-
P R E C E S S I O N , N U T A T I O N AND T H E
R E F E R E N C E S Y S T E M F O R C L O S E E A R T H
C H O I C E OF
S A T E L L I T E O R B I T S
K U R T L A M B E C K
Groupe de Recherches de GOoddsie Spatiale, Observatoire de
Paris, 92 Meudon, France
(Received 19 October, 1971)
Abstract. Expressions are given for the perturbations arising in
the motion of close earth satelites if the orbital system
introduced by Veis is used. These expressions include all terms
with amplitudes greater than 10 -8 for both long and short periods.
Resonance problems can also occur under certain circumstances.
Similar first order expressions obtained previously by Kozai are
found to contain some errors.
1. Introduction
The reference system usually adopted for describing the motion
of close Earth satel- lites is a quasi-inertial one defined by the
instantaneous equator and the equinox of an initial epoch assuming
that the precession and nutation in right ascension has occurred
along this equator. This system was introduced by Veis in 1959
(Veis and Moore, 1959; Veis, 1963) and has the advantage that the
Earth's oblateness is most nearly symmetric with respect to the
adopted equator. This system is not inertial and perturbations will
arise. Kozai (1960) gives expressions for these perturbations.
In view of the almost two orders of magnitude in improvement in
the accuracy with which satellites can now be tracked, these
expressions have to be reconsidered for not only do they contain
some errors but they are also not sufficiently precise.
2. Kinematic Transformations
If the Earth fixed system is denoted by x, y, z X Y Z the
relation between the two is given by:
and the inertial reference frame by
(;) = v, Av, ( 0 ) (u , . (la) The inertial frame is defined by
the mean equator and equinox at a specified epoch to, usually
1950.0. The rotation matrix ~ (u, v) rotates the Earth fixed system
parallel to the instantaneous axis of rotation using the
instantaneous coordinates (u, v) of the pole in this fixed system.
The matrix ~ (0) rotates this instantaneous system, rotating with
the Earth, through the siderial angle 0, to give the instantaneous
siderial system. The matrix ~ (A#, Av, Ae) represents the nutation
matrix to give the mean system at date t. The fourth matrix ~ ( x ,
v, o9) represents the precession matrix to give the mean system at
epoch t o .
Celestial Mechanics 7 (1973) 139-155. All Rights Reserved
Copyright �9 1973 by D. Reidel Publishing Company,
Dordrecht-Holland
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140 KURT LAMBECK
The nutation matrix consists of three rotations;
~(Ap, Av, Ae) (cos !)(
= - sin A# cos A# 0 0
X (i
cos Av 0 sin A v \
v ) 0 1 0
sin Av 0 cos A
o o) cosAe sinA
- sin Ae cosA
•
and since A/~ is of the order 10 . 4 second order terms should
not be omitted if a theory accurate to 10 -s (or about 10 cm in
satellite position) is sought. Ap, Av, Ae are the nutations in
right ascension, declination and obliquity respectively and their
complete expressions are given in, for example, Connaissance de
Temps.
The ~ is the part of the precession in right ascension along the
mean equator at the epoch to and co is the corresponding part along
the mean equator at date t. The v is the precession in declination.
These quantities should include the terms in t 2. The complete
matrix ~ (~c, v, o)) is given in the Explanatory Supplement to the
Astronom- ical Ephemeris and Nautical Almanac.
The Equation (la) with all the necessary nutation terms is used
to transform the terrestrial coordinates (and if necessary, any
observed directions) into the inertial
~ 2 f r a m e .
If the above mentioned quasi-inertial system )~ the
corresponding kinematic transformation is:
(i) = ( 0 ) , is used in the orbit calculations
(lb)
where (p, Ap) is a rotation about the instantaneous axis of
rotation through (k/+ + A#). That is:
cos (~ + A~) (~ + A/z)= - s i n (# + A#)
0
+ !) cos (~ + A~) . 0
3. Dynamic Transformations
The geopotential V is usually defined in the Earth fixed system
and assumed constant. In the inertial frame V is time dependent,
due not only to the rotation of the Earth, but also due to the
precession and nutation. If the inertial frame is used these time
dependencies must be introduced. There are several ways of doing
this and if numeri- cal integration is used to solve the equations
of motion it is perhaps simplest to in- troduce directly Coriolis'
theorem. This has been done by Balmino (1971). But such an approach
does not lend itself to studying the long period perturbations and
as these are the most important we will give an analytical
development instead.
