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Effect of environmental forces on the attitude dynamics of gravity oriented satellites Flanagan, Ralph Clarence 1969

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EFFECT OF ENVIRONMENTAL FORCES ON THE ATTITUDE DYNAMICS OF GRAVITY ORIENTED SATELLITES by RALPH CLARENCE FLANAGAN B.Sc. (ME) U n i v e r s i t y of New Brunswick, 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of Mechanical E n g i n e e r i n g We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA J u l y , 1969 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l l m e n t o f t h e r e -q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l -a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by t h e Head o f my D e p a r t m e n t o r by h i s r e p r e s e n -t a t i v e s . I t i s u n d e r s t o o d t h a t p u b l i c a t i o n , i n p a r t o r i n w h o l e , o r t h e c o p y i n g o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . R a l p h C. F l a n a g a n D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g The U n i v e r s i t y o f B r i t i s h C o l u m b i a V a n c o u v e r 8 , C a n a d a ABSTRACT The i n f l u e n c e of the major environmental forces on the a t t i t u d e response of g r a v i t y gradient s a t e l l i t e s i s i n v e s t i g a t e d using a n a l y t i c a l and numerical techniques. The study e s t a b l i s h e not only the e f f e c t of these forces on system performance but a l s o t h e i r r e l a t i v e importance. The problem i s i n v e s t i g a t e d i n the order of i n c r e a s i n g d i f f i c u l t y which corresponds to a systematic r e d u c t i o n i n a l t i t u d e . In g e n e r a l , the n o n - l i n e a r , non-autonomous nature of the system renders the determination of a closed form s o l u t i o n v i r t u a l l y impossible. Hence, numerical techniques are employed, i n conjunction w i t h i n v a r i a n t surfaces or i n t e g r a l manifolds, to analyse the system. For a given set of parameters, the l a r g e s t such surface defines the bound of s t a b l e motion; on the other hand, the s m a l l e s t surface that can be found ( i . e . , a l i n e or set. of l i n e s ) represents the dominant p e r i o d i c s o l u t i o n with which these manifolds are a s s o c i a t e d . The a n a l y s i s e s t a b l i s h e s the importance of p e r i o d i c s o l u t i o n s as they provide the 'frame' about which s t a b i l i t y charts are b u i l t . Furthermore, a v a r i a t i o n a l s t a b i l i t y a n a l y s i s of these s o l u t i o n s , using Floquet theory, a c c u r a t e l y determines the termination of the spikes and e s t a b l i s h e s the c r i t i c a l e c c e n t r i c i t y f o r s t a b l e motion. Phase I i n v e s t i g a t e s the a t t i t u d e dynamics of s a t e l l i t e s at high a l t i t u d e s where g r a v i t y gradient and d i r e c t s o l a r r a d i a t i o n c o n s t i t u t e the predominant torques. The approximate closed form s o l u t i o n , obtained using the WKBJ and Harmonic Bal-ance methods, was found to p r e d i c t the l i b r a t i o n a l response of a s a t e l l i t e w i t h considerable accuracy. As the s a t e l l i t e s r e q u i r i n g s t a t i o n keeping permit only small amplitude motion, the a n a l y t i c a l r e s u l t s are of s u f f i c i e n t accuracy to be u s e f u l during p r e l i m i n a r y design stages. The response and s t a b i l i t y bounds of the system, obtained n u m e r i c a l l y , are shown through the use of 'system p l o t s ' and ' s t a b i l i t y c h a r t s ' . The r e s u l t s i n d i c a t e a considerable e f f e c t due to s o l a r r a d i a t i o n on the a t t i t u d e dynamics of a s a t e l l i t e . The use of s o l a r r a d i a t i o n i n c o n t r o l l i n g the s a t e l l i t e a t t i t u d e i s explored. The optimized r e s u l t s show t h i s system to be q u i t e e f f e c t i v e , being capable of p r o v i d i n g a p o i n t i n g accuracy of 0-5° depending on o r b i t e c c e n t r i c i t y . The extension of the a n a l y s i s to the intermediate a l t i t u d e ranges, where d i r e c t earth r a d i a t i o n , i t s albedo and shadow become s i g n i f i c a n t , i s