisfies the defini¬
tion of a quasi-periodic function. )
Remark
If the original differential equations (11) are regular, it follows
straightforwardly that the near-identity transformations and secular
systems involved and even the frequencies H,V of Eq.(12) will
n n
again be regular ( cf. appendix for the proof ). Hence the final
solution for x (t) , <p (t) is regular for all initial conditions
n m—- —
in the considered range.
Applications
1) Problems of the type (11) are typical of many applications. Consider,
for example, N real perturbed harmonic oscillators
AdODtmg the complex elc-mnts s denned byn
<f - -'?2n
~
o^ * *-n '
_,
r if - -< f 7T "
»
- 62 -
the above equations are transformed to
f = 1, < y„ C a,. i, I.;?
where z,
z are to be expressed in terms of the x, x . These
P P P P
equations have the form (11), if g is analytical in z, z and
'n P P
if,for example, g ( z , z )= g(z,-z )
n p p n p p
even in the velocities z .
P
i.e. if g is
n
In the 1-dimensional case, that is to say if N= n= p= 1, the above
equation is automatically strictly suitable ( if g is even in z ).1 P
2) Consider the rotational motion of a symmetrical rigid body with mass m
perturbed by its own weight which may be assumed to be concentrated in
the center of mass at a distance s from the
origin. The center of mass is assumed to be on
the y -axis of a body coordinate system ( see
Fig.(1) ), which coincides with the principle axis
system. The body components of the external torque
is given by
where g is the constant of gravitation ( see K.
Magnus [33] or E.Leimanis [34] ) .
Let us introduce the perturbation parameter ( see Part I for notations
C =
"> 3 fm
and let uS assume that it is a small quantity. Applying Eq.(31) *f
Part I, we obtain the following differential e4uatl°ns for the pertur¬
bed elements
% = rr,u>3>
'ft
If* KaO^-haI J*-&£<?* },
- 63 -
where
Note that 0)3 is an integral of this problem.
The corresponding secular system ,obtained by averaging with respect
to (Pi ( tp2 does not appear on' the right-hand sides ) ,is given by
1 = 1,2, ?;
*? *
CN.% (2& -«i) + 2/, «£,« I
,
Jl1?* C
Uoffxf^v. (lU JR) - If**,** ],
<- C
If.1 <ct, - < "<]
-
This proves that the considered problem is suitable. By inspection ,
we observe that the problem is even strictly suitable, hence Theorem 1
is valid.
3) The complex rotational elements ax, a2, a3 of Part I, together with
the phase-variables q>i and q>2 ,constitute a system of the type (11)
in the case of a rotating earth satellite ( cf. Part III, Eq.(6) ).
If the orbital motion is pure Keplerian, these elements turn out to be
strictly suitable ( see Part III, Eq.(37) ).
4) The orbital elements 80, iSj , Sz, <53, Sk and cp of Part II,
where
<p is the independent variable, satisfy a system of the type (11) in
the J - oblatenesb problem, provided the two real elements So and &\
are replaced by the complex element 65-= 6je °. The element equations
for S2, ^3, H .iiii 6_, 3 .* strictly suitable as is seen by examining
Eq. (42) ..n Part IJ. ( : :i '-his application we may set r. = 0 in the
"non-resonance" condition appearing in the definition of suitability.
- 64 -
Without going into details, we mention that this is due to the absence
of 60 on the right-hand sides of the differential equations for the
elements and due to 9,1 = ft£ (see Part III, Eq.(32)).
The element system for the 6-elements can be shown to be of the type
(11) for any conservative time-independent perturbation problem.
Comments
1) In the canonical formulism, an equivalent to Theorem 1 can be proven
which holds true for any arbitrary Hamiltonian, provided only that
the problem at hand is suitable in an appropriate sense, such as
defined in [22J. In the present non-canonical approach, assumption
(11) in Theorem 1 has the effect of eliminating typically"non-
-canonical" characteristics such as, for example, dissipative dam¬
ping of solutions.
2) A study of the structursof averaged systems based on "amplitude" and
"phase" type variables, using the perturbation method of Hon rather
than the method of averages, is presented in [35J.
