The modifications of the semianalytical satellite theory required to include these 'weak' time dependent perturbations are described. The new formulation results in additional terms in the short periodic variations but does not change the averaged equations of motion. Thus the m monthly terms are still included in the averaged equations of motion. This contrasts with the usual approach for the strongly time dependent perturbations in which the m monthly (or m daily, if tesseral harmonics are being considered) terms would be eliminated from the averaged equations of motion and included in the short periodics computation. Numerical test results for the GPS case obtained with a numerical averaging implementation of the new theory demonstrate the accuracy improvement

Cefola, P.

Early, L.

Green, A.

Mcclain, W.

English

NASA Technical Reports Server

A SEMIANALYTICAL SATELLITE THEORY
FOR WEAK TIME-DEPENDENT PERTURBATIONS
P. Cefola,* W. McClain, ? L. Early, ? and A. Green**
Charles Stark Draper Laboratory
ABSTRACT
Previously, Semianalytical Satellite Theories based upon the Generalized Method of Averaging have
been developed for
- perturbations with no explicit dependence on time, and
- perturbations with a strong explicit dependence on time
While the assumption of time independence (TI) is exact only for zonal harmonics and for static
atmosphere density models, the assumption has also been applied successfully to develop the
averaged equations of motion for lunar-solar perturbations of satellite orbits with periods up to two
days (see AIAA preprints 78-1382 and 75-9). However, recent testing of the lunar-solar short
periodics produced via the TI assumption for the GPS orbital flight regime (12 hr period) indicates
that the relative accuracy of these short-periodics is significantly less than the accuracy of the zonal
short-periodic variations.
This paper describes the modifications of the Semianalytical Satellite Theory required to include
these 'weak' time - dependent perturbations. The new formulation results in additional terms in
the short-periodic variations but does not change the averaged equations of motion. Thus the
m-monthly terms are still included in the averaged equations of motion. This contrasts with the
usual approach for the strongly time-dependent perturbations in which the m-monthly (or m-daily,
if tesseral harmonics are being considered) terms would be eliminated from the averaged equations
of motion and included in the short-periodics computation.
Numerical test results for the GPS case obtained with a numerical averaging implementation of the
new theory demonstrate the accuracy improvement.
*Section Leader, Space Systems Analysis, Air Force Programs Department
)Staff Engineer
**CPT, ARMOR, U.S. Army
67
https://ntrs.nasa.gov/search.jsp?R=19800019892 2020-03-21T16:58:25+00:00Z
A SEMIANALYTICALSATELLITE THEORY
FORWEAKTIME-DEPENDENTPERTURBATIONS
Outline
e Review of Analytical Results _or Time-lndependent (TI) Case
• Numerical Results for Low Altitude Case w/TI Theory
• Numerical Results for High Altitude (GPS) Case w/TI Theory
m Analytical Development of Weak Time-Dependent (WTD) Theory
• Numerical Results for High Altitude Case w/WTD Theory
(Zonals, Lunar-Solar, and Solar Pressure)
• Numerical Results for High Altitude Case w/WTD Theory
(Zonals, Lunar-Solar, Solar Pressure, and 2x2 Tesserals)
68
SHORT PERIODICS
DSCULATING ELEMENT SPACE ai = _i + Eni,l(E'_) MEAN ELEr4ENTSPACE
27
/ •ai _ _FI(c'X) cni,l(E,_)dk'= 0 ai " EAi,l(E)
R = n + _F6{a,X) (i,= 1,2,3,4,5,6) _ _ n+ eA6,1(_)
ASSUFIE
al = Xio 4-_ [Xic_ cos (_) + ZI(_ sin (ok')]
o:l
BY USE OF THE GENERALIZED METHOD OF AVERAGING
Xlo = _6_6 + eAi,l(_) = _'i
2_
n- Ia---_:L--'T_, k IXio cos (o_)+ Zlo sin (ok')Io=I
DEFINE
Xi(_ Zio
Cio =- _ Dio = oT
zni,l(_,_) =_ I_Ci_ sin (ok)- _Dlo cos (ag)]
%° = o--e¢_ - \_/q6
i = 1,2,3,4,5,6
-- SHORT PERIODIC COEFFICIENTS ARE FUNCTIONS OF _HE FIVE SLOHLY VARYING MEAN ELEMENTS
AND THEREFORE SHOULD ALSO BE SLOWLY VARYING•
-- COUPLING OF THE FAST VARIABLE SHORT PERIODIC VARIATION WITH THE SEMIMAJOR AXIS
SHORT PERIODIC VARIATION.
