Resonant Orbits of Artificial Satellites and Longitude
Terms in the Earth’s External Gravitational
Potential*
A. H. Cook
(with an Appendix by H. J. Norton)
Communication from the National Physical Laboratory
“Every arrow that flies feels the attraction of the Earth.”-Longfellow,
Hiawatha.
Summary
The zonal harmonics in the Earth‘s external gravitational potential
give rise to long period or secular perturbations of the orbits of close
artificial satellites and can be estimated from such perturbations but the
tesseral and sectorial harmonics in general produce onIy short period
perturbations which are far more difficult to observe. There is thus
little prospect at present of finding the longitude-dependent parts of
the potential from the motion of an arbitrary satellite and so it is of
interest to see if in special circumstances long period or secular perturbations could arise from these parts. This paper contains a preliminary study of orbits with periods bearing some specific relation
to the period of the Earth’s rotation. For the purposes of this study,
it has been assumed that the eccentricity of the orbit and its inclination
to the equator are so small that only the first power of the eccentricity
and the square of the inclination need be retained and it is also supposed
that, independently of the longitude terms in the potential, the longitudes of the node and perigee change linearly with time.,
It is found that secular and long period perturbations can arise
and that they differ according to the parity of the difference @-q)
for the associated Legendre function P$(cos 0) in the spherical harmonic expansion of the potential. One approximate condition for
such perturbations is that
(Q-
=
1).
QWf
where n is the mean motion of the satellite and wf the Earth’s spin
angular velocity. The condition is slightly different for odd and
even (p - 4).
When @-q) is even, there are perturbations of order e-1 ( e is the
eccentricity of the orbit) in the mean anomaly and the longitude of
perigee while when @-q) is odd there are perturbations of order
* A preliminary version of this paper was presented to the 1st International Space Science
Symposium (Cook 1960).
53
54
A. H. Cook
(sin i)-1 (i is the inclination of the orbit to the equator) in the longitude
of the node and of perigee.
The simple case of the ellipticity of the equator (P22 (cos 0)) is
discussed in rather more detail and some preliminary consideration is
given to the problems of the realization of resonant orbits.
The general expressions for the perturbations, excluding terms
proportional to e2 and sin%or of higher order, are given in the Appendix
where the complete set of resonance conditions is listed. The resonances
discussed in the body of the paper are generally the most important in
that they occur for the smallest orbits but there are resonances with
smaller orbits in some special cases.
Introduction
Although the zonal harmonics in the Earth’s external gravitational potential
can be found from observations of the secular and long period perturbations of
the orbits of close artificial satellites, the parts of the potential that vary with longitude cannot be so found because in general, as a result of the rotation of the
Earth, they give rise only to short period perturbations and not to long period or
secular ones. Short period perturbations, having periods equal to those of the
rotation of the satellite in its orbit or of the Earth about its axis, or combinations
thereof, are difficult to observe, and cannot be found from observations at a single
station. It is therefore of interest to see if long period or secular perturbations can
arise in special circumstances and one naturally thinks of the possibility of resonant
orbits where the period is in some definite relation to the period of the Earth‘s
rotation about its axis. In this paper 1 attempt a preliminary study of such orbits;
I consider only the theoretical problem of the origin of secular or long period terms
and do not discuss the technical problems of putting satellites into such orbits.
Resonant orbits have also been discussed by Groves (1960).
I.
Notation and method
I suppose that the orbit can be defined in terms of osculating elements at the
ascending node and take these to be :
2.
a semi-major axis
e eccentricity
T time of perigee passage
w longitude of perigee measured from ascending node
SZ longitude of ascending node referred to axes fixed in space
i inclination of orbit to the equator.
n is the mean motion of the satellite
M the mean anomaly is n(t - T) or nt + x
is the true anomaly
r, 8,h are spherical polar co-ordinates referred to axes fixed in direction in
space with the centre at the centre of mass of the Earth and with the polar
axis the Earth’s axis of rotation
a, is the Earth’s equatorial radius
wr is the Earth’s spin angular velocity.
v
Resonant orbits of artificial satellites in the Earth’s external gravitational potential
55
Then I take a typical term of the perturbing potential to be
where 1 is the longitude measured from axes fixed in the Earth. JPq and
constants and Ps is an associated Legendre function.
Lagrange’s equations for the variations of the osculating elements are
di _
dt
I
avP,
(cot iz
na2(I - e2)*
- cosec-i
ppq are
an
In order to differentiate Vpqwith respect to the orbital elements, expressions for
r, cos 0 and h in terms of these elements are required. We have
where J y is the Bessel function of order v. Also
cos 8 = sin(w + v) sin i.