To derive the time dependent expression for V we introduce first
of all a rotating
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PRECESSION, NUTATION AND THE CHOICE OF REFERENCE SYSTEM 141
intermediate system X* Y* Z* whose Z*-axis is parallel to the
Z-axis X*-axis lies 0 to the east of X. That is:
and whose
(cos0 sin0 Y* = - s an0 c o s 0
Z* 0 0 i)(i) (!) = ~ (o) = = ~ (o) ~ (,~, o~, v) ~ (As,, Av, A~)
~ (0) ~ (u, ~) . (2)
The potential coefficients of second degree in the X* Y* Z*
system are related to the mass distribution of the Earth by (see,
for example, Hotine, 1969):
C~ 0 = 1 f X , 2 y,2 2Z,2 2]V~R 2 ( - - + ) dM
M
C~1 = f lj- 1 X'Z* d M ; S~1 = M R 2 M R 2
M M
C~2 = 1 f 4 M R 2 ( X*2 Ik
M
_ y , 2 ) d M
S~2 = 1 f X'Y* dM 2 M R 2
L
M
Y'Z* dM
(3)
the integration being carried out over all the mass elements dM
with coordinates X* Y* Z* making up the Earth. The coefficients C*
S* are in the unnormalized form and related to the geopotential V
by
g m
m l
1+ X 2 l=2 m=O
Plm (sin qg) (C'~m cos m2 + S/* sin ms
Substituting (2) into the integrals (3) would then give the time
dependent second degree gravity field coefficients of the Earth in
the X* Y* Z* system.
In most analyses of satellite orbits for the longitude dependent
part of the geo- potential it has been assumed that the principal
axis of maximum inertia and the z-axis of the Earth fixed system
coincide. In reality, they are close to each other and they change
their relative positions due to variations in the mass distribution
of the Earth caused by, for example, seasonal fluctuations in the
atmospheric mass distri- bution.
If we define a second intermediate system x* y* z* whose z*-axis
is coincident with
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142 KURT LAMBECK
the principal axis of maximum inertia and whose x*-axis lies in
the xz plane we can write:
( i ) ( 1 0 = 0 1 U* V*
) (1) - v* y* ,
1 z*
(4)
where u*, v* are the coordinates of z* with respect to z. u* is
in the direction of posi- tive x and v* is in the direction of
negative y to subscribe to the usual astronomical definition of the
pole coordinates. Note that u* and v* are not the coordinates of
the instantaneous axis of rotation as published, for example, by
the International Polar Motion Service.
In the x* y* z* system the terms C21 and $2~ are equal to zero
but in the x y z system they are given by (substituting (4) into
the integrals (3))
1 M R 2 [u* ( - x .2
M
"JI-" Z . 2 ) -I'- v*x*y* + x 'z*] dM
or, with the expressions for the other integrals (3)
C~1 -- u* (C20 - 2C22 ) + 2v*$22.
Similarly
= - (C o + 2u*S .
Since C2o ~,~, 10 -3 and C22 and 822 are ~ 10 -6 it suffices to
write C~'1 =~/*C2o, 81~= ----v*C2o , and as both u* and v*~ 10 -6 ,
C21" and S~1~ 10 -9.
We see then that as the position of the principal axis of
maximum inertia changes the C21 and $21 also change. Thus if these
terms can be observed we have a direct method of measuring the
movement of the z*-axis with respect to the fixed system x y z
(Lambeck, 1971). The amplitude of the variations in u* and v* is
unknown but certainly less than 10 -6. According to Gaposchkin
(1972) they are very small indeed because of the Earth's
elasticity.
Returning to the main problem, we carry out exactly the same
manipulations but substituting Equations (2) into the integrals
(3). The algebra becomes correspondingly more complicated but we do
not need to carry along all terms in the integrals.
If the initial epoch is 1950.0 then for a date near 1970, o9,
to, v are of the order 2 x 10 -3. Also A#, Av, Ae are of the order
10 -4. The right hand side of the integrals (3) can, as already
indicated for the example discussed above, be expanded as com-
binations of the gravity field coefficients expressed in the earth
fixed system. Of these, the largest is of the order 10-3 so that,
if we want to keep our expressions accurate to 10 -8 we should
carry along products and squares of the precession elements.