-- FOR CONSERVATIVE FORCES, ANALYTICAL EXPRESSIONS ARE POSSIBLE FOR ECio AND EDio.
69
L_ ALTITUDE TEST CASE
• EPOCHCONDITIONS: 1974, Oct. 21, I0 hrs, 24 min.
OSCULATINGELEMENTS MEANELEMENTSIPCE)
a = 6644.586 a = 6636.3797
e = .01 _ = .0106045
= 67.g_538_19 o T = 67.97090021 °
: 91,99738418 ° _ = 91.9949106 °
= 200.674168B ° _ = 200.21097331 °
M = 164.3173126 ° M = 164"77124281°
m S/_C • ATMOSPHERE
CO = 2.0 Modified Harris-Priester
Area = 1.86m2 w/FIo'7 = 150
Mass = 677okg
o FORCEMODELS
COWELL(30 second step) SEMIANALYTICAL (I day step)
J2 ..... J6 First Order: J2 .... 'J6 and Dras
and drag Second Order: J_ + J2-Drag Coupling
in the AOG
_IZSAK . Analytical Drag - J?)
?0
HIGH ALTITUDE TEST CASE
I. ASSUME A SET OF EPOCIIMEAN ELEMENTS; THESE ARE 'CONSTANTS' FOR THE
SEMI-ANALYTICAL THEORY
2. AT EPOCH, USE THE SHORT-PERIODIC GENERATOR TO PRODUCE OSCULATING
ELEMENTS
3. CONVERT TIIEOSCULATING ELEMENTS TO POSITION AND VELOCITY; THESE
ARE THE CONSTANTS FOR THE COWELL THEORY
4. PROPAGATE THE ORBIT USING BOTH THE SEMI-ANALYTICAL THEORY AND COWELL
AND COMPARE THE RESULTING POSITION _ND VELOCITY HISTORIES
TEST CASE #2 FORCEMODELS
C_oWELL SEMI-ANALYTICAL
J2'....J6 J2,...,J6 PLUS J22
LUNAR-SOLAR LUNAR-SOLAR (TI)
SOLAR RADIATION PRESSURE SOLAR RADIATION PRESSURE (TI)
"71
SEMI-ANALYTICALTIIEORY
FOR WEAKLYTIME-DEPENDENT
PERTURBATIONS
o OSCULATINGEQUATIONS
dai
_-- : _ Fi(a,k,t)
_k F6(_,k,t)= n+c
• ASSUMEbFORMS
dTi_ -_ .+
_- -- e Al(a,t) ai _ TI . _ ni{a',T,t)
d_ = n(_I) . e A6(a,t) ), = _ + _ n6(_,T,t)t
e MATCHINGEXPRESSIONSFOR dal/dtAND dk/dt GIVES
AI + _ --Bni+ -- : Fl(_,_,t), i = l.....S
_T _t
_n6 _n6 _ 3n
A6 + _ -- + _ = F6(_,_,t) - __ nl(_,_,t)
_ _t 2a
m ASSUME:
_02n Bnl--d_ = O, i = I,...,6
I
t
PHYSICALLY,THISTAKESTHE M-MONTHLIESOUT OF.THE SHORTPERIODICS
e THEN
Al _ _2n Fi(_,_,t) d_, I : I.....6
72
WTD SHORT-PERIODICS
e DEFINE
flS(_,_,t) : Fi(_,_,t) - Ai
e ASSUME
FiS(a,_,t) : _ [Xio(_,t) cos o_ + Zi_(a,t) sin o_]
o=I
ni (a,_,t) : _ -- [Mio(_,t) sin o_ - Nio(_,t) cos _]
e SUBSTITUTING INTO THE MATCHING EXPRESSIONS GIVES PDE's
I _Ni_
Xj_ = Mio - o'-E at
I @Mio
Zio = Nio +-- +__
on at
e ASSUME SOLUTION TO PDE
Mi° : Xi_ + _(I)
A(2)
Nio _ Zia + •
e FIRST ORDER RESULT
N l I I aDi'_a" (3_$ 61 B Cl'(_l sin ak-
e NOTE: Ci,a AND Di,a ARE THE COEFFICIENTS COMPUTED WITH THE TI ASSUMPTION
73
TEST CASE#2
e FORCE MODEL
COWELL SEMI-ANALYTICAL
J2,...,J6 J2,oo.,J6 PLUS J22
LUNAR-SOLAR LUNAR-SOLAR (WTD)
SOLAR RADIATION PRESSURE SOLAR RADIATION PRESSURE (WTD)
o MEAN ELEMENTS
a = 26559.5 km _ = 0o0°
e = o001 m = 0.0°