3) For the sake of completeness we mention that it is possible to genera¬
lize problem (1) and the definition of suitability such to include
real elements in addition to the complex elements x .
n
3. Redundant Elements
Assume that the possible motions of the complex elements x (t) are
n
restricted by a constraint of the type
P( xa(t). xnM ) = c,
(i4)
where P is a polynomial or an infinite series in the x and x,
and
where c is a real or complex constant which in general depends on the
initial conditions. This situation often occurs when the dimensions of
a physical system are artificially increased in order to obtain regular
elements, that is to say elements whose differential equations are re¬
gular. A set of elements which satisfy relation (14) will be called a
redundant set.
- 65 -
The following theorem states that redundant elements can, under circum¬
stances, not possibly give rise to a suitable differential system:
Theorem 2
A necessary condition for a set of elements to be suitable is that
the constraining function P(xn
x )n
can be written as a function
of the aggregates x -x alone.n n
The proof of this theorem is postponed to the end of this section.
This theorem puts us into the position of being able to detect a set of non-
-suitable elements a priori, without first having to carry out the
averaging process explicitly.
Applications
1) The complex $-set defined in Part I, Section 2, is easily shown
to satisfy the relations
A A - AA * °•
(15)
£A • AA * AA*AA = '•
According to Theorem 2, the first constraint reveals that this set
cannot ever be suitable, no matter which physical problem is con¬
sidered. ( Theorem 2 applies to each of the integrals in Eq.(15). )
2) The complex orbital Y_elements of Burdet's type, defined in Part II,
Section 2, should be avoided as is seen by examining the first condi¬
tion in Eq.(14), Part II:
tf * tf ' ft - °.
It is of interest to note that related elements ( plane polar coor¬
dinates of the Y, n=l,2,3,4 ) have been employed for the J -oblate-
n£
ness problem of the satellite by W.Flury [36] .The solution of his
resulting secular system is indeed quite cumbersome.
3) The theorem just mentioned raises no objections against any set of
complex "quatermon"-type elements oil and CX2 which satisfy the
condition
- 66 -
We recall that the redundancy relations of the rotational and orbital
elements introduced m Part I and II are of this type. Thus, in con¬
trast to the above mentioned 3- and Y~elements, "quaternion"-type
elements may be well adapted for analytical perturbation theories.
4) Without entering into details, we note that the 8 real and regular
KS-elements ( see [l8j, p.89 ) satisfy a so-called bilinear relation.
If we combine any pairs of these 8 elements to form 4 complex variables,
it turns out that the condition of Theorem 2 cannot be satisfied, i.e.
these sets are not suitable. A set of elements which does turn out to
be suitable in the J -oblateness problem has been found.
Proof of Theorem 2
Since the integral P(x ,x ) is assumed to be analytical in the x and then n n
x , we may writen
P(Xn,Xn) = IZ «£, f x\ (17)
where
r » xT; *;*, x1 - V- xJ~,
where r ,..,r and s ,..,s are nbn-negative integer components of theIN IN
vectors r and s, and where a are real or complex constants.- —
£,£
For our purposes, it is more convenient to express the integral P as
follows
m - (18)
where the coefficients b are uniquely given bym
(rn . (m,, ,/»„))Differentiation of the integral (18) yields
The average of this expression with respect to the <p-vanables gives
^{ -*%* F"(*>V * ^ *%*> } - o, (19)
- 67 -
where F is defined in Eq.(7). Inserting Eq.(18) into this expression
and applying the definition of suitability of the averaged system, we
obtain
L (4,») 4A*/,) x* - o, (20)
m
where
(£.&)* Jl, tn, *...
* -Ah1** -
By virtue of the uniqueness of the representation (20), we conclude that
Due to the linear independence of the frequencies fi we thus find, for
4A£) = o.
Hence,
which has the required property.
Corollary
Theorem 2 also holds true if the function P(x ,x ) is an integraln n
of the averaged system (8.2).
This statement follows immediately from Eq. (19) and what follows in the
above proof.
This corollary is useful, whenever integrals of the average motion are
available which do not require the averaging of the entire system.