undertaken i n phase I I . A comprehensive i n v e s t i g a t i o n was made p o s s i b l e by the determination of closed form expressions f o r earth r a d i a t i o n f o r c e s . This was accomplished through the concept of c u t t i n g plane distance r a t i o s . The a n a l y s i s shows only l o c a l v a r i a t i o n s due to earth r a d i a t i o n s without s u b s t a n t i a l l y a f f e c t i n g the maximum l i b r a t i o n a l amplitude or mainland s t a b i l i t y area. Hence, f o r a l l p r a c t i c a l purposes, d i r e c t e a rth r a d i a t i o n , i t s albedo and shadow can be neglected i n such s t u d i e s . Phase I I I i n v e s t i g a t e s the dynamics of clo s e earth' s a t e l -l i t e s i n the presence of aerodynamic and r a d i a t i o n f o r c e s , thus i v covering the remaining a l t i t u d e range. The r e s u l t s show tha t a p r e c i s e dynamic a n a l y s i s r e q u i r e s the c o n s i d e r a t i o n of both aerodynamic and d i r e c t s o l a r r a d i a t i o n f o r c e s . The i n v e s t i g a t i o n helps i n e s t a b l i s h i n g an a l t i t u d e range i n which a pure g r a v i t y gradient a n a l y s i s i s l i k e l y to be most a p p l i c a b l e . The a p p l i c a t i o n of t h i s a n a l y s i s to the r e p r e s e n t a t i v e g r a v i t y gradient s a t e l l i t e , GEOS-A, over the e n t i r e a l t i t u d e range, e x e m p l i f i e s the f i n d i n g s of the parametric study. TABLE OF CONTENTS p t e r Page 1 INTRODUCTION 1 1.1 P r e l i m i n a r y Remarks 1 1.2 L i t e r a t u r e Review 5 1.3 P u r p o s e and Scope o f t h e I n v e s t i g a t i o n . . . 9 2 EFFECT OF DIRECT SOLAR RADIATION ON ATTITUDE • DYNAMICS OF A S A T E L L I T E 12 2.1 F o r m u l a t i o n o f t h e P r o b l e m 13 2.1.1 E q u a t i o n s o f M o t i o n f o r an A r b i t r a r i l y Shaped S a t e l l i t e 13 2.1.2 D i r e c t S o l a r R a d i a t i o n T o r q u e and G e n e r a l E q u a t i o n o f M o t i o n 17 2.2 S y s t e m R e s p o n s e 21 2.2.1 A n a l y t i c a l and N u m e r i c a l A p p r o a c h . .. 21 2.2.2 R e s p o n s e P l o t s and D i s c u s s i o n o f R e s u l t s 33 2.3 L i b r a t i o n a l S t a b i l i t y 44 2.3.1 N u m e r i c a l A n a l y s i s 44 2.3.2 V a r i a t i o n a l A n a l y s i s o f P e r i o d i c S o l u t i o n s 56 2.3.3 D i s c u s s i o n o f R e s u l t s 58 2.4 R a d i a t i o n Damping and A t t i t u d e C o n t r o l . . . 66 2.4.1 Damping C o n c e p t 66 68 2.4.2 A n a l y s i s and D i s c u s s i o n o f R e s u l t s . . 2.5 C o n c l u d i n g Remarks 78 v i C h a p t e r Page 3 ATTITUDE DYNAMICS OF A S A T E L L I T E ACCOUNTING FOR EARTH RADIATIONS 81 3 .1 F o r m u l a t i o n o f t h e P r o b l e m 82 3 . 1 . 1 F o r c e Due t o D i r e c t S o l a r R a d i a t i o n 82 3 . 1 . 2 F o r c e Due t o D i r e c t E a r t h R a d i a t i o n 83 3 . 1 . 3 F o r c e Due t o E a r t h A l b e d o . . . . . 90 3 . 1 . 4 E f f e c t o f E a r t h ' s Shadow and t h e G e n e r a l E q u a t i o n o f M o t i o n 121 3.2 S y s t e m R e s p o n s e and S t a b i l i t y A n a l y s i s . . 122 3 .3 C o n c l u d i n g Remarks 139 4 ATTITUDE DYNAMICS OF CLOSE EARTH SATELLITES . . 141 4 .1 F o r m u l a t i o n o f t h e P r o b l e m 142 4 . 1 . 1 A e r o d y n a m i c T o r q u e 142 4 . 1 . 2 G e n e r a l E q u a t i o n o f M o t i o n 145 4 .2 I n f l u e n c e o f E a r t h R a d i a t i o n s on S a t e l l i t e R e s p o n s e 147 4 . 3 S y s t e m Response and S t a b i l i t y A n a l y s i s . . 152 4 .4 C o n c l u d i n g Remarks 167 5 CLOSING COMMENTS 168 5 .1 C a s e S t u d y , GEOS-A 168 5 .2 Recommendations f o r F u t u r e Work 17 2 BIBLIOGRAPHY 173 L I S T OF TABLES T a b l e Page 1 R e p r e s e n t a t i v e G r a v i t y G r a d i e n t S a t e l l i t e C h a r a c t e r i s t i c s 3 2 V a r i a t i o n o f Y w i t h S y s t e m P a r a m e t e r s 100 o 3 V a r i a t i o n o f yn w i t h S y s t e m P a r a m e t e r s 107 L I S T OF F I G U R E S F i g u r e Page 1.1 V a r i a t i o n o f maximum t o r q u e s w i t h a l t i t u d e f o r a r e p r e s e n t a t i v e s a t e l l i t e 4 1.2 S c h e m a t i c d i a g r a m o f t h e p r o p o s e d p l a n o f s t u d y . . . . „ . . . . „ . „ . . . . . . . . ' . . 