i 63°0° M = 0.0°
• OSCULATING ELEMENTS
a = 26561.56567 km _ = 359°9999657°
e = .00104842 _ = 359.8560915°
i = 63.001124° M = .1436848842°
TESTCASE#2 RESULTS
TIME (DAYS) Ax(m) Ay(m) Az(m) RSS(m)
2 -.01 .8 1.317 Io54
4 -.22 1.574 2°704 3.14
6 -.63 2.278 4D395 4,99
8 -1.31 2.866 6.274 7.02
I0 -2.14 3.601 7o801 8°85
12 -3°70 4.702 9.342 II.09
14 -5.58 5.322 I0o51Q 13o04
'74
TEST CASE#3
I FORCEMODEL
COWELL SEMI-ANALYTICAL
J2'°°_'J6 J2....'J6 PLUS J22
LUNAR-SOLAR LUNAR-SOLAR (WTD)
SOLAR RADIATION PRESSURE SOLAR RADIATION PRESSURE (WTD)
(C,S)2,I + (C,S)2,2 (C,S)2,1 + (C,S)2,2
• MEAN ELEMENTS
a = 26559.5 km n = O,O°
e = .O01 m = 0.0°
: 63.0° M =" 0.0°
• OSCULATING ELEMENTS
a : 26561.54781 km _ : 359.9999706°
e = .0DI04802 _ = 359.8538535°
i = 63.001118° M = .1459308175°
?5
Figure I. Radial Difference after 23 hours fron Eocch/Semianalytical
minus C_ell for the Lc_1Altitude Circular Test Case
_w_*_ PCS[_ICH OIFFER:-HCE
I, .3 ....... . ......... , ......... . ......... . ......... . ......... . ......... •......... • ......... . ......... .!{
1.000 .
I
I
I
O .800 .
I
0.600 . w
0 r w
I I
A 0.400 . _'_
I w w _.
F 0.200 . w;_w * *
F I ww *
I w w w_ I
C l _ w I
S I . . * I
I -O.ZOO .* * *
H I *
li I *
T -0.400 . _ *
R _ w * *
S I * *
I
-0.600 .
I ww_I
I
I.
-0.800 .
I I
I I
-l,O00 .
I.- ........ . ......... •......... •......... •......... •......... • ......... • ......... "......... "......... "_
0.0 15.00 30.00 _5.00 60.00 75.00 90.00 105.00 120.00 135.00 1SO.O0
TIHE F_QM YYMH_D HH_SS IN MIN
741022 _2100
Perturb ati ons
Cowell: 6xO gravitational field any draq: 30 sec numerical integration
time step
AOG: Ist order analytical expressions for the 6xO mravitational field,
•Zeis' expressions for Jp effects and option 7"f_r drag (48 pt
quadrature order 1 day numerical integraticn time steo
SPG: 6xO gravitational field2(7/48) and draq (7/48) to first order,
•Zeis' expressions for J2 effects,
Initial Conditions: PCE
76
Figure 2. Cross Track Difference after 23 hours from Eooch/Semianalytical
minus Cowel] for the Low Altitude Circular Test Case
_ POSZTIO_I OIFFEREHCE
I.- ........ . ......... . ......... , ......... . ......... . ......... , ......... . ......... , ......... . ......... .I
15.oo0
12.00o
5
$ 9.000 * *
w
T _ _ I
R
A _ w
C 6.000 _ *
K _ _
D Iw w
I w w
F 3.000 * _. *
F _ _
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N 0.0 *
E W * w
S I _
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H I w
I *
T -6.000 w ,t
S I * *
-9.000 . *
I • w_ * I
-I_,000 .
I
I
-15.o0o .