Products of any C21 and $21 terms with precession and nutation
elements that appear in the right hand side can be ignored as they
will be of the order 10 -12 or smaller. Products of the C22 and $22
terms with the precession elements should however be retained as
they can approach 10 -8.
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PRECESSION~ NUTATION AND THE CHOICE OF REFERENCE SYSTEM 143
With these simplifications we obtain the following expressions
for the second degree coefficients when the potential is expressed
in the inertial system.
C~o - C2o (1 - ~y2)
C ~ = C2~ + [-(v + Av)cosO + Ae sin 0 - �89 sin 0 - u] C20 - - -
21) (C22 c o s O - $22 sin O)
8~1 -- $21 -a t- [-- (Y -a t- A v ) s i n 0 + Ae cos O - �89 cos
O + v] C20 - (5) - 2v (C22 sin 0 + $22 cos O)
C~2 -- C22 -[- 2/zS22 + k v 2 C 2 o c o s 20
8~2 = $22 - 2/2C22 - �88 2C2o s in 2 0 .
For all the other harmonic coefficients, the numerical values
are the same in the two systems to within at least 10 -9 .
If we take for epoch a date closer to the time of observation we
obtain considerable simplifications in the above expressions and
only the C21 and $21 are significantly modified. That is:
C~a - C2a + AC2a = C2a + [(v + Av)cosO + Ae s i n O - u] C2o
(6)
S'~a - Sza + AS2a = SEa + [-- (V + Av)sinO + Ae cosO+ v]
C20.
4. Perturbations in the Orbital Elements with Respect to the
Inertial System
The additional potential due to AC2a and AS21 ments as (Kaula,
1966)
is given in terms of the orbital ele-
l +oo
AV2a -- F2ap(i) G2pq(e) 5'~ (7) O
p--O q=--oo
where
~ . ~ z ~ 2 1 p q - - - AS21 COS ])21pq "]- AC21 sin'})21pq (7a)
and
Y2xpq = (2 - 2p) o9 + (2 - 2p + q) M + f2 - 0 ~ )'~apq- O.
Substituting (5) into (7a) gives;
992 l pq - - [(v + Av)sin (Yzlpq + O ) - ( A e - �89 (Y2xpq +
0)] Czo -
-- 2vC22 sin (]/21pq -- O) -~" 2vS22 cos (721p~ -0).
We have dropped here the contributions arising from the u and v
as these give perturbations with periods equal to or less than one
day and have amplitudes not exceeding 20 cm (Lambeck, 1971). We
will also ignore for the moment those terms with argument ~ - 0 ( -
7" - 20) as these will give rise to perturbations with periods
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144 KURT LAMBECK
For the precession and nutation we can write the general
expressions:
so that
o sin (czjt + fit) v + Av = Vot + ~ Avj J
# ~ pot 0 Ae = ~ Aej cos (~jt +/~j)
J
~021pq "-" {Vot sin (]/21pq -~" O ) -
_ 1_~ Z ( Av~ + Ad.) oos E~t + ~j + ~ , ~ + ol + J
+ �89 Z ( Av~ - As~ [~jt +/~j - Yzxpq - O] +
+ �89 tz cos (?2,pq + 0)} Czo.
(8)
(8a)
Substituting the A V21. into the Lagrangian equations of motion
gives the following variations in the orbital elements.
