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UNLIMITED
ROYAL AEROSPACE ESTABLISMIENT
Technical Report 88068
Received for printinq 17 November 1988
SATELLITE MOTION IN AN AXI-SYMOETRIC GRAVITATIONAL FIELD
PART 1: PERTURBATIONS DUE TO J2 (SECOND OPDER) AND J3
by
R. H. Gooding
SUMMARY
Full details are given of the untruncated second-order orbital theory, ofwhich a r~sum6 (including the formulae finally derived) was published in 1983.The analysis only takes account of the first two zonal harmonics, J2 and J3(geopotential assumed), but the extension to an arbitrary zonal harmonic will becovered in a later Report (Part 2).
The principal merit of the theory derives from the extreme compactness ofthe results obtained for short-period perturbations in position and velocity,these perturbations being expressed relative to a system of spherical-polarcoordinates based on a rotating 'mean orbital plane' The use of osculatingelements is thereby avoided, and mean elements are defined so as to give thecoordinate perturbations their simplest possible form. To make the theory bothcomplete and compact, an intermediate set of elements, described as 'semi-mean',is also required.
A numerical assessment of the theory will be included in another Report(Part 3).
Departmental Reference: Space 673
Copyright
Controller HMSO London1988
UNLIMITED
LIST OF COwTmIMT
Page
I INTRODUCTION 5
2 BACKGROUND
2.1 Osculating elements and relate.; quantities 7
2.2 Lagrange's planetary equations 10
2.3 Assumed potential 14
3 MEAN ELEMENTS AND COORDINATES 16
3.1 Mean elements 16
3.2 Semi-mean elements 20
3.3 Satellite position and velocity 23
3.4 Avoidance of singularity in velocity conputation 31
3.5 Long-term perturbations and singularity 35
4 FIRST-ORDER ANALYSIS FOR J2 PERTURBATIONS 44
4.1 Perturbations in the osculating elements 44
4.2 Perturbations in related quantities 48
4.3 Perturbations in spherical coordinates 53
J22 PERTURBATIONS IN OSCULATING ELEMENTS 55
5.1 Perturbation in a (special method) 56
5.2 Porturbation in a (general method) 58
5.3 Perturbation in e 63
5.. Perturbation in i 66
5.5 Check "Ius far by perturbations in p and p cos2 i 68
5.6 Perturbation in Q 70
5.7 Perturbation in 0) 72
5.8 Perturbation in a 76
5.9 Perturbatinns In n and 1 78
5.10 Pexturba_).r. .i M 81
6 ADOPTED SOLUTION FOR J22 PERTURBATIONS 84
6.1 Secular and long-period *,rms, plur short-period carry-over 84
6.2 Pure short-period erturbation r. r 86
6.3 Pure short-period perturbations in v and u 93
6.4 PuLe short-period perturbations in b and w 100
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3
LIST Or CONTZNTS (concluded)
I P age
7 FIRST-ORDER ANALYSIS FOR J3 PERTURBATIONS 106
7.1 Exact equations for rates of change of elements 106
7.2 Perturbation in a 107
7.3 Perturbations in e, i, p and pc? 109
7.4 Perturbations in 0, # and w 111
7.5 Perturbations in p, 0, 1, M and L 114
7.6 Short-period perturbations in and u 116
7.7 Short-period perturbations in r, b and w 118
8 CONCLUSIONS 121
List of symbols 123
References 127
Report documentation page inside back cover
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___________________If
INTRODUCTION
The 'main problem' in modelling the motion of an artificial Earth satellitehas always been recognized as the representation of the perturbations due to he
low-degree zonal harmonics of the geopotential. A considerable literature has
resulted, much of which was surveyed by the present author in earlier papers1-3.
The last two of these papers gave a second-order theory* for orbits of low eccen-
tricity: Ref 2 was confined to the perturbations due to J2 , but Ref 3 covered
the effects (for low e ) of the general (tesseral) harmonic, J, ; as in
Ref 1, the formulae associated with Je. were shown to be applicable to luni-
solar perturbations when f and m are taken as negative and zero respectively.
The novelty in Refs 2 and 3 lay in the compactness that could be achieved by
expressing the short-period perturbations in terms ot a particular system of
cylindrical-polar coordinates; however, when the theory was extended to an
arbitrary elliptic orbit with formulae therefore untruncated in e ), the
corresponding system of spherical-polar coordinates was found to be more
appropriate.
The outcome of the extension just referred to has been a complete second-order theory for J2 and J3 , and a sumarized account of the new theory was
presented to the 1982 IAF Congress4. The present Report gives the full detailsof the purely analytical part of the theory, including (in particular) thelengthy expressions for the perturbations in the osculating elements as well as
the strikingly compact expressions for perturbations in the coordinates. Theoriginal intention was to include details of the hybrid (semi-analytical) compon-
ent 5 of the orbital model, the component that models the long-term evolution of
the chosen set of mean orbital elements. However, the resulting paper would havebeen unduly long, so most of this material has been held over for a later Report.Further, formulae have recently been derived that generalize the J3-results (andthe first-order J2-results) of the preoent paper to an arbitrary zonal harmonic
(Je); these formulae are unexpectedly compact and merit priority in publication.
Thus this Report constitutes Part I of an intended trilogy. Part 2(Technical Report 89022) will cover the purely analytical theory of the general-ization to JZ ; since each Jt , for t > 2 , only needs be covered to first
order, the results for different I may be superposed linearly to provide, incombination with the J2
2-results from Part 1, a complete second-order theory for
* Since all other geopotential coefficients are of order J22 , a 'second-order
theory' is second-order in J2 , but only has to be first-order in any otherJt covered.
jTR 88068
6
the zonal harmonics. Finally, Part 3 wi±.L cover the evolution of mean elements,
the topic dropped from Part 1; it is a consequence of the revision in intention
for the present paper that a large number of references to 'Part 3' will be found.
Section 2 of the Report provides sufficient background to make the paper
essentially independent of the author's previous conitributionsl-6 to the subject.
Some of the more important notation is introduced in section 2.1, but in view of
the complexity of the notation used throughout the paper, a List of Symbols is
included at the end of the Report.
Section 3 is devoted to a number of important issues, and in particular to
introducing the notion of 'semi-mean elements'. Unli..e mean elements, these are
not free of short-period variation; their utility arises from the fact that the
independent variable of Lagrange's planetary equations has to be transfo~red from
time to true anomaly, to permit an untruncated integraticn, sc each semi-mean
element differs from the corresponding mean element by a quantity proportional to
the difference between true and mean anomaly. Considerable attention is devoted
to the way in hich position and velocity are derived (via semi-mean elements)
from mean elements, since the whole ethos of the paper is to stress the import-
ance of an algorithm in which (short-period) perturbations are apFlied directly
to a conceptual 'mean' position and velocity. It is essential, of course, that
no accuracy should be lost when the algorithm is inverted, so that mean elements
can be obtained from position and velocity. This requirement is of particular
inortance at epoch, and is easier to implement than is sometimes supposed; a
method of inverting the algorithm, iteratively, was described in Ref 5. Two
aspects of orbital-element singularity are discussed to ccmplete section 3:
first, a difficilty that arises in velocity computation but not in position
computation; and secondly, difficulties in the long-term propagation of mean
elements. The latter discussion is in the nature of an introduction to Part 3.
Specific expressions for the perturbations due to J2 and J3 , constitu-
ting the meat of the Report, are given in sections 4 to 7. Section 4 covers the
first-order perturbations due to J2 , in both (osculating) elements and coordin-
ates. Section 5 derives the second-order perturbations, due to J2 alone, in
the elements, and section 6 derives the resulting perturbations in coordinates.
Finally, section 7 covers (for both elements and coordinates) the first-order
perturbations due to J3 alone; as already remarked, these are regarded as
contributing to hne second-order solution of the prol.em with both 1J2 and J3
present.
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7
At the end of the concluding section, section 8, the most important
formulae derived in the Report are referenced for convenience.
2 BACKGROUND
2.1 Osculating elements and related quantities
Our starting point is a standard set of osculating elements for an
elliptical orbit, viz a (semi-major axis), e (eccentricity), i (inclination),
11(right ascension of the ascending node), E (argument of perigee) and M (mean
anomaly), all being functions of t (time). The mean motion, n , is directly
related to semi-major axis by Kepler's third law,
n2 a3 = P , (1)
where p is the (Earth's) gravitational constant.
We also require, as an intermediary for perturbations in the 'fast'
element, M , the quantity a , known as the modified mean anomaly at epoch
(where t - 0); this is defined such that (with T a dummy variable for t)
t
M = a + f n d (2)0
and we rewrite (2) simply as
S- + (3)
since this shorthand use of I will prove very convenient. Despite its iame and
definition, a is a cuzrent variable, determined from M , in principle, by
taking the integral of n backwards from t to epoch. It is clearly less
accessible than the 'unmodified mean anomaly at epoch', which is determined (also
as a current variable) just by M - nt , but a has the advantage that we have,
direct from (2),
H - + n . (4)
An important quantity is the true anomaly, v , which is required both in
its own right and (in a particular 'mean' form) as an alteinative independent
variable to t . The derivation of v from M is given in section 3.3, and two
TR 88068 ; 1
8
other important quantities may be derived at once when v is known: first, r
the satellite's geocentric distance, is given by
p/r - I + e cos v, (5)
where p is the semi-latus rectum or 'parameter' of the ellipse, defined by
p - a(l - e2) ; (6)
second, u , the argument of latitude, is given (and defined) by
u - ) + v . (7)
We shall so often require to use the difference between the mean and true
anomalies as a distinctive quantity that it is useful to have a shorthand for it;
the quantity is traditionally known as 'the equation of the centre' and we denote
it here by m , so that
m - v - M. (8)
The elements Q, o) and M , together with some of the other quantities,
suffer certain well-known indeterminacies, or 'singularities', for orbits that
are close to being circular, equatorial or both. Dealing properly with these
singularities has been a major consideration of the study (see section 3.5 and
Part 3), but for the moment we simply introduce some further notation. As in
Refs 2 and 3, we use 4 and YJ for the quantities defined by
- e cos Co (9)
and
- e sin o , (10)
these being well-defined for a near-circular orbit that is not also near-
equatorial. However, 4 and TI are not as significant now as in the earlier
papers, which were largely written on the hypothesis of low eccentricity: it was
as a corollary of the low-e assumption that U , defined as M + Co , was also
appropriately used as a fundamental element, with 4, TI and U replacing e, C0
and M , but the use of U is not nearly so effective when the analysis is not
truncated in respect of e . In considering near-equatorial orbits, it will
TR 8S069
9
sometimes be convenient, in the same way, to use the triple (4, 71, C) to refer to
a set of quantities (sin i sin 11, - sin i cos fl, cos i) that are always well
defined (and in fact constitute the direction cosines of the orbital momentum
vector in the assumed axis system). Some other quantities - V, p and L - are
conceptually useful in the context of the circular and equatorial singularities,
but they are only fully defined at the differential level and their formal intro-
duction is postponed until section 2.2.
As an aid to the concise presentation of formulae, it is convenient now to
define seven quantities, as follows:
c - cos i , (11)
s = sin i , (12)
f 5 (13)
g 1 - f , (14)4
3 (15)h 1-3f,
q ( - 2 ,"(16)
(so that p aq2) andW - q3 (p/r)2 .(17)
It is worth remarking that e and q have the same relation to the so-called
'angle of eccentricity', , as s and c have to the inclination, i , so
that 0 (defined as sin -1 e) might have been taken as a basic orbital element
instead of e . Though there is indeed an important parallel between, in par-
ticular, e and s (of which there will be a good example in Part 3), '-ere is
an essential distinction. Thus, e can exceed unity (hyperbolic orbits) and so
remains a satisfactory parameter when 0 is no longer real (universal para-
meters, valid for all types of orbit, are considered in Ref 7). It is quite
different for s , since as i increases from Y2 to n (retrograde orbits)
s returns to zero, so that it is an inherently ambiguous quantity; further, s
fails to distinguish satisfactorily between distinct near-polar orbits, since its
derivative with respect to i is c , which is then close to zero.
The goal of concise presentation is further facilitated by the introduc-
tion of the eight families of quantities defined, in principle, by:
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I
10
cj - cos(jv + 0)) , sj = sin~jv + w)) , (18)
Cj - cos(jv + 2(0) Sj - sin(jv + 20)) , (19)
yj - cos(]v + 30)) , - sin(jv -1 30)) , (20)
rj - cos(jv + 4w) ,Zj - in(jv + 4)) , (21)
where j is any (positive, zero or negative) integer. It is to be noted,
however, that the arguments of the cj etc will normally not be the osculating
v and w themselves, but the corresponding 'semi-mean' and ) , to be
introduced in section 3.2. Thus the notation 61 etc would normally be
appropriate, but for simp.icity the tildes will be omitted unless (as in parts of
section 5) the distinction becomes important.
Final..y, it is convenient to introduce here a pair of notation conventions
that will be widely used in the long algebraic expressions that are required
en route to tke compact expressions of the final solution. The conventions
involve the use of round and square brackets, respectively, to abbreviate poly-
nominals in f and e 2 . Thus (jl, J2,j 3) stands (in the appropriate context) for
J1 + J2 f + j 3 f 2 ; (J1,J2] stands for Jl + '2 e 2 ; and, sinc3 the conveihtions
will be combined, [(11,J2), J3(J4,J5)1 stands for il + 12 f + 13 C (J4 + j5 f).
The J's will always be integers such that, in the last example, j, and j2
are co-prime, and likewise j4 and J5 •
2.2 Lagrange's planetary equations
Rates of change of osculating elements may be expressed exactly by
Lagrange's planetary equations developed in Refs 8 and 9, for example, or (with
more detail) in other text-booksI°- 12; these assume the existence of a disturbing
function, U , that expresses the perturbation pctential. The a.guments of u
are taken to be a, e, i, Q, 0) and H (with t also in principle, but not in
the resent study); then the six planetary equations are
2 aUna aM (22)
1 q2 q.Ue nae \ ( q. (23)
(c a= L---- (24)na2qs
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I au (25)
na 2 qs ai
.- u a (26)na
2 e ae q3 ail
and
-. i (I U + 2 a U (27)
na 2 e ae (2
From '27) we at once have M , in view of (4).
Before we proceed further, it is worth making explicit the conventional
assumption, which holds through the paper, that the dot notation refers to total
differentiation, with respect to time, d/dt , rather than to partial differenti-
ation, d/at . As the a/at notation refers to differentiation whilst all the
elements are held constant, ie in relation to the osculating orbit, aa/at etc
are trivially zero, and a misunderstanding could hardly arise. For other quanti-
ties, however, the distinction can be important: it is inherent in the concept
of osculating elements that i - a r/at , since d/dt = a/at when applied to
(position) coord.-rates relative to fixed axes, but in general the two derivatives
are not equal; thus, i' * av/at , for example, as appears later in this -.ection.
It will be observed, in (22)-(27), that it is only A and ?1 that have
been expressed in terms of individual partial derivatives, but the other
equations can be combined in such a way as to isolate the remaining derivatives,
and these combinations can be very useful. Thus from (6) we haje
= q2 a 2ae e , (28)
after which (22) and (23) yield
2q _U (29)
na (2
Again, combination of (24) and (29) gives
d(pc2) _ 2qc aUdt na aQ (30)
and this is an important result, since, for a disturbing potential that is
symmetric with respect to the Earth's polar axis, U is independent of Q and
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12
hence pc2 is constant (conservation of c , the polar component of the
angular momentum).
Isolation of the two remaining partial derivatives is possible if we
introduce, as in previous work 1,3, quantities 1 and p defined by
= (0+ cQ (31)
and
- a+ q*. (32)
We then have
na2e ae (33)
and
- 2 aU (34)
na a
Though i and p are not the rates of change of quantities, V and p , of the
familiar type, since c and q in (31) and (32) are not constant, first-order
analysis can be greatly facilitated1,3 by derivation of perturbations SW and Sp
from integration of (33) and (34), the assumption being that the variation of c
and q can be neglected over the period involved. (The informal use of the 6
notation here is without prejudice to the formal meaning to be attached from
section 3.1 onward.) Clearly, SW then remains well defined as i approaches
zero or r. , so long as e does not at the same time approach zero, and this is
not in general true for 6o and U1 individually; again, 8p is invariably
well defined, whereas So is in general not so as e Anoroaches zero.
We can regard p as the non-singular quantity corresponding to 6 , and
Ref 3 also introduces L , the non-singular quantity corresponding to M ; thus
L - A + qj, (35)
and again L and 8L are useful, though there is no familiar integrated 'fast
element' L . It would be wrong to give the impression that the integrated
quantities sf, p and L are so artificial as to be entirely devoid of meaning,
however; it is not hard to see, in particular, that * is the intrinsic rate of
rotation of the perigee direction within the evolving orbital plane, so that ly
TR 88068
37r
13
exists as the accumulated angle (with an arbitrary origin) swept out by this
direction. It is to be noted that, from (35), using (4) and (32), we also have
L - p+ n . (36)
In regard to the potential associated with zonal harmonics only, the right-
hand sides of the planetary equations can be expressed as closed functions of
true anomaly but not mean anomaly. It is therefore of fundamental importance to
be able to transform the independent variable from t , which is effectively
equivalent to M , to v , where v is a function just of e and M . We have,
in fact,
- v M+ ve , (37)
where the partial derivatives, denoted by vM and v. , are given in (42) at the
end of this section; also M is given by (4). We can now appreciate the
difference between the total time derivative, " , and the partial derivative,
av/at , referred to earlier; for the latter we obtain, on replacing M and ,
in (37), by n and zero respectively, and setting vM from (42),
at nW. (38)at
Transformation by (38) permits a complete first-order analysis for any
zonal harmonic; on proceeding to second order, however, we must allow for the
i term in (37), and also for the 6 term that is implied by the appearance of
A , both these terms being of first order. More awkwardly, the first-order
perturbation in (osculating) v , which is our assumed independent variable,
contains short-period terms that are therefore functionally dependent on v
itself, and this implies something of a vicious circle; further, some of these
terms - see (182) - contain the e-1 singularity factor. In second-order
analysis for J2 , all the difficulties are overcome if, anticipating the
introduction of mean elements (and semi-mean elements) in section 3, we transform
(from t) to Z (or rather 0) instead of v . We require a new relation to
replace (37) and (38), and this turns out to be
v - + o(j2, J3 ) , (39)
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~n
14
since i can be defined to include , whilst i is 0(02) (The second-
order term in (39) can be ignored, for change-of-variable purposes, since the
equation is operating on the planetary equations, which are already of at least
first order; when v is required in its own right, however, 0 must be allowed
for - cf (99)).
Application of (38), or (39), to (22)-(27) gives (for any zonal harmonic)
first-order perturbations in a, e, i, Q, 0) and 0 , as functions of . For
a complete first-order solution, we still need the perturbation in M , however,
and this is complicated by the need, through (3), for the perturbation in I , to
add to the perturbation in a . Taking 'perturbation' and 'integral' to commute,
we thus require f 8n dz , where Sn is available at once from 8a (cf (159)
0
and hence is a function of . On applying (39) again, it is immediate that
d(Sh) n. . . . ., (40)
where the right-hand side is effectively just a function of 9 , since the barred
version of (5) can be used to eliminate the implicit presence of F in W
tThus the derivation of j 8n d1 involves just the !-integration of Sn/NW.
0
Evaluation of all the partial derivatives, fox substituting in the planet-
ary equations, is straightforward if appeal is made to the following particular
derivatives as necessary:
r ae.sin vra - ' r - -acos v, r - (41)a e M q
sin v (2 + e cos v)q2 , uO) - 1 , uM - vM -W (42)U e q
2.3 Assumed potential
As we are concerned only with the J2 and J3 terms of the geopotential,
we take as the disturbing function, U , the combination of U2 and U3 defined
by
by 2 ~ J ) P2 (sin 0) (43)
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4 1 7ix
*tm
n*
15
and
7 /0% (,)3 (44)
where is geocentric latitude, R is the Earth's equatorial radius, and P2
and P3 refer to the usual Legendre polyncmials.
The Legendre argument, sin , can be eliminated from (43) and (44) by use
of the relation (which underlies the name 'argument of latitude' for u
sin - s sin u , (45)
and r can be eliminated by appeal to (5). The analysis can be significantly
condensed if we replace J2 and J3 by quantities conveniently denoted by K
and H , respectively, and defined by
3 2
and
the resultant expressions for U2 and U3 are then
p- I (3fC + 2h) (48)6p ~r 2
and
U 3 jLH (0' s (5fC3 + 12gs) , (49)12pr
the notation introduced in (18)-(20) being freely employed.
For many purposes, K and H can be thought of as constants. It is some-
times vital to recognize that they are functionally dependent on p , however, so
that, in particular, they have non-zero partial derivatives with respect to a
and e ; thus the notation K and H is appropriate when p is replaced by
its mean value, j3
It has already been remarked that for an axi-symnetric disturbing poten-tial, Ie for all the Earth's zonal harmonics, not Just J2 and J3 , the polar
component of the satellite's angular momentum furnishes a useful constant of the
TR 88068
W ,____________________
16tion, but a more important constant is provided by the energy integral. The
kinetic energy per unit mass of the satellite is given by
K.E. - - , (50)
and the corresponding potential energy by
P.E. - U(+U) (51)
hence p/2a + U is an absolute constant. It follows that if a' is defined by
1 . ! (1 ( 2aU (52)a? a --9
then a' is a totally invariant quantity to which the osculating semi-major axis
approximates. As such, it is the obvious candidate6 for the mean a that we
shall need to adopt, and it is very gratifying that this choice (5 - a') is
demanded by other considerations.
Since a' is an absolute constant, so is n' , defined by
2 a,3 - A f (53)
and we will find this of great importance6 in the analysis of M , since it
provides a fizm point of reference for the perturbation in mean motion - the
integration of Sn ha3 already been discussed in the derivation of (40).
3 MEAN ELEMENTS AND COORDINATES
3. 1 Mean elements
Let symbolize the generic osculating element, standing (in particular)
for a, e, i, 0, w, a or M . The variation of M is given by (4), but the
variation of the other C is (we are assuming) entirely due to J2 and J3 •
Each variation is made up of a series of short-period terms that are essentially
trigonometric functions of v , together with a long-term effect that is
independent of v (mod 2n) and the orbital period. The combination of all the
short-period terms in constitutes the short-period perturbation, 8
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.......
removal of which frow C gives a mean element, ( , which for most purposes
(because its variation can be plotted easily and accurately over long periods of
time) is more useful than the osculating element; thus
+ (54)
The variation of with time is given, of course, by
AL (55)dt
and and & in general have two components: one, a purely secular component
that only arises with some of the elements; the other, a long-period component
induced by the secular perturbation in 0 . This secular perturbation being a
first-order effect of J2 (and any other 'even' zonal harmonic that might be
considered), long-period variation does not arise until the second order in pure
J2 analysis, and is a cross-coupling effect in J2/J3 analysis; we shall find
that there are second-order long-period perturbations for every element but semi-
major axis. We summarize the division of into its components by writing
C - cc+ i (56)
where Csec can only have an unperturbed component when C is M
Now taec is constant, by definition of 'secular', so its integral, from
epoch to current time, is simply Csec t ; also, the integral of tp may
conveniently be denoted by At (Ref 4 used Kp). Thus the variation of
from epoch is given by
C- to + sec +At(7
There is an important distinction in torm between short-period perturba-
tions and long-period perturbations, as defined here, and it can be seen at once
from equations (54) and (57). Equation (54) indicates that W is defined quite
independently of epoch, and cannot in general be zero at epoch unless special
epochs (such as ascending nodes) are chosen to suit the particular definition of
. Equation (57), oa the other hand, indicates that A( is automatically zero
at epoch (t - 0); At is, in fact, the definite integral of tip from epoch to
TR 88068
_______________0
rz,.
i 18
t , whereas V. is essentially an Indefinite integral. Many writers have worked
with epoch-independent long-period perturbations, so that AC behaves like K,
but as the present author has remarked before3, a heavy price has to be paid, in
particular in the introduction of an intrinsically spurious singularity at the
so-called critical inclinations (where 0,ec vanishes). Nothing is lost by
having epoch-dependent A , since this simply makes the long-period
perturbations behave like the secular perturbations, sec t being necessarily
eoocn-dependent.
Implicit in the last paragraph is the understanding that a short-period
perturbation is only defined 'up to an arbitrary constant', and this is inherent
in the original (54), in the concept of an indefinite integral, in the notion of
epoch independence, and in the earlier reference to a 'particular definition of
SThe choice of a particular constant in V,, or the choice of a particular
'reference value' for & (which amounts to the same thing), is a matter of
considerable importance at the first-order level in J2 , and it is unfortunately
not always clear, from published papers, what constants the authors have chosen
or the motivation for their choice. As was remarked before3 (and more recently
in Ref 6), there are several possible rationales. Thus, the ccistants may be
chosen to make each K unbiased in time, ie such that its value is zero when
averaged with respect to t . This choice, which the use of the epithet 'mean'
rather naturally suggests, leads to awkwand constants (see equations (194) to
(198) of Ref 3), so a better one is to make W unbiased in respect of the
transformed integration variable r . The best strategy of all, however, is to
make an overall choice of constants so as to get the simplest possible
expressions in a particular set of quantities derived from the elements. This
was the strategy of Refs 2 and 3, and the present Report continues the philo-
sophy, the 'particular set of quantities' being the short-period perturbations in
the system of spherical-polar coordinates to be introduced in section 3.3.