I.- ........ ,......... .......... .......... .......... .......... .......... .......... .......... .......... .I
0.o 15,00 30.00 _5.0o 60.00 75,0o 90.00 io5.oo i_o.o0 135.0o 150.o_
TIME FROM yY_IO0 HH_,MSS IN M:[N
7_lO_Z 9ZI00
Same initial conditions and perturbations as in Figure i,
77
Figure 3. Along Track Difference after 23 hours from EDcch/Seminalytical
tnihus Co, ell for the Lo_ Altitude Circular Test Case
w_ POSITION DIFFEPE,_:CE
2.- ........ • ......... • ......... •......... •......... •......... "......... "......... " ......... " ......... .I
b, 0OO
Z
I
2
4._00 Z
L
O
H
G 3.600
T
R
A
C _ .400
K
D
I
F 1.200
F I
E "r
R
E
N -0.000 Z
C lr
E I
5 'rI
I -1.ZOO
H I Z
I
T -E.400 .w _ *w
E I *
I . w*_ W
-3.600 . *
I w . **w*
I w w
-4.800 .
-6.000 .
io- ........ , ......... - ......... • ......... •......... •......... "......... "......... "......... " ......... .2
0.0 15.00 30.00 45.00 60.00 75.00 90.00 105.00 120.00 135.00 150.00
TIME FR0.5 YTM_3D h'H,_.'I3S2H MIH
7_.102Z 92100
Same perturbations and initial ccnditions as in Figure _,
78
Figure /_ Time Ilt._tory of the Zonal Se=imaJor Axis ghort Periodic CoetEicients _ Eor th_ Low ^ltituJe
_. Circular Te_ Ca_a
I The coefficients, Cls 2 Olb 2
2_ a_d
-- _ame initial ¢ond/t/onn _nd perturb_/on_ _ in dr_; opt/on 6 of AA,_-79-133
79
Figure 5 Time Hietory of the Vr_g _cmim_Jor _ll Short Perlodla Coefficleneat for the Low A1eltudeCircular Test Case
* Tha 1_ coeffLclents_ C1,1 and Ol, _
Same Initial eondittonl an8 per_uxba_ions
_. as in drag opLion 5 of hAS-79-]33
2
2_
D
Z
._
g
Cl,
o r[ME[HAS1.
.... _ .,._ .... #. ?'m, mm l,f,m mn
80
Figure 6. Radfal Difference (Tr Theory)
_*w_ POSITION DIFFERENCE SATELLITE GPS 1111111
I.- ........ . ......... • ......... . ......... , ......... . ......... • ......... , ......... •......... , ......... .
1S,O00 .
z LUNARONLY (TI)
I
I
I
12.000 .
I
I
I
q. OOO . W *_
R I w_ *
O Z _ *
I I w *
A 6.000 . w *
O I _ w
I Z W
F 3.000 . *
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R I *
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I *** * *w
I -3.ooo * w
I ww
M I wwww*w
E I
T -6.000 .
E I
R I
S I
I I
-9. 000 ,
I I
I I
I %
I I
-12.000 ,
I
I
I I
I I
-1S.O00 .
0.0 1,50 3.00 _.50 6.00 7.50 9.00 10.50 lZ.00 13.5D 15.00
TIME FROM YYHMDO HHMMS$ IN HRS
790701 0
ORB1 FILE ON UNIT 2_, DATA RECORDS START AT 790701 0
ORSI FILE ON UNIT 81, OATA RECORDS START AT 790701 0
USER'S NOTES...,.
81
Figure 7. Cross Track Difference (T! Theory)
**_*w POSITION OIFFERENCE SATELLITE GPS iIiiI_I
I ........................................ •........... ; ............................... " ..... •.......... z
I.ooo . LUNARONLY (TI)I
I
I
I
0.600 .
c I
R I
0 Z
5 Z
5 0.600 .
I
T I
R I
A I .w_**ww*
C 0._oo . *** **w
I * ,w*
0 I * **
I I * *_* I
F 0.200 . * **
F I * #w,* I
E I * *w I
R I *w. *
I * **
_( 0.0 ,* * *
C I* * *
E I **
S I _. wW
I *
I -0.200 . * .
N I **
I w *
m I
T -0 ._00 .
E I l* **w
5 I * _,i*
-0.600 . •
I II
I
-0.800 .
I I
I I
I I
I
-1.oo0 .
