precession, nutation and the choice of reference system for close...

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 satellites 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

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 numerical integration is used to solve the equations of motion it is perhaps simplest to introduce 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

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 geopotential 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 distribution.

If we define a second intermediate system x* y* z* whose z*-axis is coincident with

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 combinations 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.

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

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


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 perturbations 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

+


~]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 expressions.

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.

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)


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 coefficients 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

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


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

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 observations 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


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.


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.

precession, nutation and the choice of reference system for close...

Documents