- 68 -
APPENDIX
In this appendix, we prove Theorem 1 which was stated in Part IV,
Section 2. In the sequel, we adopt the definitions and notations which
were employed in Part IV.
1. The Improved K-th Approximation
In the method of averages, the so-called improved K-th approximation
for the solution of the perturbation problem ( cf. Part IV, Eq.(1) )
(1)
(2)
is defined by
ft,"0- < * vMi;.*;.*?) * • E*v$W.rt)t
* *
where the varxables <p , x satisfy the secular systemm n
k - «%W> - e opcicx;)*... **"<£(&?:),
In Section 1 of Part IV, we explicitly derived the equations which
define the functions v , u ,G
,F of the improved first
m n m n
approximation ( see Eq.(4) ,Part IV ).
Rather than explicitly establishing the defining equations for any
arbitrary order of the perturbation parameter £ , it will be suf¬
ficient to apply the following algorithm with which we may conveniently
keep track of the structural behaviour of higher-order terms. Assume
that the improved (K-l)-th approximation has been established. We
may then obtain the improved K-th approximation, i.e. the functions
v(K), u(K), G(K), F(K) as follows:m n m n
- 69 -
(K—1} (K—1)
1) Insert the (K-l)-th improved approximation x , <p inton m
£ F (x ,x ,q> ) and expand with respect to £ . Denote the coeffi-
n p p m
K 1cient of £ by T
. Thenn
fK-1)2) Differentiate the expression x for the improved (K-l)-th
nif
approximation and collect terms in powers of £ to obtain, say,
K 2 2£ T
. The function V depends on quantities of the typen n
where p,j = 1,...,(K-1).
3) The functions F,u satisfy the relation
n n
which is obtained by inserting the improved K-th approximation into
the original system (1) and sorting out the terms of the order E
(K)4) The function F is obtained by the averaging principle which is
n
described in the book [3l]. Thus we have
where
"<? ' ^ / / < ? <** '*
is the averaging operator.
(K)5) The function u is uniquely defined by
n
C = r { f - C },
where I, the integration operator, is defined by
r. *; 0 - f. m{ iff}}- o,e*-r "re
*
and where f is a scalar function which is periodic in <pi
(5)
(6)
- 70 -
(K)Up till now we have determined the improved K-th approximation x
(K) (K)Once x is given, we may proceed with the determination of <p in
n m
a like manners
(K, (K) (K-l) (K— 1) (K—1)
6) Expand the function U) (x x ) + E G (x ,"5f ,m. )
mpp mp pR Jl1with respect to £ and denote the coefficient of £ by A
m
(K—1)7) Differentiate <p with respect to the independent variable t,
m
insert the (K-l)-th order secular system and denote the coefficient
K 2 2of £ by A
.A depends on expressions of the type
m m
where p,] = 1,...,(K-1).
(K) (K)8) The desired functions G
,v satisfy the relation
m m
"in+ I— ue ssr
' nm A/»,
e*t 'ft
from which we conclude that
C="f<-^h ^-if»;-/; ?. <8)
Thus F(K>, u(K), G(K), v(K> are given in Eqs. (5) , (6) , (8) .
2. The First Averaging Cycle
In the sequel, we adopt the following notations:
Let q, r, s be vectors with integer components
!_=<?* .In). £- (r<> -^l £' <X ,**),
r, ,... ,r and s,... ,s being positive.
1 N 1 N
The indexed quantities c and c will be assumed to be real
q.r,s r,s_
Constanta.
Let f be a complex function of the variables x ,x
, qi ( p= 1,...,N;p p Jo
1= 1,...,M ), and let g be a real function of the same variables.
We define
71
It is convenient to consider the following six sets of functions:
1) ttXp,\,?e) belongs to A,
if
2) HXe.X-,,%) belongs to A,
Iif
3) f(X,J„5j) belongs to A,
Mif
4) l(*p,X,,ft) belongs to B,
if
5) j6,,*,,ft) belongs to B,
Iif
6) J(H.J,,#> belongs to if
f =
MCSi I1ll
<(},p
<<%&
For the sake of brevity we suppress the arguments of f and g if no
confusion can arise.