To determine I take the zero longitude in the system of fixed axes to coincide
with the longitude of Greenwich at t = 0. The longitude AG of Greenwich, relative to fixed axes at any time t is then wrt.
Let the longitude of the satellite measured along the equator from the node
be #. Then A, the longitude of the satellite measured along the equator from the
origin of fixed axes, is
JI+ n.
But
tan$ = tan(w + v ) cos i
and for small inclinations
#
= o+v-$y2sin2(w+v)+
where y = tani (Smart 1953, p. 101). Hence
...
56
A. H. Cook
Then
2p{1+ 2 2J,,(ve) cos v M p - 1
=
’
)
f
(
.
ae r
2vJi(ve) cos vM,
and
Also,
a
-(cos
.aw
a
8)
=
cos(w + v) sin i,
-(cos 8) = sin(w + v) cos i,
ai
a
av
+
v) sin i-,
-(cos 8) = COS(W
ax
aM
a
av
+
v ) sin i-,
-(cos 8) = COS(W
ae
ae
and
,:(
A)case
= o.
Since
v = M+(ze-&@)sinM+
then
and
(Smart 19.53, p. 38)
as
- - - 1+(ze-&es)cosM+2
aM
av
_
-- ( ~ - & + ) s i n M +
ae
Lastly,
-ai- -
asz
I,
ai
- - - I -*y2cos2(w+v),
aW
-ai- - {1-~y2cos2(w+v))-, av
ax
aM
ai -- {I - $72 cos 2( w + v)]-,8V
-
and
ae
-ai-ai
ai
-aa
ae
-8
0.
tan iseczisin 2(w + v)
Resonant orbits of artificial satellites in the Earth’s external gravitational potential
57
I shall consider only orbits with small inclinations and‘eccentricities, namely
those for which tan% and e are less than 0.1 and all expressions will therefore be
carried only so far as the first order in tan% and e, except where they are to be
divided by e or sin i.
3. Ellipticity of the Equator
The potential Vzz corresponds to an elliptical equator. It is of particular
geodetic and geophysical interest and so is a useful simple case to discuss first.
The procedure that will be followed is to pick out from the differentials of
VZZthe terms of long period and to see how they combine in the expressions for the
rates of change of the orbital elements.
Thus,
avzz
-
-6sin2(w+v)sinicosisin2(1+&z)
ai
L
1.
-3 tanisec2isin2(w+v)cos 2(1+&2)
+ 3 sinz(w + a)sin 2(w + v ) sinzitan iseczicos 2(~+jg22)
1
The factor ue2/r3 is equal to
)ae;( ”();
.L
and (u/r)3 is equal to I + 3 e cos M + ... . Thus aVzz/ai contains terms of order
sin i, sin% and e sin i, together with terms of higher order. Taking these successively, and omitting the common factors (Jzz/ue)[(ue/a)3], we have :
t e r m of order sini:
-6sinZ(w+v)sinicosisin2(1+~~~).
+
This has a secular part if 2(w v ) is equal to 21, that is, if
w
+ v = w + v + SZ - w,t - iyzsin 2(w + v ) + ....
This condition cannot in fact be satisfied because h is only about 4deglday for
close satellites and so this term has no long period or secular component. The
same argument applies to the term
- 3 tanisec2isin2(w+u)cos2(1+/?22)
and to the term of order sin%, namely
3 sin2itanisec2isin2(w+v)sin2(w+v)cos 2(1+&2).
The terms of order esin i are
and
- 18esinicos isinz(w+v) sin 2(1+~zz)cosM
-9esinisec3isin2(w+v)~os2(1+,922)~os
M.
The former gives rise to a secular term if
M
= 2(1+1922).
In general this will not be satisfied exactly; accordingly let
2(1+&2) = M+T,
58
A. H. Cook
when the term may be written as
- 9e sin i cos i{ I - cos z( w + u)} (sin M cos q + cos Msin r ] ) cos M.
Then the long period part is
9
sin i cos isin 77.
- -e
2
T h e second term gives rise to a long period component if
2(z+pzz) = a(w+u)-M+T'.
I shall confine myself in this paper to terms containing sin7 or C O S T since
in general these are the largest terms, and also the condition M = z(Z+Bzz) corresponds to an orbit of smaller radius than for other conditions. Dr Norton gives
the complete expansions in his Appendix.
