Roll and attitude stability for gravity gradient satellite

Anand, D. K.

Whisnant, J. M.

English

NASA Technical Reports Server

TG-986 
APRIL 1968 
Technical Memoramhim 
ROLL RESONANCE FOR A 
GRAVITY -GRADIENT SATELLITE 
by J. M. WHISNANT and D. K. ANAND 
THE JOHNS HOPKINS UNIVERSITY . APPLIED PHYSICS LABORATORY 
8621 Georgia Avenue, Silver Spring, Maryland 20910 
O p e r a t i n g  under Contract NOW 69-0604-c wi th  the Department of the Navy 
This document has bean approved far public 
release and sole; its distribution is unlimited. 
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TnE JOHNS WOPKINS UNIVERSITV 
APPLIED PHYSICS LABORATORY 
SILVSR SPRlNO. MARYLAND 
ABSTRACT 
Previous s t u d i e s  of grav i ty-gradien t  s a t e l l i t e  a t t i t u d e  s t a b i l i -  
za t ion  showed t h a t  l a rge  roll angles  could occur f o r  a p a r t i a l l y  
shadowed o r b i t .  The cause w a s  determined t o  be a resonance e f f e c t  
due t o  s o l a r  r a d i a t i o n  pressure.  Here an expression for t h e  solar 
torque f o r  roll on a dumbbell-shaped satel l i te  is  presented. The 
amplitude of t h e  torque i s  shown t o  be a f'unction of t h e  angle  
between t h e  s a t e l l i t e - s u n  l i n e  and t h e  normal t o  t h e  o r b i t  plane.  
For c i r c u l a r  o r b i t s ,  an expression i s  der ived t o  determine f o r  what 
p o s i t i o n  of t h e  sun r e l a t i v e  to t h e  o r b i t  plane t h e  resonance e f f e c t  
i s  a maximum. 
shadowed as a funct ion of sun-orbi t  o r i e n t a t i o n  is determined. The 
For o r b i t s  of modest e c c e n t r i c i t y ,  t h e  amount of o r b i t  
pers i s tence  of the resonance e f fec t  fo r  retrograde o r b i t s  i s  discussed 
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THE JOHNS HOPKINS UNIVERSITY 
APPLIED PHYSICS LABORATORY 
SILVER S P R I N G .  M A R Y L A N D  
TABLE OF CONTENTS 
Abstract . 
List of Illustrations 
INTRODUCTION . 
ANALYSIS . 
References.  
Acknowledgment . 
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LIST OF ILLUSTRATIONS 
Figure Page 
1 Solar Torque for  Roll During One Nodal 
Period . 3 
2 Twice Orbital Roll Forcing Function for a 
Partially Shadowed Orbit 6 
3 Shadowing for  an Eccentric Orbit a 
4 Time Variation of the Resonance Effect . 9 
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THE JOHNS HOPKINS UNIVERSITY 
APPLIED PHYSICS LABORATORY 
SILVER SPRING. MARVLAND 
ABSTRACT 
Previous s t u d i e s  of  grav i ty-gradien t  s a t e l l i t e  a t t i t u d e  s t a b i l i -  
za t ion  showed t h a t  l a rge  r o l l  angles  could occur f o r  a p a r t i a l l y  
shadowed o r b i t .  
due t o  so la r  r a d i a t i o n  pressure.  
torque f o r  r o l l  on a dumbbell-shaped sa t e l l i t e  i s  presented.  
amplitude of t h e  torque i s  shown t o  be a func t ion  of  t h e  angle  
between t h e  s a t e l l i t e - s u n  l i n e  and t h e  norrllal t o  t h e  o r b i t  plane.  
For c i r c u l a r  o r b i t s ,  an expression i s  der ived t o  determine f o r  what 
p o s i t i o n  of t h e  sun r e l a t i v e  t o  t h e  o r b i t  plane the  resonance e f f e c t  
i s  a maximum. 
shadowed as a func t ion  of sun-orbi t  o r i e n t a t i o n  is  determined. 
pe r s i s t ence  of t he  resonance e f f e c t  f o r  re t rograde  o r b i t s  i s  discussed.  