11 2.1 Geometry o f s a t e l l i t e m o t i o n . 14 2.2 D i r e c t s o l a r r a d i a t i o n and s a t e l l i t e g e o m e t r y . . . . . . . . . . . 19 2.3 V a r i a t i o n o f f u n c t i o n s F and J w i t h s y s t e m p a r a m e t e r s : (a) K.=1.0 . . . . . . . . . . 25 (b) K i =0 .8 26 (c) K.=0.6 27 2.4 Bounds o f d e s i g n p a r a m e t e r s f o r J =1/3' . . . . 30 ^ ^ max ' J U 2.5 E f f e c t o f r a d i a t i o n p r e s s u r e on l i b r a t i o n a l r e s p o n s e o f a s a t e l l i t e as o b t a i n e d u s i n g a n a l y t i c a l and n u m e r i c a l methods: (a) s m a l l a m p l i t u d e i m p u l s i v e d i s t u r b a n c e ; V (0)=0.1 . . . . . 34 (b) s m a l l a m p l i t u d e a r b i t r a r y d i s t u r b a n c e ; i|> (0)=2.9° , (0)=0.05 35 (c) i n t e r m e d i a t e r e s p o n s e i n e c c e n t r i c o r b i t ; e=0.1 36 (d) l a r g e a m p l i t u d e r e s p o n s e showing t h e i n f l u -e n c e o f i n c r e a s e i n s o l a r p a r a m e t e r ; c=0.3 . 37 2.6 C o m p a r i s o n o f i n t e g r a l m a n i f o l d c r o s s - s e c t i o n s a t 9=0 f o r c o n d i t i o n s c o r r e s p o n d i n g t o t h o s e i n F i g u r e 2.5 39 i x F i g u r e Page 2.7 S y s t e m p l o t s s h o w i n g t h e maximum l i b r a t i o n a l a m p l i t u d e and a v e r a g e p e r i o d f o r a r a n g e o f e c c e n t r i c i t y as a f f e c t e d by: (a) s o l a r and i n e r t i a p a r a m e t e r s 41 (b) s o l a r a s p e c t a n g l e and i n i t i a l c o n d i t i o n e 42 (c) i n i t i a l d i s t u r b a n c e IJJ(O) and ty' (0) 43 2.8 T y p i c a l l i m i t i n g i n v a r i a n t s u r f a c e s : (a) e=0, c=0.5, K ^ l . O 47 (b) e=0.1, c=0.1, K i=1.0 48 ( c 1 ) e=0.1, c=0.5, K i=1.0 49 (C2) s t a b l e i s l a n d s f o r 2.8 (c^) 50 2.9 F a m i l y o f l i m i t i n g i n t e g r a l m a n i f o l d c r o s s s e c t i o n s a t 0=0 f o r a r a n g e o f e a t K^=1.0: (a) c=0, 0.1 52 (b) 0=0.3, 0.5, 0.75 53 2.10 S t a b i l i t y c h a r t s s h o w i n g t h e e f f e c t o f s o l a r p a r a m e t e r a t K\=1.0: (a) c=0, 0.1 54 (b) c=0.3, 0.5, 0.75 55 2.11 S t a b i l i t y c h a r t s s h o w i n g t h e e f f e c t o f i n e r t i a p a r a m e t e r a t c=0.1; 1^=1.0, 0.7, 0.5 60 2.12 0 - i n d e p e n d e n t s t a b i l i t y r e g i o n s f o r r a n g e o f e a t c=0.1, K.=1.0 61 1 2.13 V a r i a t i o n o f c r i t i c a l e c c e n t r i c i t y w i t h c f o r ra n g e o f K, 62 2.14 R e p r e s e n t a t i v e s t a b i l i t y c h a r t s f o r K.=1.0 and 0.7 w i t h optimum v a l u e s o f c 64 2.15 V a r i a t i o n o f c r i t i c a l e c c e n t r i c i t y w i t h <j> f o r K.=1.0, 0.7 65 1 X F i g u r e Page 2.16 T y p i c a l l i b r a t i o n a l d e c a y i n p r e s e n c e o f s o l a r damping 69 2.17 S t a b i l i z i n g i n f l u e n c e o f s o l a r damping . . . . 70 2.18 S y s t e m r e s p o n s e s h o w i n g u p r i g h t c o n t r o l due t o s o l a r damping 71 2.19 S y s t e m r e s p o n s e s h o w i n g i n d e p e n d e n c e o f l i m i t c y c l e f r o m i n i t i a l c o n d i t i o n s 72 2.20 O p t i m i z e d p l o t s s h o w i n g v a r i a t i o n o f \p and \i w i t h d e s i g n p a r a m e t e r s : (a) K i=1.0 74 (b) K i=0.8 75 2.21 Optimum r e s p o n s e o f a s p r i n g - m a s s v i s c o u s damper 61, 62 77 3.1 V a r i a t i o n o f d i r e c t s o l a r r a d i a t i o n p r e s s u r e w i t h a n g l e o f i n c i d e n c e . . . . . . . 84 3.2 D i r e c t e a r t h r a d i a t i o n and o r b i t a l g eometry . . 85 3.3 V a r i a t i o n o f d i r e c t e a r t h r a d i a t i o n p r e s s u r e w i t h a l t i t u d e and o r i e n t a t i o n . . 