In Ref 3, explicit arbitrary constants were introduced into the expressions
for K derived from first-order J2 analysia, so that the chcices adopted by
particular authors could be conveniently quoted and compared. This involved a
generality that will not be repeated in the present paper, and in section 4 we
use the 'best' constants from the start. In the present section we must briefly
consider just one of these constants, the one associated with semi-major axis,
and here the obvious (and best) choice recovers the energy constant a' intro-
duced in section 2.3, this being sometimes referred to as the semi-major axis of
Brouwer 13 . It was used at RAE until the advent of the program PROP14"16, for
which the decision was taken - mistakenly with hindsight - to follow the
TR 88060
t19Smithsonian Astrophysical Observatory by switching to the semi-major axis of
Kozai17.
The choice of a also bears upon the choice of n since this must at
first sight be tied to i by a barred version of (1), in which case (53) demands
that n - a' . Kepler's third law does not have to be carried over in this
unmodified way, however6, and in spite of the vital role played by n' , there is
a simpler and more natural choice of H To arrive at this choice, we set
to M in the generic (57), getting
M 0 + Met + Am (58)
where
Msec sec Sec59
and
- A + A. (60)
Here I is the shorthand quantity introduced by (3), and we can therefore write
t
I - n't + J(n - n')d (61)0
From (1) and (53) we can relate n - n' to 8a I - a - a') and then integrate
using (40). (The notation 8n has not been used for n - n' , slnce it more
appropriately applies to n - H .) The integral in (61) has a secular component,
(d')t say, such that n' + dn' constitutes Lec " Then the natural definition
of 5 is given by
-M - ( + n' + dn' (62)
If is chosen in this way, it becomes part of the task of orbit analysis to
express Kepler's third law in an appropriately modified form6 such that f can
be correctly derived from i
We shall find, in section 4, that in the first-order J2 analysis f is
identical with n' , so that the Kepler law is valid for mean elements without
modification - in fact it is (53). In section 5, however, we shall find this to
TR 88069
o__M
20
be no longer true at the s cond-order level; fi differs from n' , and the
modified Kepler law is givn by (299).
It is natural, and convenient, to extend the concept of 'mean' from the
standard orbital elementi to other quantities, eg v and r . The basis for
definition is that the ,xpression for a mean quantity in terms of mean elements
is the same as the ex 'ession for the original quantity in terms of osculating
elements. Thus the ejuations of section 2.1 hold true after the addition of bars
to all quantities, th'a only (possible) exception being (1) - for the reason that
has just been given. As an example, (5) yields
1r - +i cos , (b3)
after which the quantity W, anticipated at (39i, becomes available. (Obviously,
; and i are n~t like the & in being free of short-periodic variation!)
3. 2 Semi-leau elelents
An important complication arises in regard to the derivation and use of the
mean elements, & , and their smooth (free of short-period variation) rate of
clange, t ; it relates to the change of variable from t to v when
integratinq the planetary equations, (27)-(27). The integration of secular and
long-period terms in dt/d, leads to long-term variation that is 'smooth in I
rather than 'smooth in t ' , whilst thf integration of short-period terms leads
to a pure Poisson series (i. to a combination of trigonometric terms with
arguments linear in r and o ). We can think of this long-term variation as
applying to a semi-mean element , rather than to the true mean element, the
relation between the two being of the form
- + (64)
here Fm is the 'mean equation of the centre' defined in accord with (8), wnilst
t is essentially of the form C/H as we shall see. (The qualification'essentially' is because the precise definition we adopt, as most useful in
practice, involves only the secular component of t, being given by
equation (67), but this is irrelevant for the moment.) To justify (64), st:ppose
that C has been introduced to denote the long-term component of dC/d,
defined from the right-hand side of the planetary equation for t after the
change of variable. This integrates (at least for a purely secular long-term
variation) to C - C0 , given by ( - ) But this may be rewritten as
TR 8806e
21
M + + - ) , whore the first term is no% smooth in t and is to be
identified with t This term constitutes - 0 so we have derived (64)
in the torm specifying variation from to to t the second term may be
thought of as an induced -carry-over' term.
It follows from (54) and (64) that
- - -,(65)
where in principle (until has been precisely defined) (65) expresses the
pure-Poisson component, SA say, of the short-period perturbation, to which
must be added to give the complete 5 . Now we are concerned with a
relation (operating to second order) of the form
= + ( /n)i + 8 (66)
in which the right-hand side reduces to a two-term sum if each component of the
middle (induced) term is combined with either & (contributing to & ) or 5.
In first-order J2 analysis, the only induced terms to arise are those associated
with 6sec and Rosec (because of the identity between n' and R), and it is
natural to incorporate them with 11 and 6) , respectively, leaving just the
pure Bp8 to be amalgamated in the form of perturbations in spherical
coordinates. On proceeding to second order (as in section 5), we find this
first-order policy to have been essential (and not merely 'natural*), at any rate
for w , since awkward carry-over terms can only be avoided if 6) , not @,
appears in the arguments of the first-order terms. It is then obviously sensible
for h and & to include the second-order terms induced by hs,, and WsC,
as well as the first-order ones, and it will be found in section 5 that there
also has to be a second-order term in A , induced by Asec + da ' from (62);
there is no effect from n' (the dominant term in (62)) because this integrates
directly to n't , without any need to transform the variable from t to v .
The induced effects of the long-period variation could be treated in the same way
(and would probably have to be if the analysis were being tdken to third order),
but as Cp arises for every element (except a ) it is simplest if the entire
long-period component of the middle term of (66) is combined with 8 rather
than ( ; the result.ng quantity may be denoted by , and it is the that
are amalgamated into coordinate perturbations.
TR 88068
22
Thus the precise definition of is
SC/ , (67)
except that M29C must be replaced by i - n' when C is M . With this
definition, we can rewrite (66) as
.C +C . (68)
We also require the time derivative of (64). With m defined by (8), v
given by (39), M taken as i , and C taken as constant, this derivative is
given at once in the form
- + i W- 1). (69)
Now C is given by (67). Also, it is legitimate and convenient to replace
by V , since they are the same to O(J2) and C is 0(32) , so we can rewrite
(69) as
W c + ;f . (70)
except that (70) needs the extra term n'(1 - W when C is M
Explicit results for the perturbations in the orbital elements, associated
with J2, J22 and J3 respectively, are derived in sections 4, 5 and 7 of this
paper, and it is convenient to represent the three sets of expressions in terms
of a compact notation involving the suffixes 1, 2 and 3. Thus, for the secular
perturbations we write
A 2 . ( ^
t - i t, + i 2 ( ; 3)(11
(all the C3 in fact being zero); for the long-period perturbations we write
(since the , are all zero)
TR 88068
23
-p ( 2,,p + H 3,p (72)
and for the pure short-period perturbations we write
8pt - K + K 2 + H 3 •(73)
it then follows that
2-K C + ( 2 + 2,2p fi) + i1(t 3 + 3, p) (74)
it being immaterial whether we write mi or m in (74).
3. 3 Satellite position and velocity
Given a set of mean orbital elements at epoch, the unimaginative and
laborious way of computing the satellite's position and velocity at time t is
to incorporate the appropriate perturbations, long-term and short-period, in each
element, and then determine position and velocity frcm the resulting osculating
elements. The algorithm for determining position from osculating elements was
given before2,3 (see also Ref 7), but for completeness it is given again here,
with the necessary extension for generation of velocity, after which we return to
the use of mean elements. The standard system of geocentric coordinates
(x, y, z), is assumed, with x measured towards the vernal equinox and z
towards the north pole. Then the algorithm for position and velocity from
osculating elements ( ) is as follows:
(i) the eccentric anomaly, E , is found by solving Kepler's equation 8,19
E- e sin E - M (75)
(ii) v is found from one of the two equivalent formulae (apart from the
ambiguity of quadrant in the first formula, which is automatically resolved
if the Fortran ATAN2 function is used)
tan v q sin E (76)cos E - e
and
tan V 77-e tan )iE ; (77)
TR 88068
24
( (iii) U is obtained from (7), and r from either (5) or the equivalent formulaIu
r - a(l - e cos E) ; (78)
(iv) n is obtained from (1), and then (38) gives v , with which we can
identify a (the dot notation is legitimate, because we are referring to a
frozen osculating orbit, for which there are no perturbations, so that
dv/dt - av/at and du/dt - au/at);
(v) i is obtained from one of the equivalent formulae
r = naeq -1 sin v (79)
and
- ne (a2/r) sin E (80)
(vi) x, y and z are obtained from the doub:.e coordinate transformation
expressed by the matrix formula
(x y z)T - R3 (-41) R1 (-i) (r cos u r sin u 0 )T , (81)
where Rj( ) describes rotation about the jth axis, so that
R,(e) 1 0 0 etc (82)
0 cos 0 sin e
0 -sin e Cos )and T in (81) denotes transposition;
(vii) x, j and i are obtained by a-differentiation of (81), which gives
(i j )T R3 (41) R,(-i) (i cos u - r6 sin u sin u + ru cos u 0)T .
............................. (83)
If M0 , the conceptual value of the mean anomaly at epoch (though defined at
time t ) is the starting point, rather than the mean anomaly at t itself, then
there is of course a preliminary step given by M - M0 + nt
TR 88U68
25
When we start from a set of 0 (as opposed to o ) and apply (57),
followed by (54), to obtain the ( from which the foregoing algorithm can be
applied, then much the most involved part of the procedure arises witn the
computation of the K to substitute in (54) (or in practice the 8 to
substitute Jn (68)). Hence the overall procedure will be greatly simplified if
we can operate the position-velocity algorithm in a modified form, starting (at
time t ) from the & (-r alternatively the 3 instead of the The
modified algorithm would then lead to '(semi-)mean position and velocity', after
which it would (hopefully) be possible to add greatly simplified combinations of
the 8 directly to the position and velocity components.
It was in pursuance of this philosophy that Kozai17 obtained first-order
short-period perturbations, Sr and Bu , associated with J2 , in a much more
compact form than the perturbations, Sa, 8e, &o and SM , that they displaced.
These perturbations (8r and Su) were, in effect, the perturbations in a pair o!
two-dimensional polar coordinates (r and u), the plane of the coordinates beirg
simply the osculating plane of the orbit. Much more recently, it has been
pointed out by the present author2'3 that it is logical to complete Kozii's
approach by making thq coordinate system three-dimensional, rotating but with a
standard geocentric origin, with a reference plane that is now th3 'icnan oroital
plane' instantaneously defined by i and TI rather than i rnd D . (it is
the 'seri-mean orbital plane', based on a rether than l , that we strictly
require, but this may be regarded as an unnecessary complication at Lhis point.)
On this basis we can interpret a set of mean orbital elements at time t a.3
defining trio position of a 'mean satellite', having mea) anomaly M in a 'moan
orbit' defined (instantaneously) by a, e and V ; the mean orbit lies in (and
moves with) the mear, orbital plane. This is a heipful pcint of view for
geometric visualization, but care is needed. 'hus tne nean satellite itself ha.
a set of evolving osculating elements, and these cann.t Ye identified with the
mean elements of the actual satellite, in particular recause the onculatiig
elements of the mean satellite are not (in genera]) fre-e of short-ptriod
variation - the underlying explanation of this is chdt the instantaneous mean
elements only define the position, not the velocity, of th nean ate~lire*.
* For satellite motion i:& a uniiormly rotating Keplerian e)l pse in a unifo-mlyrotating orbital plane (non-conservative example, in ge, ira!), mean elementscan be defined such that the 'mean satellite' actually coinc.ides wiU.h the truesatellite; then i identifies with i (osculaLt.nq) at the '&Fxes', but. notwith i at the 'nodes'. (This example underies ,he cnrcdc.e of i in firnt-order J2 analysis - see sectioa 1.)
TR 88068
-~ ~ " ~ ~ S -* -' n4
tr .. ,~ ~ l A'A - - A~ - h4A2
26
Refs 2 and 3 described a completion of the Fozai approach in which the axissystem is referenced by cylindrical polar coordinates, the coordinates within the
mean orbital plane (corresponding to r and u ) being designated r' and u ,
and the out-cf-plane coordinate c . It would have been just as satisfactory to
have used spherical polar coordinates, however, and in the complete (untruncated)
analysis presented here it turns out that the spherical system is superior. The
radial coordinate of the (actual) satellite in this system then naturally reverts
to r , and the other two coordinates will be denoted by b and w , the
(vector) relationship of r, b and w to rectangular coordinates, X, Y and
Z , based on the mean orbital plane as XY-plane (with X towards the node),
being
() - (rC3Osb c o s w ) (84)
(Z \r sin w
The directions of the axes of X and Y are not well defined for a near-
equatorial mean orbital plane, but it will be found in due course that (so far as
position is concerned) this causes no difficulty. The relationship of the new
spherical system (r, b, w) to the (obsolete) cylindrical system is immediate,
since in the old system we have (X, Y, Z) - (r' cos u', r' sin u', c) , but in
view of the close relationship between the out-of-plane coordinates b and c
(since sin b - c/r ) it is worth an explicit observation that, whereas c was
the third cylindrical coordinate, b is the second spherical one. (Clearly,
aio w - u' and r2 - r'2 + c2 .)
The position of the mean satellite is conceptually given by coordinates
r, b and w , but it is really the 'semi-mean satellite' we are concerned with,
so the reference point for (short-period) perturbations must actually be expressed
as (Z, b, w). Moreover, since Z is zero for the semi-mean satellite, we have
from (84) that
b-0 (85)
and it is also clear that
w - u. (86)
Since the whole point of this approach is that we apply (68) to coordinates,
rather than elements, the spherical coordinates of the actual satellite are given
by
TR 88068
27
(r, b, w) - (, b,w)+(r, b, 6w), (87)
with a natural extension of the notation that (68) introduced. From (87) we also
have ((i, r, b) -w ) + ( 6, 4,8) ,(88)
on the understanding that Sr (etc) denotes d(8r)/dt , rather than 8(dr/dt);
the distinction arises because (in particular) i is not identical with r
assuming the latter to be defined by converting the right-hand side of (80) to
semi-mean elements.
We can now focus on the specific objective of this section, namely, the
derivation of position and velocity at time t from mean* elements at epoch
without any reference to osculating elements. The derivation of mean elements at
t is necessary as a preliminary step, carried out in principle by application of
(57); the practical difficulties in this step are considered in section 3.5 (and
further in Part 3), so we may assume here a starting point consisting of the six
Sat t . We confine ourselves initially to the derivation of position, and
start by applying steps (i) and (ii) of the algorithm that was given for oscula-
ting elements; the result is the 'mean true anomaly' v. Since Ei is now avail-
able, (64) gives the appropriate , viz ?1, 6 and M ; then a repeat of steps
(i) and (ii) (only done to reflect the change from R to A ), followed by (iii),
leads to v, r and a . At this point, in view of (85) and (86), we have the
semi-mean satellite position (r, b, w) ; on the assumption that formulae for
8r, 6b and 6w are available, it follows that (87) yields (r, b, w), and then
(84) gives (X, Y, Z). We finally require the formula corresponding to (81) in
step (vi) of the algorithm, and thJ may be written
(x y z)T - R3(- ) R (-i) (X Y Z) , (89)
since there is no distinction between i and i.
It is a straightforward matter to express the generic first-order formulae
for 46r, b and 8w in terms of the S , and we now do this in advance of the
specific first-order J2 analysis of section 4 and the J3 analysis of section 7.
• To avoid confusion, it should be noted that the basic (reference) elements aremean, not semi-mean; in application of the 'position-velocity algorithm',however, the first requirement is that the & be superseded by the , and it1s the latter that define the spherical-coordinate system.
TR 88068
28
(Second-order formulae are much harder to express, however, and their derivation
is deferred until they are needed, in section 6.) Our starting point is the
identification of the two formulae for (x y z)T , viz (81) and (89).
Using also (84) in the 3dentification, we get
(c0s b cos w\ R 1() R3 R} R,- 1 cos U . (90)
cos b sin W) sin u)sin b 0
Now
R3(d) R31-) 3 R 3-811) , (91)
exactly, but first-order approximation is required to complete the reduction,
which leads to
b - sin u 6i - a cos u 8Q (92)
and
w - u + c . (93)
In view of (85) we can write 6b in place of b in (92), and in view of (86),
which can be subtracted from (93), we also get
8w - 6u + c 6( (94)
For the formulae we are seeking, it only remains to express &r and bu
in terms of the 8 , and the required expressions are immediate from (41) and
(42). We thus obtain (generically)
br - (r/a) 8a - (a cos v) 6e + (aeq -1 sin v) 6M , (95)
8b - (sin u) 61 - (s cos u) (96)
and
8w - 8M + c 6f + q-2 sin v (2 + e cos v) je + W8M. (97)
In practice, of course, the coefficients in (95)-(97) are to be interpreted as
semi-mean quantities and not as osculating ones.
TR 88068
29
It was remarked, earlier in the section, that no difficulty in position
computation arises from the indeterminacy (for near-equatorial orbits) of the
X-axis (origin for w ) in the rotating system of spherical coordinates, and this
needs to be justified before we pass on to velocity (for which there is a
difficulty - to be dealt with in the next section). The underlying principle
here is that a set of standard elements (osculating, mean or semi-mean) can
always be safely used (for position computation), so long as their values are
consistent. All this means is that no matter how badly defined the elements
), and M are individually, no accuracy is lost so long as they are such
that their non-singular combinations (cf the quantities V -nd L of section 2)
retain full accuracy.
Given a consistent set of &0 (mean elements at epoch), the first step in
the computation of satellite position at time t is, as already noted, the
application of long-term perturbations, to produce at t . This must be done
with full allowance for the possibility of singularity, as described in
section 3.5, after which we have (at time t ) a consistent set of mean elements
and hence a consistent set of semi-mean elements. If C1 is not well defined in
this set, then the X-axis in the semi-mean orbital plane must also be ill
defined; but this will not matter, assuming that the value of a) , to be added
to v to give a ( - ) , is consistent with C . In other words, the
coordinates (X and Y) given by (84) will only be arbitrary to an extent that is
precisely compensated for in the application of the correspondingly arbitrary
fA in (09). This is true even in the extreme case - which can arise in practice
as will be seen in section 7.4 - when Q becomes infinite, but such an infinity
inevitably has a more profound effect on velocity, which is why the velocity
formulae to be presented at the end of this section are only provisional.
Thus a nominal indeterminacy for w in (i, b, w) is dealt with easily
enough. For the perturbations (6r, 3b, 6w) the situation is even better, since
these are well determined at all times. This is essentially because of the &I
term in (94), which reflects the fact that 6w (unlike 6u ) is tied to the same
reference direction as w itself (Kozai's 6u is free of 'circular singularity'
but not of 'equatorial singularity').
We now proceed to the formulae for k, j and i , to complete the algorithm
for position and velocity, bearing in mind that the formulae are only
provisional. The adjustments to avoid infinities are given in the next section.
We have to cover the three remaining steps - (iv), (v) and (vii) - in the
adaptation of the original algorithm to the use of (semi-)mean elements. It
TR 88068
30
follows from the way in which is defined that we can invoke (41) again, as in
the derivation of (95), to obtain r ; since i - 0 we get:, - ---
( ( q sin v M - (a7COS)e . (98)
Clearly b is zero. Finally, (42) can be invoked again, with (86), to give w
the result being
0 WM + sin e (2 + e cos v) e. (99)
The absence of a term in 2 (cf (97)) is due to the point just made in relation
to 8w and w ; £ is allowed for separately (ds we are about to see) via the
motion of the (semi-)mean orbital plane, whereas M is resolved directly into
contributions 8b and 8w
From (r, b, w) we obtrin (, b, w), by appeal to (88), (8i, 8b, 8 )
being given by differentiation of (6r, 8b, 8w). The resulting short-period
expressions involve combinations of v and (0 , including u ( - v + 0) , but
first-order representations of these are adequate for our second-order analysis,
so the singularities associated with long-period perturbations in a and CO
cause no difficulties in these expressions.
It remains to obtain a formula having the same relation to (83) as (89) has
to (81). Differentiating (89) directly, we get
- R3 (-5) R1 (I)A + R3'(-l R,(i) }X\ + R3 (-2 R(i/\
...... (100)
where R1 (-01 - dR -6)/d - 0 0 0 (etc) . (101)
0 -sin 0 - cos 0
0 cos 0 -sin 0
TR 88068
Now from (84) we have
x - Cos b co~i w - sin b cos w - jin w \ ,(102odco b sin w - sin b sin w cos w r
s\n b cos b 0 r wco b
Also, since i c- 0 we have
i- ip, (103)
whilst
-fl, P + W0sa (104)
by (70). It may be observed that the coefficient of i in (100) can be
expressed %s (z sin I - z cos fl Y - jZ)T , whilst the coefficient of
l can be expressed as (- y x 0)T
3.4 Avoidance of singularity in velocity computation
The formulae at the end of the last section being unsuitablc for universal
velocity computation as they stand, because of possible singularity, we must
start by identifying the potentially infinite quantities. They are 0 and M
in (99), which propagate into 4 in (102), and the compensating Li in (100).
The quantity M is potentially singular through expressions for Mel,
that contain e as a factor - see (399) in particular. This causes no problem
in (98), where M occurs with a multiplying e , but in (99) the potential
infinity is real. It is convenisnt to deal with the two formulae in the same
way, however, and we seek to replace M by as close a non-singular equivalent as
possible.
From the special case of (70) when C is M
M n' + W(i; - n') + M , (105
with only the last term potentially singular; this can also be written as
R+(4 1) (F n + Mp (106)
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32
We now recall that L , given by (35), cannot be singular, so we introduce a new
quantity, N (designated L in Ref 4), that can be regarded as a modified
version of the mean motion (though it is not f&ee of short-period variation, let
alone constant); it is defined by
N( -i)( - n') + ('p (107)
Then from (106) and (35) we have
- N - Ip" (108)
Hence, since e s a 0 , (98) can be rewritten as
r- (ii sin ) N - (i cos ia- (,Bsin P (109)
All terms here are free of singularity, the last term (like the original M term)
being so because of the i factor.
Our real concern is to rewrite (99), and this introduces the complication
that (as we have seen) the quantity on the left-hand side, viz w , itself has
the potential singularity, so that a modified quantity must be expVessed. From
(70), with C taken as o, and (108), we have
- I~NqijjeP + *(110)
in which the last term is potentially singular for (both) circular and equatorial
orbits, but even the term in W~p is no longer singularity-free. However, by
addition of 6 to both sides of (110), we may write
and now the right-hand side is well behaved, since
1 -2 { 2 cos + il + cos2 0) ' (112)
TR 88068
33
It ...nediately fol.' s that the modified (non-singular) version of (99) that we
require is
w' im + as*)+ t + q sin (2 + ecos ;) Lp
2 cos + (1 + CoS 1 s3]....... (113)
where w is defined by
w (114)
(It is remarked that 0' is the appropriate quantity to use as the derivative of
, when differentiating the trigonometric arguments of the individual terms of
6r, Sb and 6w to provide Ur, 8b and Sw ( (86) suqgests it is w rather
than w' that is required, but w is not singularity-free; also, the difference
between w and can be neglected in 5i etc, since the resulting effects
are O(J23, J2J3).)
Since w', rather than w, is the non-singular quantity, it follows that
and Y , as given by (102), should be replaced by quantities computed from 1'
rather than i , where 4' is obtained from addition of 64 to w' instead of
to w . This will make the last term of (100) non-singular and, as we shall see,
in such a way as to make the middle term non-singular as well. What we are doing
to (102) amounts to the creation of errors -r Z Qp cos b sin w and
r E Qp cos b cos w , in X and Y respectively, that can be corrected in (100)
by the additional term
r Z -p cos - sin 9 sin 11 cos b sin w
sin fC c cos b - C os - cos b cosw
0 0
on expanding R3 (- 6) R 1-1 , the multiplier of (X Y ZI)T But the
additional term can be rewritten as
TR 88068
FE ! S sin oo 0 ' cos bcosw
- cos sinQ 0 cos b sin w
-i 0 0 sin b
which is in just the right fom for combination with the 6p component of themiddle term of (100), which can itself be written as
r 3sin L - cos h coo a Co b Co W
c0s-3 sin 1 sins1 ( cos b sin w
0 0 0 sin b
On combination, and in view of (84), we get
s p sin CI 0 C03
C Co 1 0 sin ) (This can be rewritten as s P R - ; the appearanceof 5 , effectively replacing 6 as a multiplier of fp , makes the expression
fully non-singular, and its form is such that it can be gathered back into the
last term of (100).
Only the secular component of Q in (100) now remains and it follows at
once, in view of the simplificttions indicated by the last sentence of
section 3.3, that the equation can at last be rewritten, using (104), as
(X) = i~zsinfi + (D Y) + R 3(~ R~ I)(' 15
where X', Y' and Z' are non-singular versiono of X, Y and Z , given by
- icos b cos w - r sin b (b cos w -i ) - r4' cos b sin w, (116)
- cos b sin w - r6 sin b sin w + r&t' cos b cos w (117)
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and
- r sin b + r cos b (b - p Cos w) ; (118)
here w' , as indicated after (114), is given by
+ +Cflp(119)
3.5 Long-term perturbation@ and singularity
We have seen how to get from the mean elements, , at tie t , to the
components of position and velocity in the normal (equatorial) axis system,
without any real problem associated with singularity. It remains to consider
such difficulties as exist in the propagation of the mean elements themselves,
from their values (&o) at epoch, this having been described in section 3.3 as
'a preliminary step'. The schematic propagation formula is (57), and the only
difficulties turn out to be with the long-period terms, A . One difficulty
relates to the general method of evaluating each AC as a well-behaved integral
of Cp , and the others relate to the way in which the circular and equatorial
singularities are dealt with. W3 give a general consideration here of how the
difficulties can be overcome, reserving the full details, for O(J , ) pertur-
bations, to Part 3.