The cause was determined t o  be a resonance e f f e c t  
Here an expression f o r  t h e  solar 
?'he 
For o r b i t s  of modest e c c e n t r i c i t y ,  t h e  amount of  o r b i t  
The 
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APPLIED PHYSICS LABORATORY 
SILVER SPRING. MARVLAND 
In t roduct ion  
&e- and post-launch s t u d i e s  of t h e  geodetic s a t e l l i t e  (GEOS-A), 
launched i n  November 1966, ind ica ted  t h a t  large roll angles  could occur 
when t h e  o r b i t  w a s  p a r t i a l l y  shadowed [l]. An examination of t h e  s o l a r  
r a d i a t i o n  pressure  fo rc ing  func t ion  f o r  roll showed t h a t  it had a com- 
ponent with twice o r b i t a l  frequency. Since t h e  n a t u r a l  frequency of roll 
f o r  a gravi ty-gradien t  s a t e l l i t e  i s  a l s o  twice o r b i t a l ,  t h e  occurrence of 
resonance i s  c l e a r .  Furthermore, t h e  amplitude of t h e  roll l i b r a t i o n s  w a s  
shown t o  be a func t ion  of 
ear th-sun  l i n e  onto t h e  e q u a t o r i a l  plane and t h e  l i n e  of nodes. The 
purpose of t h i s  note i s  t o  der ive  an expression f o r  t h e  value of s1 
which maximizes t h e  twice o r b i t a l  component of t h e  r o l l  fo rc ing  func t ion .  
s? , t h e  angle between t h e  p ro jec t ion  of t h e  
Ana lys i s  
The s o l a r  forc ing  func t ion  cp i n  roll f o r  a s a t e l l i t e  whose geometry 
i s  s i m i l a r  t o  t h a t  of a dumbbell i s  given by [l] as 
cp = cp s i n  i s i n  61 (1) 
0 
where i is  t h e  o r b i t a l  i n c l i n a t i o n  and t h e  s a t e l l i t e  i s  i n  t h e  s u n l i t  p a r t  
of t h e  o r b i t  with t h e  ear th-sun  l i n e  ly ing  i n  t h e  e q u a t o r i a l  plane. 
is  a func t ion  of t h e  s o l a r  flux and phys ica l  p rope r t i e s  of t h e  s a t e l l i t e  
as w e l l  as a weak func t ion  of t h e  s a t e l l i t e ' s  a t t i t u d e .  For s m a l l  l i b r a t i o n a l  
cpo 
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THE JOHNS HOPKINS UNIVERSITY 
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angles ,  it may be considered constant.  If the  sun i s  allowed t o  have a 
dec l ina t ion ,  6, , then  t h e  forc ing  func t ion  is  
where 17 
t h e  normal t o  t h e  o r b i t  plane.  
zero r i g h t  ascension of t h e  sun, i t s  d i r e c t i o n  cosines a r e  (cos b S ,  0, s i n  a s ) .  
The d i r e c t i o n  cos ines  of t h e  normal t o  t h e  o r b i t  plane are then  given by 
( s i n  i s i n  hl, - s i n  i cos SI, cos i) s o  t h a t  
i s  i n t e r p r e t e d  t o  be t h e  angle between t h e  earth-sun l i n e  and 
Using an e q u a t o r i a l  coordinate system and 
cos r( = [ s i n  i s i n  R cos 6 + cos i s i n  b S ] .  
S 
( 3 )  
(4) 
If t h e  o r b i t  i s  p a r t i a l l y  shadowed then  
'Po cos r( 
0 s a t e l l i t e  i n  shadow 
s a t e l l i t e  i n  sun l igh t  
cp = [  
as shown i n  Figure 1. 
The amplitude of  t h e  second harmonic when equation ( 4 )  i s  expanded by 
a Four ie r  s e r i e s  i s  
( 5 )  
where @ i s  t h e  amount of  o r b i t  shadowed i n  rad ians  and 
of s a t e l l i t e  l a t i t u d e .  
8 i s  t h e  argument 
Analysis of equation ( 5 )  with reference t o  equation (3)  ind ica t e s  t h a t  
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t h e  va r i ab le  i n  ( 5 )  i s  h2 f o r  any given s a t e l l i t e ,  o r b i t ,  and epoch. It 
is  t o  be noted t h a t  s ince  @ is  a funct ion of , it a l s o  i s  a funct ion of 
0, as w i l l  be  shown. 