91 3.4 E a r t h a l b e d o and o r b i t a l geometry 92 3.5 Geometry o f p h y s i c a l s y s t e m f o r e a r t h a l b e d o s t u d y i n Case I I I 97 3.6 C u t t i n g p l a n e g e o m e t r y and i t s e f f e c t on l i m i t s o f i n t e g r a t i o n i n Case I I I 99 3.7 Geometry o f p h y s i c a l s y s t e m f o r e a r t h a l b e d o s t u d y i n Case IV 104 3.8 C u t t i n g p l a n e g e o m e t r y and i t s e f f e c t on l i m i t s o f i n t e g r a t i o n i n Case IV 106 3.9 Geometry o f p h y s i c a l s y s t e m f o r e a r t h a l b e d o s t u d y i n C a s e V: (a) S a t e l l i t e and t e r m i n a t o r p l a n e s n o t i n t e r s e c t i n g i n t h e s p h e r i c a l c ap I l l (b) S a t e l l i t e and t e r m i n a t o r p l a n e s i n t e r -s e c t i n g i n t h e s p e r i c a l c ap 112 x i F i g u r e Page 3.10 V a r i a t i o n o f e a r t h a l b e d o f o r c e w i t h s a t e l l i t e a l t i t u d e and a t t i t u d e : (a) A s=0 114 (b) A s = 3 0 ° 115 (c) ^ = 6 0 ° 116 (d) A s = 9 0 ° 117 3.11 V a r i a t i o n o f e a r t h a l b e d o f o r c e w i t h s a t e l l i t e a t t i t u d e and s o l a r a s p e c t a n g l e : (a) A l t i t u d e = 500 m i l e s 118 (b) A l t i t u d e = 3000 m i l e s 119 3.12 E f f e c t o f shadow on l i m i t i n g v a l u e o f ty' (0) f o r s t a b i l i t y i n c i r c u l a r o r b i t : (a) c=0.1; K i=1.0, 0.7, 0.5 123 (b) K i=1.0; c=0.1, 0.3, 0.5 124 3.13 E f f e c t o f e a r t h r a d i a t i o n s on l i b r a t i o n a l r e s p o n s e : (a) e=0.1, c=0.3, K i=1.0 126 (b) ,e=0.1, c=0.1, K i=0.65 127 (c) e=0.2, c=0.3, K i=1.0 128 (d) e=0.2, c=0.4, 1^=1.0 129 (e) e=0.2, c=-0.3, K i=1.0 13,0 3.14 S y s t e m p l o t s s h o w i n g t h e maximum l i b r a t i o n a l a m p l i t u d e and a v e r a g e p e r i o d f o r a r a n g e o f e c c e n t r i c i t y as a f f e c t e d b y : (a) s o l a r and i n e r t i a p a r a m e t e r s 132 (b) s o l a r a s p e c t a n g l e and i n i t i a l c o n d i t i o n 6 133 (c) i n i t i a l d i s t u r b a n c e s (0) and ty1 (0) . . . . 134 (d) a l t i t u d e f o r a g i v e n s a t e l l i t e g e ometry . . 135 X l l F i g u r e Page 3.15 S t a b i l i t y c h a r t s s h o w i n g t h e e f f e c t o f s o l a r and i n e r t i a p a r a m e t e r s : (a) K i=1.0; c=0.1, 0.3 137 (b) K i=1.0; c=0.5, 0.75 K.=0.7, c=0.1 138 l 4.1 Response p l o t s i n a e r o d y n a m i c r e g i m e s h o w i n g e f f e c t o f e a r t h r a d i a t i o n s : (a) e=0, c=0.3, 1^=1.0, A l t = 5 0 0 m i l e s . . . . . 148 (b) e=0, c=0.3, 1^=1.0, A l t = 4 0 0 m i l e s 149 (c) e=0.1, c=0.5, K i=1.0, A l t = 5 0 0 m i l e s . . . . 150 4.2 S t a b i l i t y c h a r t s i n a e r o d y n a m i c r e g i m e s h o w i n g e f f e c t o f e a r t h r a d i a t i o n s 151 4.3 S a t e l l i t e r e s p o n s e p l o t s s h o w i n g r e l a t i v e i m p o r t a n c e o f e n v i r o n m e n t a l f o r c e s : (a) e=0, c=0.3, 1^=1.0, A l t = 5 0 0 m i l e s 153 (b) e=0, c=0.3, K i=1.0, A l t = 4 0 0 m i l e s 154 (c) e=0, c=0.3, K i=1.0 / A l t = 3 0 0 m i l e s 155 (d) e=0.1, c=0.3, K i=1.0, A l t = 5 0 0 m i l e s . . . . 156 (e) e=0.1, c=0.3, 1^=1.0, A l t = 4 0 0 m i l e s . . . . 157 (f) e=0.1, c=0.5 f K i=1.0, A l t = 4 0 0 m i l e s . . . . 158 4.4 S t a b i l i t y c h a r t s s h o w i n g r e l a t i v e i m p o r t a n c e o f e n v i r o n m e n t a l f o r c e s 160 4.5 S y s t e m r e s p o n s e s h o w i n g v a r i a t i o n o f \> and T w i t h e, c, and a l t i t u d e 162 av ' ' 4.6 V a r i a t i o n o f w i t h a l t i t u d e f o r g i v e n s a t e l l i t e g e ometry 163 4.7 S t a b i l i t y c h a r t s i n p r e s e n c e o f e n v i r o n m e n t a l f o r c e s as a f f e c t e d by a l t i t u d e : i (a) c=0.1 164 (b) c=0.3 165 (c) c=0.5 166 x i i i F i g u r e Page 5.1 S t a b i l i t y c h a r t s f o r GEOS-A sh o w i n g (a) d i r e c t s o l a r r a d i a t i o n e f f e c t a t h i g h e r a l t i t u d e s 169 (b) a e r o d y n a m i c e f f e c t a t l o w e r a l t i t u d e s . . . 170 ACKNOWLEDGEMENT The a u t h o r w i s h e s t o e x p r e s s h i s g r a t i t u d e t o Dr. V . J . Modi f o r h i s e n c o u r a g e m e n t and g u i d a n c e t h r o u g h o u t t h e s t u d y and p a r t i c u l a r l y f o r h i s h e l p d u r i n g t h e c r i t i c a l i n i t i a l and f i n a l s t a g e s . Thanks a r e a l s o g i v e n t o Dr. J . E . N e i l s o n and D r . R.C. B r e r e t o n f o r t h e numerous d i s c u s s i o n s i n c o n n e c t i o n w i t h t h i s p r o j e c t . The n u m e r i c a l work i n t h i s t h e s i s was c a r r i e d o u t i n t h e C o m p u t i n g C e n t e r o f t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a . The use o f t h e s e f a c i l i t i e s i s g r a t e f u l l y a c k n o w l e d g e d . T h i s p r o j e c t was s u p p o r t e d ( i n p a r t ) by the Defence Research Board o f Canada, Grant No. 0201-04. L I S T OF SYMBOLS A e f f e c t i v e a r e a o f t h e s a t e l l i t e A e a r t h a r e a c o n t r i b u t i n g t o s a t e l l i t e f o r c e e ^ A . ( i = l , . . . 