(a) General method of evaluating the
As remarked in section 3.1, the 9 arise, by definition, from components
of t in which c , but not v , is an explicit argument, the point here being
that w has a first-order secular variation due to J2 . For the general zonal
harmonic, Jg , there are trigonometric terms in ke for all values of k , up
to 4 - 2 , having the same parity as t , so that for J3 we just have (as we
shall see in section 7) terms in cos o and sin C . If the elements are to be
propagated over long periods of time, it is obviously desirable that the J3
analysis should reflect the secular variation of (o in the integration of these
terms, even when the analysis is only taken to the first order in J3 • (The
question of the precise meaning of the terms 'first-order', 'second-order', etc,
was considered in section 9 of Ref 3 and that discussion will not be repeated in
detail here.) On taking J2 analysis to second order, we have similarly (as we
shall see in section 5) terms in cos 2o) and sin 2(0 , integration of which
should again allow for the first-order variation in w . On this basis (and
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36
with J3 regarded as second-order in J2) we derive the complete formal* second-
oro 'r solution that we require; it will be found that we need to incorporate some
terms that (as Ref 3 argues) may formally be regarded as part of the third-order
variation.
To specify the problem precisely, we write
t t
Ck - fcos kI d and Sk- J sin k dT , (120)
0 0
where o , written &C to avoid ambiguity, is given by
" o(121)
observing that & is not the same as ) (except at 1 0) , because it does
not include the long-period term Aw . Then the problem is to have formulae for
evaluation of Ck and Sk that are 'universally computable' in giving accurate
values for all values of sec this includes (in particular) zero, 0)sQ
being zero (as far as the first-order J2 formulae are concerned) at the so-
called critical inclinations given (as we shall see from (152)) by g - 0 .
The formal evaluation of the definite integral Ck is, of course, given by
(sin k - sin k&0)/k(sec , but this reduces to the indeterminate 0/0 , rather
than the determinate t cos kw0 , as Wsec tends to zero. To avoid this
situation, we follow, with modifications, the course taken by Merson1 4, who
introduced the bounded functions F1, F2 and F3 , given by
F(0) - 0-1 sin 0 , (122)
F2 (0) _ - (1 - cos 0) (123)
and
F 3(0) - 0-(3 - sin 0) (124)
* The errors in the formally complete second-order solution can only be regardedas truly o(J 2
2, J3 ), ie O(J23, J2J3) , if the timescale is sufficiently short,ie t must be o(32-1) ; generally speaking, a formal solution of j'th orderin J2 will produce errors that are O(J2
j"l t) in the long-term variationover time t . This point was not made in Ref 3.
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~37
For values of 0 that are 'far enough from zero' these functions can be
computed, without loss of accuracy, from their defining expressions, whilst for
values near zero they can be computed from a small number of texma of their
power-series expansions, which are then rapidly convergent, the limiting values
of F1 (0), F2 (8) and F3(e) being 1, 2 and 6 respectively. (Merson's
functions are equivalent to three of the functions introduced by Stumpff 20.) We
also define, for convenience, Vo given by
0V)O 0 t. (125)
sec
Merson's formula for Ck was derived on the basis that
sin ko)t - sin kC0 = sin k(Co + V(O) - sin kC
sin kVm cos kV0c - - cos kV)) sin ko 0 , (126)
so that he obtained (with different notation)
Ck - {F(kV%) cos ki0 - kVo) F2(kVw) sin k@0 t (127)
However, it is more natural to write
sin k;t - sin kF 0o 2 sin k(; t - F0) cos Y2k(@ 0 + )
- 2 sin )WkV0) cos k , (128)
since we then have the simpler expression
Ck {F 1 (Y kV)) cos kt} t ; (129)
the corresponding expression for Sk is
Sk {F IIkV%) sin k(t t (130)
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38
Use of (129) and (130) removes all difficulty associated with the general
evaluation of the A4 , except that for each of AG and A(m it is necessary to
*" allow for a further term that is important though formally only of third order3.
* These terms are induced in the first-order effects of Qsec and &sec by the
second-order Ae and Ai , since asec and asec are functions of 6 and I
The computation of these terms involves the integration of Ae and Ai , so we
require universally computable formulae for the (definite) integrals of Ck and
Sk - specifically of S2 fcr J22 analysis and C for J3 analysis.
The formal expression for the integral, from T - 0 to T - t , cf Ck
(as a function of T ) is - (cos k t - cos k@)0)/(kbsec)2 - (sin kro0 /k(sec)t
We can write this as
2 sin ) kV) sin kGt 2 sin AkVw cos kwo - sin k;t
2 k0osec se_
and then rearrange the terms to give
2 sin Y4kV% cos k t sin k t " k - 2sin kVo))k I t 2 O)SC 2 sin k%
kosec (kSoc)
From this we obtain at once the required result, viz
t2d [FO k 6) cos k; -;kVo) F3 (Yk%) sin k t (131)
0
similarly
t
I Sk d, - YF( k%) sin k0 + )ik% F (hk%) cos k.)] t (1 2)
0
When the induced contributions to Mf and AD) are allowed for, the
propagation of the should (apart from the question of singularity, still to
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39
be discussed) give accurate results over quite long periods of time. As
indicated by the last footnote, however, there will be an increazing need, in
modelling the long-term variation as the number of revolutions from epoch builds
up, for terms that are formally of the third order. Without going to these
terms, accuracy can in principle be improved if a long interval is split into
short intervals, as a form of 'rectification' involving subsidiary epochs. This
brings in the hybrid (or semi-analytical) component of the orbit generator,
referred to in section 1 and to be described in Part 3. Ephemerides, generated
with and without rectification, will be compared in Part 3, and assessed against
ephemerides produced by pure numerical integration; this assessment proviaes the
ultimate justification for all the formulae developed in the prezent section of
the present Report.
(b) Singularity avoidance for near-circular orbits
It will be found from equations (389) and (399) that C03,p and M3,9p , as
defined by equation (72), are infinite when e is zero, this being a manifesta-
tion of the circular-orbit singularity that has been referred to several times.
However, the phenomenon is merely a consequence of the use of elements that
purport to define the position of perigee accurately (relative to the ascending
node, in regard to w , and relative to the satellite's (mean) position, in regardto M ) in circumstances where the concept of 'perigee' cease3 to be meaningful.
If the elements e, o and M are replaced by the euantities 4 and i , defined
by (9) and (10), and U ( - M + (0), then the difficulty vanishes.
It is not a satisfactory procedure to work throughout witn , T and U
as elements, however. For one thing the analysis becomes much more complicated,
particularly (as we ahall see) in respect of U . For another, 4 and 7
themselves become undefined when the orbit is equatorial but not circular, but we
defer all questions of the equatorial singularities until sub-section (c).
Finally, it is essential that the dominant (first-order) component of the secular
perturbation, Osec t , be applied to CO directly, since the perturbation occursprecisely and naturally (and hence necessarily non-singularly) in this form.
Fortunately, the effuct of working throughout in terms of & , q and U can be
obtained by appeal to their differential relationships to e , W and H
without irtroducing them explicitly at all. Thus, since 5 AW is finite for
all e , and may be thought of as a single entity to be operated with instead of
Ao , we compute and 6 (the mean values of e and 0 at time t ) via
and 0 , where was introduced, at (121), to cover secular perturbations
alone, and (defined on the same basis) is equal to go ; the formulae
required are
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40
.cos F - (+ e) cosa- (e Am) sinO) (133)
and
sin F - (+ e) sin + (SA) cos 10. (134)
(Recovery of e ±rom e coo and e sin • is trivial; recovery of (0
involves, in Eortran, the use of the ATAN2 function, unless e is zero, in which
case the arbitrary value of zero may be adopted for ) .)
The value of M still has to be found, but this is easy, if the non-
singular AU is known, since it is only necessary to match M to AU . Thus
the formula in terms of AU is
H H+ (;) + AU - 0,(135)
in which any indeterminacy in 1 , as derived from (133) and (134), is properly
reflected in the preservation of a fully accurate U .
The fact that e and @ are legitimately derived in such a simple manner
will be considered further in Part 3. It is assumed, of course, that Ae etc
are evaluated, following sub-section (a), by integration of ep (etc); ap is
given, via equation (72), by e2,lp and e3,tp (assuming only J2 and J3 to
be operative), and these are obtained in sections 5 and 7. The essentidl point
about the first-order Taylor expansions (133) and (134) is that they effectively
allow for first-order A4 and Al faithfully, without contamination by second-
degree combinations of As and AW that would introduce intolerable error when
Aw is unbounded; a geometrical way of looking at this is that the formulae
'straighten out' the effect of an e AO which, near the origin of the ( , f)-plane, has a pronounced curvature.
It was indicated in section 2.1 that, for analysis in which there is no
e-truncation, U is not a natural parameter to use, and this is because (as we
shall see) the direct addition of AM and AO does not lead to simple
expressions. Simple (and hence natural) expressions are given by AM + q AwO,
or AU' say, and on this basis (135) should be replaced by
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41t
M (M+ + AU + au,- (136)
but an awkward point now arises, viz that when Am is large (because 5 is
small) an ambiguity of 2x in the recovery of i from (133) and (134) can lead
to an error of 2g(1 - ) in the derivation of M from (136). There is a
similar difficulty in relation to the inclination singularity, and a double
difficulty when the singularities occur together, ab we shall see in sub-sections
(c) and (d). It is enough to remark, in the present Report, that the procedure
proposed in Ref 4 for resolving these ambiguities was inadequate, and new
resolution criteria will be given in Part 3.
(c) Singularity avoidance for near-equatorial orbits
The situation for near-equatorial orbits closely parallels that for near-
circular orbits, 13,ep and 03,fp being infinite when s is zero. It is now
the concept of 'node' that ceases to be meaningful, and with direct orbits the
difficulty would vanish if we replaced the elements i, C1 and (0 by
(paralleling , 1 and U ) the quantities , 7 ard W , equal to
sin i sin Q , - sin i cos fl and 0 + 0 , respectively. The use of Z has
been traditional in celestial mechanics, where orbits are almost invariably
direct, but it is inappropriate for retrograde orbits, for which it would be
necessary to redefine i6 as 0) - f ; furthermore, as with the addition of AM
and Am to form AU , the perturbations in W and Q do not naturally either
add or subtract directly. Both difficulties (different r~gimes and unnatural-
ness) disappear if we use the generalized quantity V, defined at the differ-
ential level by (31), as the replacement for w .
As with 4, 1l and U' , it is only necessary to use , 7 and
implicitly in the relations corresponding to (133), (134) and (136), viz
sin 1 - sin (I +Ai) sin 5 + (i Af) cosfl , (137)
-cos - -sin Ji+Ai) cosfl + (iAQ) sin (138)
and
c - ( 5+c) + A -f B5 (139)
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42
As with i Ao , j Af is to be regarded as a single entity in (137) and (138).
Clearly, i and b can be recovered at once from these equations, but a
difficulty arises in the recovery of i from s , since near-polar orbits have
to ba covered as well as near-equatorial ones. The inaccuracy in then deriving
from i has already been referred to (in section 2.1) and an extreme
situation could arise in which the computed value of i became greater than
unity. The problem is easily dealt with, however, at the expense of a minuscule
amount of extra computing, by making use of E , given by cos (i + Ai) , as well
as i ; this amounts to operating with all three components of the unit vector
( , , ) . (See also Ref 7.)
It is worth remarking that the quantities 4 and n turn out to be 'less
natural' than the corresponding 4 and I , associated with e and Co , in that
the perturbation expressions lack the homogeneity (in regard to occurrences of
the factor c ) that might have been expected. This can be seen, for example,
when (149) and (151) are combined, to give 41 and ?I, and (as will be
remarked again in section 4) the inhomogeneity is apparent. Two pairs of
quantities exist that are homogeneous, however, viz tan 4i cos Q and
tan Y i sin Q , the pair appropriate for direct orbits, and cot 4i cos 11 and
cot Yji sin 11 , the pair appropriate for retrograde orbits. These quantities
belong to the two sets of 'equinoctial elements* introduced by Broucke and
Cefola 21 , one set being fully non-singular for direct orbits and the other set
fully non-singular for retrograde orbits. (The remaining equinoctial elements
are a, e cos co, e sin i and M + Zi , Fo being, as defined earlier, (0 ± Q
as appropriate.) An obvious advantage in redefining 4 and n to be a pair of
equinoctial elements would be that no C would then be required, but the need
for two different pairs is not very satisfactory. Further, there is really no
disadvantage in the 'unnaturalness' of our 4 and n , since for explicit
formulae (associated with J2 and J3 ) we are sticking to Ai and 9 Al ,
and 7 only being (implicitly) introduced in the general transformations (137)
and (138). We do require explicit formulae for AW , however, to avoid singular
expressions, so it is only here that homogeneity is desirable; it seems ironic,
therefore, that equinoctial elements are 'natural' where it does not matter, but
'unnatural' ( i) and M + i ) where it does matter.
There is a more basic respect in which the parallel between the eccen-
tricity and inclination singularities breaks down, since the variation of (0
constitutes the very essence of the A , by definition, the variation of Q
being irrelevant. Thus the study of e and 9) (or and i ) is esson-
tially self-contained, whereas the study of i and f cannot be divorced from
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43
. (In another sense, of course, the reverse of this is true, as i and
define the orbital plane, and its node, independently of 0) , whereas 0) only
defines the perigee relative to the node.) The importance of the distinction
should be clearer when these studies are pursued in Part 3.
(d) Avoidance of all singularity
Putting together the results of the previous sections, we can now summarize
an approach that in principle avoids all singularity. The conceptual U' , of
sub-section (b), must be replaced by L , where L is given by (35), and our
first step is the evaluation of the five specific long-period perturbations Ae
Ai , 5 AD , 5 Ay and AL , using the expressions (of type 42,ep and 43,tp)
to be derived in sections 6 and 7; the evaluation is based on (129) and (130),
with (131) and (132) also invoked for the induced contributions to AQ and
Aw ; since these contributions are always non-singular (free of s and
divisors) they can most conveniently be incorporated directly with the purely
secular generation of B and ) (required together with that of H ).
The values of i and [2 can now be obtained, in principle from (137) and
(138) but with 6 used as well as , as has been described. We next, again
in principle, derive S and * from versions of (133) and (134) in which 0
has been replaced by f , with i set to )+ crl ; then 6 is obtained from
(139), re-expressed simply as = * - 5. The arbitrary nature of AV imposes
no difficulty in the use of cos and sin I , since (133) and (134) are
preserved under the addition of an arbitrary quantity to 7 , and in fact it is
convenient to use versions of these formulae in which FO is replaced by
+ T ( - [), instead of ' , since 6) in then obtained directly; however,
this does not get round the difficulty associated with the 2n ambiguity in
[ - , and we return to this in Part 3.
It remains to derive M . The formula for this is a modified form of
(136), viz
F1 + + AL - q1 (140)
and again the only difficulty is the one associated with the 2x ambiguity in
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44
4 71XT-O".DU AUa, TSiZ rM J2 ln aTzoNS
4.1 Peztubations In the osaulatuq elements
Substitution of the potential given by (48) into the planetary equations,
(22)-(27), gives exact equations for the variation of the osculating elements
(excluding M ); expressed with the help of the Cj and Sj families given by
(19), these equations are
S(. 4 {f 15eS 3 + 4S 2 - eS1 ) + 4eh sinv} (141)2q5 r
%3- . {f(5eS 4 + 14S3 + 12eS 2 + 2S 1 - eS 0)+ 4h(e sin 2v + 2 sin v)}S8q3
....... (142)
j = 3~ () 3sCS (143)
q 3
/p% c (C2 - 1) ( 144)
- Kn /0%r3 If (5eC 4 + 14C3) 2e(4- 72) C2 - f(2C1 -eCo)8 eq
3
+ 4h(e cos 2v + 2 cos v) + 2e(6 - f
...... 145)
and
0 n - 2 (fr) {f(5eC4 + 14C3 _ 18eC2 - 2C + eC0 )8 eq 2 (
+ 4h(e cos 2v + 2 cos v - 3e)) (146)
The independent variable is now changed from t to (or strictly ), via
(39), after which the first-order integration of each equation is straight-
forward. It is noted that v (in one form or another) acts as the only variable
in each integrand, being involved in three different ways: (i) oxplicitly;
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{ 45
ii) via the Cj and Sj (for j * 0 ) by definition; and (iii) via /i and
equation (63). For ease of expression, we will from now drop bars (and tildes)
in - , etc, wherever this should cause no confusion. (To first order, the
results of this section are correct without the bars, but bars must be added to
all first-order expressions when we extend to second order.)
The results of the integration are as follows, when expressed in terms of
the notation introduced by (71) and (73), by using the bracket conventions of
section 2.1, and after adoption of the best set of integration constants (as
discussed in section 3.1 and to be Justified in section 4.3):
2- -- aq {3f(e 3Cs + 2eC4 + 3e[4,1]C 3 + 42,3C 2 + 3e[4,1]C 1 + 6e2 C0 + e
3 C1'a 24
+ 4h(e3 cos 3v + 6e2 cos 2v + 3e[4,11 cos v + 2[2,3]) }, (147)
f - { f (3e2C5 + 18eC4 + (28,17]C3 + 60eC 2 + 3[4,111C 1 + 18eC0 + 3e2C 1)48
+ 4h(e 2 cos 3v + 6e cos 2v + 3[4,1] cos v + 10e) } , (148)
1+
i I - C so(eC3 + 3C2 + 3eC1 + 3) , (149)
c (150)
1 " Ic(eS +3S2 +3eS 1 - 6e sin 0 (151)
A 2g, (152)
t 1 -i 32f
W I e8 {3e + 18efS4 + [28f, - (8,-39)]S 3 - 12e(2,-5)S21 48 432
-3 [4f, (8, -15)] S1 - 18efS 0 - 3e2fS-
+ 4e h sin 3v + 24eh sin 2v + 618h, (14,-17)] sin v, (153)
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II46
Er" (154)
and
a 1- q If (302S + leeS + [28,-13S - 36eS - 3[.I, 7)S
- 18eS o - So
+ 4h(e 2 sin 3v + 6e sin 2v + 3[4,-5J sin v) }. (155)
A certain inconformity will be observed in (153), in comparison with (148)
and (155) in particular, in that we do not have the same neat division into
'pure f-terms', involving the Cj , and 'pure h-terms', involving cos jv .
This inconformity, propagated from (145), follows from the presence of the term
in au/ai in (26), and the natural partner of (148) is not (153) but (166), ie
the expression for Yj given in the next section, since (33), the equation for
does not contain 8U/aJ . (For further comnents on this, see Part 2.)
It only remains to obtain the perturbation in M (effectively, to supersede
the perturbation in 0 ), the basis for the analysis having been given in
section 2.2. As explained in section 3.1, this in principle involves the
derivation of a modified version of Kepler's third law, but it turns out that no
modification is needed in practice; this is fortuitous, and it does not apply
for any other Jf . Thus we require nothing but the formula for M1 itself.
The starting point is (40), and to use this we require a suitably expressed
formula for n1 , the derivation of which calls for some preliminary discussion
of (147), the formula for a, .
As explained in sections 2.3 and 3.1, the integration constant in (147),
part of which appears in the 3f-component of the equation and part in the
4h-component, is such as to make the mean element i identical with the exact
constant a' given by the conservation of energy. But a' is related to
(osculating) a by the exact equation (52) which, on substitution for the
potential given by (43), gives
a, aq-2 () (3fC2 + 2h) (156)
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47
This is in non-standard form, because of the factor (p/r) 3 , but expansion of
this, by use of (5), at once confirms (147) and incidentally explains the
manifest symetry there of the Cj terms - the 3f-component of the integration
coantant, viz 6e2C0 , is clearly essential for this symetry, to match the term
6e 2C4 Of the two (non-standard) expressions, intermediate in powers of p/r
thit emerge as (156) expands to (147), the first is the one corresponding to the
required formula for nj , but the second has an application as well, to be mat
in section 5.9. The first expression is
a, - I (- ) { 3f (c3 + 2C2 + eCj) + 4h(l + e cos v)} (157)
and the second is
a1 - -aq -2 {3f (e2C4 + 4eC3 + 2[2,11C 2 + 4eC1 + e2C0)
+ 4h(e2 cos 2v + 4e cos v + [2,1]) } (158)
To each of the four expressions for a, there is an immediately
corresponding expression for ni , given by
n - - (!al (159)
on the provisional assumption that the n we require is indeed such that the
Kepler law is satisfied without modification. The reason why (157) leads to the
most apposite expression for nj is that the factor (p/r)2 then cancels out
when Sn (derived from this expression) is substituted in (40), W in (40)
being defined by (17).
Thus we have
n1/nw - - q{3f(eC3 + 2C2 + eC) + 4h(I + e cos v)} , (160)
which, in view of (40), is in the right form for further integration. The
integration leads to a secular term, as well as short-period terms, and the
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48
resulting expressions, with the usual notation involving the symbol introduced in
(3), are given by
S- -qh (161)1
and
- -q {f (es 3 + 3S2 + 3+ S ) + 4h e sin v} (162)
The secular perturbation in M is now given by the combination of (154)
with (161), and the short-period perturbation by the combinatio,. of (155) with
(162). The former combination gives
M, . 0 (163)
as anticipated, validating the provisional assumption about Kepler's third law
for mean elements, whilst the latter combination gives
M- = 42 e-
S- -e q{f(3e2S5 + l8eS4 + [28,-1]S3 - 3[4,5]S - 18eS0 - 3eS)
+ 4h(e 2 sin 3v f 6e sin 2v + 3[4,-l] sin v) } . (164)
4.2 Perturbation* in related quantities
For reference, we give formulae for perturbations in a number of quantities
related to the standard osculating elements covered in the preceding section; we
start with the conceptual V, p and L , whose rates cf change are given by
(33), (34) and (35). We have
- h , (165)
- I - f (3e 2 S 5 + leeS 4 + t28,11]S 3 + 36e3 2 - 3 I4,_7]SI
- 18eS0 - 3e2S.. 1 )
+ 4h(e sin 3v + 6e sin 2v + 314,3] sin v) } , (166)
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49
- 2hq (167)
P1 1 ~q f (eS, + 3S2 + 36S1) + 4he sin v} (168)
- hq
(169)
and
L . I q{f(es3 + 3S2 + US1 ) + 4he sin v} (170)
In practice, of course, the V and p results were obtained before the 0 and
a results, (152)-(155) being derived from (150)-(151) and (165)-(168). Further,
it will be noted that p, = - 21i and Ll -- 1k (and likewise Pi 1 - 21i and
Li - ); these are not coincidental results, and it may be seen from Refs 1
and 22 (see also Part 2) that there is a general first-.order result, for the
zonal harmonic Ja , given by Sp- (f + ) 61 and 8L -- (2t - 1) 8.
Next, we can use (147) and (148) to derive p! , since, from (28)
P1 - q2 a - 2ae ei (171)
Thus
p1 {P{f(eC3 + 3C2 + 3eC1 ) + 2h 1 (172)
We can employ the last two equations in reverse, to obtain a more compact (but
rnoin-standard) expression for el based on the most compact of the four formulae
for a, , viz (156). This expression, which can be confirmed by direct reduction
from (148) and which closely corresponds to the first-order expression for 8e
given by Kozai 17 , is
6- 3(13e1 .~ {(.D)(3fC, + 2h) - q2 [f(eC3 + 3C2 + 3eCj) + 2%]}(13
(Though compact, the expression is unfortunate in having the factor e- ; as
can easily be seen, however, this cancels out when the bracketed terms are
expanded.)
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oz
50
Again, since
(pc2 )I - C2 P1 - 2psc i1 , (174)
(149) and (172) lead to
(pC2 ), 2 .pC(1 - 3f) (175)
IThis is a (firsg-order) constant, as it must be, since (as was remarked in
section 2.2) pc? is an exact constant for a geopotential that involves only the
zonal harmonics; it represents the polar component of the satellite's angular
momentum. The reason that (pc2)1 does not actually vanish is that the
constant. _x integration for a, e and i are not (optimally) such as to force
this to happen.
Perturbations in 4 and l1 , defined by (9) and (10), can be obtained
using (148), (152) and (153), but it would actually be unhelpful to use (152) to
introduce quantities 4 and A because this would involve the conversion of a
straightforward (purely secular) & into an apparently long-periodic variation
(cf remarks in section 3.5(b)). However, the pure short-period perturbations can
usefully be expressed with the help of the Yj , Cj , cj and sj families,
given by (18) and (20) (and so far unused); in terms of these, (148) and (153)
lead to
- 6 {3e2fT5 + 18efy4 + 2 [l4f, -(2,-9)]y 3 -, 12e(1,-5)y 2 - 3e 2 (4,-13)y1
+ e 2 (9,-7) c 3 + 36e(1,-1)c 2 + 6 [2 (4,-5), (10, -11)] c1
+ 2e(20,-21)c 0 - 9e 2 (4,-5) c_ }
..... (176)
and
I {3e2fG5 + 18efG4 + 2 (14f, -(2,-9)]O3 - i2e(l,-5) 2 - 3e 2 (4,-13)Gi - 5e2fs
+ 12e(l,-3)s 2 + 612(4,-7),3(2,-3)]s1 + 2e(20,-39)s 0 - 3e2 (12,-13).R 11.