For c i r c u l a r  o r b i t s  Pa t t e r son  121 gives  a formula for t h e  t i m e  t h a t  a 
sa te l l i t e  spends i n  t h e  sun. 
i s  used t o  ob ta in  
This expression,which neglec ts  penumbra e f f e c t s ,  
F I  1 s i n  7 
-1 p = I T - 2 s i n  
where 
2 1/2 F = (1 - 1/a ) 
and a i s  t h e  semi-major a x i s  i n  u n i t s  of e a r t h  radi i .  
S u s t i t u t i n g  t h e  above i n t o  ( 5 )  y i e l d s  
2 2 1/2 
( s i n  7 - F ) ] 2F 
s i n  7 
[ c o s  7 2 Cp(2e) = - 'PO 
T I  
Since w e  are i n t e r e s t e d  i n  maximizing (7 ) ,  w e  set dq/dG = 0 and using 
equation (3) obta in ,  a f t e r  some a lgebra ,  
(7) 
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The above equat ion provides t h e  nodal angle  t h a t  w i l l  g ive t h e  l a r g e s t  
r o l l  l i b r a t i o n  with twice o r b i t a l  frequency. A s  an example, consider  a 
sa te l l i t e  which has  a = 1.2,  i = 74 The v a r i a t i o n  of t h e  
"twice o r b i t a l  r o l l "  amplitude f o r  a p a r t i a l l y  shadowed o r b i t  i s  shown 
i n  Figure 2. 
0 and bS = 0. 
For an e c c e n t r i c  o r b i t ,  Escobal [ 3 ]  gives t h e  shadow equat ion as a q u a r t i c  
i n  t h e  t r u e  anomaly which cannot i n  genera l  be solved a n a l y t i c a l l y .  
expressions are der ived t o  r e f l e c t  t h e  f i r s t - o r d e r  e f f e c t s  of e c c e n t r i c i t y ,  
E. For t h e  case 
c o n s t r a i n t s  on t h e  a p p l i c a b i l i t y ) , t h e  arguments of l a t i t u d e ,  eI and eo , 
a t  which t h e  s a t e l l i t e  e n t e r s  and ex i t s  from t h e  e a r t h ' s  shadow satisfy 
Here 
= 0 ( t h i s  s impl i f i e s  t h e  a lgebra  without imposing 
6 S  
(9) 2 1/2 cos 8 cos s2 - cos i s i n  0 s i n  Lz = -(1 - l/r ) , where 
r 
a t  8 = TT i s  given by 
i s  t h e  magnitude of t h e  rad ius  vec tor  t o  t h e  s a t e l l i t e , a n d  f o r  per igee 
P 
Expanding t h e  right-hand side of (9) and r e t a i n i n g  only terms of f irst  order  
i n  e c c e n t r i c i t y  y i e l d  
cos e E = F+-- 2 1/2 2 F  (1 - I/' 1 a 
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THE JOHNS HOCKINS UNIVERSITY 
APPLIED PHYSICS LABORATORY 
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p (degrees) 
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where F has  been defined previously.  S u b s t i t u t i n g  (11) i n t o  (9) and 
solving f o r  f3 ( =  e o  - e1), we obta in  
-1 2F 2 2 1/2] 
@ = s i n  [y ( A  - F ) 
where 
A 2 = ( c o s  + - E l2 + (cos i s i n  62) 2 . 
F a  2
Using t h i s  expression i n  ( 5 ) ,  t h e  amplitude of t h e  second harmonic becomes 
Although (13) i s  similar t o  (7),  s e t t i n g  dcp/dhl = 0 no longer y i e l d s  a 
closed-form so lu t ion .  However, any of s e v e r a l  numerical  techniques are 
app l i cab le .  Figure 3 shows f3 = @(a) using both (6)  and (E') and compares 
them with f3 obtained by numerically so lv ing  Escobal ' s  equation. 