(d (~a) )
dt /2 l pq
( d (6e)~ at ,S~,,,,, (d (~i))
dt 72 1 pq
(d (6co))
dt / 2 1 p q
(d (~s
dt ,]2 l pq
_- F21p (i) Gzpq(e) ( 2 - 2p + q) 5e~lpq
(1 - e2) ~/2 = ~ v ~ . ( ~ ) ~ ( ~ ) x
e
x [(2 - 2p + q) (1 - e2) 1/2 - ( 2 - 2p)] 5~'~ap~
--?'l ( ~ ) 2 F2ap (i) G2pq (e) [(2 - 2p) cos i - 13
sin i (1 - e2) 1/2
_cot
= " (1 - ~ ) ~ "
dFzlv(i) di
G2pq (e) +
(1 - e 2 ) 1/2 + F21p(i)
e
dGzpq (e); 5P21pq + de
+ - ~ n C2o sin 2i
(1 - e2) 2 ~i+
+ 3ne C2o (1 -- 5 cos 2 i)
(1 - e 2 ) 3 ,Se
- - F / (_~)2 1 dF21v(i) G2pq(e)
sin i di (1 - - e 2 ) 1/2 "-'qP21pq
- 3n C2o (1 - e2) z 6i+
( t ) 2 + 6n C2o cos i e (1 - e2) 3 ~e
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PRECESSION, NUTATION AND THE CHOICE OF REFERENCE SYSTEM 145
where
(d (fM)) = n (~)2 F21p (i) [ (1 - dt /21pq e
e 2) dG2p q (e)
de + 6G] ~P21pq "-F
9 . + ~ C2o
sin 2i (1 - e2) 3/2 5 i -
e 49--n C2~ (1 - e2) 5/2 (3 cos2i - 1) 5e
~?mpq ~Plm pq
~]) Impq "
(9)
The terms on the right hand side containing the fe and 5i are
due to the interaction of the perturbations in the variations in e
and i with the secular perturbations in co, O and M due to C2o. The
5e and 5i are obtained by integrating the second and third of the
above equations assuming that the only time dependent variables are
the c~t and the Q, co, M. For the combination of coefficients 2 - 2
p + q=0, the perturba- tions are a maximum when q = 0 ; that is
when p - 1 and the second equations of (9) indicates that
d(fez~pq)/dt=O. For the other possible combinations ofp and q (p=0
, q= 2; p = 2, q = - 2 ) d (fe)/dtoce and the interaction of fie
with the secular variation due to C2o is of the order e 2 and can
in most cases be neglected. The expressions are given in the
appendix, Equations (A-l) with (A-2).
We can derive the second part of the short periodic terms due to
A V2 ~ in the same way as before but with
A
~9~ 2vC22 sin (~21pq- O) -Jr- 2v822 cos (~21pq- 0).
The gravity coefficients C22 and 822 a r e of the order 10 - 6
SO that it will be sufficient to take only the principal precession
term: That is y= rot. The expressions for the variations in the
elements are identical to (9) but replacing 5: with 5p, and 5:*
with 5e* = OSe/aT.
Integrating Equations (9) gives the following perturbations in
the orbital elements referred to the inertial reference frame.
fa21pq
fe21pq
fi21pq
= 2B (ae) a (1 - e 2) F21 p (i) G21p (e) x
x ( 2 - 2p + q ) ( r + ~)21pq
= B (ae) F21p (i) G2pq (e) [(2 - 2p + q) (1 - e2) 1/2 - (1 --
eZ) 5/2
X ( 0 -~- ~ ) 2 lpq e
= B (ae) F21 p (i) G2p q (e) (1 - - e 2 ) 3/2
sin i X
(2 - 2p)] x
x [(2 - 2p) cos i - 13 (0 + ~)2~vq
-
-.
~.
'~--
~
8 ~
• x
~ +
~ +
~
~ ~
I ~
+
I I O I
.,X.,
! I
r ~.,
,.,,,.
X
,....~
" ~,,,,,
,. +
~,.,,,.
.
X
X
+ i.,,
a
i,.a
+
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PRECESSION, NUTATION AND THE CHOICE OF REFERENCE SYSTEM 147
~]21pq
1[121p q
_ �89 (Avj + z . j )
J
sin K] ( Avi + - � 8 9 r~ L] 1 ~ , C22
C20 ~f21pq dt = - 2Vo C2o
J
+ 2Vo
X
x ~ _ + (Y _ 20)2 cos (?* - 20) +
$22 ['t cos (?* - 20) _ 1
C~o ~ 9* - 20 (9* - 20) ~
l f , , C22[tcos(?*-20) ~'21pq dt = 2Vo . ,
C2o C2o ? - 20
+ 2Vo
sinLj
sin (?* - 20)1
sin (?* - 20)1
- (5~_20)2 J +
$22 I t sin (?* - 20) cos (?,* - 20)]
C2o 9 " - 2 0 + ~ - - ~ J '
where we have omitted the subscript 21pq for the ?*. Also Kj =
O~jt + flj + ~ lpq I~j - OQ + ~lpq
c i = ~jt + & - ~ ~ L j = ~j - 9~1~
?~lpq = (2 - 2p) co + (2 - 2p + q) M + O
"* = (2 - 2p) 6) + (2 - 2p + q) 3 ) / + ~Q ~)21pq
For the short period terms ( 2 - 2 p + q)# 0 and in the divisor
of the ~ we have terms ( 2 - 2 p + q)n where n is usually between
10 and 15 revolutions/day. The maximum amplitudes for the short
period terms occur when q=0; that is p = 0 or 2. Thus p* will be at
least 20 times smaller than the corresponding long period terms.