...... (177)
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51
Form-lae for 41 and 711 cre not given, because (effectively due to the
c-factor in il ) il and Ci do not combine in a natural (homogeneous) way.
This was remarked in section 3.5%c), where it was also xemarked that, for the
equinoctial elements, il and 01 do combine homogeneously. The resulting
formulae, for both pairs of equinoctial elements, are:
(tan hi cos Q sin i cos {3 cos 0 - 3e cos (v + Q) + 3e cos (v -( )
6(1 + cos i)
+ e cos (3v + Q + 2()) + 3 cos (2v + Q f 2(0)
+ 3e cos (v + Q + 2(0) } , (178)
(tan Y~i sin Q) - sin i cosi {3 sin Q - 3e sin (v + f) - 3e sin (v-0)1 6(1 + cos i)
+ e sin (3v + Q + 2(0) + 3 sin (2v + Q + 2(0)
+ 3e sin (v + Q + 20))} , (179)
(Cot C03sin i cos L 13 cos Q - 3e cos (v - Q) + 3e cos (v + Q)1 ) 6(1 - Cos i)
+ e cos (3v -Q + 20) + 3 cos (2v - + 20)
+ 3e cos (v - Q + 2(0) } (180)
and
(cot ) i sin Q) -sin i cos L {3 sin 0 + 3e sin (v - 0) + 3e sin (v + Q)1) 6(1 - cos i)
-e sin (3v - L + 2)) - 3 sin (2v -0 + 2(0)
-3e sin (v - Q + 2w) } (181)
For the remaining quantities covered in this section, it would be even less
helpful to consider long-term perturbations than for and 1 , but pure short-
period perturbations can again be usefully expressed. Thus, v, can be derived
from (148) and (164), using (42); the result is
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i 52
f -(3e2 S + 18eS4 + (28,11]S3 + 32eS2 - [12,-51S,
48 5 3
- 18eS o - 3e2 S-
+ 4h(e 2 sin 3v + 6e sin 2v + (12,1] sin v) } (182)
From v, and 1 , the latter given by (153), the important expression for uI
can be obtained; it is
u= - i 2 {2e(1,-I)S3 + (6,-7)S2 + 2e(3,-5)S1 - 4e(5,-6) sin v} . (183)
The expressions for 41, Ti and ul can be used for a good overall check
that also invoves the expression for r, in the next section. Thus, from the
formula
e cos v - cos u + sin u (184
it follows that
(e cos v)I = cos u 4 Ti sin u - u ( sin u - i cos u) , (185)
which evaluates to
Ce cos v). = 2 {f(3e2C 4 + 16eC 3 + 2(0,7]C 2 4 32eCI + Ile2C
+ 4h(e 2 cos 2v + 8e cos v + [6,11) } ( 86)
The factor (1 + e cos v) can be extracted from the right-hand side of (186) and
the resulting expression leads, in view of (5), to
(p/r) I . .-L (p/r) {f(3eC3 4- 1C 2 + lec ) + 4h(e cos v + 3) } (187)
This result ties up immediately with p, , given by (172), and r1 given by
(188).
A certain property may be observed in the structure of the various
expressimns given in sections 4.1 and 4.2. For a fully non-singular quantity,
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53
the property for tj is that the coefficient of the sine or cosine of
jv + ku C j'v + ko) , with j' - j + k ) contains (at least) the factors e1I 1
and s'k l if it were not so, a change to non-singular variables (involving
a cos v, e sin v, s cos u and s sin u ) would result in uneliminated singular
terms, via a negative power of e or 3 . The property operates in the first-
order pcrturbation analysis for an arbitrary zonal harmonic, and it is related to
the so-called d'Alembert characteristic1 2. It may be seen to apply here with
al, Pl, L1 and pl .
The property does not apply directly to el and i! , because the elements
e and i , though non-singular in themselves, do not have well-behaved rates or
change at their associated singularities. (This is easily seen by considering
the relation e2 42 + q2 , in particular.) Thus it is eel and sil , rather
than el and il , that have the property. For quantities that are intrin-
sically singular, the effect on C_ is naturally more severe, the property being
satisfied for s2 , s2e2 I1, e2al, e2Ml, e2)1, 8341, S31l, e
2v, and S2U, .
in the next section it will be found that rl, b, and wl also hdve the
property, this being an essential aspect of their merit of '-ourse. Finally, for
the second-order formulae to be developed in sections 5 and 6, where mu<iification
of the direct pcoperty relates to second-order rates of change, the property will
be found to hold, in particular, for a2, e3e2, sSi2, P2, s4D2, s4e 4012, e4M2, r,,
b2 and %2 •
4.3 Perturbations in spherical coordinates
As explained in section 3.3, the complicated expressions for short-period
perturbations in the osculating elements (as given in section 4.1) can be com-
bined into compact and convenient expressions for perturbations in the system of
spherical coordinates (r, b, w). Expressions for pure short-period perturbations
can be obtained from the six basic j of section 4.1 by substitution in
(95)-(97). The results are worth writing with unabbreviated sines and cosines,
in view of their importance, and in terms of explicit mean and semi-mean
elements; thus
r = .(cos 25 -2) ,(188)
1 - 3 {sin (U + ) - 3 sin )} (189)
and
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i- , t~~' - ~ -~*
544S.54 S
w - 1 -{f[sin 26 + 4e sin (5 + 0)] + 8eh sin ,} (190)1 12
The independent check of (188), based on (187), has been mentioned. There
is no obvious check of (189) or (190), however - verification that (94) is
satisfied proves little, since (94) was used directly in the derivation of (97).
We can now justify the claim that we are -sing the best set of integration
constants* for the l . With any other choice of constant fo: a, , (188) would
(via (95)) contain a complicating term in r/a . With any other constants for
el and M1 , (188) would also contain a term combining cos v and sin v , and
there would be considerable complication in (190). With any other constants for
il and Qi , (189) would, .ikewise, contain a term combining cos u and
sin u . Finally, any other constant in 0) would be transmitted directly into
(190).
On differentiating (188)-(190), we have expressions for r1, b, and wi
immediately. Thus
- i - 1 . f sin2i , (191)3
~ {i+)Cos 0 i+ )-3wcos@} (192 ,
and
- cos 2 + 26 (+) cos + @) ] + 47ao' cos (193)
For a purely first-order solution w, can set ) = 0 and v = EWn in these
expressions. For incorporation in the second-order solution, however, we must
allow for the first-order variation in 6) ; we deal with u just by identi-
fying it with w' , as given by (113) - cf the remark after that equation.
* Some of the 'constants' are represented by two distinct terms in the l , in(147), eg, there is a term in CO , which contributes a first-order constant toKal , whilst the final term may be regarded as a 'pure constant', making a con-tribution that is constant to second order. In the J2
2 analysis, similarly,
2 may contain three distinct 'constant' terms, of which the first two are'less constant' than the third.
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55
Equations (188)-(190) lead at once to 5r, 6b and Sw , since there are
no 1,p in (74). If we want the complete short-period perturbations (8r, 6b
and 8w ), we must allow for the carry-over effects of £. and F01 The
resulting formulae are given by
8r - Kr1 , (194)
8b - K(b + cos (195)
8w - K(w 1 + .FQ (196)
The full expressions can easily be shown to be equivalent to the first-order
terms of (366)-(368) of Ref 3, so long as an error in (366) ibid is corrected by
changing ... {( 2 cos 2a ... into ...Kp{ (f cos 25 ... . A second error in
(366) ibid is worth remarking, viz that ... + 4(4 - 12f - 19f 2 ) ... should
read .., + (16 - 48i + 23f 2 ) ... ; this pair of errors (in the same equation)
are, apart from rather obvious ones in equations (35) and (81), the only errors
known in Ref 3. (The quoted equations from Ref 3 are from the 'Note added in
proof'; this was written after the author suddenly registered the superiority of
a' as mean semi-major axis, having started by recommending an 5 only 0(Ke2)
different from Kozai's 9 .) From (194)-(196) we can get the corresponding
expressions for 8i, 8b and 8w . For the reason indicated in section 3.2,
however, the first-order solution with carry-over effects included is not a good
basis for development of the complete second-order solution; so it is best to
regard (194)-(196) as of only academic interest - the important equations in this
section are (188)-(190) and (191)-(193).
5 J22 PERTURBATIONS IN OSCULATING ELEMENTS
Before proceeding to the second-otder solution for the J2-only field, it
is worth stressing again that the first-order solution given in section 4 should
in general be understood to be expressed in terms of semi-mean quantities and not
osculating ones. Thus the arguments of the trigonometric functions should be
interpreted as if tildes were carried, and the coefficients of these functions as
if bars were carried, there being no distinction, for a, e and i , between
mean and semi-mean elements. The omission of bars and tildes was a matter of
vi ial convenience, and was in any case legitimate when second-order terms were
not under consideration.
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i 56
We can summarize the first-order solution for the generic osculating
element as follows. For .0} and 0 only, there is a secular perturbation given
by Kilfit (cf(71)), that propagates the mean element t , together with a carry-
over effect, KlA , that (added to t ) yields the semi-mean element tThere are no long-period perturbations, but for each element there is a pure
short-period perturbation given by K(I (of (73)).
The second-order solution for t comprises the first-order terms just
summarized and the second-order terms now to be developed. The understanding
about the omitted bars and tildes in the first-order terms now becomes vital, but
(because we are not concerned with third-order terms) it will be legitimate (and
certainly easier on the eye) if bars and tildes are omitted in the new terms
wherever possible. In addition to the second-order secular and short-period
perturbations specified by C, and C2 respectively, long-period perturbations
will now be making their appearance. Onco the expressions for the 2, are
available (of (72)) - and they are summarized in section 6.1 - the J22-component
of each g is effectively known, since the method of generating it was given in
section 3.5(a). Further, it will be found that these components of the A
(unlike those associated with J3 ) are all non-singular. The carry-over effects
of the t2, p must not be forgotten; they are incorporated in the 68 (of (74))
aid the amalgamation of the 6C into 8r, 6b and 8w is the subject matter of
sections 6.2 to 6.4.
The basic idea in deriving expressions for the 2' t2, p and C2 is to
'bootstrap' on the first-order solution by substituting it in the right-hand
sides of the planetary equations and then re-integrating. A special procedure,
free of integration, is possible for C - a , and this is developed in
section 5.1, paralleling the derivation of (147) via (156). We then proceed, in
section 5.2, to an effective rederivation of a2 (A2 and a2,ep do not arise)
by the general procedure, and exteikd it to the other elements in sections 5.3 to
5.10. It will be found, in sections 5. and 5.10, that i can no longer most
conveniently be identified with n' .
5.1 Perturbation in a (special method)
We want a2 such that
a + a K q 2(p i) (33 + 2h) + K a (197)32 2'
the starting point for which is the exact equation (52), which leads (still
exactly) to (of (156)) 88TR 88068
a a' + aKaq-2(p/r) 3 (3fC2 + 2h) (198)3 2
On comparing (197) and (198) it follows that there will be five sources of termscontributing to a2 ; four of these sources are essentially the same as four of
the five listed in Refs 2 and 3, but source (ii) of these no longer arises now
that we have unreservedly identified a with a' , and it is replaced by a
source that we could previously neglect (due to e-truncation). The sources are
as follows: (i) the variation of p in K , which could easily be overlooked;
(ii) the variation of e in q-2 (this is the new source); (iii) the variation
of (p/r)3 ; (iv) the variation of i , which affects f and hence h ; and
(v) the variation of u in C2 . The contributions of these sources are as
follows, with bars and tildes suppressed in line with the general principle laid
dovin:
(i) - *3 q 4 (p/r)3 (3fC2 + 2h)p ;
(ii) jaeq-4 (p/r)3 (3fC2 + 2h)e I ;
(iii) aq - 2 (p/r) 2 (3fC2 + 2h) (p/r)1
(iv) 2aq -2 (p/r)3 sc (C l)i
pr)c( 2 -1
(v) -2aq "2 (p/r)3 fS2u1 *
Here pl, el (p/r)l, il and ul are given by (172), (148), (187), (149) and
(183) respectively. In using these quantities, we are effectively identifying
8 with 5. (as appropriate), this being iegitimate as a 2 is to be
multiplied by R2 ; the appearance of C2 , as opposed to C2 , in (197)
validates the use of ul (associated with 8u and 3,u rather than Su ).
It is vory convenient that (187) expresses (1/r)1 in a form containing
(p/r) as a factor, since this ensures that each source of a2 contains (p/r)3
as a factor. On multiplying out the remaining factors in each source, and
combining the resulting expressions, we eventually derive the required result,
viz that (with bars and tildes omitted)
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58
2 -1--q "44 l~ecov)44[3f 2 { re37 + 18er6 + 15e[2,1!r, + 414,11]r,
o + 1e 2 3e[46,-lj r 3 r 1822 + 3e3Fr,
" 12f{(2e3 C 5 + 12e 2hC 4 + e[(2 (22,-31), - (14, -17)]C 3
+ 8h[3,2]C 2 - e[2(2,3), - (34,-39)]C 1
+ 12e 2hCo0 + 2e 3hC_j
" 21e 2(8h 2 + 9f 2) (e cos 3v + 6 cos 2v)
+ 3e[2(24,-56,49), -(8,8,-J7) ] cos v
+ 2 [2(20h2 + 3f 2 ) + 39e2f2]
...... (199)
the bracket conventions of section 2.1 are used in this result.
5.2 Perturbation in a (general method)
This section serves as a prototype for the application of the general
'bootstrapping' procedure to one orbital element after another. The fact that it
permits a rederivation of (199) acts as a check on the correctness of the general
method, and also on the algebraic reductions involved (for a ) in both methods.
The starting point is tne exact equation (141), converted into an equation
for da/dv by use of (39). To have a valid basis for second-ord-r analysis it
is essential that we distinguish carefully between barred and unbarred quantities
in this conversion, the required form of the equation being
- - Ka(n/ ) (M3/q5) {(p/r)4/(O/)2} Qa (200)
dv
where
Qa - f(5eS 3 + 4S2 - eSI) + 4eh sin v (201)
To express da/dC as a function of mean quantities only, we must eliminate
the osculating quantities a (and the basically equivalent n ), i, e, v and
0 , and we do this by substituting the first-order solutions for these five
quantities into the right-hand side of (200), via a first-order Taylor expansion
relative to the corresponding (semi-)mean quantities. A special situation arises
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4 F-
in regard to to (implicit in Q. via S3, S2 and S1 , and explicit when each
Sj is replaced by sin(jv + 20)), and this requires preliminary consideration.
First-order analysis effectively involved the identification of barred and
unbarred quantities in (200) and (if we ignore for the moment the complication
caused by the resulting factor (p/r)2 ) the interpretation of each Sj in (201)
as (zero-order) sin(j- + 2i) , with • taken as constant in the integration
with respect to ; the result of the integration was then naturally taken as
* -(l/j) cos(j, + 2@) , with i evaluated (after the integration) to allow for
6) . In the second-order analysis we require a replacement for Sj that is
first-order correct, ie given by sin{j(N + Kv) + 2(9 + Kw)} , but in the
first-order component of the extended solution we will still not be making proper
allowance for the variation of 6 , since we want to go on taking the integral
in this component to be - Cj/j ; this means that there is a source of second-
order perturbation (for each Sj in Q. ) additional to the direct Taylor
effects associated with v, and 0i (viz given by jK Cj v, and 2K Cj j,
*respectively). It is this additional source of perturbation that constitutes the
'special situation' we are considering, being associated with the error in taking
the integral of Sj to be - Cj/j when (to second order) it is really
-Cj/(j +2K ) . But
1 2 + 0( ) (202)j + 2 ij j 2
so the error can be rectified by introducing the 'carry-over' term (2KI/j2) Cj
Since this term is the integral (with respect of i ) of (-2K6 1/j) §j
(neglecting terms that would carry over to third order!), we can cope with the
'special situation' by supposing that there is an '6) contribution' to da/d ,
as well as the obvious 0) contribution. This & contribution is such that each
Sj in Qa would (apart from the complication of the (p/r)2 factor that has
been ignored so far) make a contribution to da/d proportional to (-2Koj/j) Sj
as well as the contribution proportional to 2K Cj 0i,
At first sight we can only deal with the complication of the (p/r)2 factor
by multiplying each Sj by this factor, interpreted as (1 + e cos v)2 , to
yield a set of terms in Sj.2, Sj-1 , Sj, Sj+ 1 and Sj+2 , and applying the general
procedure to the individual terms. This is exactly what we have to do with the
v and (o contributions to da/dT , but for the 6) contribution, which we are
currently concerned with, we can escape the complication by making use of a known
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60
expression for a, directly. The argument of tha last paragraph effectively
shows (on multiplying through by -j ) that to each term of a, in the form of a
multiple of a particular Cj there corresponds a carry-over contribution to
da/da consisting of the same multiple of 2K2 to Sj . (The same principle applies
to the analysis for every element, , of course; when a multiple of Sj , as
opposed to Cj , is involved in C, ' arising from the appearance of S/j in the
first-order integration of C , then the second-order carry-over to d /d7 is
that same multiple of -2K2 ;I Cj .) The important point is that the factor
2K2 to, is independent of j , and this means that the argument is not affected by
an unexpanded power of (p/E) in a, ; indeed, we can take the 6 contriuution
(to da/d ) from the simplest possible expression for a, , viz (156). (The
general principle, being independent of j , applies even when j = 0 t)
With a self-explanatory notation, we may now list the six contributions to
the zecond-order component of da/dv as follows, dropping all bars and tildes,
even from ; (it is remarked again that the Da contribution covers the
variation in n as well as a 1:
D(da/dv) -K 2q -2(p/r)2 Qa (203)4 a 1
D (da/dv) - - K 2aq 2(p/r)2 sc(5eS3 + 4S2 - eS - 6e sin v) i i (204)
De (da/dv) -- K2aq-2 (p/r) {Oa[9eq-2(p/r) + 4 cos v] + (p/r)Qae}e , (205)
where 0ae = 5fS 3 - fS + 4h sin v , (206)
DV (da/dv) - hK 2aq-2 (p/r) 14eQa sin v - (p/r),Qav)V I (207)
where Qav = f(15eC 3 +
8C2 - eC ) + 4eh cos v , (208)
D (da/dv) - - K2 aq - 2 (p/r)2 f(5eC3 + 4C 2 - eC1 )01 (209)
and
D.ida/dv) - 4K2aq-2 (p/r)3 fg S. . (210)
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61
The six contributions are most conveniently dealt with by grouping into thefour terms that contain Qa as a common factor (the Da-term, two De-terms and
one Dv-term) and the five other terms (one associated with each of Di, De, Dv,Do and D6 ). Seven of the nine terms contain (p/r)2 (at least) as an
explicit factor, the reason that only the first power of (p/r) appears in theother two terms being that these are the el and v, components of what is
really a term in (p/r)l ; but (p/r)1 itself contains p/r as a factor, being
given by (187); thus (p/r)2 is an overall factor.
The combination of the two groups of terms may L. denoted by D2 (da/dv)
and we find after some tedious algebra that
2 -4 2D2 (da/dv) K - Kaq (p/r) x
x [f2 {15e4Z8 + 102e 3Z7 + 6e 2 (38,1111 6 + 2e[103,128]Z 5 + 4 (16,86,3] 1 4
+ 2e[73,32]1 3 + 6e 2 [6, 1 ] 1 2 - 6e 31l - 3e4 L
+ 8f{4e4hS6 + 26e3hS + e2(3(30,-43), 4(5,-6)]S65 4
+ e[6(16,-23), (2,-9)) S3
+ [24h, 6(2,-5), (34,-39)]S - e(2(4,-J), -(22,-27)]S II2
+ e2 (2,3), -2(8,-9)]S 0 - lOe3hS - 2e4hS_2 }
+ 2{e3(8h2 + 9f2 ) (e sin 4v + 6 sin 3vi
+ 2e 28(8,-20,19), -(8,8,37)] sin 2v
+ 2e((64,-176,14S,), -4(2,-2,-11)] sin v} ]
..... (211)
It is not obvious how we can integrate (211) without first expanding the
factor (p/r)2 , but we are in the happy position of knowing what we expect the
answer to be. Rather than attempt to integrate (211), therefore, we differenti-
ate (199) with respect to v , tusing the fact that, eg,
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62
a{+ecosV)C 1 Y} - - +0 cosv)'{e(j +3)Sj+1 + 2j S + ej - 3)Sj_ }
...... (212)
We find, after some further tedious algebra, that K2 da 2 /dv , derived in this
way, is identical with D2 (da/dv) , as given by (211). This completes the check
that the special and general methods yield the same result.
In sections 5.9 and 6.2 we shall need the expression for a2 with (p/r)2,
rather than (p/r)3 , as a factor. This can be obtained at once from (199), of
couree, and will be given here for completeness. In section 5.5 we also need the
expression with no p/r factor at all, but this is so lengthy that it seems best
to suppress it. (We also suppress the version of a2 that has a single factor
p/r ; finally, it is easy to show that a further such factor cannot be removed
from (199), which is on this basis the simplest. possible expression for a2 .)
The expression required in sections 5.9 and 6.2 is:.
a2 aq -4 (l + e cos v)2 x288
X [3f 2 {3e'r 8 + 24e3F 7 + 6e2 (11,3- 6 + 4e(19,23Jr 5 + 2(16,82,7]r 4
+ 12eC9,5] r 3 + 2e2
141,1Jr 2 + 24e3rI + 3e4 r 0 }
+ 24f{e4hC6 + 8e3 hC5 + e2 [(34,-49),-(6,-7))C 4 + 4e[2(7,-0),-(0,1)]C3
+ [24h, 2(18,-29), (10,-ll)JC2 + 4e[2(1,-3),3(4,-5)C I
+ e 2(10,-21),3(6,-7)C 0 - 8e3hC 1 + e4 hC_2}
+ 2f{e 3 (8h 2 + 9f2) (e cos 4v + 8 cos 3v)
2+ 2e [3(40,-104,103),-(8,24,-69)j cos 2v
+ 4e[(112,-288,243),-3(0,16,-45)] cos
+ [8(20h2 + 3f ),6(24,-56,75),-3(8,8,-37)].
...... (213)
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63
S . 3 Partuxbation in I
The starting point is given by converting the exact equation (142) into an
equation for de/da , making use of (391. The result is
- -qK(n/) q. { (p!r) 3/(g/i) 2}Q , (214)
where Qe - f(5eS4 +14S 3 + 1 2eS +2S -GSc ) 44h(e sin 2v + 2 sin v). (215)
We proceed as in section 5.2, listing six contributiors to the second-order
component of de/d . With self-evident notation they are:
D (de/dv) - - K 2 (p/r)Qea (2161
8 12
D (de/dv) - -1K (p/r)sc(5eS + 14S + 12eS + 2S - eS
- Ge sin 2v - 12 sin v)i , (217)
S2{Qe(7eq(218)D (de/dv) - -(K{Q[7eq 2 (p/r) + 3 cos v] + (p/r)o}e,
where Qee - f(5S4 + 12S 2 - S0) + 4h sin 2v , 1219)
Dv(de/dv) - K 2J{3eQe sin v - (p/r)QevV , (220)
where Qev 2f (lOeC4 + 21C3 + 12eC 2 +C 1 8h(e cos 2v + cos v), (221)
2
D (de/dv) - -K (P/r)f(5eC4 + ItC 3 + 12eC2 + 2C1 - eCo) ( 1 (222)
and
Dz(de/dv) K2fg 13e 2 S + 18eS + (28 + 17e 2 )S + 60eSo12 5 4 3 2
+ 3(4 + lle 2 )S + 18eS + 3e2S (223)
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64
As in section 5.2, the six contribution~s may be grouped into four terms with
Qe as a comuon factor and five other terms. The first two Qe termns (and only
these) are found (af ter subs3titution for a, and el to involve the f actor q- 2
but their combination yields a cancellingq factor q2 30s q-2 does not appear in the
overall combination. Six of the nine terms contain the factor p/r and it can be
extracted, as in section 5.2, from a combination of two of the others, but it does
not occur in the Da) term, so p/r cannot b~e factored out of the overall
combination. Some lengthy algebra eventually leads to:
D (de/dv) 1 -K 2e- X2 1536
x [f 2{15e, E10 + 162e3 l~ ---' 8e2 [82,13]Z8 + 14e[84,53]E 7 + [784,1712,391E6
+ 20e[54,l]E 5 - 8[56,58,53]E 4 - 12e[66,103]t 3
+ (48,-624,-371)Z 2 + 2e[36,-91]X1I - 18e 3 E1 - 3e 4 E-}
+ 8f {4e 4hS 8 + 42e 3 hS1 + e?[l72h, -(110,-141))S,+ 5e[64h, -3(38,-49) )S5
+ [224h -8(94, -123) ,-(338,-~459)1S 4
- 2e[4 (14,-23), 5(78, -113)]S 3 - 2[80h, -2 (26, 9), 3 (10, - 23)] S)
- 6e[16(3,-4),-(98,-103)]SI - 8e q 2 (14,-15)S 0
- er48h, (l58,-16S)]S_ e e2 (52h,3(18,-19]S- 2
- 8e 3hS-, - 2e 4 hS-4 }
+ 2{le3(8h 2 4 9f2) (e sin 6v + 10 sin Sv)
+ 4e 2 [2 (40h 2 + 37f 2 ), - (24, -8, -31))1 sin 4v
+ 2e(36(Oh 2 + 5f 2, -(296, -312,-165)] sin 3v
2 2
-4e(2(88h2 _ 21f2) (168, -08.-207) ) sin v} (224)
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65
The term in S0 is independent of v and represents our first source of
long-period variation; the absence of a corresponding term in 2I will be
noted, and this applies to every element except Y (see section 5.8). We may
write, in the notation of section 3.2,
-eq f(14 - 15f)S0 , (225)
the contribution to -p being given (cf (72)) by K2 n e2, . This leaves the
short-period carry-over term given (cf (74)) by K2 e2, p m . The method of
integrating e2.tp with respect to time, to give the J22 component of Ae , was
discussed in section 3.5; see also Part 3.