Proceeding in this way, the terms proportional to COST or sinq in all the
differentials are identified and listed in Table I. Now from the equation for
0,it follows that the part of 0 proportional to s i n 7 is
Similarly, all the parts of the rates of change of the orbital elements proportional
to sin 77 or cos 7 are picked out and listed in Table 2.
Table
I
Factors of terms proportional to sin r ) m cos 7 in derivatives of V2.2
r] = 2(z+pzz)-M
Order of term
Derivative
with respect
e
I
Y2
to
a
-
27
- -esin
-
q
2a
e
3
- -sin7
-
2 sinzisin q
2
X
-
21
-e cos 9
-
2
w
-
9e cos q
-
i
-
-
-
9
--e
2
All terms to be multiplied by
sin icos isin
Resonant orbits of artificial satellites in the Earth's external gravitational potential
F'
v)
8
2 4
!I
F
P
C
F
.*c
.A
-I
v)
P)
I
I
x
c4
81.
.C
I
i
c
'u
0
P
.
I
U
8
I
B
F
I
c)
2
rl
h,
'cf
'X
B
1%
59
60
A. H. Cook
Let us now look in more detail at the condition
2(1+/322) = M+v.
Since w = M+2esinM+ge2sin2M+
M+q
= 2{w
+ !2 + M -
...,the condition is
wrt+ zesin M - kyzsin 2(w
Suppose, as is the case with close satellites, that
linear functions of time, so that
w = wo+
+ M ) + ...}+ 2/322.
and Q are very nearly
w
chtfsmall periodic terms
and
!2
=
!20+fit+small periodic terms.
Also, since M = n t + x ,
nt + x +v =
2{wg
+ QO+ nt+ x - wrt+ zesin M + cit + ht - &y2sin:2(w+ M)) +
+ 2p22 +periodic terms.
Thus we must have
n+2(&+h)
= 2wr
as the condition for resonance.
Then
v
= 2(wo
+ Qo +&z) +x + qesin M-iyzsin
+
2(w M ) .
T h e small terms with argument M are periodic with speed n.
We then see that the magnitude of the secular contribution to h, which is
proportional to sin 7, is determined by the value of x, that is, by the position of
the satellite at zero time.
In particular, siny = o if
x
= j77-2(wO+Qo+p22)
(j
= 0, I, ..*)
(ignoring the small terms in y2 and e). But the longitude A, of the satellite at t
is w0+ Szo + x (for very small inclinations) and so
=
o
A, = jr - wo - QO- 2/322.
Similarly, for a maximum effect, positive or negative
7T
A, = (2j+1)--wo-Qa,-2/322.
2
It follows from these considerations that if J22 and p z z are both unknown, they
cannot be determined from a single orbit.
However, if v changes slowly with time because the resonance condition is not
fulfilled exactly, sin r) will pass through the values + I and - I , so that J22 could
be found from the amplitude of the periodic variation of h. Further, fizz could be
found from the satellite position at the maximum.
The variation of h is only of order eJ22 and so scarcely likely to be observed.
It can be seen from Table 2 that the largest variations are in and ci, and are of
Resonant orbits of artificial satellites in the Earth’s external gravitational potential
61
order e-lJ2z. Terms proportional to e-1 are a novel feature of these resonant
orbits, they arise from differentiation of (a/r)3 and sin 2(Z+/322) with respect
to e. Since the terms in and ci, are of opposite sign, the net effect on the longitude of the satellite is zero, but motion of perigee should be observable. The
next largest term is that in d of order Jz2. All other terms of are order eJ22 or
J22 sin%.
Two limitations on the analysis must be emphasized. First, the inclination of
the orbit has been restricted by assuming that tan2i is less than 0.1,and no attempt
has been made to discover what features are characteristic of orbits with large
inclinations. Secondly, it has been assumed, particularly in writing down the
condition for resonance, that the variation of w is dominated by the secular part
such as arises with close satellites from the principal second harmonic in the Earth’s
gravitational field. But, for a satellite with a period roughly twice that of the
Earth’s rotation, namely at a distance of about 4 Earth’s radii, the perturbations
due to the Sun and Moon are important and the validity of this assumption of a
dominant secular term may require study.