4 ) c o n s t a n t s d e f i n e d i n e q u a t i o n s ( 2 . 2 4 ) , ( 2 . 3 4 ) , 1 (2.35) A + , B ,... c o n s t a n t s d e f i n e d i n e q u a t i o n s ( 2 . 3 2 ) , (2.36) A f i r s t moment o f a r e a a b o u t t h e Y - a x i s yy s AD a e r o d y n a m i c B^ ( i - 1 , . . . 1 5 ) c o n s t a n t s d e f i n e d i n e q u a t i o n s ( 3 . 2 0 ) , (3.24) a e r o d y n a m i c d r a g c o e f f i c i e n t a t ^=0 C \ ( i = l , . . . 7 ) c o n s t a n t s d e f i n e d i n e q u a t i o n s ( 2 . 3 3 ) , (2.36) D^, D ^  r e l a t i v e c u t t i n g p l a n e d i s t a n c e s ( F i g u r e 3.9b) DSR d i r e c t s o l a r r a d i a t i o n DER d i r e c t e a r t h r a d i a t i o n EA e a r t h a l b e d o F i n e q u a l i t y f a c t o r d e f i n e d i n e q u a t i o n (2.23) , F, (P, ) s a t e l l i t e f o r c e ( p r e s s u r e ) due t o d i r e c t s o l a r ds ds j• . • r a d i a t i o n F (P ) s a t e l l i t e f o r c e ( p r e s s u r e ) due t o e a r t h a l b e d o r a r a ^ F (P ) s a t e l l i t e f o r c e ( p r e s s u r e ) due t o d i r e c t e a r t h r e r e ,. . . ^ r a d i a t i o n G i r r a d i a t i o n on t h e s a t e l l i t e due t o an e l e m e n t a l e a r t h a r e a GG g r a v i t y g r a d i e n t I ,1 ,1 p r i n c i p a l moments o f i n e r t i a a b o u t t h e X ,Y •, xx yy zz „ ^ . • -t s s J 2 Z s ~ a x e s , r e s p e c t i v e l y J i n e q u a l i t y f a c t o r u s e d i n e q u a t i o n (2.26) xvi constant defined i n equation (3.6) mass d i s t r i b u t i o n parameter, A /I yy yy i n e r t i a parameter, (I -I )/I c xx zz ' yy constant defined i n equation (3.14) distance from the s a t e l l i t e center of mass to the elemental earth area aerodynamic torque on the s a t e l l i t e moment c o e f f i c i e n t for aerodynamic torque defined i n equation (4.8) moment and force c o e f f i c i e n t for d i r e c t solar radiation defined i n equation (3.1) moment and force c o e f f i c i e n t for earth albedo defined i n equations (3.15), (3.20), (3.24), (3.25) moment and force c o e f f i c i e n t for d i r e c t earth radiation defined i n equations (3.8), (3.10) center of force normal pressure on a s a t e l l i t e due to aerodynamic forces shear force on a s a t e l l i t e due to aerodynamic forces generalized forces on the s a t e l l i t e distance from the center of force to the s a t e l l i t e center of mass earth radius solar constant solar energy per unit time incident upon a unit area i n c l i n e d at angle a , S|Cos a| s a t e l l i t e center of mass speed r a t i o , ( s a t e l l i t e velocity)/(average molecul velocity) k i n e t i c energy XVI1 a m b i e n t t e m p e r a t u r e , °R t e m p e r a t u r e o f t h e e l e m e n t a l e a r t h a r e a , °R s a t e l l i t e s u r f a c e t e m p e r a t u r e , °R p o t e n t i a l e n e r g y 4 s a t e l l i t e v e l o c i t y , ( V c = 2 . 6 x l 0 f t . / s e c . ) p r i n c i p a l body c o o r d i n a t e s w i t h o r i g i n a t t h e c e n t e r o f mass, Z g - a x i s o f symmetry s a t e l l i t e a b s o r p t i v i t y e a r t h a l b e d o 3 s o l a r p a r a m e t e r , {r ( l + p - x ) S K }/yc r P a s p e e d o f l i g h t r a d i a t i o n f o r c e on t h e s a t e l l i t e a r e a dA o r b i t e c c e n t r i c i t y a n g u l a r momentum p e r u n i t mass s a t e l l i t e a l t i t u d e , ( h 1 = r e f e r e n c e a l t i t u d e = 460 m i l e s ) ° s a t e l l i t e mass mass o f p h o t o n s i n c i d e n t on a u n i t a r e a p e r u n i t t i m e u n i t v e c t o r n o r m a l t o t h e e l e m e n t a l e a r t h a r e a u n i t v e c t o r n o r m a l t o t h e s a t e l l i t e d i s t a n c e between t h e c e n t e r o f f o r c e and t h e s a t e l l i t e e l e m e n t a l mass 'dm' d i s t a n c e f r o m t h e c e n t e r o f f o r c e t o p e r i g e e d i s t a n c e f r o m t h e s a t e