The integration of the rest of (224) is immediate, except that we have to
incorporate the right 'arbitrary constant'. This is naturally expressed via
terms in r0 and CO , together with a 'pure constant' (see the footnote in
section 4.3) that is a function of e and i alone, the 'coefficients' being
chosen to suit the derivation of r2 in section 6.2. Anticipating the r2
analysis, we derive (from the integration) the required result for 02 , viz
e2 -- L e-1 x9216
X [f2 {9e'r,( + 108esr 9 + 6e2 182,13]fl9 + 12e[84,53]r 7 + (784,1712,3916
+ 24e[54,l]r 5 - 12156,58,53]r 4 - 24e[66,103]r 3
+ 3148,-624,-3713r2 + 12e[36,-9i]FI + 6e2 (66,-35ir0
+ l08e3r_1 + 9e4r-2}
+ 4f{ 6e hC8 + 72e hC7 + 2e2[172h,-(l10,-141)]C 6
+ 12e[64h,-3j38,-49)]C 5 + 31224h,-8(94.-123),
-(338,-459)C 4
8e[4(14,-23),5(78,-13)]C - 12[80h,-2(26,9),3
3(10,-23) IC2
- 72e[6(3,-4),-(98,-103)]Ci +
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66
+ [288h,32(40,-33), (3002, -3255) Co
2+ 12e[48h, (158,-165)IC_1 + 6e (52h, 3(18,-19) C_2
3 4
+72e hC_3 + 6e hC_ 4
+ 2{e3 (8h2 + 9f2) (e cos 6v + 12 cos 5v)
+ 6e2 [2(40h; + 37f2), -(24,-8,-31)] cos 4v
+ 4e(36(8h 2 + 5f 2), -(296,-312,-165)] cos 3v
+ 3[16(24h 2 + 7f 2), -16(72,-136,33),
-(216,120,-551), cos 2v
- 24e[2(88h 2 - 21f 2), (168,-88,-20711 cos v
- 218(72h 2 + 29f 2 ),6(264,-600,251), (632,-168,-1051)1
...... (226)
5.4 PerturbatIon in i
The starting point, given by conversion of the exact equation (143), a5 in
earlier sections, is
7 - K(n/n) (Z 3 /q') { (p/r)r3/( / )2}Q, (227)dv
where Qi - scS 2 . (228)
We proceed as usual, listing six contributions to the second-order
cozonent of di/d , viz
Da (di/dv) - 7-K a-'(p/r)Qia1 , (229)
2D (di/dv) - - K (p/r)(i - 2f)S 2i1 , (230)
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67
De(di/dv) - - K2{7eq-2(p/r) + 3 cos v}Q e. , (231)
D v(di/dv) - K2{3e0 sin v - 2(p/r) scC2 }v1 , (232)
D0,(di/dv) - - 2K2 (p/r) scC2 0 1 (233)
and
Di.(di/dv) - 3K scg (eS3 + 3S 2 + 3eS1 ) . (234)
There are four terms with Q1 as a common factor, and four other terms.
As in section 5.3, the q-2 factor in the first el term disappears when it is
combined with the al term. The overall combination is given by
D (di/di) - Ksce2(2,5)1, + 10e(l,2)1 5 + 212,3,5), (4,3)]Z 4 + 6e(3,2)E2 48 3
+ e2 (6,1)Z2 - 2e'(26,-33)S4 - 20e(5,-7)S3
+ 412(5,-3), - 3(2,-3))S 2 + 4e(31,-33)S I + 2e 2(14,-15)S0
- 4e 2(3,-7) sin 2v - 8e(3,-7) sin v}. (235)
The term in So is, as in section 5.3, a source of long-period variation,
given by
= 1,e2i 2- a c(14 - 15f)S, (236)
with the inevitable short-period carry-over term. The remaining terms of (235)
lead to the following formula, in which the term in Co and the 'pure constant'
are chosen to suit the derivation of b2 in section 6.4:
1
2 288
x {e2(2,5)r 6 + 12e(1,2)r 5 + 3(2(3,5),(4,3)Fr4 + 12e(3,2)P 3 + 3e 2(b,1)r 2
- 3e 2(26,-33)C, - 40e(5,-7)C3 12(2(5,-31,,-3(2,-3)]C3 2
+ 24e(31,-33)C1 + 4e2 (9,1)C0 - 12a (3,-7) cos 2v
48e(3,-7) cos v + 2(3(17,-25),-(90,-9l)] } .237)
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68
5.5 Check thus far by perturbations in p and p cos 2 i
The absolute constancy of pc2 . for a disturbing function associated with
only the zonal harmonics, has been referred to in sections 2.2 and 4.2. Now that
we have second-order solutions for a, e and i , we can derive the
corresponding solutions for p and c2 , and use their combination to check that
pc2 is indeed constant to second order.
Since there is no long-period variation in a , whilst the long-period
variation in e is given by (225), it follows at once that the long-period
variation in p is given by
P. p2 f(14 - 15f)S 0 (238)
We obtain P2 from the second-order identity
+ Kp + R p2 " ( a + a ) (1- + )- 2Ke - K (1 + 22eW (239)
which gives (on dropping bars as usual)
F2 - q2a2 - 2aee 2 - el(ael + 2eal) (240)
From (147) and (148) we get
ael + 2eal 1 .8 aq -2 [f{3e[l,3](eC5 + 6C4 ) + (28,133,19]C3 + 12e[13,7]C21 4832
+ 314,55,1]C 1 + 3e[1,3] (6C + eC 1)
+ 4h{ 1,3] (e 2 cos 3v + 6e cos 2v + 3(4,1] cos v)
+ 2e(13,71}] . (241)
We use (148) again, to obtain the product of (241) with el , then subtract
the result from the combination of a2 and e2 specified by (240); the required
form for a2 was suppressed in section 5.2, it will be recalled, whilst e2 is
given by (226). We find, after the subtraction, that most of the terms have
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69
cancelled out, and a factor q4 , needed to convert aq-2 in (241) into p , can
be extracted from all that remain. The resulting expression for P2 is:
p2 " - p [f2 {e2r - 3[2,5]r, - 48er - 33e 2r2 }
- f{3e2 (26,-33)C4 + 8e(32,-41)C 3 + 24h[2,3]C 2 - 72e(8,-9)CI
- 36e 2 (8,-9)C0 }
- 2{6ef(4,-5) (e cos 2v + 4 cos v) + [(8,-2 4,1),3(8,0, -15) 11 ]
...... (242)
From (236) and (238) it is immediately confirmed that there is no long-
period variation in pc2 . Also, expansion of cos2 (i + Ri1 + K i2 ) leads to
(c2) 2 = (2f - l)i2 - 2sci 2 (243)
from which it follows that
(Pc2) 2 c2p2 - 2psci2 + i {p(2f - 1)i1 - 2scp,} (244)
From (149) and '172) we get
p(2f - 1)i - 2scpl = -. psc{(1,2) (eC3 + 3C + 3eC1) + (11,-18)} . (245)
Using (149) again, for the product of (245) with i1 , and adding the result to
the required combination of P2 and i2 , given by (242) and (237) respectively,
we eventually cbtain the final result we seek, viz
(pc2)2 - - .3 pc2 {e2f(63,-82)C0 - [4(1,-6,2), (12,-50,13)] 1. (246)36
TR 88068
;i ., =".,eiJ . ;J ; ;,l 5- ,;i . -'-,; ,'',,' ' ', ---. me m . e m m - _ m. 1 .e - -. , , M
70
The absence of v-dependent terms in (246) completes our check of the
constancy of pc2 ; on inverting our standpoint, we have a powerful check on the
exprGssions for a2, e2 and i 2 . (In practice, a number of errors were made in
the e2 analysis and were located via this check.) Expressions (175) and (246)
provide the second-order relation between the mean and osculating values of
pc2 , connected (to second order) by
2 ._..F2 +-2PC )i (pc) 2 (247)
In the light of (247) and the absolute constancy of pc2 , (246) may appear
as a paradoxical result, since the first-order variation of Co (interpreted as
cos 26 , with & varying secularly) induces third-order variation in R2 (pc2)2.
But the second-order variation of p and f in (pc2)i , given by (175) with
bars added, induces third-order variation in K(pc2)1 , and this may be regarded
as cancelling the variation in K2(pc2)2 . However, the teal resolution of the
paradox stems from the observation that, in the present analysis, it is2meaningless anyway to regard a , in (247), as contant to third order, since we
have not carried the definitions of (semi-)mean elements beyond the second order.
5. 6 Perturbation in 1
The starting point, from the exact equation (144), is
d - K(n/) (j 3/q3) { (p/r) 3 /( /E) 2 )Q , (248)
where % = (l - C2) . (249)
The six contributions to the racond-order compon:ent of df/di may be
listed as usual, viz
D(d/dv) - T (pr) aI (250)a 2Dfli
D1 (dD/dv) - K (p/r)s(l - C2)ii , (251)
D (dl/dv) - - K 2.f7eq- 2 (p/r) + 3 cos v}Oje 1 , (252)
TR @806hb
i 71
Dv(df/dv) - X2 {3e% sin v - 2(p/r)cS2 }v 1 , (253)
2D (dI/dv) - - 2K (p/r)cS2 01 (254)
and
D. (dO/dv) = - I K2cg(eC3 + 3C + 3eCZ) (255)
There are, as in section 5.4, four terms with Qn as a common factor, and
four other terms, the combination (free of the necessity for a q-2 factor, as
usual) being expressible by:
D2 (dl/dv) - K2c X2 48
x {e2(4,3)r 6 + 10e(2,1)r 5 + 214(3,1), (8,-l)]r 4 + 6e(6,-l)F3
+ e2 (12,-5)r 2 - 52e2 (1,-1)C - 8C(11,-8)C3
+ 8[(8,-15), -(3,1)]C 2 q 8e(17,-30)CI
+ 4e 2(7,-15)C0 + 2e2 (4,5) cos 2v
+ "2e(2,1) cos v + 2(4(1,-1), (4,5)] }
..... (256)
The final term in (256) is responsible for the first appearance of second-
order secular variation; thus, it is given oy
^2 -- c[4(l - f) + e2 (4 + 5f)] (257)
The term in CO in (256) leads, as in earlier sections, to the long-period
variation given by
- e c(7 - 15f)C (258)2,tp 12 0(2)
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72
The remaining terms of (256) lead to the following formula for f 2 , the
'constant' term in So being chosen to suit the derivation of b2 in
section 6.4:
11 12 288
28
X {e 2(4,3)E6 + 12e(2,1)E5 + 3(4(3,1),(8,-l)]E 4 + 12e(6,-1) 3
+ 3e 2(12,-5)Z - 78e2 (1,-)S4 - 16e11,-8)S3243
+ 24[(8,-15),-(3,1))]S2 + 48e(17,-30)S1 + 2e2 (18,-19)S0
+ 6e2 (4,5j sin 2v + 72e(2,1) sin v}
...... (259)
5.7 Perturbation in o
The starting point, from the exact (145), is
d) Ke (n/fi) (3/q3){ (p/r(/ / 12}Q , (260)Sdv 8 0
where CIO- f(5eC + 14C 3 + 2eC - 2C + eC 0 + 4e)
4h(2eC2 - e cos 2v - 2 cos v - 3e) . (261)
Equat.on (261) has a certain inconformity, in comparison with (215), similar to
that remarked upon in the first-order analysis in relation to (153) (propagated
from (145)) since we do not have the neat division in which the f-terms involve
praerie]y the Cj and the h-terms involve precisely the cos jv
The six contributions to the second-order component of dw/d may be
listed as usual: thus
TR 88068
Da(d/dv) - a7 e (p/r)Qa1 , (262)
I2D1 (dO/dv) = -e-(p/ r) s c 5eC 4 + 14C 3 + 14eC - 2C + eC0
- 2(3e cos 2v + 6 cos v + 7e)]i,
(263)
D (dwt/dv) 2 K2e- { [7eq-2 (p/r) + 3 cos v] - 2e- (p/r)Q }ei , (264)08 0)W
where Qe = f(7C3 - C1) + 4h cos v , (265)
Dv (d(O/dv) -K e'f{3eQ sin v + 2(p/r)Q}v , (266)
where Q_ _ f(lOeS4 + 21S3 + 2eS2 - S 1) + 4h(e sin 2v + sin v - 2eS 2 ) , (267)
D (dw /dv) -1 K2e-'(p/r) {f(5eS4 + 14S 3 + 2eS2 - 2S3 + eS3) - 8ehS 2}o, (268)1o 4
and
D. (d(f/dv) K - Ieg {3e2fC5 + 18efC + (28f,-(8,-19)]CCo 12 4 3
- 12e(2,-5)C 2 - 3[4f, (8,-151]C I - 18efC 0 - 3e2 fC-1 } (269)
There are four terms with QeO as a common factor, ard five other terms,
the combination (free of q-2 as usual) being expressible by:
D2 (dw/dv) - K e { 15e 3ro + 162e 2 r 9 + 8e[82, 15Ir8 + 14184,671F 7 }
22 2+ (784f ,244Sf ,- (64,16,-277)]1' + 20o[106f ,-(16,0,-69)]F,5
2 2_ 8e (2(24,-4,-0'c:, (32, -20,-23)r 4 - 12e[18f2, (48,-32,-35)I 3
[48f 2,208f 2, (192,-176,-9)ir' - 'f22[36,l91r 18e 2 r +
TR 88068
I& L 0 -1 ."
74
+ 8f4e hfC8 +42e hfC7 + e 2f[172h,-3(14,-17)]C 6
+ 5ef[64h,-(34,-39)]C 5 + [224hf, -12f (2, 3),
(104,-242,159) ] C4
+ 8e(2f(21,-31), (22,-17,-5)]C 3 - 4e 2 (2(16,-138,141),
-(12, 22, -33)]C 2
2
+ 8e(f(4,-3),-(34,-173,150)]C 1 + 2e (2f(14,-15),
- (28, -158, 135)] C0
+ ef[48h, (110,-129)]C I + e 2f[52h, (34,-39)]C 2
+ 18e 3 hfC_3 + 2e 4 hfC_4 }
+ 2{e3(8h 2 + 9f 2 ) (e cos 6v + 10 cos 5v)
+ 4e 2 [2(40h 2 + 37f 2), -(8, 8,-37)] cos 4v
+ 6e[12(8h2 + 5f 2),-( 2 4 ,2 4 ,--137 )] cos 3v
+ (16(24h 2 + 7f2),-32(4,8,-41),-(264,-136,-301)] cos 2v
+ 4e(2(8h 2 + 43f 2),-(256,-400,9)] cos v
4e 2 (2(64, -180, 95), (56, -36, -45)] .
...... (270)
The final term in (270) represents, as in section 5.6, the secular
variation, which is thus given by
2. [2(61 ) + e (56 - 36f - 45f 2 )] (271)
The term in CO leads, as usual, to the long-period variation given by
- 4-L [2f(14 - 15f) - e2 (28 - 158f + 135f
2)]C 0 . (272)
R 48 0
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75
The remaining terms of (270) lead to the following formula for Co2 , the
'constant' terms in ZO and So being chosen to suit the derivation of w2 in
section 6.4:
-= -X--e- X4608
x [3ef 2 {3e3E1 0 + 36e 2E9 + 2e[82,15]Z8 + 4184,67]E7
+ [784f2 ,2448f2,-(64,16,-277) ]Z 6 + 24e16f 2,-(16,0,-69) 1 -5
- 12e [2(24,-4,-103), (32,-20,-23)1Z4 - 24e[18f 2 , (48,-32,-351E 3
2 2- 3[48f , 208f 2, (192,-176,-9)] Z2
- 3ef2 1436,19]Z1 + 2e[66,-l]t 0 + 36e 2EI + 3e 3T- 2 }
+ 8ef {3e3 hS8 + 36e 2 hS7 + e[172h,-3(14,-17)]S 6 + 6164h,-(34,-39)]S5}
+ 4{31224hf,-12f(2,3),(104,-242,159)]S
+ 32e[2f(21,-31), (22,-17,-5)]S 3 - 24e 2 12(16,-138,141),
- (12, 22, -33)] S2
+ 96e[f(4,-3),-34,-173,150)]S - [288hf,16f(1l6,-147),
(144,-418,311) ]S }
- 24ef { 2 (48h, (110, -129)] S_1 + e F52h, (34, -19) IS_2
+ 12e 2 hS_3 + e 3 hS_ 4}
+ 2{e 3 (8h + 9f (e sin 6v + 12 sin 5v)
+ 6e212(40h 2 + 37f2 ),-(8,8,-37)] sin 4v
+ 12e[12(8h2 + 5f2 ),-(24,24,-137)] sin 3v
2 2
+ 3(16(24h 2 + 7f2),-32(4,8,-411,-264,-136,-301)1 sin 2v
+ 24e(2(8h + 43f 2 )-(256,-400,9)] sin v}]
...... (273)
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76
5. I Vezturbation ina
The starting point, from the exact equation (146), is
_ Ke'(n/R) (3 /q 2) { (p/r1/ ((/)2}Q (274)
where Q% - Z(5eC 4 + 14C 3 -18eC 2 -2C 1 + eC0 ) + 4h(e cos 2v + 2 cos v- 3e)
...... (275)
The six contributions to the second-order component of da/dv may be
listed as usual; thus
.(d/dv) -K a e Iq(p/r)Qoa, 76(276)
D (da/dv) = -I K2e q(p/r) s c f5eC4 + 14C 3 - 18eC2 - 2CI + eCO4 4
- 6(e cos 2v + 2 cos v - 3e)1i I , (277)
te(da/dv) - -. K 2e- q{Q [6eq- 2 (p/r) 1 3 cos v. - 2e-(p/r)Q }e , (278
where Qoe - f(7C 3 - C) + 4h cos v ( - QW) , (279)
Dv,d(y/dv) _ K2e-Iq{3eQ0 sin v + 2(p/r)Qlv}v1 , (280)
where QOv f(lOeS4 + 21S 3 - 18eS 2 - S) + 4h(e sin 2v + sin v) , (281)
I%(da/dv) - 4K2e-q(p/r) f{5eS4 + 14S 3 - 18eS2 - 2S 1 + eSo}O) 1 (282)
and
L. (d(7/dv) - K2e-Iqfg{3e2C5 + 18eC + (28,-131C - 36eC212 54 32
- 3[4,17]C 1 - 18eC0 - 3e2C } (283)
There are four terms with Q0 as a common factor, and ive other terms,
but 'because the first De term has coefficient 6 instead of the usual 7) it is
TR 88068
77
L not possible this time to extract the factor q2 when this term is combined withthe D. term. The overall combination is expressible by:
D (dCF/dv) 1 K1 2Ke-2q'l x
x If 2{ 15e'(2,-1jr 10 + 162e 3 2,-l]r 9 + 2e 2 [656,-208,-97,r 8
+ 2e[1176,350,-7371 r7 + (1568,4112,-3814,-21]r 6
+ 4e(1060,-704,-203]r 5 + 60e 2(16,-32,-5]r4
- 36e[12,8,3511". - (96,368,350,473]r 2 - 2e(72,2,121]r,,
- 18e r 0+ Te 3[2,-lr + 3e 4 r2,-1]r_2}
+ 8f{4e'h(2,-l]C 8+ 42e3 h(2,-l]C 7
+ e 2(344h. -8 (32, -45), (86, -105)C C6
+ e[640h, -30 (22,-29), (382,-453)]C 5
+ (448h,-8(34,-33),2C298,-327),-(418,-531)JC4
+ 2e(16(21,-31),4(60,-61),-(582,-749)]C3
+ 2e 2 (16(17, -21) ,-4 (118, -147) ,-(10, -63)] C2
+ 2e[8 (4,-3), -8(95,-111), (506,-531))C
+ 4e 2 (2(14,-15),-4(44,-51),(106,-111)ICo
+ e[96h,2 (86,-93) ,-9(26,-31)] CZ
+ e (104h,1S,-3(22,-25)]C_
+ 18e h[2, 1]C3 + 2e 4 h(2,-11C4 1 +
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78
+ 2{e3 (8h2 + 9f')2,-11 (e cos 6v + 10 cos Sv)
+ 2e 28(40h2 + 37f 2),-8(24,-56,45), (24,56,-143)) cos 4v
+ 6e(24(8h2 + 5f 2),-2(72,-120,1), (40,72,-253)] cos 3v
+ (32(24h2 + 7f 2 ),-16(40,-40,-103),
2(312,88,-1331),-(504,-744,101)1 cos 2v
+ 4e[4(8h 2 + 43f2 ),4(76,-156,-17),
-(488,-1176,689)] cos v
+ 2e 2 16 16,-56, 31),-16 (16, -88,95),- (280, -328, -79)1 }
...... (284)
The final term at once yields (with the usual notation) &2 , and the long-
period variation is given by the terms in r0 and CO . However, there is
little point in giving the explicit expressions, or in giving the formula for the
short-period 02 , since 0 (from its definition in section 2.1) is merely an
intermediate element that has to be analysed as part of the analysis for M . The
existence of a non-zero long-period term in r0 , in addition to the usual term
in CO , is another aspect of the ephemerality of 0 , and it will be found in
sections 5.9 and 5.10 that in the M-analysis the r0 term is cancelled by an
equal and opposite term from the quantity J defined by (3).
5.9 Perturbations in n and S
The second-order perturbation in n , like the first-order perturbation, is
purely short-periodic and may be obtained from the perturbation in a by use of
the appropriate version of Kepler's third law for mean elements. The generalizedA A
second-order form of this law, with (non-dimensional) off-sets Ai and 92 to
suit the definitions of n and , may be written
-2-3 A 2An a - + KR1 + K 2) , (285)
but we have seen in section 4.1 that the natural first-order definition of a
and H leads to the taking of 01 to be zero. There is no reason to expect 4A2
to be zero as well, however, and it will appear in section 5.10 that, with 5
TR 88066
.7
defined to be identical with the exactly constant a' , and n chosen to second
order such that the secular variation in M vanishes,
A2 -T4q3(8 - 8f - 5f2 (286)
A (
If we write, for convenience, aj and nj for aj/a and n1/n ( 1
and 2), it follows that
2_ 2- A 2A 3 RA+-2A2
n2 a3 - n2a3 (i + KA1 + K a2) (l+ Kn + K n22 (287)
From (1), (285) and (287), we have at once that
A A A= - (3a1 + 2n (288)
and
A k2 A A A2 A A42 6'a 1 +6a In I+ 3n1 -3a 2 - 2n 2 (289)
Equation (288) just leads to (159), on the basis (from section 4.1) that
A A 3 Ai 0 ; the new material is in (289), which with n1 set to --7a, yields
n 2 2 42 (5- ~(~24a 2 (290)
The appropriate expression for a, is given by (158). When this isA A
squared and combined with the expression for a 2 given by (213), then for AL2
given by (286) it is eventually found that
- 1 -4 2n2 -nq (I + e cos v) X2 768
x [3 f2{36r 8 + 24e 3r 7 + 12e 2 8,-l]r6 + 8e[22,-l]P 5 + 2156,32,17]r 4
+ 24e(2,5]r 3 + 4e2(8,131r 2 + 24e3 + 3e4 1o} +
TR 88068
80
+ 24f{e 4 hC6 + 8e 3 hC5 - 2e2 18(l,-l),-(22,-29)]C 4
- 4e[4(4,-5),-(30,-4l)]C 3 - [16h,-32(3,-4),(1C.l',)C2
+ 4e[4(8,-9),-3(6,-5)1C 1 + 2e2 18(5,-6 ), -(26,-27) ]C0
+ 8e hC_ + e 4hC}-1 -2
+ 2{e3(8h 2 + 9f2 )(e cos 4v + 8 cos 3v)
2- 4e 296f(l,-2),-(56,-72,-3)] cos 2v
- 8e[2(32,-48,-27),-3(40,-88,45)1] cos v
- [8(40h 2 - 33f 2 ),-96(4,-16,15),-3(72,-88,-13)} } ]
+ -- nq (8 - 8f - 5f48
...... (291)
The second-order perturbation in M , to be derived in section 5.10, is
given by the combination of the perturbation in G with the perturbation in
the integral of n ; since (ii section 5.8) we only took the analysis as far as
D2 (dG/dv), hzwevo-, what we need here is D2 (dJ/dv), to combine with D2 (dG/dv) to
give D9 (dr/dv) .