The assumption of Theorem 1, Eq.(11), Section 2, Part IV, is easily
shown to be equivalent to
Fn ^ A tn, & 8 (9)
It is our goal to prove that, for all p = 1,2,... ,
c'f A<?> A
(10)
(11)
The proof is based on the following facts, which are not difficult to
verify:
Statements
I. If f or g belongs to a certain set mentioned above, then so does
its complex conjugate.
II. f • A =*> H{{ j « A„, £{ i } = Aj .
in. 3 « e -> m\i) * 8M, i{ i} = sz.
IV. Let f„ « Ar/
3m 6 SI (»*l ,H. r» '% .*) then
(a) f Uni\.T„) 6* => f ( f/> <%,*<>, ft) , f/**.*?.?,). %, * %(*•.*,. f< ) )
tb) y^B,i»fj«* => 9f f.(*,M). £(*,*,*), %, +u(*,r*.,n))r % ( *,, xA f«) * tf
.
- 72 -
V. if fi * At, fx >J^ , ?, « 3X
, 9i « tf„ then
(a) #, * 6/|' TiM "*< ?£* •"
In the case p = 1, the results(10),(11) follow immediately by
applying statements II and III to Eq.(4) of Part IV.
Assume, now, that Eqs.(10),(11) are valid for p = 1,...,(K-1). By
straightforwardly applying the algorithm described in the previous
section, it will be shown that Eqs. (10) ,(11) continue to hold true for
p = K, by virtue of statements II to V.
According to statement IV, (a), the expression
* * *
considered as a function of x , x,
cp, ( a= 1,... ,N ; b= 1,.. . ,M ) ,
a a b
is a member of set A,
hence
n,1 e A. d2)
Due to statement V, (a), and Eq.(4), we obviously have
l~? 6 A.
(13)
Thus we conclude from Eqs.(12) , (13), (5) , (6) and statement II that
K « AM. U» * AI
.
Thus Eq.(10) is valid for p = K .
(K)Since the structure of x is now given, we may proceed to determine
(K) (K)the structural properties of G
,v . Using statement IV, (b), we
m m
immediately find
Al « 6.
(14)
Application of statement V, (b), to Eq.(7) yields
- 73 -
Am & 8.
(is)
Thus, by applying Eqs.(14),(15),(6) and statement III, we obtain the
required result (11) for p = K .
*
Suitable Elements : If the differential system for the elements x
n
is suitable, we may introduce the intermediate elements y ,¥ defined
n n
in Part IV, Section 2. They were shown to satisfy the following differen¬
tial system
-7F ~tf(hV <16>
where T = e-t and where fl is defined in Part IV.
J17)
3. The Second Averaging Cycle
If suitability is assumed, we may apply the averaging method to the
intermediate system (16),(17). The structur of the resulting secular system
can be established in the same manner as in the foregoing section.
Let e be a unit N-dimensional vector whose n-th component is equal_n
to 1 whereas the remaining components vanish. As in the previous section
c are assumed to be real constants, whereas now g is a real function
£'-
of the intermediate variables y ,y ,¥ and f is a complex function
P P P
of these variables.
Let us define the following classes of functions:
1) Uft,%,%) belongs to An,
if f = / J^_ CM l^T <?' "'"
t
( £» - L- S - £n )
2>f( &.?„#) belongs to a",
if f = IZ C^ f jl <?' *"",
J'( in =r- i- §> )
3) HV y„, Yl>) belongs to a",
if f - IZ1 CLl 1~ 1~t-l'Sn
74
4) ft y^y*, V>) belongs to B,
if J » XZ C,, i^V4 e
r.r"" *
5) <](fyj,,fp) belongs to B,
if J = < £1 Crj J^f* ^' '"
.
6) ff»-7s. fp) belongs to Bu ,if <J ' Y~ Cr,r llVl
ML
"
The perturbation terms on the right-hand side of the differential
equation (17) for y are members of the set A,whereas those on
n
the right-hand side of the equation for T vanish. In contrast ton
the preceding case in Section 2, the differential equation for yn
now belongs to its own individual set, A
The properties of the above sets are similar to those of the preceding
section. Statements I,II still hold true, while statements II, IV and V
now read
II. J* A' -» *{i } fe <, *(*} * Az
IV. If f, « -tj, <}a « &x (» "ft .*) then
(a) f e/i/ => f( hl*rf,.%). £(*»!»%) . V, * ?.CV **«))
- f, C *»,*/>, Vi) * A*.