The s e l e c t i o n  of t h e  co r rec t  o r b i t a l  parameters becomes p a r t i c u l a r l y  
s i g n i f i c a n t  when we consider  a re t rograde  o r b i t .  In such an o r b i t  t h e  
and t h e  precession of t h e  sun (which i s  p62 sense of nodal precession 
about 0.985 degrees/day) i s  t h e  same. 
and i , a , e  could be so se l ec t ed  t h a t  t h e  magnitude of nodal precession 
Furthermore, Po = f ( i  ,a,  E )  [ 41, 
becomes equal  t o  t h a t  of t h e  sun. If indeed t h i s  happens and t h e  o r b i t  i s  
p a r t i a l l y  shadowed t h e  condi t ion  of r o l l  resonance can p e r s i s t  near  i t s  
maximum value f o r  long per iods 
prograde o r b i t s  t h e  precess iona l  sense of hl i s  opposi te  t o  t h a t  of t h e  
(Figure 4 ) .  We note ,  however, t h a t  f o r  
t h e  sun,and r o l l  resonance w i l l  not p e r s i s t  f o r  long per iods s ince  t h e  
f r a c t i o n  of shadowed o r b i t  i s  i tself  going t o  vary.  
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THE JOHNS HOPKINS UNIVERSITY 
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REFERENCES 
1. Pisacane, V. L . ,  Pardoe, P. P . ,  and Hook, B. J. ,  " S t a b i l i z a t i o n  
System Analysis and Performance of t h e  GEOS-A Gravity-Gradient 
S a t e l l i t e , "  Jou rna l  of Spacecraf t  and Rockets, Vol. 4, No. 12, 
Dec. 1967, p. 1626. 
2. Pa t t e r son ,  G.  B., "Graphical Method f o r  Predic t ion  of Time i n  
Sunl ight  for a Ci rcu la r  Orbi t ,"  ARS Journa l ,  V o l .  31, No. 3, 
March 1961, p. 441. 
3.  Escobal, P. R . ,  "Orbi ta l  Entrance and Ex i t  From t h e  Shadow of t h e  
Earth,"  ARS Journa l ,  Vol. 32, No. 12, Dee. 1962, p. 1940. 
4. King-Hele, D . ,  Theory of S a t e l l i t e  Orbits i n  an Atmosphere, 
Butterworths,  London, 1964. 
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ACKNOWLEDGMENT 
The au thors  would l i k e  t o  acknowledge a h e l p f u l  d i scuss ion  
with Robert E .  Jenkins  of t h e  Applied Physics Laboratory. 
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R E P O R T  D A T E  7 a .  T O T A L  N O .  O F  P A G E S  
April 1968 11 
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DOCUMENTCONTROLDATA- R B D  
7 6 .  N O .  O F  R E F S  
4 
3a. C O N T R A C T  O R  G R A N T  N O .  
R 011 Resonance for a Gravity -Gradient Sat ellit e 
9a. O R I G I N A T O R ' S  R E P O R T  N U M B E R I S )  
4 .  D E S C R I P T I V E  N O T E S  ( T y p e  of report and  i n c l u s i v e  da tes )  
1 1 .  S U P P L E M E N T A R Y  N O T E S  
5 .  A U T H O R 1 S )  (Firs t  name, midd le  i n i t i a l ,  l a s l  name)  
1 2 .  S P O N S O  R I N  G MI L I  T A R  Y A C T I  V I  T Y  
NA SA 
J. M. Whisnant and D. K. Anand 
NOW 62-0604-c 
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th is  report)  
I 
I O .  D l S T R l E I U T l O N  S T A T E M E N T  
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3 A B S T R A ~ T  
Previous studies of gravity -gradient sat ell it e attitude st abili zat ion showed 
that large roll  angles could occur for a partially shadowed orbit. 
determined t o  be a resonance effect due t o  solar radiation pressure. 
sion for the solar torque for roll  on a dumbbell-shaped satellite is presented here-  
in .  
satellite-sun line and the normal t o  the orbit plane. For circ.dar orbitsS, an ex- 
pression is derived to  determine for what position of the sun relative t o  the orbit 
plane the resonance effect is a maximum. 
amount of orbit shadowed a s  a function of sun-orbit orientation is determined. 
persistence of the resonance effect for retrograde orbits i s  discussed. 
The cause w a s  
An expres- 
The amplitude of the torque is shown t o  be a function of the angle between the 
For orbits of modest eccentricity, the 
The 
UNCLASSIFIED 
Security Classification 
I D  1 N O V 6 S  1473 
UNCLASSIFIED 
Security Classification 
14. 
K E Y  WORDS 
Gravity -gradient st a biliz at i on 
R e  sonance 
Orbit shadowing 
UNCLASSIFIED 
Security Classification 
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Roll Resonance for a Gravity-gradient Satellite

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