An approximative model is formulated for assessing these perturbations that significantly affect long term modal motion of sun synchronous orbits. Computer simulations with several independent computer programs consider zonal and tesseral gravitational harmonics, third body gravitational disturbances induced by the sun and the moon, and atmospheric drag. A pendulum model consisting of evenzonal harmonics through order 4 and solar gravity dominated nodal motion approximation. This pendulum motion results from solar gravity inducing an inclination oscillation which couples into the nodal precession induced by the earth's oblateness. The pendulum model correlated well with simulations observed flight data

Duck, K. I.

English

NASA Technical Reports Server

LONG PERIOD NODAL MOTION OF SUN SYNCHRONOUS ORBITS 
Kenneth I. Duck 
Goddard Spac;. Fleht Center 
Greenbelt. Maryhnd 
The bun synchronous orbit has been used since the early 1960s for almost every meteorologi- 
cal satellite launched as well as for the two Earth Resources Technology Satellites 
(Landsat-1 and Landsat-2). This well known orbit concept makes use of the earth’s oblate- 
ness to induce a pmcession of the orbit line-of-nodes in order t o  maintain a fixed angular 
1 .entation of the orbit plane relative to the mean sun. The sun synchronous orbit, even 
though it has been used for many years, will continue to be used for most earth observation 
and remote sensing applications because: 
0 The satellite passes through each latitude paint at the same local time thus 
ensuring similar ground lighting conditions on each pass and consequently faciii- 
tates data comparisons. 
The average spacecraft solar array angle of incidence to the sun remains within a 
fixed boundary thus ensuring the availability of electric power. 
The orbits (altitude and inclination) can be chosen such that the majority of the 
earth’s surface can be mapped with near northsouth contigums swaths in a fixed 
period with repeatability . 
The long-period perturbations which disturb sun synchronous orbits have not been the 
subject of detailed investigations in comparison t o  geosynchronous orbits where the litera- 
ture abounds with material. The disturbances acting on sun synchronous orbits has not been 
examined in detail for several reasons: 
0 With the exception of the Landsat satellite, previous sun synchronous vehicles 
have not had orbit adjustment capability to fight the perturbations. 
The spacecraft hardware design lifetimes ( 1 to 2 years) have been sufficiently 
short that the perturbations could be neglected without significantly degrading 
mission petformapce. 
Prior to the use of inertia1 guidance on the Delta launch vehicle, injection uncer- 
tainties were sufficierrtly large so as to  mask any perturbation effects present. 
0 
The assessment of the perturbations acting on sun synchronous orbits becomes more signifi- 
cant when longer lifetime spacecraft are developed as anticipated over the next 10 to 20 years. 
https://ntrs.nasa.gov/search.jsp?R=19760003087 2020-03-22T19:28:33+00:00Z
With improvements in hardware technology and reliability and the on-orbit refurbishment 
capability associated with the advent of Space Shuttle, operational satellite lifetimes could 
easily exceed 5 years. This increase in lifetime is almost a reality today as most earth 
orbiting spacecraft launched in recent years have exceeded their design life. 
The object of the study documented here was to  determine which perturbations significant!y 
affected the long term nodal motion of sun synchronous orbits and then construct an approx- 
imate model which described the phenomena observed. Many computer simulations were 
made with several independent computer programs to  assess the relative effect of various 