Now (291) displays n2 as a sum of many terms, all with (I + e cos v)2
Aas a factor, together with a final term that emanates from 92 . Apart from a
complication to be dealt with in the next paragraph, the 'sum of many terms'
gives an immediate contribution to D2 (dJ/ Iv), since (with bars omitted as usual)
d2 n 2 n2q3
dZ nW 2 (292)
by (39) and (17). Thus the factor (I + e cos v) 2 in (291) is conveniently
cai.elled and the required contribution from the sum of many terms is given on
replacing nq "4 (1 + e cos v) 2 by K2 q 1 . This just leaves the 'final term',
which contributes L K2 q3 W - (8 - 8f - 5f ) , since av/aM - W48
TR 88068
; i 81
The complication in the 'sum of rany terms', referred to in the last
paragraph, arises because D2 (W/dv) has to cover some carry-over terms from nj
as well as the direct terms associated with n? . The carry-over terms arise
from the first-order secular variation in ( , and constitute a final
manifestations of the 'Dio source' of terms introduced in .ection 5.2 and
encountered in all the subsequent sections. Here, the origin of the terms is the
first-order evaluation of 11 by v-integration of (160). As in section 5.2, to
compensate for the neglect of the term (-2R&1/j2)Sj , when replacing each
integral of Cj by Sj/j , we need a contribution to D2 (dJ/dv) given by
Iqf(4 - 5f) + + and the effect of this on the conversion of (291)4 C3 3 2 3e 1
into the expression required for D2 (dJ/dv) is to replace the terms in C3, C2
and C, (within the '24f bracket') by
-4e[2(4,-5),-(22,-31)IC 3 + [16(5,-6) ,-8f,-(10,1) )C2
+ 4e[2 (28, .-33) ,-3 (14, -15) ]Cj
5. 10 Perturbation in X
The expression for D2 (dM/dv) is given at once by the combination of (284)
with the expression for D2 (dJ/dv) obtained from (291) as indicated in the last
section. The result is:
D2 (dM/dv) - 1 K2e-2q- I2 1536
X [f 2 {15e'12,-1]r10 + 162e 3 (2,-l]r 9 + 16e[82,-26,-1I]F8
14e(168,50,-95]F? 4- (1568,4112,-3238,-931' 6
+ 20e[212,-88,-43]r 5 + 96e2 [17,-16,-I][f
108e[4,0,5]r - [96,36P,158,161ir 2 - 2e(72,2,491r
+ 18e,[2,-l]r_ I 3e4 2,-l]r 2}
+ 8f{4e 4 h[2,-lC8 i 42e3h(2,-lC 7
+ 2e 2[172h,-4 32, -45), (,5,-57)}C 6
+ (448h,-8(4,-33),2(250,-279),-(154,-183)]C4 +
TR P.9O68
682
+ 2e[16(21,-31),4(36,-31),-(318,-3 7 7) C 3
+ 8e2(4 (32,-39), -(118. -141),-5 (2,-3)] C2
+ 2e[8(4,-3),-8(11,-12),(2,9)] IC1
+ 8e2 q4 (1 4,-15)C0 + e[96h,2 (86,-93 ,-3 (62,-69))C_
+ 2e 252h,8,-3(10,-11)]C_ 2 + 18e h[2,-l)C 3
+ 2e 4 h[2,-l]C_ 4 }
+ 2{e 3 (8h 2 + 9f2 )[2,-1](e cos 6v + 10 cos 5v)
+ 8e 22(40h2 + 37f 2),-2(24,-56,45), (8,8,-29)] cos 4v
2 2+ 2e[72(Sh + 5f ),-6(7?,-120,1), (184,24,-543)] co3 3v
2 2+ (32(24h 4 7f ),-16(40,-40,-103),2(312,-296,-563),
-(56,-168,125)) cos 2v
+ 4e[4(8h 2 + 43f 2),4(12,-60,37),- (8,-120,149)] cos v
- 16e2 q 4 (8,- 8 ,-5)1]
+ -L- KI2q3W- (8- 8f - 5f 2
48
...... (293)
AThe basis for the choice of 92 defined by (286), which deternined the
final terms of both (291) and (293), should now be apparent, since the
combination of the final term in (293) with the last of the preceding collection
of terms is LK2q 3(8 - 8f - 5f 2
(W"I
- 1) , the mean vilue of which (with
respect to v ) is zero oy (39). The penultimate terr of (293) in fact defines
^M2 , vi z
= -9I q 3 (8 -8f -5f 2 ) , (294)
TR 88068
83
Aand 92 was set to 2A2 to allow for the resulting secular effect, as
anticipated in the last section and (before that) in section 3.1. Though ourA
choice of 92 in principle allows for A2 , making i and Misc identical as
required, we must not overlook the short-period carry-over effect of M2 . This
can be allowed for by applying (64) with C - M .
As forecast in section 5.8, (293) contains no term in r0 . The term in
CO leads, as usual, to the long-period variation given by
M q 3-qf(14 - 15f)C0 . (295)2,2p 240
The remaining terms of (293) lead to the following formula for M2 , the
'constant' terms in E0 and So being chosen to suit the derivation of r2 in
section 6.2:
M2 921 - [f 9e'[2,-11Y, + 108e3 (2,-l] + 12e 282,-26,-ll]Z
84 9-- 216 ~ Le , 10 9 Oe 8
+ 12e[168,50,-95]I 7 + [1568,4112,-3238,-931E 6
+ 24e[212,-88,-43]E 5 + 144e2 [17,-16,-l]E - 216eC4, 0, 5,
- 3(96,368,158,161]E - 12e[72,2,49]E 12e2 (66,-34,13]E,2
- 108e3 [2,-l]E_, - 9e4 2,-1]Y 2}
+ 4f{6e 4h2,.-l]S8 + 72e 3h(2,-l]S7
+ 4e 2172h,-4(32,-45), (46,-57)]S 6
+ 12e [128h, -6(22, -29), (86, -105) ] S5
+ 3(448h,-8(34,-33),2(250,-279),-(154,-183)]S 4
+ 8e(16(21,-31),4(36,-31),-(318,-377)l S 3
+ 48e 2 (4(3,-39),-(118,-141),-5(2,-3)]S2
+ 24e[8(4,-3),-8(ll,-12),(2,9)S 1
TR 80068
|I
84-
- [576h, 16(214,-267),-2(802,-963),-(830,-861)|S 0
- 12e[96h,2(86,-93),-3(62,-69)]S i - 12e 2f52h, 8,-3(10,-1)] _2
- 72e 3h[2,-l]S_3 - 6e4h[2,-l]S_ }
+ 2{e 3 (8h 2 + 9f' ) [2,-1] (e sin 6v + 12 sin 5v)
+ 12e212(40h2 + 37f2),-2(24,-56,45,,(8,8,-29)] sin 4v
+ 4e[72(8h 2 + 5f2 ), -6(72,-120,l), (184,24,-543)] sin 3v
+ 3[32(24h2 + 7f2),-16(40,-40,-103),2(312,-296,-563),
-(56,-168,125)] sin 2v
+ 24e[4(8h 2 + 43f2),4(12,-60,37),-(8,-120,149)] sin v} .
....... (296)
6 ADOPTED SOLUTION FOR J22 PZRTURBATIONS
6.1 Secular and long-period terms, plus short-period carry-over
Solution of the J2 problem to second order amounts to the extension of the
first-order solution of section 4 by additional terms. The terms expressing the
long-term variation in the mean elements are compact and small in number and,
since they are not easily to be found among the much longer expressions of
section 5, it is convenient to repeat them here. Though the r2,fp are actually
all non-singular, propagation of the will nevertheless be via %2,Zp and
L2,tp (rather than (02,tp and M2,2.) , as explained in section 3.5 and Part 3,
so we include here also the expressions for these.
Each r2,ep induces a short-period carry-over (cf (74)), and these are
dealt with by combining them into components of the perturbations in the
spherical coordinateb, as indicated in section 3.2; the resulting expressions
(components of 8r, 6b and 8w ) are included in the present section. The
pure short-period components, representd by r?, b2 and w? , ere derived in
sections 6.2 to 6.4.
e onl, secular terms are in Q and W . They are given by K 2 t and
K nR02t where, to repeat (257) and (271) with bars added,
TR 88068
!
-A
85
.' = L [4( 1 - f 2) + 2 (4 + 5f)] (297)2 24
and
- [2(64 - 180f + 95f ) + (56 - 36- 42 (298)
A secular perturbation in M is avoided, essentially because, foilowinj (285)
and (286), Kepler's third law is used in the modified form
2 3 -3n a -24 K (8 i - 5f2) (299)
However, M2 , given by (294), induces a second-order carry-over contribution to
the semi-mean M , just as b2 and (- ao, to h and 6) respectively.
The long-period perturbations are given by the integration of the various
j2 f t2,Ip , following the methods indicated in se-tion 3.5 and Part 3. The
expressions for the C20tp are given by (225), (236), (258), (272) and (295),
repeated here as
. - f (14 - 15f) sin 2@ , (300)e2,fP 24
i 2,P - 2c (14 - 15f) sin 2) , (301)
-- e _2 (7 - 15f) cos 2C, (302)
2,tp 12
02,9 - --- [2i(14 - 15F) - F2(28 -158i + 135f 2 )] cos 2w (303)48
and
M 2, - f(14 - 15f) cos 2@ . (304)2, p 24
TR 88068
86
From (302)-(304) we get the expressions to replace (303) and (304) in practice,
viz
. - (2 + 5i 2) (14 - 15F) cos 2- (305)V2,tp -48-
and
7 -2-L - e f (14 - 15f) cos 2C, (306)L2, P " 48
The short-period effects induced by (300)-(304) are combined into spherical
coordinate perturbations by use of (95)-(97). The results may be expressed, in a
new but obvious notation (in which the carry-over r2.0 etc will be added
dizectly to the r2 etc to be developed in sections 6.2 to 6.4), as
1-pef(14 - 15f) mS1 , (307)
1 2
b2 - 4" e scm [ff 1 - (28 - 45f)c ] (308)
and
w2 --ef (14 - 15f) m (eC + 4C - 4eC) (309)2o 48 2 1 0
(In equation (43) of Ref 4, the factor e2 s c is missing in the terms of b c .)
6.2 Pure short-period perturbation in r
Since r is a function of the three elements a, e and M , with first-
order partial derivatives given by (41), evaluation of r2 in principle entails
the combining of nine terras from a Taylor expansion, three of them involving a2,
e2 and M2 through first-order derivatives, whilst the other six involve the
second-degree products of al, el and MI through second-order derivativel.
Thus the analysis is potentially much more complicated than for v2 and b2 (to
follow in sections 6.3 and 6.4), each of which involves only a pair of orbiLtal
elements (not three) and hence only five Taylor terms (two plus thtee). A
somewhat different approach was followed, therefore, 'tailored, rathei than
'Taylored'; it is based on the use of eccentric anomaly, r being the :implo
function of a, e and E given by equation (78).
We need expressions for El and E2 , defined to repteoent pure short-
period perturbations in the usual way, and these can be obtained from (75). We
R 66068
87II
-2replace by I + KRe + K 02) in this equation, and likewise for M and E
'I -2then expand sin E (and thence e sin E) to O(K ) * and can thus identify
expressions equivalent to MI and M2 Rearrangement of these expressions,
with algebraic simplification through the introduction of v where possible,
gives us
El q -e sin v + q- 2 M(1 + e cos v) , (310)
where e, q and v are of course to be read as barred, and
E2 - (a/r)M2 + q-1 e 2 sin v + )a(a/r)E1 [el (cos v + cos E) - q- eM1 sin v]
.......(311)
Equations (310) and (311) can now be used, as required, in the expressions
obtained by expansion of (78). At the first-order level this expansion gives
r -. (r/a)al - ae, cos - eE1 sin E (312)
from which the use of (310) leads at once to a formula for rl that is
equivalent to (95). At the second-order level the expansion yields
- (r/a)a2 - ae2 cos E + aeE 2 sin E - a 1(eI cos E - eE sin E)
+ )aEI(2e 1 sin E + eE cos E) , (313)
from uh-Vi.h zh ie of f3101 and (311) leads to
r - ncr - ,- sin v)
+ z.q-2 [ 1 sin v Iq- (i + e co, v)] X
X fe'4 sin v + e sin 2-) + eq M (e + 2 cos v + e cos 2v)]
aeq M2 sin v - ae. cos v . 314)
2 2
Sa __
I The five terms, of varying complexity, in (314) have been presented in an
order that reflects the way in which they are most naturally combined. TO start
Iwith, we combine the first two terms. Since (r/a) is simply q2(1 + e cosv-,
the first term is given by (213) with the overall coefficient replaced by
288 tq (1 + e cos v) . Also, (148) and (164) give
2C 2
eCOS v -eq 'M sin V f . f(3eC + l2eC3 + 2[5,41C2 + l2eC1 +3e C)
+ 4h(e2 cos 2v + 4e cos v + 3)},
.....................................(315)
the product of which with (158) yields an expression for the second teirm of
(31-1), with (l + e cos v) as a factor. To avoid another long expression here
(and likewise at some other points of section 6) we give the result in a skeleton
form in which only the 'end terms' are quoted; it is
- ~ 28P + o v 3f {3 r 8 + ... + 3e4 r0}I + 8fh{3e4 C6 + ... + 3e C-2
2 e4 [8h 2 + cos 4v + . + 8h2 12,22,1) + 3f' 220,74,11}
The first two terms of (314) can now be combined, whereupon a considerable
sirplification occurs since ten pairs of erms cancel out and a factor 4q2 ,ail
be extracted from the remaining terms. The result is worth expressing in f -Ill;
thus
eI ) a v- ae(eM cos v - eq 1 M e s+ v)
22 (12 e 2ecos 2v +4e cos v + 31},
, 11 wth(I + e Cos v1, a3f r +atr Toaod nter lon 2qe pr e so here
O f{e 2 (7,-9)C4 + e(16,-21)C3 + r(2(2.,-3),-(4, -3flC,
I ooe(20,-21)C 2 ' e(1l,-12)C 0
-2e(8,O,--15) cos 2v- + 8-(32,-48,-3 cos v
|+1 2e4 8 2 + 92] cs 4v ... + [(16,2), (82,O,l)]
Ta 86068I
fI
Th is w em f111cnnwbecmieweepnacnieal
siIiiainocr ic e ar ftrscne u n atr4 2 ~beetatdfo h eann em.Th euti ot xrsigi Il
89
Turning to the third term of (314), we find that
0e sin v + 1 m(1 + a cos v)
-= _ 1.e-I [f {3e2S5 + 6e[3,-1134 + (28,-13]S + 16eqS2 _ 3[1.11S,5 4438
- 6e[3,-I]S 0 - 3e2S.}
* .
+ 4h{e 2 sin 3v + 2e[3,-1] sin 2v + (12,-7] sin v
... (317)
and
e,(4 sin v 4 e sin 2v) + eq - M1 (e + 2 cos v + e cos 2v)
- -L [f {3e2S6 + 6e[3,1]S + 2(14,191S4 + 2e[25,11]S36448
22- 48q2S, - 2e[41,-5]S 1 - 612,9]S 0 - 6e[3,1]S. 1 - 3e2S.2
+ 4h{e' sin 4v + 2e(3,1] sin 3v + 2'6,5] sin 2v + 2e[ll,1] sin vi]
...... (318)
Multiplying the product of these two expressions by a further factor YVaq -2 , we
get the required 'third term'; in skeleton form the resulting explession i-
183 ae q 2 [ f {ge"F + + 9 4 r, + h - ... *h 3e-C 4.
0 4 3 2 2 2-) - 5f
2e (8h + cos 7v + .. - e27168h2 + 7 -( 2 97f
2 2 *-4 Oh 25f )IlJ
* This expression coones well witbh the fourth term of (314), given by m0-tilpiylng
the expression for M4 , . (296), by .e-I sin v ; th. result of that
multiplication, in skeleton form; is
T 8608
a
90
1 ,8-l-2 [ 9e',(2,-lr + + 9e4 (2,_-r} 18432 L' *-
+ 4f{6he4 12,-l]C 9 + ... + 6he4 [2,-1]C_5}
+ 2{e4 2,-i] (8h + 9f2) cos 7v + - 24e(4 (8h? + 43f 2),4 (12,-60, 37),
- (8,-120, 149)]1].
The virtue of combining the third and fourth terms of (314) is that a common
factor q2 emerges, to cancel the existing q-2 in both terms; the resulting
expression (in full) is
- 1 ae-1 [f2{9e4 r, + 108e 3r10 + 3e2 (164,29]r 9 + 24e[42,31]r 8
"944,-322]-
+ [784,2204,-267]r 7 + 36u[64,-35]r.6 [112,-1944,-325]F
_ 16e(98,25)F 4 - (528,-328,-347]r 3 - 12e[96,-55]r
+ 9[16,-164,-i9]r1 + 24e(18,-41]r 0 + 3e 2 [132,-.67]r r
+ 108e 3 F + e 41 }
{6e4 hC + 72e 3hC8 + e [344h,-(214,-2-13) jC
+ 24e(32h,-3(i8,-23] C6 + [672h,-4 (478, -609), t558,-645)C 5
+ 4e[8(i0,-13), (798.-925)j' 4 - (288h,-16(436,-519),
-(2, -!47) 1 C.,
a + 16e[2(38,-4l),-(138,-139)]C 2 - (672n,8(9i4.-1047),
I (!198,-1413) :C,
- 36e[8(10,-i3), (50,-59)]C 0 + [288h,4(398,--381,,
5 (1C= -105) ict
+ 24e[24h, f82,+8e7)C . f 3e 1!~, (1I0,-
1+4- 72e 3hC- - h +
TR 8A068
--- 5
a1 . .
I
J91
+ ±{e4 1[8h2 + 9f2 ] cos 7v + 12e 3 1 2 + 9f2] cos 6v
+ e2[12(40h 2 + 37f2),-(136,-24,-213)] cos 5v
2 +52)+ 8e[18(8h + 5f2),-(136,-120,-123)] cos 4v
+ [48(24h 2 + 7f 2),-12(248,-424,5), (232,-312,-81)] cos 3v
2 2- 4e[48(16h - 9f ),-(232,-696,351)) cos 2v
- [16(72h 2 + 37f 2),-24(104.-152,-61), (104,-312,-97)] cos v
+ 8e(10(24h2 - 37f 2 ),(8,264,-275)j}]
The terms in r1i, r-l, C1 and C-1 here are, of course, determined by the
apparently arbitrary choice of 'constant' terms in E0 and So that was made in
(296), but we are now close to seeing why the particular choice of coefficients
was made.
We are left with only the last term of (314). MultipJying the expansion
for e2 , viz (226), by -a cos v , we obtain for this last term, quoting in
skeleton form only,
-ae 2 COS v - ae- 1 [f 2 {9e4r1 1 + ... + 9e4 r 3 }21843211-
+ 4f{6he4 C9 + ... + 6he4C_5}
+ 2{e4(8h2 + 9f
2 ) cos 7v + .. - 24e[2(88h2
- 21f2
(168, -88,-207)) 1]...... (319)
When the full version of (319) is combined with the long preceding expression, a
tremendous simplification occurs, since no less than 20 pairs of terms cancel out
and a factor l6e can be extracted from the .emaining terms. The result, which
represents the last three terms of (314), is worth giving in full, being
TR 88068
92
15 a 'f2{2463r, + 120e2r, + e[185,-17] r5 + 16[5,-8ir 4 - e[181,131]i 3
- 264e2r2 -72e3r,}
- 4f{16e 3 (7,-9)C5 +48e 2 (10,-13)C4 + e[(538,-717), (86,-147)]C3
+ 32[(l0,-13);-2(6,-5) C2 - 48e[12(1,-1l), 5(1,-l) IC1
- 48e 2 (14,-15)C0 - 16e 3 (11,-12)Ci}
- 16{e 3 (8,0,-15) cos 3v + 3e 2 (16,-16,- 11 ) cos 2v
+ 3e[16(2,-4,1),(8,0,-15)] cos v + [(48,-144,77),8(4,0,-7)]) ]
We can now make the grand combination of this last axpression, which
represents all but the first two terms of (314), with (316), which (on
multiplying out the factor 1 + e cos v ) compatibly represents the first two
terms. In the process, another 12 pairs of terms cancel out and a common factor
can be extracted, which has the effect of changing an external factor from
'a' into 'p' the final result is
1 [E2 {er5 + 16r4 11ersr2 - -- kL i.5+ 6 lr
1152 4 3}
- 4f{e(38 - 51f)C3 - 32(6 - 7f)C 2 }
+ 16(16h 2 - 13f] (320)
As one of the definitive formulae of the paper, this has been written without use
of the bracket conventions; its brevity, in comparison with the individual
components of (314), is striking, and it is likely that there is some physical
significance in this compression.
It should now be clear how the coefficients of r0 and Co , together with
the 'pure constant', were chosen in (226), and likewise for the coefficients of
LO and So in (296) already referred to. They were triginally set as algebraic
unknowns, and carried through the complete analysis for r2 . The five unknowns
were then derived, by solution of linear algebraic equations, so that the
coefficients for rI, r-1, Cl, C-1 and cos v in (320) should all be zero.
TR 88068
t 93
. $ ls b e t-pearJi -ne Iu mtla Is v 44 a
An approach similar to that use4 for TZ W&d also be possible for v2
but in this case the direct apprach. c&ly slightly v ified. is siwp1er Thestarting point is (42). ich gives the partial eri&1tativ*es v. and wt that
occur in the first-or r foumia
V. - (3 - w N. -321)
The corresponding fovala for z v,%_ l* t tS. dr'%5& !.-= tlZ,
Taylor expansion as rmazked in se -ti 6.2. but ation of v. and -t
with respect to M 13 Met iinedtat*. wb*Zreas with rt3P*-- V Lt 13. 30
is preferable to start with the follovwiq six-te*zr fa-- -la"
V2 ( VV 2 ,vN * w . * e, * V te v v .322)
here ve - q4 sin v [q 2 cos v + 2(2 + e cos v)) (323)
v . - q-2[e sin2v - cos v (2 + e cos )] , (324)
vM, .q-5 (1 + • cos v) [2q 2 cos v + 3e(1 + e cos v)] (325)
and
VMv - - 2eq -3 sin v (1 + e cos v) . (326)
(The hybrid nature of these partial derivatives must be noted; thus v. is the
e-derivative of v with M held constant, but vee is the e-derivative of v.
with v held constant.)
It is convenient, in substituting these second-order partial-derivative
expressions into (322), to split their components into three combinations of
terms. The first combination is given by the first half of each of vee, Vev
and VMe , together with the whole of vMv ; the terms involved are precisely
the ones that arise when the differentiation of v. and vM , given by (42), is
restricted to the variation of e cos v , and this first combination of termr
factorizes in a convenient way. The second combination involves the second half
Of Vee and vev , and the third combination involves the second half of vje.
TR 88068
f .
94
We now express the working formula for v2 , corresponding to (314) for r 2 , as
a sum of five terms, of which the first, third and fourth represent the three
combinations of partial-derivative terms just described; the second term of the
'working formula' covers the e2 term from (322), whilst its final Larm covers
the M2 term. Thus we get
v2 - q-2 (el cos v - ev sin v) {e1 sin v + 2q' M1(1 + e cos v)}
q2e2 sin v (2 + e cos v)
* Aq 4el (2ee sin v + q2v cos v)(2 + e cos v)
+ % eq- e1MI(I + e cos v)2 + q'3 M2 (l + e cos v) (327)
For the first term in (327), we have, from (148) and (182),
cos v- av sin v - 2.[f{3e 2C4 + 16eC3 + 2[10,7]C + 32eC1 4 l1e 2C0 }
+ 4h{e 2 co s 2v + Be co s v + [6,1]1
(328)
whilst from (148) and (164) we have
e0 1ir v + 2q 1 Ml(I + e cos v) -
19 e - [ f - 3e3S6 + 30e 2S5 + 10e[10,-I)S4 + 2[56,-5]$3 + 48eq 2S
- 618,9)S1 - 6e[14,1]S 0 - 30e 2S_1 - 3e3S_21
+ 4h{e 3 sin 4v + 10e2 sin 3v + 6e(6,-l] sin 2v + 2[24,-7] sin v)]
...... (329)
Multiplying the product of these expressions by a further factor Yq-2 , we get
the required 'first term'; the result, in skeleton form, is
TR 88068
95
1 -;.j- 2 902 51:1 + 5S .-+ 33e + 8fh-3
+ 2{e5(8h 2 + 21f') sin 6v +
+ 4(16(72h2 + 25f 2 ),2(168h2 + 457 ),-(104h2 + 33f sin v}]
For the second term of (327), we multiply out the exansion for e2 viz
(226), by q-2 sin v 42 + e cos ,) The result, in skeleton form is
1 ---: -q-2 [ f g2 {95 + - 9e 5 -4 } -+ 8f{3ehS10 + - 3e hS 6}368642
+ 2o5 (8h + 9f ) sin 8v + ...