This means that we will not have to carry along all the nutation
terms in the short period ex- pressions.
If we make the approximation:
~ ~B (ae) cos/ and lmpq-2110, then for the inclination
5 i 2 1 1 o ---- Vot sinf2 + V0 cos f2 -
�89 X (Avj + Aej) o9+.(2
J
cos ( ~ / + & + ~ ) -
cos (~jt + / b - a) +
+ �89 - t 2 cos ? + ~ sin ? (lOb)
which, taking only the principal precession and nutation terms
is equivalent to Kozai's expression.
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148 KURT LAMBECK
The perturbations due to AC20 , AC22 and AS22 can be derived in
the same way as above. Inspection of Equation (5) indicates however
that these perturbations will be smaller so that we can ignore the
nutation terms. Now
where
and
~ 2 2 p q --" 2# [ S 2 2 C O S ('~22pq - - 20) - __ _L 2 - C22
sin(y~zpq 20)] + 2v C2o cosy~2,q
= 3 2 ~9~ zv C2o cos [ ( / - 2p) o9 + ( l - 2p + q) M] ,
* = ( l - 2p)co ~22pq "71- (1 - 2p + q) M + 2Q
* = ( l - 2p)co + ( l - 2p + q) M. ])20pq
The perturbations are:
where:
(~a22pq + (~a2op q = 2 B ( a e ) a (1 - e2) 2 x x F21p(i) G2pq
(e)(2 - 2p + q) (022pq + 02opq)
6e22pq + 6e2opq = B (ae) F21p (i) G2pq (e) x x [(2 - 2p + q)(1 -
e2) 1/2 -
(1 - e2) 5/2 x ( 0 ~ + 0~o~)
e
( 2 - 2p)] x
6i22pq + 6i2opq = B (ae) Fzlp (i) (1 --e2) 3/2
G2pq (e) sin/ x
(~ (_D 2 2 p q --[- (~ (_D 2 o p q
6~'~22pq "F" 0Q20pq
x [ ( 2 - 2p)cos i - 1] (0z2pq + ~tzopq)
= B ( a e ) [ - (1 - e2) 3/2 G2pq (e) dF21p(i)
F21p(i) (1 - e 2 ) 5/2
x c o t / + e
di
dG2.q (e)]
de * * t
dF 2,p (i) (1 - - e2) 3/2 = B (ae) G2pq (e)
di sin i x
x
x
(~M22pq "n t- 6M2opq =B(ae) (1 -e2)216G2pq(e ) - (1 -e2)dG2pq(e
) 1 e de
* t x Fzap(i ) (0~2pq + 02opq) + 6M221o
22pq
~l~2pq =
.o + Fcos_ * ( 2 "" L
1 2 [ - s i n y * ( 2 ~0 ~r 71- 2VO
2t ~in 7"
X
(11)
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PRECESSION, NUTATION AND THE CHOICE OF REFERENCE SYSTEM 149
t 2) sin y* ]3 ,2
) t2 sin y* q ~'2 2t ] 2
cos T v o 20pq
cos 1 20pq
( l l a )
The (~092210 , (~ '~2210 and bM~21o a r e the interactions
between the perturbations in bi22pq and the secular variations in
o9, g2 and M due to C2o. They are given in the Appendix, Equations
(A-3).
There will be no long period terms in the semi-major axis, and
those in the eccentric- ity will be smaller by e than those in the
other elements. In general it will suffice to consider only q= 0.
The short period terms will have amplitudes at least ~1 of the
amplitudes of the corresponding long period terms. The terms due to
A C2o will give rise to secular perturbations when both 2 - 2 p + q
= 0 and 2 - 2 p = 0 : That is when q = 0, p = 1. In this case the
expressions for ~rl2Opq and ~t *2opq are not valid. These secular
perturbations are
d (~(_D2 010 - -
dt 9 . 2.,2n (ae) 5 cos --ffVot n ( 1 - 2i) ~ ~V2ot2Co
d 6f22olo =
dt 9. 2.,2n (ae) i ,.~ -~v~tZI'2 ~ V o t D COS
d 6M2010 =9_~vo,,2,2u,_, (ae) (1 - e2) 1/2 (3 c o s 2 i - 1 ) =
3,,2,22v0, ~,*A;r dt
where 69, .(2 and 3;/are secular perturbations caused by C2o.