l l i t e c e n t e r o f mass t o t h e s a t e l l i t e a r e a dA t i m e c u t t i n g p l a n e d i s t a n c e s ( F i g u r e s 3.6, 3.8) i n e r t i a l c a r t e s i a n c o o r d i n a t e s w i t h x', z' i n t h e o r b i t a l p l a n e ( F i g u r e 3.2) s a t e l l i t e p o s i t i o n a n g l e measured f r o m p e r i g e e X V I 1 1 <f> solar aspect angle measured from perigee <JJ i n c l i n a t i o n of the s a t e l l i t e ' s longitudinal axis to l o c a l v e r t i c a l a angle of incidence of the d i r e c t solar radiation flux on the s a t e l l i t e 3 angle between the l o c a l v e r t i c a l and the s a t e l l i t e l i n e of sight to the elemental earth area 3^ angle between the unit normal n^ and the s a t e l l i t e l i n e of sight to the elemental earth area 32 angle between the unit normal n2 and the s a t e l l i t e l i n e of sight to the elemental earth area 3^ angle between the unit normal n2 and the l o c a l v e r t i c a l m maximum value of 3 Y longitude of the elemental earth area as measured from perigee 6 shadow factor 6 1 o f f s e t between the center of mass and the center of pressure of the s a t e l l i t e e earth emissivity e perturbation element about a periodic solution P A angle between the unit normal n^ and the l o c a l v e r t i c a l X maximum value of A m A g solar aspect angle measured from the l o c a l v e r t i c a l M g r a v i t a t i o n a l f i e l d constant Mc proportionality constant, c=±ycif; 1 p s a t e l l i t e r e f l e c t i v i t y p (p 1) atmospheric density at al t i t u d e h (h 1) * o o ^ 1 o o a angle between the position of the sun and the unit normal n^ a 1 Stefan-Boltzmann constant XIX a surface r e f l e c t i o n c o e f f i c i e n t for tangential momentum transfer a' surface r e f l e c t i o n c o e f f i c i e n t for normal momentum transfer T s a t e l l i t e t r a n s m i s s i b i l i t y T average l i b r a t i o n a l period as a function of the o r b i t a l period a) reference angle defined i n Figure 3.9 (b) WQ reference angle f (2ir-ijj-a>) Dots and primes represent d i f f e r e n t i a t i o n with respect to t and 6 , respectively, unless otherwise defined. 'Altitude' i n the text denotes perigee distance. 1. INTRODUCTION 1 .1 P r e l i m i n a r y R e m a r k s F o r many s p a c e a p p l i c a t i o n s , s u c h as c o m m u n i c a t i o n , w e a t h e r , m i l i t a r y a n d e a r t h r e s o u r c e s s a t e l l i t e s , i t i s n e c e s s a r y t o m a i n t a i n a f i x e d o r i e n t a t i o n r e l a t i v e t o t h e e a r t h . O f t h e n u m e r o u s m e t h o d s p r o p o s e d f o r s t a t i o n k e e p i n g , g r a v i t y g r a d i e n t s t a b i l i z a t i o n h a s g a i n e d much a t t e n t i o n p r i m a r i l y due t o t h e p a s s i v e n a t u r e o f t h e s y s t e m . T h e e a r t h ' s n a t u r a l s a t e l l i t e , t h e m o o n , p r o v i d e s an e x c e l l e n t e x a m p l e o f s u c h a t t i t u d e c o n t r o l . I t p r e s e n t s o n l y 59% o f i t s s u r f a c e t o t h e e a r t h and e x e c u t e s s m a l l d y n a m i c l i b r a t i o n s i n l o n g i t u d e a n d l a t i t u d e . T h e r o t a t i o n o f t h e moon a b o u t i t s p o l a r a x i s i s s y n c h r o n i z e d w i t h i t s o r b i t a l r o t a t i o n . T h e l u n a r g l o b e i s a t r i a x i a l e l l i p s o i d w i t h i t s l o n g e r a x i s c a p t u r e d b y t h e e a r t h ' s g r a v i t a t i o n a l f i e l d . T h i s i s g r a v i t y g r a d i e n t s t a b i l i -z a t i o n a t w o r k . T h e k e y t o t h i s s t a b i l i z a t i o n p r i n c i p l e i s t h e f a c t t h a t t h e g r a v i t a t i o n a l f i e l d v a r i e s o v e r a s a t e l l i t e r e s u l t i n g i n an a p p a r e n t s h i f t i n g o f i t s c e n t e r o f m a s s . T h u s , an o r b i t i n g , n o n - s p h e r i c a l b o d y w i l l t e n d t o a p r e f e r r e d o r i e n t a t i o n w i t h r e s p e c t t o t h e e a r t h . " G e t t i n g a c o m m u n i c a t i o n s o r w e a t h e r s a t e l l i t e t o p o i n t c o n t i n u a l l y t o w a r d t h e e a r t h i s a t r i c k y b u s i n e s s , e s p e c i a l l y i f s h o r t - l i v e d a c t i v e s y s t e m s s u c h as j e t t h r u s t o r s a r e r u l e d o u t . B u t g r a v i t y g r a d i e n t s t a b i l i z a t i o n d o e s t h e j o b n i c e l y , and i n a g e n t l e way t h a t t a k e s l i t t l e o r no e f f o r t t o c o m p l y 2 w i t h t h e i n e v i t a b l e laws o f p h y s i c s . T h a t ' s why i t s f u t u r e i s so b r i g h t . 