- 16[4(216,-648,56,),24(44,-124,77),(648,-312,-907)1 sin v}]
Combination of the full versions of the first two terms of (327) now q-ves the
following complete expressioq:
S - e q- 2 f2{ 9e 51 2 4 144e4 E + ")e3 (154,19]E + 24e 2 (124,61]1Z36864 1 13
+ 2e[2408,3592,39]Z 8 + 8(392,2040,133]E;
+ 2e[7824,2470,-85911 6 + 32[98,267,-314J E
+ 64e[46,-251,-68]- - 32[.8,25,41b]Z 3 - 6e(688, 882, 343Z 2
- 24[24,192,-49!E - 6e[312,184,-151. - 24e 284,-111Z_
6e3[138,-13]E- 2 - 144e 4 . 3 - 9e 5 1}
+ 8f{3ehS 1 0 + 48e4hS9 + e
3 (316b,-(98,-123)'S8
+ 4e2(268h,-(234,-291) ]S7
+" e[1872!h,-4 (670, -807) ,- (526,.-717) ]3
+ 4 336h.-2(130,-93 ,-(782,-1 I %5 ] S5 +
: TIP. 880#
96
+ e(16(266,-391),-2(914,-1787), -(346,-687)3S 4
44
-4e(8(78,-77),-(5074,-5785),-(422,-459flS 2
- l6[36h,2(28,-23),-(518,-625)]S 1
+ 3e[16(26,15),2(2066,2235),3(42,71)]So
- 4 (l44h, 2(766, -969), (2210, -2367)] S1
- e(1296h,1080 (6, -7, (1418, -1503)]S_2
- 2 3- 4. (228h, (494,-561)jS_3 - e (300h, (l90,-213)]S 4
t 48e 4 hS_5 - 3e 5 hS_6 }
+ 2{e~ (8h2 + 9f)( sin By + 16 sin 7v)
+ 2e 3[6(72,-2l6,235),-C56,24,-171)] sin 6v
+ Be 2[24(16,-48,49),-(152,24,-465)j sin 5v
+ 2e[24(120,-360,337),-8(280,-24,-925),-(392,-24,-661))
sin 4v
+ 8[24(24,-72,61),-4(168,-24,-817),-(904,-504,-?211)] sin 3v
+ 2e[16(72,-216,671),-2(4808,-5592,-4277),-(3368,-840,-1215)
sin 2v
+ 32[6(24,-72,61),-(36C,-984,89),-8(47,-39,-40)1 sin v}]
For the third term of (327) we require (to be multipl~ed out by iq-4 el)
the following result, derived using (148) and (182) again:
TR 88063
97
!2
(2 + e cos v)(2ee sin v+ q2v 1
. - .L *1 [f{3e3(1,-31S. + 30e2[l,-3]S6 + *(10C,-283,-511S 5192
+ 4[28,-53,-89]S 4 + 2e(122,-287,-45]S 3 + 64(1,3,-4]S2
+ 10e[6 ,3 5 ,1151 - 4(12,-37,-89)S0 - e(84,-251,-67]S_1
- 30e2[1,-3)S_2 - 3e3 (1,-31S.3)
+ 4h{e 3 11,-3] sin 5v + 10e2 [1,-3] sin 4v + 3e(12,-35,-3] sin 3v
+ 4[12,-31,-191 sin 2v + 2e(18,-85,-31 sin v)] .
...... (330)
Again, the fourth term requires 3eq "' M, (1 + e cos v) 2 to be multiplied by this
same factor, *q-4 el . It is natural to combine the two terms, therefore, the
result of the combination being expressible, with the factor Yq-4 included, as:
- e q-4e1 [f{3e 3S7 + 30e2S6 + e(100, 53, -36]S 5 + 4[28,85,-56]S 4384
+ 2e [290, -119, -66] S3 + 64[1,6,-7]S 2 - 2e[42,17,46)S1
- 4112,53,-81S 0 - e(84,37,-41S- 1 - 30o 2 S.2 - 3e 3S 3}
+ 4h{e 3 sin 5v + l0e 2 sin 4v + 3e(12,5,-4] sin 3v
+ 4[12,23,-16] sin 2v + 2e[90,-49,-6] sin v}
On substituting for e from (148) and multiplying out, we get, in skeleton
form,
- 1_ e 1 q-4 [f 2 'el ... 9e 5 4 } + 8fh{3e5 S1 o + ... - 3eS_6 }36824
+ 2{e 5 (Sh2 + 9f 2 ) sin 8v + ... + 16(12(24h 2 + 7f 2),
2(984h2 + 643f 2),-8(116h2 - 43f2),-(184h2 + 427f 211 sin v)]
TR 88068
VIA.
I
This leaves just the last term of (327). On multiplyin the expansion forM2 , viz (296), by q-3 (1 + a C0 v) 2 , we obtain for this last term, in
j skeleton form,
q-3M2 ( + a coS v) 2
- - e-2q - 4 [3f 2{3e6[2,-1]2 12 + 3e6 2,-1 4ZS864 14
+ 8f{3e6 h(2,-l]S1 0 + ... - 3e 6h[2,-]S_6 }
+ 2{e6(8h2 + f2) [2,-1] sin 8 +
+ 16e(48(16,-48,61),-24(0,64,-121),
6(64,16,-253),-(8,-312,453)] sin v}i . (331)
On combining the full versions of (331) and the preceding expression (that
represents the third and fourth terms of (3271) together), a common factor q2
can be extracted, for cancelling, and the resulting expression, in skeleton form,
is
Seq "2 [fq 2 { 9e6E1 * ... - 9e6 1. } + 8f { 3e 6hSo + ... - 3e6hS 6 I36864
+ 2fe6(8h2 + 9f21 sin 8v +
+ 32e[6(40,-120,183),-(744,-1464,307),-2(44,-60,97)] sin v}]
The grand combination of the full version of this expression, which
represents the last three terms of (327), with the earlier expression (given in
full) for the first two terms, is now possible. A further (common) factor of
8q2 emerges in the combination, and ten terms cancel out completely. The final
result is
TR 88068
99
i v2 ., V02 X
1 + 108 + + 120(84:67]Z + [784,2448,1911E 6
21* + 16e(159,77Z 5 + 4e (482,25]Z4 . 48e(9,-8)Z3 - (144,624,-5)Z2
- 12eE36,19]Z1 - 6e 2 (66,-13;o - 108e 3 E 1 - Se 4 Z 2}
4 4f{604 hS8 + 72e 3hS7 + 2e2[172h,-3(14,-17)]S 6
+ 12e(64h,-(34,-39) 3S5 + [672h,-36(2,3),-(94,-153)]S4
+ 8e(8(21,-31), (130,-139) 1S3 + 2e2 32(77,-92),(230,-243)1S 2
+ 32et3(4,-3),(79,-91)]5 - [288h,16(116,-147),-(266,-303)]S o
- 12e(48h,(110,-129)]S_1 - e2 [52h,(34,-39)]S.2
7e3 S 4 4- 72eh. 3 -6ehS_4
+ 2{e (8h 2 + 9f 2 ) (e sin 6v + 12 sin 5v)
+ 6e 2 2(40,-120,127),-(8,8,-37) ] sin 4v
+ 12e[12(8,-24,23),-(24,24,-137)] sin 3v
+ (48(24,-72,61),-96(4,8,-41),-(312,-168,-515)] sin 2v
+ 8e(6(8,-24,61),-(144,-96,-257). sin V}].
...... (332)
With the formula for v2 to hand, the formula for u2 may be obtained at
once, simply by incozporation of the formula for *2 , 1* (273). The simplifi-
cation is now dramatic, since 20 pairs of terms cancel in the combination, and a
factor 2e2 can be extracteO from the remaining terms. On the debit side, it is
no longer the case that all the Z-terms contain f2 as a factor, whilst the
S-terms contrin f , but this property is restored when we finally proceed to w,
in section 6.4. The result for 112 is given by
TR 88068
100
u .-Lo [{e 2 (32,8,-43)Z6 + 4e(48,0,-53)Z5 + 8(2(l8,-3,-17), (24,-15,-11)]Z2 2304 Z4
+ 12e(48,-32,-191 3 + *2 (288,-264,-ll)z 2}
- 4{2e2(78,-158,81)S4 + 4e(Se,-19,119)S 3
- [8(4,-106,55),-(144,34,-53)]S 2 - 16e(102,-440,359)S 1
- 4a2 (18,-19,l1)SO}
+ 4{e2(120,-60,-97) sin 2v + 8.(156,-276,71) sin v} (333)
The coefficients of Ej, Sj and sin jv , for Ii I 2 , in both (332) and
(333) are coloured by the choice of the 'constant' terms (for e2 and M2) in
(226) and (296). that was made to suit r2 , 8o it is noteworthy (and gratifying)
that (333) contains no terms in ZI, T-1 , - 2, S-1 or S-2 • No tarm in Y0
occurs, because it has been avoided by the choice of to coefficient (for (0)2) in
(273); there is still a term in So , however, because the coefficient, in
(273), was chosen to avoid a term in W2 , not u2 .
6. 4 Fuze short-period perturbations in b and w
We now have expressions for the pure second-order perturbations in r, u, i
and 1 , given (in the fo-m of r2, u2, i2 and 12 respectively) by (320),
(333), (237) and (259). These are the four fundamental quantities of Kozai17 and
it remains, in pursuance of the philosophy of section 3.3, to compress u2, i2
and S12 into formulae for b2 and w2 ; in combination with r2co, b2c o and
w~co as given by (307)-(309), the spherical polar representation of seconid-order
short-period perturbations will then be complete.
Our starting point is (90), which resulted from the identification of two
formulae for (x y Z)T , one in terms of osculating elements and the other in
terms of semi-mean elements. Making use of (87) and (91), we have, for the three
components of (90),
cos(b + Sb) cos(; + 6w) - cos S0 cos u - c sin SQ sin u , (334)
cos(B + 8b) sin(; + 8w) , 6 sin SO cos u + (Cos i - CE(1 - COS 60)1 sin u
...... (335)
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I : :01
and
ain(b +8b) - i - nl.8 cos u + {sin i + ci(l -CO c Z11)] sin u . M056
As these formulae include the long-period carry-over term- of the short-
period perturbations, we have (ignoring J3 I of course, in the definition of the
81 - ki l + K (i 2 + i2 , p )
(and similarly for 5f0); also
fib - ibi + i (b2 + co(3 8Kb+~ 2 b2 c ) (338)
(and similarly for 6w ). We have dealt with carry-over terms separately,
however, so we can interpret (334)-(336) as applying to pure short-period
perturbations, ignoring i 2 ,tp in (337) and b2co in (338). We also replace u
in the three formulae by a + Ru1 + R2 u2
It is now straightforward, in principle, to expand each side of the three
-2formulae to O(K ) and then to identity the expressions that emerge as
coefficients of R . (Identification of the coefficients of R merely leads
again to the formulae (96) and (94) that were used in the generation of bj and
wl , given by (189) and (190).) We naturally make use of the generic second-
order formula
sin( 0 + K91 + K20 2 ) - sin 0. (1 - %6K2O2 ) + CoS 0 (Ke 1 + K202) (339)
and the corresponding cosine formula.
On this basis we derive from (336), since it is now permissible to omit
bars from all the terma,
b a sinui -12 2+ hc sin1ua2+ u1(cosui i + 3 sinu f1 ). (340)
TR i8069
II
11 102
The first two terms of this formula are given by (237) and (259), ant their
combination is given by
sinui - s cosu0 2
- s c(2(1 - f) (e2 sin(5u + 2v) + 6e sin(5u + v) + 3(3,2] sin 5u576
+ 18e sin(4u + (0) + 9e' sin(3u + 2(0))
+ (a2(6,-13)05 + 4e(15,-29)O4 + 6C(21,-41),3(2,-7)]( 3
+ l2e(15,-53)a2 + 54e2 (1,-1)G1)
- (3e2 (32,-41)s3 + 8e(ll,-18)s, - 4(3(9,-11),2(27,-35)31
- 120e(13,-22)s0 - 4e2 (21,-37)s _ )};
...... (341)
the coefficients of a1, s and s-1 here are determined by the choice of
'constants' made in (237) and (259), these choices being now close to
explanation.
The first five terms of (341) are of the form sin(jv + 50)) , but there is
no point in having a special notation for these as they all disappear when the
third term of (340) is combined with (341). This combination reduces to
6sc e2(32,-39)5 + 8e(18,-25)a4 + 6[4(6,-11), (16,-31)]03576543
+ 24e(6,-25)G2 - 9e 2(20,-23)s3 - 56e(2,-3)s2
+ 813(6,-7),4(11,-13)]s1 + 120e(14,-23)s 0 + 32e2 (3,-5)s }
It remains to evaluate tie last term of (340). From (149) ant (151) we get
cosui1 + 3 sinu 1 - 1sc (2ec2 + 3c1) , (342)
and then multiplication by (183) yields the expression required On. combining
this expression with the preveding expression for the first three terms of (340),
we finally derive our result for b2 . As with r2 , given by (320), we express
TR 88068
103
this without use of the bracket conventions, its brevity being almost asremarkable as for r 2 ; thus,
_ - Sc [f{e2a5 + 40e0 4 + 2(48 + 13a )Y3 + 360e02}576
- {e 2 (140 - 177f)s3 + 40e(8 - 9f)s 2 + 8e(120 - 229f)s 0 }]
...... (343)
(In equation (43) of Ref 4, the factor s c was unfor.unately not included.)
This result is free of terms in 61, 31 and s-1 , which would arise if
the right choice had not been made for the coefficient of CO and the 'pure
constant' in (237), and the coefficient of So in (259). In practice, of
course, the 'right' choice was made by setting algebraic unknowns in (237) and
(259); these were carried through the analyss for b2 and then determined such
that the coefficients of 01, s and s- would be zero.
To derive w2 , we must return to (334) and (335), prepared to use either
or both of these equations after elimination of cos(S + 6b) or substitution for-2
its expansion as (1 - AK b 12 ) . Again recalling that the carry-over terms have
already been dealt with, we can express these equations, after the aforesaid
substitution and an expansion of the right-hand side, as
;0s w cosu -K( sin u l1)
+ {sin u (3i 11Q1 - _2) - 4 cos u ( 12- b1 )} (344)
and
sin w sin u + K(6 cosu )
2 2 -22 2+ K cos u 22 - h sin u (i + c a b (345)
It is now convenient to write, as in Refs 2 and 3,
- w - u - O(K), (346)
since (339) and the corresponding cosine formula then immediately give
TR 88068
104
_22Cos w Cos u - a, sin u - (2 s0n U) (347)
and_2
sin w -1sin u + Cos u + K (2 coos u - AV 2 sin u) , (348)
equations that may be identified with (344) and (345) respectively. In both
identifications the terms in K simply lead to (94) in the form
V) - cO , (349)
-2but the coefficients of K lec.d to two alternative expressions for V2 given
by
()2 - 02) sin u - -i 1fisin u - h (b1
2 - j12 + 1)12) cos u (350)
and
W2 - B2) cos u = (bi2 2 12-12+ 1 sin u (351)
In Refs 2 and 3 it was thought best to develop (350) and (351) separately, show-
ing how the same formula for V2 - C2 could be derived from either, but the
most natural procedure is to multiply the equations by sin u and cos u
respectively, and thel add them together. When this is done, we get, with the
terms in b12 (and also U12 ) cancelling as was inevitable,
'2 - c02 - - Y4f{2s(1 - cos 2u)iQ, + sin 2u (i2 - fa12) }, (352)
since it is now legitimate to drop all bars.
The terms on the right-hand side of j352) can be obtained at once from
(149) and (151). Thus, direct expansion of products gives
TR 88068
f105
ill, -L1 - fUl 2 E6+ 6eE5 + 3E3,21Z4 + 18OE + 9e2 E
- 6e2S4 -2eS 3 + 6[3,-2]S 2 + 36eS 1 8e 2S0 - 36e sin v} , (353)
12 1 2+902r2+1 6
- f(1 - f){e 2r6 + 6er5 + 313,2ir 4 + 8er 3 + 9er 2 + 12eC 3 + 36C2
+ 36eC1 + 6.2 cos 2v + 24e cos v + (27,10]} (354)
and
1l 72J1-( f) {e 2 r6 + 6er 5 + 313,2]r + 18eF 3 + 9e2r
1 726 5432
12eC4 - 36eC3 - 24eC 2 + 36eC1 + 36e2C
+ 30e2 cos 2v - 24e cos v - [9,46] } (355)
and the last two of these give
1- f 1 2 - f(1- f) {e 21r6 + 6er 5 + 3[3,21r 4 + 18er 3 + 9e2r236
-6e2C4 - 12eC3 + G[3,-2]C 2 + 36eC1 + 18e
2C0
+ 18e2 cos 2v + 9[1,-2] } (356)
It is now natural to combine the cos 2u component of the first term of (352)
with the whole of the second term, the result being (with the factor -Y4
included)
.._ f(1 - f) (8e 2S4 + l2eS3 - 24e S2 - 36eSI + 24e2 sin 2v + 48e sin v)S1441
after which the full evaluation of (352) yields
ii- {coi 6e5 + 3{3,2]E + 18eZ + 9e2 2 + 2e2 S2 2 144 6 S 4 3 2 4
+ 18[1,-2]S 2 + 182 S0 + 24e2 sin 2v + 12e sin v}. (357)
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106
It now follows from (357) and (259) that
(2 1 f)te2(4,5)Z6 + 24e(l,l)Z5 + 3[2(6,5), (8,3)]E 4 + 24e(3,1)Z3288
+ 3e2 (12,1)Z2 - 2e2 (39,-41)S4 - 16e(1.,-8)S 3
+ 12[(16, -27), -2(3,4)]S 2 + 48e(17,-30)S I + 2e 2 (18,-1)S 0
+ 6e2 (4,13) sin 2v + 48e(3,2) sin v}.
....... (358)
Finally, from (358), (346) and (333) it follows that
w 2 [ [f 2{3e2E6 + 20eZ5 + 16(2 + e 2 )z 4 + 36eZ. - 13e 2 E212304
+ 4f{2e2 (2 - f)S 4 - 4e(46 - 55f)S 3
- (8(23 - 26f) + e 2(14 - 39f)]S 2 + 16e(158 -179f)S 1 }
- 4{e2(72-168f+59f 2 ) sin 2v + 8e(l1'O-264f+95f2) sin j
...... (359)
As with (320) and (343), this has been deliberately written without the use of
the bracket ionventions. (In Ref 4 the wrong sign was atcached to all the 4f
terms.)
7 rIRS-ORDER ANALYSIS FOR J3 PERTURBATIONS
7. 1 Eact eoquations for rates of change of elements
The first-order analysis fcr J3 is very similar to that for J2
(section 4), the main differences being that (as for all the odd harmonics) there
are no secular terms in the solution and that (as for all Je with Z > 2)
there are long-period perturbations induced by J2 • Our starting point is the
set of exact equations for the variation of the osculating elements, given by
substituting (49) into the planetary equations, (22)-(27). Wcrking with (33) and
(34), rather than (26) and (27,1, to keep the analysis (inevitably more laborious
than for J2 ) as simple as possible, we get
TR 38068
107
ik !n. ) {5f(7ey +M6a -V + 129(5c 2 + 2cI - 3ecO) , (360)j " 12q
i .t AIL .@4 5f(Vey,+ 20, + lY 3 + 4
48q
+ 12g(5ec 3 + 12c 2 + 6ec 1 - 4co - 3ec-.) (361)
D (Sf7 + 4gc1 ) , (362)
4q 3
. .n ( 4) {5 a, + (4- 15f)s}, (363)4q 3 s
Hns () 4 {5f(7eo 5 + 20C4 + 8e 3 - 402 + eel)48eq3
+ 12g(5ea3 + 1232 +8es1 + 4s0 + 3es ) (364)
and
3 2" (5f63 + 12gs1 ) (365)•3q2
Integration of these equations is quite straightforward, on changing the
indegendent vaTiable from t to v (or rather C ) as in section 4. Fxom the
integration of each t we obtain, in the notation of (72) and (73), the required
expressions for 3,tp and 3 . Results for (0, 0, J , and hence M , are
available at once from those for a, ., y and p , and the complete set of 3
and C3,1P are developed in sctions 7.2 to 7.5. Formulae fcr r3, b3 and w3are obtained in section 7.7, based on the expressions derived for v3 and 113
in section 7.6. In Part 3 we make use of the ,3.gp in our study of the long-
term variation of the mean orbital elements, as parturbed by J2 and J3 , with
singularity very much to tha fore.
7.2 Perturbation in a
As in *ections 4.1 and 5.1, the best way to evaluats a3 is by the special
method that appeals to the exact energy constant a' . On this basis, (52)
leads, for the potential given by (49) alone, to
TR 81068
lb0ala' - 1 + Hs(1 + e cos v)4 q 2 (5f03 + 12gs) (366)
63
Thus
a3 - -aq2s (P/r)4 (5fa3 + 12gsl) (367)
by combining successive powers of p/r ( I 1 + e cos v ) with 03 and
s1 , we can, as in section 4.1, write down progressively more complicated
alternative expressions for a3 . These expressions, all of them necessarily
symmetric in both their 0 components and their s components, are
a laq42s (P/ r) 3{5f (ea4 + 203 + e02 ) + 12g (es + 23, + es0) },(368)
3 1.-2 cpr(feo 4 46203
a3 2a4 (p/r)25fle2a 4 4ea + 2(2,1103 + 4e62 + e 20 1 )
+ 12g(e 2 s3 + 4es2 + 212,1)s + 4es0 + e 2s-) } (369)
a3 -". aq2s(p/r){5f(e 3o 6 + 6e 2o 5 + 3e(4,110 4 + 4[2,303 + 3e[4,110 2
+ 6e2o1 L 300)
+ 12g(e 3s4 + 6e2s3 + 3e[4,11s 2 + 4[2,311 + 3e[4,1]s 0
+ 6e2s_ + e3s 2}
...... ( 370)
and
a3 - 9 q62a5f e 7 + 8e 3 + 4e216,110 + 8e[4,310 4 + 218,24,3103
+ 8e(4,3]O 2 + 4e 216,1101 + 8e300 + e 4OC_
+ 12g(e 4s5 + 8e3s4 + 4e2 16,1]s3 + 8e[4,3s 2 + 2(8,24,3]si
+ 8e[4,3)s0 + 4e2 16,11s 1 + 8e 3 s_2 + e4s ) }
...... (371)
TR 88068
109
We shall use (369) for the development of M3 via 13 , (370) for the develop-ment of r3 and (371) for the development of P3
The final expression, (371), can also of course be developed from (360) by
the general method now to be used for e3, i 3, etc.
7.3 Pertubation* in e, i, p and pe2
From (361), with the independent variable changed to v by use of (38), we
get
dev .192 Hs{5f(7e3Y7 + 48e 276 + 4e[27,8175 + 16[5,9174 + 42e[4,l]73
+ 1611,6172 + 4e[3,4lly - e 37 )
+ 12g(5e 3c5 + 32e2c4 + 4e[17,4c 3 + 1613,4]c 2 + 14e[4,I]cI
- 16q 2c0 - 28ec 1 - 16e2 c-2 - 3e3c3 ) }.
...... (372)
The term in co leads to the long-period variation specified by
e 3, - - q2 gs cos (0 , 373)
whilst the rest of (372) yields, on integration,
3 - .19- s3 f (5e3a7 + 40e 2G6 + 4e[27,8)o5 + 20(5,914 + 70e[4,1]0 3
292
+ 40[1,6102 + 20e[3,4]a1 + 40e 2a 0 + 5e3G 1 )
+ 4g(3e3s5 + 24e2s4 + 4e[17,4]s 3 + 24(3,4)s2 + 42e[4,1 1
+ 24 (1,6]s 0 + 84es_1 + 24e 2s_2 + 3e 3s. 3 ) };
...... (374)
the coefficients of 0 and s0 in (374) have been chosen to make r3 , given
by (408), as simple as possible.
TR 880ES
I ... . . ' .. _ _ _ _ _
110
From (362), in the sams way, we' get
,. - .L R~l. .- , + 2 12 ,1JT3 + 4e, + e,)
+ 4g{e 2c 3 + 4ec + 2[2,1c. + 4ec 0 + e 2 c 1 )}. (375)
The term in co leads to
3,tp ego cos (0 , (376)
whilst the rest of (375) yields
i "- c f(3e2 s5 + 15e4 + 10(2,1303 + 30ea2 + 15e2013 48 5 43 2
+ 4g(e 2s 3 + 6es2 + 612,1]s - 3e2s 1) - 6e(4,-15)s0}. (377)
The coefficient of 3O in (377) has been chosen to make b3 , given by (410), as
simple as possible; the fact that it is a multiple of 4 - 15f , rather than
4 - 5f as for the other s coefficients, is bound up with the fact that in
a3 ,to be given by (385), the coefficient of co is a multiple of 4 - 5f
whilst the other c coefficients are multiples of 4 - 15f . (See also Part 2.)
From (373), using (28), we get
P3, "2pegs cos (0 , (378)
and from (371) and (374) we get
p3 .L psf3e2a_, + 15e 4 + 10(2,1]3 + 30eG 2 + 15e
201).P3 2413
+ 4g (e 2a 3 + 6es2 + 6[2,1]s1 + 18eS0 0 3e2s- ) }" (379)
TR 88068
, !
ff
We now have our usual check in terms of the quantity pLo that is an absolute i
constant oftemte.Zuto8(376) and (378) yield !
(PC 2)3p - 0 ,(380)
whilat (377) and (379) yield
(pc 2 ) 3 - Ypesc 2 (8 - 15f)a0 : (381)
(381) is 'constant' (to order zero), which completes the check, non-zero because
the constants in 03 and i are not chosen by concomitant criteria.