(Vo t)2 is of the order 10 -8 for t = 1 yr and these perturbations
are negligibly small. For example, an error of 10 - 9 in C20 would
give rise to a secular perturbation in f2 of the order
d 10 - 9 dr2
dt (6f2) = 10 -3 dt
or about two orders of magnitude larger than the secular terms
introduced by the choice of reference system.
In summary, if we use the inertial reference frame and keep the
geopotential coeffi- cients time independent we apply the
transformation (la) as well as introduce the perturbations in the
elements given by Equations (10) and (11). That is, for an element
e
= +
Pq
If we choose this reference frame it will obviously be simpler
to perform directly the dynamic transformations (5) rather than
compute explicitly the perturbations.
5. Orbital Perturbations with Respect to the Quasi Inertial
System
Often the quasi inertial system discussed above is used for the
orbit calculations. The elements then refer to the instantaneous
equator and a mean equinox defined at a
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150 KURT LAMBECK
Z
1
t /
. , / ~ / I , I n s t a n t a n e o u s / / / ~ ~ . equator
Mean equinox / / I /i / ~ r at epoch t o / / ~ / / /
x ~ O ....... ---~ ...... - - - ~ c o ~Mean equator at
~~----~2"~-~ . ~ ~ / / epoch t o
Fig. 1.
specified epoch ( t ransformation 1 b). The geometric
relationship between the elements in this orbital system (indicated
as ~:) and the elements in the inertial system is indi-
cated in Figure 1. That is, denoting the angle between the two
equatorial planes by
~/and the longitude of the ascending node of the equator of date
by No.
i - f - Ai = ~/cos (f2 - No) - �89 sin 2 (f2 - No) co t /
f2 - ~ -- Af2 = - r / co t i sin (f2 - No) + ~2 sin (f2 - No)
x
x cos (f2 - No) (cot 2 i - �89
09 - 6) - Ao) = r/cosec i sin (f2 - No) - r/2 sin (f2 - No)
x
x cos (f2 - No) cot i cosec i.
The r / and No are related to the precession and nutat ion
according to:
(13a)
1 / s i n N O = v + A v + #Ae, + ...
o (~j t + fl j) = rot + ~ Av ~ sin (~j t + fl j) + #ot ~ Aej cos
J
- ~ cos No = Ae, - # (�89 + Av ) + ...
o ( ,jt + �89 = s Aej c o s o sin (o~jt + flj). - #ot ~ Av j
(13b)
The equivalent first order expressions used by Kozai (1960)
contain errors in sign. If the epoch of 1950.0 is selected the
terms in (13a) containing r/2 have an amplitude
of about 4 x 10 - 6 and cannot be neglected. However, it will
suffice to take only the
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PRECESSION, NUTATION AND THE CHOICE OF REFERENCE SYSTEM 151
principal precession element for these second order terms. That
is, with (8), we obtain:
A i = rot sin f2 - 2 (Vot) 2 cos2 f2 cot i
0 (czjt -~- f l j + ~"~) + - ~ Z (Avj + A~ ~ ~os J
o _ A~O)~os (~jt + ~ j - ~) + ~ Z (Avj J
1)+ Af2 = rot cos f2 cot i - �89 (rot) 2 sin 2f2 (cot 2 i 2 O +
�89 c o t / 2 (Avj + Ae ~ sin (ejt + flj + (2) +
J
o _ A e O ) s i n (~jt + f l j - f2) + � 8 9 (Avj J
Aoo = - Vot cos f2 cosec i + 1 (Vot) 2 sin 20 cot i cosec i
--
--�89 cosec i Z (Av ~ + Ae ~ sin (o~jt + flj + ~ ) - J
- �89 cosec i ~ (Av ~ - Ae ~ sin (eft + / l j - f2). j (14)
The perturbations in the elements 4, ~, f, &, .0 and M
referred to orbital reference frame are then given by:
6a = Z (6a21pq + 6az2pq) Pq
Pq
Pq
Pq
~ = Z ( ~ , ~ + ~ ~ ) - A~ Pq
Pq
(15)
The summations are carried out for, p = 0 , q= - 2 ; p = 1, q =
0 ; p = 2 , q=2 . The orbits published by the Smithsonian
Astrophysical Observatory refer to this quasi inertial system and
contain the perturbations given by Equation 15.