= J. (*>,*,,,%) * 8.
V. If $„ e A?, fn 6 A^ (/>,/! -1, ,*), ?r* Sr, yt* tf^then
(a, %*>* A*.
(b) 11cJX,
^**""' -fc 6
•jfrz * *• ay, * *
Taking these properties into account while carrying out the algorithm
described in the first section of the appendix, we immediately obtain
* *
a secular system for the new variables y ,1" which is of the type
n n
- 75 -
"*- fl &'jl?(y;%<) e e„
,da,
< £ -7 z= c^.0 . >.;.
The last equation may be written as follows
where
w^ tff) H1 .-«
(19)
is a real function of the aggregates y*-y* .
n n
Thus the resulting secular system (18), (19) has the form which was
stated in Theorem 1 in Part IV.
Obviously, the "frequencies" Q,V are regular.
n n
- 76 -
REFERENCES
1. A. Schoenflies: "Einfache Ableitung der Parameterformeln fur Bewegungen
und Umlegungen", Rend. Circ. Matem. Palermo t. 26, 1910, p.329.
2. F. Klein: Mathematical Theory of the Top, Lectures delivered at the
Princeton University 1896, Charles Schribner Sons, New York, 1897.
3. L. Euler: "Problema Algebraicum ob Affectiones Prorsus Singulares Memo-
rabile", Novi. Com. Acad. Sc. Petropolitanae 15 (1770), 1771. pp.76-106.
4. A. Caley: "Sur Quelques ProprietSs des Determinants Gauches", Journal
f. r. u. a. Mathematik 32, 1846, p.119.
5. O. Rodrigues: "Des Lois Geometriques qui Regissent le Deplacement d'un
System Solide dans l'espace", J. de Math. 5, 1840, pp.380-440.
6. W»R. Hamilton: "On Quaternions, or on a New System of Imaginaries in
Algebra", Proc. Roy. Irish Acad. Vol. Ill, 1847, pp.1-16, Communicated
11. Nov. 1844.
7. A. Caley: "Results Relating to Quaternions", Phil. Mag. Vol. 26, 1845,
pp.141-145.
8. Ueber den Stand der Herausgabe von Gauss Werken, Nachrichten der Kgl.
Ges. der Wiss. zu Gottingen, Heft 1, 1898.
9. B. Riemann: "Ueber die Flachen von Kleinstem Inhalte bei Gegebener
Begrenzung", Art. 8, Ges. Werke, 2.Aufl., p.309.
10. F. Klein: Vorlesung uber das Ikosaeder, Kapitel II, § 1.
11. G. Darboux: Lecons sur la Theorie Generalsdes Surfaces, 1. Edition 1887,
Gauthier-Villars, Paris.
12. F. Klein, A. Sommerfeld: Ueber die Theorie des Kreisels, 1. Edition
1910, B.G. Verlag. Stuttgart.
13. E. Cartan: La Theorie des Spineurs. Paris, Hermann et Cie, 1938.
14. L. Euler: "Nova Methodus Motum Corp. Rig. Determinandi", Novi Comm.
Petropolitanae, 20, 1775, p.208. St. Petersburg.
- 77 -
15. M. Vitms: "Uniform Theory of a Rotating Rigid Body with Dynamical Sym¬
metry", Communicated at the NATO-Institute of Dynamical Astr., Cortina,
1972.
16. G.E.O. Giacaglia, W.H. Jefferys- "Motion of a Space Station", Cel. Mech.
Vol.4, No.3/4, Dec. 1971.
17. C.A. Burdet: "Le Mouvement Keplerien et les Oscillateurs Harmonics",
Journal f. reme u. ang. Mathematik, Bd 238, pp.71-84, 1969.
18. E.L. Stiefel, G. Scheifele: Linear and Regular Celestial Mechanics.
Berlin, Heidelberg, New York. 1971.