..................................... ._
_ ........ . ......... . ......... . ......... . ......... , ........ . , . .
I.o 1.SO S.O0 &.50 6.0_ 7.50 9.00 _0.50 IZ.O0 13..50 15.00
TIME FROM YYMMOO HHh_I$S IN HRS
790701 0
ORB1 FILE ON UNIT E_p DATA R_CORD$ START AT 790701 0
ORBI FILE ON UNIT 81, DATA RECORDS START AT 790701 0
USER'S NOTES.....
82
Figure 8. Along Track Difference (TI Theory)
*_*_* POSITION DIFFEREf';CE SATELLITE GPS 1111111
I.- ........ . ......... . ......... . ......... . ......... . ......... . ......... . ......... . ......... . ......... .I
zs.ooo LU_'JARONLY (TI)
20.000
A
{.
0
N
G 15. 000
T
R
C 10,000
K I
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F 5.000 ww, **
E w* _*
E *** * T
N O. 0 **
C _ww.w.w*** w,W 'rE I
S * I
Z
l -5°000 *
rI * Z
m
M
E *
T -10.000
S wW WW,
w_
-15,000 * *
,w. w.
-20.000 _'
N
-Z5.000 w***_*
0.0 1.50 3.00 _,.50 6.00 7.50 9.00 10.50 lZ.O0 13.50 " 15.00
TI_,E FROH YYMHDD HHIiHSS IN HRS
790701 0
ORBI FILE ON UNIT 2¢_ DATA RECDR05 START AT 790701 0
ORB1 FILE ON UNIT 81, DATA RECORDS START AT 790701 0
USER'S NOTES.....
w
83
Figure g. Along Track Difference (WTDTheory)
ww_ww POSITION DIFFERENCE SATELLITE GPS 7907010
I.- ........ . ......... . ......... . ......... . ......... . ......... . ......... . ......... ° ......... • ......... .I
0.55000 .
i LUNARONLY (WTD)I
I
I
0.44000 .
A I
L I
0 I
N X
O 0,33000 .
R I * *_
A I * *
C 0.2_000 . *
K I w w
O I * w
F 0.II000 . w
F I *
E I * *
E !w_ w* * _. **
H -0.00000 ._ w* _ . _.
C I *www* * W *
E I _ w
S I * *
I N W
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ww
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R I *
S I *
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I w
Z
I w
-0.44000 . w
I w * I
I * * I
I * * I
-0.55000 . www*_
Z.-........ °......... ,......... .......... .......... ,......... .......... ,......... ,......... ,......... .
0.0 1.50 3,00 4,50 6.00 7.50 9.00 10.50 1Z.O0 13.50 15.00
TIME FROM YyH_O0 HHMHS5 IH HRS
790701 0
ORB1 FILE ON UNIT 24, DATA RECOROS START AT 790701 0
ORB1 FILE ON UNIT 81, DATA _ECORD5 START AT 790701 0
USER'S NOTES_..'..
84
Figure I0o Along Track Difference (WTDTheory)
*www_ POSITION OIFFEREHCE SATELLITE GPS 7907010
I.- ........ .......... .......... .......... .......... .......... .......... .......... ,......... .......... .I
1.75000 .
i LUNARONLY (WTD)
1._0000
A
L
0
H
G 1.05000
T
R
A
C 0.70000
K
O
I
F 0.35000
F
E
R
E
H 0.00000
C
E
S
I -0.35o00
H
H
E
T -0.70000
E X
R
-l.OSO00 _ _ w w_
_w_ _ ww
w
ww w
-I,_0000 * **
** w
* w
ww _*
-1.7500o wwww_www
I.- ........ .......... .......... o......... .......... •......... •......... •......... •......... •......... .I
61.50 63.00 64.50 66.00 67.50 69.00 70.50 7_.00 73.50 75.00 76.50
TIME FROM YYM,MOO HHHMSS IN HRS
790701 0
ORB1 FILE OH UNIT Z4, DATA RECORDS START AT 790701 0
ORBI FILE ON U_IT 81_ DATA RECORDS START AT 790701 0
USER'S NOTES .....