6. Resonances
Small divisors occur in Equation (10) when either:
o r
( 2 - 2p)69 + if2 = 0
(2 - 2p) & + ~ = __+ ~j.
(16a)
(16b)
The major perturbations occur when p = 1 and the Equations (10)
are consequently not valid for polar satellites. For polar orbits
the amplitudes of the terms in the
perturbations in 6092110, 6f221~o and 6Mz~lo containing ff2~aq
are zero and the first two terms of ~'211q and ~,~axq are replaced
by v o s i n [ ( 2 - 2 p ) o o + O ] . Similarly for
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152 KURT LAMBECK
and ~*. For p - -0 or 2, the numerators of the expressions (10)
are proportional to e 2
and the amplitudes will be small unless the resonances become
very pronounced. That
is when 4- 263 + ~ - 0 or, with:
when
63 ~ 3.55 (5 COS 2 i - - 1 )
.(2 ~ - 6.70 cos i ~
i ~ + 5 6 ~ o r + 6 9 ~
~
The more interesting resonance occurs for the condition (16b).
That is when the motion of the orbital plane becomes commensurate
with the nutation. Thus Equation (10) shows that the number of
terms that have to be carried along in the nutation expression (8)
depend not only on their amplitudes but also on their periods. A
near resonance arises, for example, for the satellite GEOS 2, whose
{2 is about 178 day - t and close to the mean motion of the
nutation term whose argument is 2l~-2D~ and whose Av-Ae=O".O018
(see for example, the Explanatory Supplement to the Astro- nomical
Ephemeris and the American Ephemeris and Nautical Almanac). This
very small term nevertheless causes a perturbation in the
inclination of about 0'.'03 (Equa- tion 10b). These resonances are
of interest as they could provide a means of observing directly the
amplitudes of some of the nutation terms. At present the theory
used for the nutation is that of Woolard (1953) and is based on the
assumption of a rigid earth except that for the principal nutation
term a value derived from astronomical observa- tions is used.
Fedorov (1961) has also analysed the astronomical data for the
semi- annual and fortnightly nutation terms. Discrepancies between
these observed values and the rigid earth values exist due to the
earth not being perfectly rigid. The orbital resonances of greatest
interest are those for the combination c~-O ~ 0 since Av~ - A e and
Av + Ae ~ Av -Ae (see Equation 10). Exact resonances should be
avoided as these will give secular perturbations that will be
difficult, if not impossible, to separate from other secular
perturbations. A satellite with inclination of 107 ~ 15' would, for
example, resonate with the semi-annual nutation term. GEOS 2 lies
close to this inclination and Equation (10b) gives for the
perturbation in inclination:
i ,5 i l - �89163 A v - As 1.80(0.507 + 0.552)
=�89 x =5'.'3 1 . 9 8 - 1.80
with a period of about 2000 days. Similarly, a satellite with
inclination about 117 ~ would resonate with the nutation term of
122 day period. For example, a satellite with i= 115 ~ would
exhibit a perturbation in inclination of:
15il- �89 x 2.83 (0.019 + 0.022)
2.83 - 2.95 = 0'.'4
with a period of about 3000 days. The important monthly and
fortnightly nutation
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PRECESSION, NUTATION AND THE CHOICE OF REFERENCE SYSTEM 153
terms unfortunately cannot cause resonances since I2 (max) is
about 7 ~ day -I and (monthly term) ~ 13 ~ day- 1, e (fortnightly
term) ~ 26 ~ day- 1. The nutation terms could also be determined
kinematically from the analysis of satellite orbits in exactly the
same way as is possible for determining the polar motion.
Appendix
The interactions between (~e21pq and the secular variations in
co, f2, M due to C2o are given by
6co = 3B (ae) (1 - - 5 COS 2 i ) ef (1 - e 2) 6e dt ef
6f2 = 6B (ae) cos i 6e dt 1 - - e 2
Q
(ae) (3 cos 2 i - 1) e
(1 - - e 2 ) 1/2 ,f 5e dt, (A-l)
where
f fe dt = B (ae) FG [-(2 - 2p + q) (1 - e2) 1/2 (2 - 2p)] x
(1 - e 2 ) 5 / 2 f X l/I21pq dt.
e
Only the long period terms are of importance so 2 - 2 p + q=0.