X
4. Resonant orbits for a general potential
In this section I concentrate on determining the resonance condition for a
disturbing potential of general form,
Differentiation of the factors a-P, ( a / r ) p and sin q(Z+ppq) is quite straightforward. The differentials of ( a / ~ ) pwere given in Section 2 and those of
sinq(Z+/3ppa)
are just
qcos q(Z+Ppq) - -,
(a:,
-, -, - 1.
ax e: ii)
a:
The associated Legendre function Pi (cos 0) is
(sin0)q
dptq
2pp! d(cos 0)pfq
(cos2e - I )P
which may be written as
(sin e)q
C a,(cos e)p-q-2r.
r=O
t p - q - I ) according as Cp - q )
the upper limit in the summation being *(p - q) or C
is even or odd. ar is
(- )’(2p - zr)!
zw!(P--.)!(p- q- 2r)!.
If we ignore terms in sin%or higher order, we may write
dP$
-d cos 0
and
-xxcose
if (p-q) is even
if (p - q) is odd,
62
where
A. H. Cook
x = (p2+p-q2),
Y =
+ q+
(-)*(p-g-yp
2”{+@ - q - I)}!{+@
I)!
+ q + I)}!
and
y = +(pZ+p - 9 2 - I).
We also ignore terms of
e2
or sin% in the other factors of the differentials of
Vpq.Thus we write
1+2
CJ,(ve)cosvM
=
I+ecosM
and
x (cosM+zecos2M-esinM).
With the foregoing restrictions on e and sin% terms in the differentials of V,,
and in the rates of change of the orbital elements are given in Tables 3(a) and (b)
and 4(a) and (b).
It will be seen that the behaviour differs according to whether (p-4) is even
or odd. When (p-q) is even, there is a resonance condition analogous to that for
the Vzz term, namely
q(l+lS,d
or
=
M+r
+ + a) = qwr,
(q- 1)n q(h
with 7 equal to
q(wo
+ Qo+/3,,) + (q-
I)X
+ 2qe sin M -
iqyzsin z(w + M)+ ... .
But when ( p - q ) is odd, the corresponding resonance condition is
q(l+lS,B,,)
= w+v+C,
that is,
x + nt-
q[wo t no+
+
+
wrt+ 2e sin M + ht ht- iyzsin 2(w v ) +ppB,,l
&+ x+ nt+ zesin M + 5.
= wo+
Hence
(q-I)(n+h)
while
=
q(wr-h)
5 is equal to
(q-
I)(WO+X)
+ q(Q0+/3,~)+ 2(q-
1)esin M - iqy2sin z(w + M ) .
T h e complete set of resonance conditions is given by Dr Norton in the Appendix.
In addition, a whole series of other resonances occurs. (u/r)p+l may be
expanded in a series of terms such as ev sin vM and in general these terms give
rise to resonances when
4M+rv)
= q(l+P,B,,)
Resonant orbits of artificial satellites in the Earth's external gravitational potential
P
r-
P)
I
+
+
W
Y
4
-ia
bi
II
I
F
P
I
pl
N
I
3
-(9
h
I
P
r
8
9)
P
h
CI
$1
c
F
P
.r(
PI
3
9)
h
M
8
h
c(
+ +
I
I
I
4
rw
0
Y
-.
63
A. H. Cook
n
P
+
-f
a
W
I
+-
SJ,
W
Y
m
ck
8
II
SJ,
N
x
I
I
I
x
n
LP
sdl
m
sdl
8
*CI
c
.
I
m
?.
h
CI
+
-4:
I
D
I
."c:m
."mc
.CI
I
a4
x
I
m
r(N
I
Resonant orbits of artificial satellites in the Earth's external gravitational potential
65
F
a
8
I
I
z
9)
a,
P
v
-i
I
-
4
I
I9)
x
-.
+
Ia
I
F
C
F
.3
C
.3
F
I
0
d
I
.u
a
I
I
rn
4
I
h,
N
I
I
+w
N
I
:
Y
9
Pa
E
.D
'9)
'X
x
n
66
A. H. Cook
+
*
hl
-
N
+
8
-+
H
3
+I
h
N
I
ca
w
I
+"
3m
iz
I
h
n
Y
I
0
Ir)
hl
JJI
r(
I
x
I
8
67
and if e is close to I, many of these terms may be important, whatever the value
of i. Such resonances may therefore be important with highly eccentric orbits.
Furthermore, P$ (cos 0) may be expanded in powers of sin(w + v ) sin i and
terms of order greater than sin% will give rise to resonances, but the situation
is now more complex because of the dependence on sin%' which has also to be
taken into account in the expression for 1.