combinations of perturbations. Typical of the perturbations included in the simuIations were 
zonal and tesseral gravitational harmonics, third-body gravitational disturbances induced by 
the sun and moon, and atmospheric drag. i t  was observed that a model consisting of even- 
zonal harmonics throuph order 4 and solar gravity dominated the nodal motion. It was further 
observed that for long runs the orbit inclination and orientation of the line-of-nodes ex- 
hibited an oscillating behavior each having the same period. For all the cases run, the inclin- 
ation amplitude was very small (always less than 1 degree); however, the nodal motion could 
be quite large. Due to  these observations, it was feft that a resonance existed between the 
inclination and the nodal motion. The mean daily rate of change in inclination due to solar 
gravity (in radians/mean solar day) was found by analytic averaging to be 
2 
di YS - = 16200 -(I t cos iJ2 sin i sin 2 Qoc 
dT r )  
where 
= mean motion of earth about the sun 
r )  = mean motion of satellite about the earth 
is = solar obliquity 
a, = o'clock angle, the angle between the longitude of the ascending node and th: mean 
sun used computing local time 
i = orbit inclination 
The precession rate of the line-c>f-nodes due to zonal harmonics through order 4 (assuming 
a circular orbit) is 
83 
where 
J,, J, = zonal gnvitationnl coefficients 
Rc = mean equatorial radius of the earth 
a = orbit mimajoraxis 
Differentiating fi, substituting di/dt, and noting that 
sia  =!Lis 
where Js z 0.9856 '/day, then 
n: 
52- = - 16200 - (1 + axis)' sin2 i (A -38  cos2 i) sin 2 Qm 
n 
- -  
where A, B are A and B in radianslmean solar day. 
Examination of Qoc shows that for altitude regions where drag can be neglected, and 
assuming that sin2 i is approximately constant, 
.. 
Roc * k sin 2 sla 
a form of the familiar pendulum equation. It is known that systems which are characterized 
by the above equation can ex3ibit libration or circulatory characteristics. Both characteris- 
tics can be observed on the phase plane plot (figure 1) made for ITOStype orbits. It is 
observed that this system has stable equilibria which correspond to orbits whose line-of-nodes 
lie through the 6:OO a.m. and 6:OO p.m. points. The pendulum equation can be solved 
analytically using elliptic functions with a typical solution for a quarter cycle oscillation 
being 
1 2m cn2 F 1 -rn sn2 (67') 1 2 52, =-cos-I -1 t 
L J 
84. 
where 
m = 1/2(1 +cos2 R-) 
o <T<P, /4  
Note that P, denotes the libration period and noro denotes the value of o’clock angle where 
hoc = 0. -The libration period, P, is 
4 K(m) PL = -a 
where K (m) denotes a complete elliptic integral of the first kind. Figure 2 shows the 
o’clock angle libration period for ITOS-type orbits (h = 1489 km, i = 101.9’). It is seen that 
the libration period increases from 26 years as the reference o’clock angle moves from one 
of the stable equilibria toward one of  the unstable ones. Investigation of the libration 
period has shown that the minimum libration period lies between 22 to 30 years for sun 
synchronous orbits between 200 and 2000 km. 
0.a 
0- 
0.03 
0.m 
0.01 
0 
-0.01 
-0.31 
X.03 
-0.01 
annm.iaa KU 
RCLINAT*)nlOl.Y29 
o 30 a so im 1x1 la n o  za nu a0 m 1~ 
I I 1 1 1 1 I 1 1 1 I I 1 
12:m 1400 1000 1100 20:00 2200 M:00 a00 01:OO W O O  W:00 1000 1200 
LOWL TIME Of ASCENDING NODAL CROSSING 
I I I I I 1 1 1 I I I 1 1  
N:OO a 0 0  WOO 0I:W MOO 1000 1200 1440 1100 lt00 m:00 2200 N:W 
LOCAL TIME Of DESCENOIM NODAL CROSSING 
OTLOCK ANGLE.Ooc .D€O. 
Figure 1. O’clock angle motion in phase space for ITOS-type orbit. 
85 
" r 
0 
Y 
P 
F? 
I 
f 
< r Y
MTITUIW-1- KM 
YCLINATIm 101.950 
I I 1 1 I 1 1 I 1 I 1 1  20 
o 30 m JO 1 z o i 1 5 o i m n o m z m 9 o m 3 p  