Also for q=0, 6e=O and the only long period terms occur when p = 0
, q = - 2 ; p = 2 , q=2. For O21pq we need only take the two
principal terms due to the precession: That is,
v~ 7" rot sin 7* q cos O21pq "-" ,~, ( ] ) * ) 2
with
7" = (2 - 2p)co + f2
and
f 2Vo 7" v~ �9 ~r dt = (~*)3 sin -- t ( ] ) , ) 2 cos 7* -=
~21pq.
Thus
f fe dt = - 2B (ae) (1 - - e 2 ) 5/2
e [F21oG2o.2~21o-2 F212G2o2UP2122-[
(A-2)
and substituting this into the above expressions for &o,
6f2, 6M gives the additional terms that should be added to the
right hand members of Equation (10).
-
154 KURT LAMBECK
The right hand members of Equations (11) contain the terms re),
8f2, 6 M due to the interactions between bi22 + 6i2o and C2o. That
is
j ~ 6o9 = a45 B (ae) sin 2i 6i dt 6f2 = - 3B (ae) sin i . f 6i
dt
6 M = 9 B (ae) sin 2i (1 - 16 e2) ~/2 f fii dt
where
(8i22pq + 8i20pq ) d t
= B (ae) FG (1 - e2) 3/2 [(2 - 2P)sinCOSi i - 1] ,f (O22pq +
020pq) dt.
^, The ~122pq is given by Equation (lla). The first term - ~ o 0
2 1 p q / V o has only short periods and we can ignore it. The part
giving rise to long period perturbations is
where
! 2 [ sinT* ( 2 +
~)* --- ~) 22pq -- (l - 2p) o9 + 20,
since 1 - 2 p + q=0 for long period terms. The maximum amplitude
occurs when q=0; thus p = 1, and Y22~o=2f2, O20pq is also given by
Equation (lla), but Equation (11) shows that the only long period
terms in 6i2opq occur when p=0,2 or when q= + 2. Thus we can
neglect the interactions between the 6i and C20. Consequently
and
.f (O22pq + O20pq) dt = f 02210 dt - ~2210
�89 2 Z~2 i 2-~ t 2 sin 20 2/,23 cos 20
' = lSB2(ae) c o s i F G ( 1 - e 2 ) 3/2 02210 t~0)2210 8f2'22~0
= 3B2 (ae) FG(1 - e2) 3/2 022~0
~ M 2 2 1 o ---- -~B 2 (ae) cos i FG (1 - - e2 ) 2 (P2210
(A-3)
References
Balmino, G.: 1971, 'Choix d'un rep6re absolu en g6odesie
spatiale', Tech. note, Groupe Rech. Geod. Spatiale, Meudon.
Fedorov, E. P.: 1961, Nutation and Forced Motions of the Pole
(translated by B. Jeffreys), Pergamon Press, London.
Gaposchkin, E. M.: 1972, in P. Melchior and S. Yumi (eds.),
'Rotation of the Earth', IAU Symp. 48, 128.
-
PRECESSION, NUTATION AND THE CHOICE OF REFERENCE SYSTEM 155
Hotine, M.: 1969, Mathematical Geodesy, Essa Monograph 2, U.S.
Department of Commerce, Washington, D.C.
Kaula, W. M.: 1966, Introduction to Satellite Geodesy, Blaisdell
Publ. Co., Watlham, Mass. Kozai, Y. : 1960, Astron. J. 65, 621.
Lambeck, K.: 1971, 'Determination of the Earth's Pole of Rotation
from Laser Range Observations
to Satellites', Bull. Gdod., No. 101. Veis, G.: 1963, 'Precise
Aspects of Terrestrial and Celestial Reference Frames', in The Use
of Artificial
Earth Satellites for Geodesy (ed. by G. Veis), North Holland
Publishing Co., Amsterdam. Veis, G. and Moore, C. H.: 1959,
Smithsonian Astrophysical Observatory Differential Orbit
Improve-
ment Program, in Astronautics Information, Seminar Proceedings,
Tracking Programs and Orbit Determination, Jet Propulsion
Laboratory, Pasadena, California.
Woolard, E. W.: 1953, 'Theory of the Rotation of the Earth
Around its Center of Mass', Astronomical Papers, XV, Part I.