Thus, in general, resonances will be obtained whenever
Resonant orbits of artificial sate lites in the Earth's external gravitational potential
or
vn
=
q(n - wr) approximately,
n = - Q"r
q-v
the exact condition differing slightly for @ - q ) odd or even.
Since the same value of the ratio q/(q- v) is obtained for an infinite set of pairs
of q and v, it is clear that in general an infinite set of harmonics contributes to the
rates of change of elements at any one resonance. Also, when v is I and q is quite
large, all the resonances cluster round the value
n
= wr
and there will be a sort of band spectrum with this frequency as a point of accumulation, contributions to the rates of change of the elements again coming from an
infinite set of harmonics.
The situation is much simpler when the eccentricity is small for then only the
first order resonance gives terms that are not very small.
5. Realization of orbits
It is evident that if we consider only the terms in Tables 2 and 4,the orbits
should have small eccentricities and inclinations so that the perturbations of
order e-1 and cosec i may be as large as possible. However e and i cannot be
too small or the effects, the motions of o and R,will not be observable since it will
become impossible to define perigee and the nodes observationally. The periods
giving such resonances range from 12h for q = 2, to 24h. Some of these and
the corresponding semi-major axes are listed in Table 5. The velocity of a satellite
Table 5
Periods, semi-major axes and velocities for E =
4
T
a
""12for
UE
h
Earth's
radii
2
I2
4 '2
4
3
4
5
16
18
5 '1
5 '5
5 '75
4'4
4-6
4'7
4 '9
19
10
22
co
24
6'2
6-65
resonant orbits
5'04
.
in its orbit is the most relevant quantity in considering the actual realization of an
orbit and in Table 5 the velocities U E at eccentric angles of 7712 are given since
these are independent of eccentricity (this velocity is (fMe/a)*, Me being the
68
A. H. Cook
mass of the Earth and f the gravitational constant). The important fact shown
in Table 5 is the small range of velocities corresponding to the resonant orbits.
This may be shown in another way. If ~ U is
E the departure of U E from the
resonant value, the period T, of the long-period motion proportional to sin 9
or COST is
zn
a
~~
3(4-1) 8uE'
We then find the following values of T,:
6 . 6 10-3km/s,
~
T, = Iood;
6 . 6 Io-Zkm/s,
~
T, = Iod;
SUE = 2.85 x Io-zkm/s, T, = Iod;
for q = 4,
~ U =
E 1.06~
Io-Zkm/s, T, = rod.
for q = 10,
Clearly it will be more difficult to put a satellite into resonant orbits corresponding to the higher values of q.
From other points of view, the observation and analysis of resonant orbits may
not be too difficult. Whether or not the velocity may be adjusted within such
close limits so that T, may be chosen to be quite different from other periods such
as the revolution of the node or perigee or the Moon's orbital period and so on,
the actual value of U E can be measured and T, identified and perturbations with
this period can be separated out. There will be no non-resonant contributions to
perturbations with this period and therefore it is not necessary to have a highly
exact theory of the effect of the Earth's oblateness and of the Sun and the Moon,
in order to determine longitude terms.
It would be an advantage to have an orbit of small inclination and small eccentricity, always provided that perigee and the nodes may be defined by observation.
Some recent results of Kozai & Whitney (1959) show that the effects of the
Sun and Moon will have to be studied rather carefully. Kozai & Whitney found
that the eccentricity and perigee distance of Satellite 1959 82 (the apogee distance
of which is 25000 miles) are considerably perturbed by the Sun and Moon and
although the period does not appear to be greatly affected, it may be that if a
resonant orbit were subject to such large perturbations, the resonance condition
would be destroyed. The perturbations computed by Kozai & Whitney are proportional to the eccentricity of the orbit and one may therefore expect that if the
eccentricity is small, the perturbations due to the Sun and Moon may not disturb
the resonance conditions. A numerical investigation might help to elucidate the
matter.
for q
= 2,
SUE
=
~ U =
E
Acknowledgments
I am indebted to Dr H. J. Norton for correcting a number of errors in the
draft of this paper and for indicating some simplification.
This paper has been prepared as part of the research programme of the National
Physical Laboratory and is published with the permission of the Director of the
Laboratory.
National Physical Laboratory,
Teddington,
Middlesex :
1960 September.
Resonant orbits of artificial satellites in the Earth’s external gravitational potential
69
References
Cook, A. H., 1960. Proc. 1st Int. Space Sci. Symposium, Nice. (Amsterdam:
N. Holland Publ. Co.)
Groves, G. V., 1960. Proc. Roy. SOC.A, 254, 48.