1 , 1 U n f o r t u n a t e l y , a g r a v i t y s t a b i l i z e d s a t e l l i t e c o r r e c t l y p o s i t i o n e d i n i t s o r b i t , d e v i a t e s w i t h t i m e f r o m t h i s p r e f e r r e d o r i e n t a t i o n . T h i s i s due t o p e r t u r b i n g e n v i r o n m e n t a l f o r c e s s u c h as r a d i a t i o n and a e r o d y n a m i c p r e s s u r e s , g r a v i t a t i o n a l and m a g n e t i c f i e l d i n t e r a c t i o n s , and m i c r o m e t e o r i t e i m p a c t s . A l t h o u g h M a x w e l l d e v e l o p e d t h e e l e c t r o m a g n e t i c t h e o r y i n 1872 and e x p e r i m e n t a l v e r i f i c a t i o n t h a t l i g h t waves e x e r t a p r e s s u r e was a c h i e v e d i n t h e e a r l y t w e n t i e t h c e n t u r y , r a d i a t i o n was n o t r e c o g n i z e d as an i m p o r t a n t p e r t u r b i n g f o r c e u n t i l c e r t a i n d y n a m ic b e h a v i o r o f e a r t h s a t e l l i t e s c o u l d o n l y be e x p l a i n e d by a c c o u n t i n g f o r t h i s p r e s s u r e . I n f a c t , f o r o r b i t s above t h e e a r t h ' s e f f e c t i v e a t m o s p h e r e (~ 500 m i l e s ) , r a d i a t i o n p r o v i d e s , 2 i n g e n e r a l , t h e m a j o r d i s t u r b i n g t o r q u e s . S h a p i r o e t a l . w h i l e s t u d y i n g t h e o r b i t a l p r o p e r t i e s o f West F o r d d i p o l e s , f o u n d s u n l i g h t p r e s s u r e p e r t u r b a t i o n s t o be p e r h a p s t h e most i m p o r t a n t o f a l l . The need f o r t h e p r e c i s e d e t e r m i n a t i o n o f t h e s e f o r c e s and t h e i r i n f l u e n c e on s a t e l l i t e a t t i t u d e r e s p o n s e i s t h u s e v i d e n t . A p r e l i m i n a r y m a g n i t u d e s t u d y h e l p e d e s t a b l i s h t h e a l -t i t u d e bounds w i t h i n w h i c h t h e s e e n v i r o n m e n t a l f o r c e s become s i g n i f i c a n t f o r t h e d y n a m i c a l a n a l y s i s o f a s a t e l l i t e . I t was f o u n d t h a t : (a) above 6000 m i l e s a l t i t u d e , o n l y d i r e c t s o l a r r a d i a t i o n n eed be c o n s i d e r e d ; (b) b e l o w 6000 m i l e s , d i r e c t s o l a r and e a r t h r a d i a t i o n s , 3 as w e l l as t h e e a r t h a l b e d o c o n t r i b u t i o n s h o u l d be i n c l u d e d ; (c) b e l o w 500 m i l e s , a e r o d y n a m i c f o r c e s s h o u l d a l s o be c o n s i d e r e d . 3 F i g u r e 1.1 shows, f o r t h e r e p r e s e n t a t i v e c o n f i g u r a t i o n d e s c r i b e d i n T a b l e 1, t h e moments due t o t h e s e f o r c e s on a g r a v i t y s t a b i l i z e d s a t e l l i t e as a f u n c t i o n o f a l t i t u d e . The r a d i a t i o n t o r q u e s g i v e n a r e f o r u n i t a b s o r p t i v i t y , c o n s e q u e n t l y , t h e i r m a g n i t u d e c o u l d c o n c e i v a b l y be d o u b l e t h a t shown. N o t e t h a t t h e p l o t , t h o u g h s u b s t a n t i a t i n g t h e above a l t i t u d e b o u n d s , i n d i c a t e s t h a t t h e y a r e i n d e e d c o n s e r v a t i v e . T h i s was n e c e s s a r y s i n c e t h e i n t e g r a t e d e f f e c t o f a f o r c e c a n o n l y be d e t e r m i n e d t h r o u g h t h e d y n a m i c a l a n a l y s i s o f t h e s y s t e m . TABLE 1 REPRESENTATIVE GRAVITY GRADIENT S A T E L L I T E CHARACTERISTICS S a t e l l i t e GEOS-A Moments o f I n e r t i a : I , I xx yy 615.3 s l u g f t . 2 I zz 20.8 s l u g f t . 2 P r o j e c t e d A r e a 13.1 f t . 2 O f f s e t between t h e c e n t e r o f mass and c e n t e r o f a r e a 5.75 f t . F i g u r e 1.1 V a r i a t i o n o f maximum t o r q u e s w i t h a l t i t u d e f o r a r e p r e s e n t a t i v e s a t e l l i t e 1. 