Using (28) in reverse, we can get an expression for e3 that is somewhat
more compact than (374) but, like (173), is unfortunate in carrying the factor
e ; the expression, based on (367) and (379), is
63 1 e-1s{4(p/rd(5f0 3 + 12gs 1 ) - q2 f(3e 20 5 + 15e04 + 1012,1]0 ,
+ 30ee 2 + 15e 26 1 )
- 4q2g(e 2s3 + 6es 2 + 6[2,1)s 1 + 18es 0 - 3e2s) }- •
...... (382)
7.4 Vertuxaatiene in 0, If and 0)
From (363) we get, after the change of integration variable,
j& - -L Hc " {5f (e 2oS + 4e, + 2(2,1103 + 4W02 + 2 01 )dv 163
+ (4 - 15f) (e233 + 4es 2 + 2[2,11I + 4es 0 + o 2 83 1 ) }. (383)
I $8068
_ _ _ _ _ _
171
112
The term in so lead* to
I iD - Ve(4 - 5lif)c 1 sin a (384)3,4 p
whilst the rest of (383) yields
1 -. -- c - {f(3e1 + 15e7 + 1o2,l]y + 30e + -Se 2y)3 48 +. 1S4 + '3 + 2 1
+ (4 - 15f) (a2c3 + 6ec2 + 612,11c 1 - 3e2c1) - 24egco}
....... (385)
The coefficient of cc is chosen, in conjunction with the coefficient of so in
(377), to make b3 as simple as possible; we have already remarked on the
effective swapptng of the coefficients 4 - 5f and 4 - 15f , as between (377)
and (385), and the full explanation for this (in the context of the general it )is reserved for Part 2.
From (364) we get, after the change of integration variable,
d . Hes{5f (7 30 48. 2G6 + 2e154,1110 5 + 1615,6]a4 + 24e(4,1]Gdv 192 7 63
- 16[1,-3)a2 - 2e16,-510, + e 3 0_i)
+ 12g(5e 3s5 + 32e 2s4 + 2e134,91s 3 + 1613,§]s2 + 24e84,1]s1
+ 16[1,41s + 14e12,1)s 1 + 16es2 + 3e3s-3 }.
...... (386)
The term in so leads to
V3, p qs (1 + 4e 2 1 sin O0 , (387)
whilst the rest of (386) yields
j , Tk 88068 "- " I
%i y J _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _
r
I113
IKV3 a . e. {f(S.i 407.f+ 2,(54.111,,+ 20(5,.61 + 400[4,11
- 40[11-3172 - 100(6,-Sly - 40 .2 _ 5-
3 ¥ ' 1 )
+ 4g(3e3 c 5 + 24e*c4 + 2e*34,9]c 3 + 243,51c 2 + 72e(4,1]c 1
-24c 0 - 42e[2,11c. - 24e 2c 2 - 3e3c 3) };
....... (388)
'he coefficients of 7o and co in (388) have been chosen to make w3 , given
by (413), as simple as possible.
Expressions for 003,gp and 103 are now irmdiate, by subtraction of
cD3 ,tp from V3,tp and of c0 3 from V3 . These expressions, which are
necessarily moee complicated than for 13,4p and V3 because differing
combinations of f are involved, are
0)3,tp e-1 a- [4fq - e2(4 - 35f + 35f1 ] sin ( (389)
and
03 -131 f192 " (e3fy + 40e2f16 + 2e[54f,-(6,-17)ly + 20[5f,-3(1,-3)1y
1275 4
- 40e(2(1,-3), (11,-2)17 3- 40(f,3(1,-2)f 7 2
- 10(e6f,(6,-11)]y - 40e 2fo - 5e3fy1)
+ 2 (6e3fgc5 + 48e 2fgc 4 + e(136fq,-(8,-74,75)1c 3
+ 12(12fg,-(4,-39,40)]c 2 - 12e(2(4,-43,45), (4,-31,30)]c
- 48gtf,-(l,-l)]c 0 " 3et56fg,-(8,-66,65)]c I - 48e 2fqc 2
II
. . . . . . . . . . . . . . . . . . .....(390)
'I rTR UMog
._ _......._
i-*.~.114
7. S Pex ltuzatofsIS P, , Ila" L
Fron (365) we get, after the change of variable,
-.. Hqs { 5f 2 + 4e. + 212,1103 + 4e 2 + e2)dv 6 5 4
+ 12g(e 2 ., + 4es2 + 2[2, 1]s + + e 2 s 1 ) } (391)
The term in so leads to
=P " 8eqsg sin (, (392)
whilst the rest of (391) yields
-{qs{Sf(3e2Y + 15e74 + 10(2,11y3 + 30ey 2 + ISe2 )P3 90 5 3 0es
i + 12g(5e 2c3 + 30ec2 + 3012,lc, + 12ec - 15e 2c 1 ) }; (393)
the coefficients of yo and co (the former being zero) have been chosen to
produce the appropriate coefficients (of 7 and co ) in M3 given by (400).
Expressions for 03,tp and G3 are obtained, in view of (32), by
subtraction of qC3,tp from P3,,p and qV3 from P3 , the results being
a3 p - 1 qsg (1 - 4e2) sin O (394)
and
3 p1 q1 3 5(1e + 120e02Y2800 qs 5f (15e7 6 + 6e(54,-5]T 5 + 60(5,-2]y4
2880
- 408[4,51]3 - 120[1,51y 2 - 30e16,llly - 120e2y0 -15ey_
+ 12g(1563C5 + 12082c4 + i0e134,-7]c 3 + 360q2c2
- 120e[4,5]c 1 - 24[5,16]c 0 - 30e[14,-9]c_ - 120e 2c2
- 15e c3) }.
...... (395)
TR 88068
-- ___ iz~-1
T jJ
19 +115
correspeia~g to (159), together wLth (369), to derive
n 1..6 , .(2 + 4.6, 2 (2,1)03 4062 q { ,e2U+ )
~~+ 12g(e2s3 + 4es2 + 2(2,11s1 + 4e6o + e2~) } . (396)
In view of (40), we now get
-3 - 3eqsg sin 0 37
and (on integrating the remaining terms)
13 24 S(3e2¥5 + l5ey4 + 10(2,1173 + 30e72 + 15e271)240o57 )3 1
12g(52c3 + 30oc + 30[2,1]c1 + 12ec o - 15ec ) j (398)
These results could have been obtained directly from (392) and (393), since
13,8p " P3,P and 13 " " f P3 from the general formula (quoted in
section 4.2) with I set to 3; the coefficient of co in (398) has been taken
to conform with this relation between P3 and 13 . (Taking different constants
in 13 and -A P3 would provide greater flexibility than we need; this was
tacitly assumed also for 12 and -+2p2 in section 4.)
The formulae for a and I can now be combined, the results o-C the
combination being that
Mp - . e-lq38g sin 0 (399)
and
M LI q4f(5 2* o(4 2N3 " - - e "qsjf15.,v 7 + 12o, y + 6e(54,1)75 + 60(5,1'4 + 80eq 3
- 12011,2]y - 30e(6,51y - 120e2yo - 15e3¥.
+ 12g(3e3 c5 + 24e 2c + 2eC34,-1Jc3 + 72c2 + 48eq2c.
- 24[1,21c 0 . 6e14,-3 1 cl- 24e 2c 2 3 3 c 3 ) }. (400)
TI-2 3)
TR :.
116~I
The perturbation in L need WAC detain us long. Thus, the formulae forL3.4P and L3 follow at once ftcm (.hose for P3,8P ad P3 (or IMP ad 13
since (from the general formula of section 4.2, with f set to 3)
L3,p A3,S31, .-1 p 3eqsg sin e (401)
and
L - 8 3 -- 1.(402)
there being little point in quoting L3 in full.
7. 6 Short-period perturbations in V and u
It is recalled, from (74), that the (first-order) J3 contributioa to 81is given by i (;3 + C3,ap r). Also, 8v and Ju may be expressed in terms of
8 by means of (42). Thus, from 03, () and M3 we may derive expressions for
v3 and U3 , whilst from e 3,tp, 03.1p and M3.4p w, derive expressions for
the quantities that, with an obvious notation referring to carry-over effects (cf
section 6.1), we may denote by v3co and U3m . We have no need for a quantity
v3,1p , such that V3co - v 3 ,tp M , because the long-period perturbations are
applied (in a non-singular manner) to the orbital elements, and not to derived
quantities.
From (42) it follows that 8v can be developed as the sum of three
components:
(A) 2q-2 (e sin v + eq 1 82M cos v);
(B) heq"2 (5e sin 2v + eq"1 5M [cos 2v + 3)); and
(C) q41 8M .
We darive the three components separately for both the 3 aad the 3,tp , and
then combine.
In combining the appropriate multiples of e3 sin v and M3 C03 v to form
Component A, eight pairs of) terms cancel, the result being
. q-2e [f{15e3 6 + 6002y5 + e(28,47)y - 90q 2y3 - 5e[28,-13]7 2
60e 2 , - 15e3Yo) +
TR 86068
4 i
] 117
+ 12g{303 c 4 + 120203 - *4,-19]c2 - 24q2c1 - 3[4,13c 0
-12.c- 1 -•3 c 2
Rather more terms cancel, in a similar manner, in forming Cocpcnent B, the result
being
,. q-2 a [f{3Oefty6 + 4e(37,-7]y + 15[13,-31y + 96*q27 3 - 15[1,91-,2283
- 20e[1,517 1 - 30e 2Yo}
+ 24g{302c4 + 6e[3,-11c 3 + 3[8,-3)c2 + 160q2cI - 15e2Co
- 6e(3,-1]c I - 3e2c2 }]
Four pairs of terms cancel in the combination of Components A and B, and a factor
q2 emrges to cancel the q-2 , the result being
a [f{2 e2y + 1390Y4 + 12(15,8173 + 265eY2 + 100271 }
+ 48g{302c 3 + 14cc2 + 4(3,21c 1 + 6c 0 - 3e2c.1 ]
Component C is given at once by (400), of course. It then follows, from the
overall combination, that
-L 1 s [f{153y, + 120e 2y 6 + 2e(16 2 ,31]¥5 + 2(150,16914
+ 8e[55,14]¥3 - 10(12,-2912 - 10e10(,-57, - 120o270
- 15,ety }
+ 12g(303 c5 + 24e 2 c 4 + 2e(34,11]c 3 + d(9,14]c 2 + 16019,11c 1
-24c 0 - 60[14, 1)c1 - 24e 2 c2 - 3e 3 c_3 ) ]...... (403)
Tt 80
I I
From 03.8p and H3,p , it follows in the am way that Component A of the
long-period carry-over is -2sgial , Component S is - 3 esg 1(82 + 3so) , andComponent C is -e 1 q2 s 9 a so . Putting these together, we get
Vc - - 3esgm(e 2s2 + 4es + 12,1]s) (404)
For Su , we simply feed in the effects of 03 and (3,4p . Thus
u -l- s " {f (2e2(9,-lo); + e(90,-101); + 4[5(6,-7), (15,-16)]3288
+ 5e(36,-43)y2 + 10e2 (9,-14)yI)
+ 6 (e2 (4,-15,10)c3 + 2e(12,-61,50)c 2
+ 2[6(4,-31,30),(12,-85,80)3c1 - 6e(4,-9,5)c0
- 3e 2 (4,-31,30)c_1) }
...... (405)
and
u - 4efgs 2 + 16fgs + a(8,-66,65)s 0 (406)
7.7 hot-period perturbations in r, b and w
We can now complete the J3 analysis formally, by derivation of the
quantities required in the spherical-coordinate representation, viz r3, b3 and
w3 (pure short-period terms), together with r3co, b3c o and w3co (long-period
carry-over terms).
From (95) it follows that
r 3 - (r/a) a3 - (a cos v) e 3 + (aeq "I sin v) M3 . (407)
Eight (pairs of) terms cancel in the combination of the terms in e3 and M3
the combination being
TR 88068-I .o
119
144-a a If (15e + 90e + e(176,49164 + 15(7,13)63 + Se[32,13162
+ 90e2 1 + Ise 300
+ 12g(3e3s4 + 18e2s3 + e(32,131s2 + 1212,3]s! + 3e[16,-1]s
+ l8e23_1 + 3e33_ 2 ) }
On combining, further, with (r/a)a 3 , given by (370), nine more terms disappear,
with the terms in S1, s-, 01 and 0.1 vanishing as a result of the choice of
constants in (374) and (400), and a factor q2 can be extracted from the rest.
The result is
r 3 - 1Lp ps f (4ea4 + 1503 + 20e0 2 ) + 48eg(s 2 - 3 0) } (408)144 (2III From e3,Ip and M3,1p , given by (373) and (39M), we have, similarly,
r3co - psgm COS U . (409)
Next we require b3 , and it follows from (96) that
b3 - (sin u) i3 - (s cos U) 03 (410)
Combination of the f-terms in (377) and (385) is immediately possible, giving
A- c f (3e 2C4 + 156 3 + 10(2,11C 2 + 30eC1 + 15e 2Co )48 4°
The combination of the g-terms in (377) with the (4 - 15f) term in (385) is
• only a little more involved, and leads to two sets of terms, viz
1 c (2 - 5f) (e2 cos 2v - 3e cos v - 3(2,1])12
and
TR 66068
t! 2
-To'cf (e2 C4 + 6eC + 6(2,11C 2 - 3e2C0)
Lastly, the chosen so term in (377) combines with the chosen co term in (385)
to give
_ ce [5fC, + 2(2 - 5f) cos v]
precisely the combination required to cancel the terms in C1 and cos v
already generated. Combination of the four sets of terms yields
b - If (2e 2 C + 15eC + 20 (2 + e 2]C - 30e 2C )
3 48 4 3 2 0
+ 4(2 - 5f) (e 2 cos 2v - 3(2 + e2]) } (411)
From i3AP and 13 .1p , given by (376) and (334), we have, similarly,
b -3co e c m ( 5fS1 + 2(2 - 5f) sin vJ . (412)
Finally, we require w3 and W3co . Adding u3 , given by (405), to
c03 , with 03 given by (385), we have
w3 " s{f(2e 2y5 + i + 4 (5 + e2 y3 + 35ey2 + sojy,)
- 24g (e2 c3 - 2ec2 - 2(18 + 7e2 ]c1 + 9e2c 1 }. (413)
Equation (413) may also be obtained by direct addition of (403) and (388),representing v3 and 1#3 respectively. Yet another derivation is possible
without using either u3 or V3 ; we return tu the sum of the 'A' and 'B'
components of v3 derived in section 7.6; if to this sum we add q-1 L3 , as
expressed by (402) via P3 or 13 , instead of q-1 M3 , then we have w3 atonce. From U3c o and ci13 ,P , or from V3co and V3,Ap , or most directly from
the 'A' and 'B' components of v3c o together with q-1 L3,tp , we obtain,
similarly,
TR 88068
121
W3c - - s 2(es + 4s1 - 7e O 0 (414)
It will be observed, from (408), (411) and (413), that r3, b3 and w3
contain five, six and nine terms respectively. Similarly, from (188)-(190), rl,
bl and wl contain two, two and three terms respectively. As an introductie,n
to Part 2, we now give the number of terms associated with the general J
noting that the quantities generically denoted by C in the present Report willIhave to be relabelled C2 in Part 2, where the analysis is taken to 'first
order' only, but covers every 4 . Odd and even values of Z have to be dealt
with separately.
For odd * , the number of terms is t(9 - 3/2) + h for r,
( 3/-2) 4 3/2 for bg , and f2 for wt . For even t , the number is
- h) + 1 for both rg and bt , and t2 - 1 for wl . That the last
figure is t2 - 1 , rather than t , is somewhat fortuitous, being due to the
fact that (as we 3hall see in Part 2) the coefficient of S+ 1 , ie of
sin(( + 1)v + 2o), which arises only when f is even, always vanishes. Thus
the absence of an S3 term in (190), which seemed strange when the analysis was
originally performed, was not a fluke occurring just for J2 •
a CONCLU3SIONS
The main function of this Report, Part 1 of a projected trilogy, has been
to provide full details of the untruncated second-order orbital theory, involving
J2 and J3 only, of which the outline was given in Ref 4. The principal
novelty of this theory lies in the reduction of second-order J2-induced pertur-
bations to very compact expressions in a special system of spherical-polar co-
ordinates based on a mean orbital plane. First-order J3-induced perturbations,
regarded as second-order in the overall theory, have been expressed in the same
way, the results being particular cases of formulae (for the general Jj ) that
will be developed in Part 2. (Postscript: now available as TR 89022.)
The special coordinate system cannot be used in the treatment of secular
and long-period perturbations, but this is of no consequence as so few terms are
present in the basic expressions for perturbations in the standard alliptic
elements. The J3 perturbations in these elements suffer from singularities,however, and the Report has included introductory material on the treatment of
these singularities. Additional formulae relating to the long-term evolution of
an orbit, subject to J2 and J3 , will appear in Part 3 of the trilogy, which
will also include numerical results from an assessment cf the ovexall accuracy of
TR 88068
1#'
122
the second-order model for motion under J2 and J3 Over long periods of
time, such that the total angle described by the satellite (in true or moan
anomaly) approaches J2- 1 in order of magnitude, the errors in the model
inevitably increase from third order to second order, Is typically (for close-
Earth satollites) from centimetres to decametres.
The trilogy of Reports is to be viewed as part of a continuing study. The
current limitations of the theory (for Earth satellites) may be summarized as
follows: only perturbations due to the geopotential have been considered (though
lunisolar perturbations were addressed in a precursor5 to the present theory);
tesseral harmonics have not been covered (though the effects of J2,2 on a low- I
eccentricity orbit were considered in Ref 5); the zonal harmonics have been
restricted to J2 and J3 , except in Part 2 of the trilogy; the analysis has
been taken to second order only, implying only first order in J3 ; and the
build-up of long-term error, as indicated in the last paragraph, cannot be
avoided. In spite of these limitations, however, the present Report serves as a
significant step towards the goal of an analytic (or semi-analytic) orbit
generator that is much more efficient than numerical generators of comparable
accuracy.
To facilitate a reference back to the formulae that summarize the formulae
developed for the generator covering J2 and J3 only, we conclude the Report
by recording the relevant equation numbers. The modified version of Kepler's
third law is given by (299). Secular variation rates are given by (150) and
(297) for 0 , and (152) and (298) for 0 . Long-period variation rates are
given by (300)-.(306) for J22 effects; for J3 effects, the equations are (373),
(376), (384), (387), (389), (392), (394), (39'1), (399) and (401). The carry-over
effects on coordinates are given by (307)-(309) for J22 , and (409), (412) and
(414) for J3 . Last, but far from least, the short-period effects on coordi-
nates are given for J2 (first order) by (188)-(190); for J22 , by (320), (343)
and (359); and for J3 , by (408), (411) and (413).
TR 88068
......_____________________ _________ ________________
!123
List or SUMLOS
a semi-major axis (osculating)
a' energy-based fixed-mean semi-major axin
b latitude-like coordinate of (r, b, w) system
V c cos i (also: obsolete cylindrical coordinate)
cJ cos(jV + (0)
IcJ cos(Jv + 2w)
Ck cos k& dc0
dn' defined such that n' + dn' constitutes tsec
D used in context of J22 contributions to 4/d
e eccentricity
E eccentric anomaly
f sin 2i
F Merson's functions, defined in equations (122)-(124)
4
h I - f2
H iJ (R/p)32 3
i orbital inclination
j arbitrary integer (usually associated with v)
it geopotential (zonal harmonic) coefficient of degree Z
k arbitrary positive integer (usually associated with (o)
K J2 (R/p) 2
2
£positive integer (suffix for J), usually 2 or 3
L non-singular quantity such that L - M + q4
m v - M (also: suffix, in JIm , in section 1)
M mean anomaly
n mean motion
n' energy-based fixed-mean mean motion iN modified mean motion (nL in Ref 4)
TR 88068
124
LZS! Or $MWLS (eontiAued)
p semi-latus rectum ('parameter' of ellipse)
Pt Legendre-polyncmial function of degree t
q 4(1- e2)
Q; used as factor of d/di in J22 analysis
r geocentric radius vector, coordinate of (r, b, w) system
r' obsolete cylindrical coordinate
R equatorial radius of the Earth (6378.14 kta)
Rj rotation matrix for jth axis (j - 1, 2, 3)
Rj derived rotation matrix
s sin i (of c; and sj, Sj, Sk by analogy)
t time (measured from epoch)
T affix for matrix transposition
u argument of latitude (v + 0)
u' obsolete cylindrical coordinate
U M + 0
U' quantity such that U' - M + qC6
U disturbing function U2 and U3 as particular Uj)
v true anomaly
w longitude-like coordinate of (r, b, w) system
w +S
W q-3 (p/r) 2
x, y, z usual (equator based) geocentric coordinates
X, Y, Z geocentric coordinates based on the mean orbital plane
geocentric latitude (ie declination)
YJ ~Cos (JO + 3ro)
ri cos(JQ + 4o)
total short-period perturbation (in )
TR 88068
-1 -~- - -~~-- -
125
LIST OF -313U0L (continued)
6, pure (Poisson) short-period perturbation
perturbation relative to
• A long-period perturbation (from epoch)
VC te t (used when C is 0)
C generic orbital element (osculating)
' Cmean C (notation used in i etc also)
- AC (when t is o, in particular)
C semi-irean C
tp long-period component of
Cseo secular component of
tsec/ii
C0 value of t at epoch (likewise t, more usually required)-2 -I
Ci cofactor of K, K, H (j- 1, 2, 3) in 8A
jjp cofactor (likewise) in tp/H
tj cofactor (likewise) in
C see
e sin 0)
1) see
0 arbitrary angular quantity
IEarth's gravitational constant (398 600.4 km3/s2)-2
0i, 02 cofactors of K and K in Kepler's third law (modified)
e cos 0
€, 1, C sin i sin 11, - sin i cos , cos i
p quantity such that - d + q*
a quantity such that M sie +
S j, Ej as tj, i , but sines
T %time (dummy variable for t in integrals)
) w- u
TR qS06%
126
SLIST Or STSOMLS (conalUded)
* quantity such that e - sin *quantity such that 6 - & + ce
i argument of perigee
see
!*
right ascension of the ascending node
0bracket convention for polynomial in f
I bracket convention for polynomial in e2
S806
& 1
4t
I -
127 --
No. Author isAte *t
1 R.H. Gooding Satellite motion in an axi-symmetric field with an'
application to luni-solar perturbations.
RAE Technical Report 66018 (1966)I2 R.R. Gooding Second-ordor perturbations due to J2 , for a low-
eccentricity earth-satellite orbit.
RAE Technical Report 79100 (1979)
3 R.H. Gooding A second-order satellite orbit theory, with compact
results in cylindrical coordinates.
Phil. Trans. Roy. Soc., A, 299, 425-474 (1981)
4 R.H. Gooding Complete second-order satellite perturbations due
to J2 and J3 , compactly expressed in spherical-
polar coordinates.
Acta Astronautica, 10, 309-317 (1983)
RAE Technical Report 83033 (1983)
S R.H. Gooding On the generation of satellite position (and
velocity) by a mixed analytical-numerical
procedure.
Adv. Space Res., 1, 83-93 (1981)
RAE Technical Memorandum Space 311 (1982)
6 R.H. Gooding On mean elements for satellite orbits pert-irbed by
the zonal harmonics of the geopotential.
Adv. Space Res., 10, 279-283 (1990)
RAE Technical Memorandum Space 37.1 (1989)
7 R.H. Gooding Universal procedures for conversion of orbital
elements to and from position and velocity
(unperturbed orbits).
RAE Technical Report 87043 (1987)
8 T.E. Sterne An introduction to celestial mechanics.
London, Interscience Publishers Ltd (1960)
9 A.S. Roy Orbital motion.
Bristol, Adam Hilger Ltd (1978)
2
128
aiuJsmS (ee~Uem)
e. et~h.: 5Vtitle, eta
10 B.C tlmr An Iatzed-ory traetie an dynamica" astronomy.Ne York, Dower Publications (1960)
(ifum CUP (191)1
11 w.N. isrt Ceoloaial mechanics.
Loodon, etO, Lo4mansa, Green & Co (1953)
12 D. momwe: Ithods of oelestLal mechanics.
a.m. Clenme0e Now York and London, Academic Press (1961)
13 D. a-ouver Solution of the problem of artificial satellite
theory without drag.
hAtron. J., 64, 378-397 (1959)
14 R.N. Nerson The dynamic model of PROP, a computer program for
the refinement of the orbital parameters of an
earth satellite.
RAN Technical Report 66255 (1966)
15 R.H. Gooding A PROP3 users' manual.
R.J. Taylor PAZ Technical Report 68299 (1968)
16 R.N. Gooding The evolution of the PROP6 orbit determination
program, and related topics.
RAN Technical Report 74164 (1974)
17 Y. Kosai The motion of a close earth satellite.
Astron. J., 64, 367-377 (1959)
1s R.N. Gooding A monograph on Kepler's equation.
A.W. Odell RAN Technical Report 85080 (1985)
19 A.W. Odell Procedures for solving Kepler's equation.
R.R. Gooding Celestial Mach., 3S, 307-334 (1986)
20 K. Stumpff Weue Formln und Kilfstafoln zur
4 phemeridenrechnung.Astron. Nachichten, 275, 108-128 (1947)
21 R.A. Broucke On the equinoctial orbit elements.
P.J. Cefola Celestial itch., 3, 303-310 (1972)
TR 8806R
_____
+,-.. _-___
129
aavaauuc& (concluded)so. Author tlte, eto
22 R. . Gooding An analytical generator for drag-f moe satellite
orbits.
an ftcbnical Report 30015 (1960)
JVA
4