19. S.P. Altman: "A Unified State Model of Orbital Trajectory and Attitude
Dynamics", Cel. Mech. Vol.6, No.4, p.425. Dec. 1972.
20. R. Broucke, H. Lass, M. Ananda: "Redundant Variables in Celestial Mec¬
hanics", Astron. and Astrophys. 13, pp.390-398, 1971.
21. H. Goldstein: Classical Mechanics. Addison-Wesley, Reading. Mass.,
1951.
22. G. Scheifele,E.L.Stiefel: Canonical Satellite Theory based on Indepen¬
dent Variables Different from Time. Report to ESRO 1972.
23. R.H. Lyddane: "Small Eccentricities or Inclinationjin the Brower Theory
of the Artificial Satellite", Astron. J. Vol.68, No.8, p.555, 1963.
24. W.T. Kyner: "A Mathematical Theory of Orbits about an Oblate Planet",
J. Soc. Indust. Appl. Math. Vol.13, No.l, p.136, 1965.
25. G. Colombo: "On the Motion of Explorer I around its center of mass"
Torques and Attitude Sensing in Earth satellites. Acad. Press Inc. N.Y.
26. V.V. Beletskn. Motion of an Artificial Satellite about its center of
mass. Translation from Russian by Z.Lerman, Israel Program for Scien¬
tific Translations Ltd., distributed by Olbourne Press, London.
27. R.I. Holland, H.J. Sperling: "A First-Order Theory for the rotational
Motion of a tnaxial Rigid Body Orbiting an Oblate Primary", Astron. J
Vol.74, No.3, p.490, 1969.
28. J.W. Crenshaw, P.M. Fitzpatrick. "Gravity Effects on the Rotational Mo¬
tion of a Uniaxial Artificial Satellite", AIAA Vol.6, p.2140, 1968.
- 78 -
29. F.L. Chernousko: "On the Motion of a Satellite about the Center
of Mass under the Action of Gravity Moments", PMM Vol.27, No. 3,
p.474, 1968.
30. J.E. Cochran: "Effects of Gravity-Gradient Torque on the Rotational
Motion of a Triaxial Satellite in a Precessmg Orbit" Cel. Mech.6,
1972.
31. N.N. Bogoliubow, J.A. Mitropolski: Asymptotic Methods in the Theory
of Non-Linear Oscillation. New York. 1954.
32. P.F. Byrd, M.D. Friedmann: Handbook of Elliptic Integrals for Engineers
and Physisists. Springer, Berlin, 1954.
33. E. Leimanis: The General Problem of the Motion of Coupled Rigid
Bodies about a Fixed Point. Berlin, 1965.
34. K. Magnus: Kreisel, Theorie und Anwendung. Springer-Verlag Berlin
1971.
35. U. Kirchgraber, M. Vitins: "The Structureof the Averaged System in the
Theory of Non-Linear Oscillation" to appear m Cel. Mech.
36. W. Flury: Integrationstheorie von Krylow-Bogoliubow und Gestorte
Keplerbewegung. Thesis ETH, No.4358, 1969.
- 79 -
BILDUNGSGANG
Am 27. April 1947 wurde ich in Bad Rehburg, Deutschland, geboren. Nach
Aufenthalten in Schweden und Kanada, wo ich die Primarschule und die
Intermediate Highschool besuchte, liessen sich meine Eltern 1961 in
Wettmgen, Schweiz, nieder. Hier trat ich m die Bezirksschule und an-
=!chliessend in die Kantonsschule der Stadt Baden ein, welche ich mit
der MatuntSt C im Herbst 196b verliess. Ira gleichen Jahr begann ich mem
Studium an der Abteilung fur Elektrotechnik, an der ich im Dezember 1970,
nach emer Diplomarbeit bei Herrn Prof. Dr. E. Stiefel, das Diplom eines
Elektromgenieurs erworben habe. Seither arbeite ich als Assistent am
Institut fur Angewandte Mathematik, wo mir Herr Prof. Dr. E. Stiefel die
Gelegenheit gab, diese Dissertation zu schreiben.