85
Fiqure II. Radial Difference/Semianalytical minus Cowell for the GPS Test Case
=''_ PCSITICH OIFFEQE_CCE
i 1........ . ......... . ......... . ......... . ......... . ......... . ......... . ......... . ......... . ......... .I
0.35_03 .
i
0._$000 . _ w
I
0.213_0 . * *
D
L , m
I
F 0.07000 _ *
F
E
E *
N -0.00000
C w I
E w * I
I -O.070QO _ W * *_
T -0.1_000
E W
R *
-O.ZlO00
-O._80Q_
-0.35000
I.- ........ .......... .......... ,......... -......... •......... •......... -......... -......... •......... -i
1_.00 13.50 15.00 16.50 18.00 19,50 _.00 _Z,SO 2_.00 _S.50 27,00
TI_ FROM YY_t_O HH_HSS IN HRS
79070_ 0
Perturbations
Cowell: 6xO field, lunar-solar, solar radiation pressure: 300 sec
integration_time step
AOG: Ist order analytical expressions for 6xO field and lunar-solar pt
mass effects, Zeis's J_ expressions, numerical solar radiation
pressure effects (48 p_ quadrature order): l day integration time
step
SPG: Ist order weak time-dependentmodel for 6xO field (7/48), Zeis's J_
expressions
Initial Conditions: [EPC] "1
86
Figure 12. Cross Track Difference/Semi_analytical minus C_Jell for the
GPS Test Case
_=_ POSIT_OH Q[FFE_ED;CE
I.- ........ • ......... •......... . ......... - ......... - ......... • ......... • ......... " ......... " ......... .Z
5.0_C03 i
I
_.cc0oo
c
5 3.00030 w
T
R w
A w_
C _.00_00 _
w
O ww
F 1.00000 _ w
F ww _ I
R ww * I
E w_ I
N O.OOOC3 _
S I ww
w
I -1.00000 wW w
H w_
E
N -_.00000
T
M
Z
T -3.00000 _LE
R
ww I
-_.00000 _
• w* I
_ _ _ 0 G 0 0 0 .
........................... . ......... . .................. . ......... . ......... • ......... . ......... .Z
i2.OO 15.50 15.00 16.50 18.00 19.50 ZI.QO ZZ.50 2_.00 Z5.50 27.00
TIHZ FROH yy_OD HHH_SS IN HRS
790701 0
Same initial conditions and perturbations as in Fi(lureII.
87
Figure 13. Alono Track Difference_eminalytical minus Covlellfor the _PS
Test Case
_--_" PosIrIO_I OIFFEP_CE
! .53003
1.tO_O0
A
L
0
G 0.90000
T
C 0.60000
K
0
F 0.30000
F
[
E
_I 0.00000
C
E *
T -0.60000
-0.90000 * ** *
m _F_
W mm
ww _w
I
-l._OOOO
-1.50000
I-~ ........ -......... •......... -......... •......... •......... •......... •......... •......... •......... .i
12.00 13.50 15.00 16.50 18.00 19.50 _i.00 22.50 _.00 _5.SO ZT.00
TIM_ FROM YYHM30 HHHMSS I_lHRS
190?Ol 0
Same initial conditions and Per_cu.bationsas in Figure II.
88
Figure 14. Radial Difference/Semianalytical.minus Cowell for
GPS (Test Case #3)
*_ POSITIC._ O!FFE_EHCE
1.50 .
I
I
I
1.20 . _ _ _ _
I
Z *
l *
0.90 .
R I
A r *m
0 I
! l
_-, 0.60 . N
t I ,N
F 0.30 . e
E I
N _.00 .e
S I _
Z -0.30 .
I
_- I _ •
T -0.60 .
$ I *
I
-1.ZO .
'r
-1.50 .
0.0 _'.50 5.00 7,50 10.00 1Z.50 15.00 17.50 ZO.O0 Z_.50 Z5.(
T_.E ?_0.'I"(Y_.Ot3 iIII_,_SSIN OYS
790701 Q
ORal FILE ON _IT _ 0ATA _ECO_OS ST_T _T 79070Z 0
0_51 FILE ON UHI7 81, DATA _ECORDS START AT 790701 O
USEr'S _0TES .....
89
Figure 15. Cross Tnack Difference/Semian_lytical minus C_;ell for
GPS (Test Case #3)
_ ........ . ......... . ......... . ......... . ......... . ......... o......... . ......... . ......... . ......... .I
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A semianalytical satellite theory for weak time-dependent perturbations

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