Kozai, Y. & Whitney, C. A., 1959. Res. in Space Sci., Sp. Rep. No. 30, p. I .
Smart, W. M., 1953. Celestial Mechanics. (London: Longmans.)
Appendix
H. J. Norton
A complete set of resonance conditions
I n this appendix it is shown that many resonant orbits are possible besides those
given in Section 4of the paper. I n general they will either be further from the Earth
or else give rise to smaller perturbations in the orbital elements. Some resonant
orbits can be found, however, whose mean distance from the Earth is less than
4ae, but this can only occur with values of q greater than 2.
The angles 71, 772, ..., 7713 and [I, 52, ..., [12 are defined, in Table 6, in terms
of quantities which have been given earlier. If any one of these angles is small, a
resonance condition is obtained except for certain special values of q. The appropriate resonance condition in each case is given in the second column of Table 6.
The quantities 771, 772, ..., 713 apply when p - q is even; while [I, 52, ..., r;12 are
applicable for odd values of p - q. The resonance conditions given in Section 4
correspond to 773 and 52.
When q takes the values 3,4, 5, 6 or 7, one of the resonance conditions may be
satisfied for orbits with a mean distance from the Earth’s centre which is less than
4ae. For example, if q = 4 then 77 will be small when
n
=
4wr-2&-4Q.
A satellite with this mean motion will describe an orbit at a distance equal to about
two and a half times the radius of the Earth. Thus these other resonance conditions will be useful when trying to determine high-order terms in the expression
for the Earth‘s potential.
The equations for the derivatives of the potential with respect to the orbital
elements are given in Tables 7 and 8. Since aV,,/ai is always divided by s i n i
in the formulae I (d),(e) for the time derivatives of !2 and o,terms of order sin%,
which occur when p - q is even, have been included. Elsewhere terms of order
greater than el e sin i or sinzi have been neglected.
The first-order terms in Tables 7 and 8 correspond to orbits whose average
radii are not less than 4 4 . Any resonant orbits closer to the Earth, and these can
only occur with values of q greater than 2, have secular terms of smaller order.
It will be noticed, however, that aV,,/ae is divided by e in the formulae I (c), (e)
for the time derivatives of x and w , and aV,,/ax is divided by e in the time derivative of e. Thus perturbations of order unity arise when 775, 5 6 and 510 are small.
If values of q greater than z are being considered, resonant orbits derived from
these quantities would have radii less than 4ae.
A. H. Cook
Table 6
Resonant orbits of artificial satellites in the Earth’s external gravitational potential
Table 7
p -q even
av,, = K[q cos 71 +Dle +DZsin%],
-
an
avpq
aw
- = K[qcos 71 +Ele +Ezyz +E3 sin2i],
a vpq
- = Ksini[Ficosi+Fasec3i+F3ecosi+F4esec3i+F5sin2isec3i],
ai
Fl =: f x [ - 2 sin 11+sin 1/10 +sin 1/11],
F a = fq[ -sin 1/10 +sin 7111,
F a = ~x@+I)[-zsin1/a-zsin~3+sin~e+sin1/7+sin?e+sin79],
F4
= t q ( p + ~ ) [-sin?a+sin1/7-sin1/~+sin?s],
Fa = ?,qx[z sin 1/10 -2 sin 1/11-sin
1/12
+sin 9131.
71
A. H. Cook
72
Table 8
U
A1 =
t [-cos
AZ = #(p+I)[
51 +cos
-COS
la],
55 +COS 1 6 +COS 1 7 -COS
581;
a V’_
, - Ksin i[B1 +&el,
_
ae
D1 = h[sin
51 -sin
la],
DZ = #q(p + x)[sin 55 -sin
56 -sin 57 +sin
bs] ;
av,,
-- - Ksini[El+Eae],
a,
EI = t [ ( q+- I ) sin 41 -(q - I) sin ta],
EZ = +~ ) [ (+qI ) sin 5s -(q - I ) sin 16 -(q -1) sin C7 +(q + I ) sin 581;
a V’,
-= KIF~cosi+Faecosi+F~sinZicosi+F~sin~isec~i],
&J
ai
Fl = #[ -cos
Fa
+
= $(p
11 +cos 591,
I)[ -cOS
66 +COS l 6 +COS 57 -COS
Fa = * y [ g cos 51 -3 cos l a -COS
F 4
=
&[ -COS
51 -COS
SZ+COS
T 3 +COS l 4 ] ,
58 +COS {d].
f~],