) : : : : : : : : : : : : : : : ! : : : : : : : I  
INITIAL OTLOCI *LE 
w o o W 1 4  16 18 20 22 N 2 4 6 8 1 O w o o W  
LOCAL TIM€ OF ASCEWDUIG NOD€ CROs5)MG-WURS 
1 : : : : : : : : : : : : : : : : : : : : : : : i  
MIDNIGHT 2 4 C 8 10 woo(y 14 16 18 20 24 MlMlffiHl 
LOCAL TWE OF MSZIIMNG NOD€ EROESWi-HRIRS 
Figure 2. Libration period for ITOS3vpe orbit. 
The pendulum analogy has been compared with both simulation and flight spacecraft data. 
Figure 3 compares the approximate solution with several simulations having various combin- 
ations of perturbations. The reference orbit in this comparison is the ITOStype (h = 1489 km, 
i = 101.9') having an ascending nodal crossing at 3:OO p.m. local time. It is seen that there 
is excellent agreement between the pendulum analogy and those simulations which include 
zonal harmonics and solar gravity. 
The remaining figures (4  through 9) compare the o'clock angle time history gercrated using 
the pendulum model with that obtained from Brouwer mean elements for several flight 
spacecraft. In examining these comparisons one observes that thcrc are two curves repre- 
senting approximate nodal drift propagation. The dashed curve is the propagation calculated 
using a set of orbit elements at the initial epoch for the particular satellite. The solid 
curve uses an iterated value of inclination to improve the agreement. This approach was 
taken because of the large relative uncertainty in measuring the inclination. It is seen in all 
cases that excellent agreement is obtained. In examining these data, note that the ESSA-2 
solution (figure 4) is a case where the motion lies in the circulatory region in phase space 
while the other wlutions lie in the libration region. The ESSA-8 solution (figure 7) exhibits 
the turnaround expected for the libration motion. The final observation to be noted is for 
Nimbus-5 (figure 9) where the node is located near one of the unstable equilibria. There is 
still good agreement for this case where the disturbing acceleration is near zero. 
In conclusion, the nodal motion of sun synchronous orbits has been investigated and found 
to exhibit the characteristics of a pendulum. This pendulum motiorr results from solar 
86 
gravity inducing an inclination oscilktion which couples into the nodal precession induced 
by the earth's oblateness. The pendulum model has been compared with simulations and 
flight data with excellent correlation observed. 
r 
Figure 3. Approximate ulution. 
- PENWLVM IVCROXIMATKYI 
111' i ITERATED INITIAL lPr* INATION 
BAS0 ON €SA2 FLlCnT EL MCNTS A 
Figure 4. ESSA-2 pendulum solution comparison. 
87 
X 
0 
54 
- PENDULUM UNK)XIYATION - PENWLW UCROXIMATION 
WITH ITERATE0 INITIAL IWCLINATWW 
M S i D  (rN T11105Y FLIGHT ELEMENTS A 
r’ 
0 190 200 JOO 400 500 
TIME-DAIS AFTER 2/Mo 
Figure 5.  Tiros-M pendulum solution comparison. 
Figure 8. ESSA-7 c idulum solution comparison. 
88 
- - - - --PENDULUM AWROXIYATION  CENDULUY AIFROXIMATION 
WITH ITERATED INITIAL INCLINATIDN 
W E D  ON ESSAI FLKinT E LEYENTS A 
130 
TIME DAVS AFTER 1/16'60 
Figure 7. ESSA-8 pendulum solution comparison. 
__- - -  mNWLUM AIFROXIMATIW 
- PENDULUM A?WDXIMATIDN 
WITH ITERATED INITIAL INCLINATION 
A WSED Oh'NOA.&l FLIGHT ELEMENTS 
132 8 
1 9 . 6  
1 9 . 4  
1u.i 
0 loo 100 y*l uy) 600 (100 700 
TIME DAVS AFTER 10/1C72 
J 
Figure 8. NOAA-2 pendulus-- solution comparison. 
89 
--- --- PENDbCJY APPROXIMATION 
- PENDULUM IV’ROXIMATKJN 
WITH ITERATE0 INITIAL INCLINATION 
m’6 r 
A RASEDONNIMWSSFLIGHT FLEMENTS 
\ 
\ 
\ 
d ‘. 
m.1 
1 1 I 1 I J 
1Q) 2m 90 4w 500 ma 0 
TIME.DAVS AFTER 12RU72 
Figure 9. Nimbus5 pendulum solution comparison. 
90 


Long period nodal motion of sun synchronous orbits

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