2 L i t e r a t u r e Review A s u r v e y o f t h e p e r t i n e n t l i t e r a t u r e r e v e a l s c o n s i d e r -a b l e i n t e r e s t i n g r a v i t y g r a d i e n t s t a b i l i z a t i o n . H owever, t h e d y n a m i c a n a l y s i s o f t h e s e s y s t e m s a c c o u n t i n g f o r t h e p r e -d o m i n a n t p e r t u r b i n g f o r c e s h a s g a i n e d r e l a t i v e l y l i t t l e a t t e n t i o n . The p i o n e e r i n g work, f o r p u r e g r a v i t y o r i e n t e d s a t e l l i t e s 4 was c a r r i e d o u t by K l e m p e r e r (1960) who o b t a i n e d t h e e x a c t s o l u t i o n f o r p l a n a r l i b r a t i o n s o f a d u m b e l l s a t e l l i t e i n 5 c i r c u l a r o r b i t , and by B a k e r (1960) who f o u n d p e r i o d i c s o l u -t i o n s o f t h e p r o b l e m f o r s m a l l o r b i t e c c e n t r i c i t y . S c h e c h t e r ^ (1964) a t t e m p t e d , w i t h l i m i t e d s u c c e s s , t o e x t e n d K l e m p e r e r 1 s s o l u t i o n t o n o n - c i r c u l a r o r b i t a l m o t i o n by p e r t u r b a t i o n m e t h o d s . 7 Z l a t o u s o v e t a l . (1964) a n d , more r e c e n t l y , B r e r e t o n and M o d i ^ ' ^ ' 1 ( ^ (1966 , 1967, 1968) s u c c e s s f u l l y e m p l o y e d n u m e r i c a l m e t h o d s , i n v o l v i n g t h e u s e o f t h e s t r o b o s c o p i c p h a s e p l a n e , t o a n a l y s e m o t i o n i n t h e l a r g e f o r o r b i t s o f a r b i t r a r y e c c e n t r i c i t y . 11 12 T h ey a l s o i n v e s t i g a t e d t h e c o r r e s p o n d i n g p e r i o d i c s o l u t i o n s ' (1969) and e x t e n d e d t h e a n a l y s i s t o n o n - p l a n a r m o t i o n i n a 13 14 c i r c u l a r o r b i t ( 1 9 6 8 ) . B r e r e t o n (1967) h a s p r e s e n t e d an e x c e l l e n t r e v i e w o f t h i s w o r k . The r e l a t e d p r o b l e m o f s l o w l y s p i n n i n g s a t e l l i t e s i n a 15 g r a v i t a t i o n a l f i e l d was i n v e s t i g a t e d by Thomson (1962) f o l l o w e d by Kane e t a l . 1 ^ (1962) u s i n g a l i n e a r i z e d a n a l y s i s . They o b t a i n e d a s t a b i l i t y c h a r t i n t e r m s o f s p i n and i n e r t i a 17 p a r a m e t e r s f o r c i r c u l a r o r b i t a l m o t i o n . Kane and B a r b a (1966) a t t e m p t e d t o a n a l y s e t h e m o t i o n i n an e l l i p t i c o r b i t u s i n g 18 F l o q u e t t h e o r y w h i l e W a l l a c e and M e i r o v i t c h (1967) s t u d i e d t h e same p r o b l e m , w i t h q u e s t i o n a b l e s u c c e s s , u s i n g an a s y m p t o t i c a n a l y s i s i n c o n j u n c t i o n w i t h L i a p o u n o v ' s d i r e c t method. M o d i and N e i l s o n i n v e s t i g a t e d r o l l d y n a m i c s o f a s p i n n i n g s a t e l l i t e u s i n g t h e W K B J 1 9 (1968) and n u m e r i c a l 2 0 (1968) methods. The c o n c e p t o f i n t e g r a l m a n i f o l d s was s u c c e s s f u l l y e x t e n d e d t o t h e 21 s t u d y o f t h r e e d e g r e e s o f f r e e d o m m o t i o n i n c i r c u l a r o r b i t s ( 1 9 6 9 ) . The l i t e r a t u r e on s p i n n i n g s a t e l l i t e s has b een r e v i e w e d 22 by N e i l s o n (1968) i n some d e t a i l . P r i o r t o 1960, e x c e p t f o r o c c a s i o n a l a r t i c l e s on s o l a r 23 s a i l i n g ( 1 9 5 8 ) , n o t much a t t e n t i o n was d e v o t e d t o r a d i a t i o n 24 p r e s s u r e . I n 1958, R o b e r s o n p r e s e n t e d an o r d e r o f m a g n i t u d e e s t i m a t e o f t h e r a d i a t i o n e f f e c t . However, i n 1960, t h e o r b i t s o f t h e E c h o and V a n g u a r d s a t e l l i t e s e x e m p l i f i e d t h e i m p o r t a n c e o f r a d i a t i o n p r e s s u r e c o n s i d e r a t i o n s . The l a r g e number o f p a p e r s t h a t f o l l o w e d may be c a t a l o g u e d i n f o u r g e n e r a l d i v i s i o n s F i r s t i s t h e c o n t r i b u t i o n o f t o t a l r a d i a t i o n f l u x , b o t h d i r e c t and i n d i r e c t , t o t h e t h e r m a l b a l a n c e o f a s a t e l l i t e . The work by Camac e t a l . 2 5 ( 1 9 6 0 ) , C u n n i n g h a m 2 6 ' 2 7 ( 1 9 6 1 ) , K r e i t h 2 8 (1962) D e n n i s o n 2 9 (1962) , H r y c a k 3 0 (1962) , and L e v i n 3 1 (1962) , i n -d i c a t e s t h e i m p o r t a n c e o f t h i s c o n s i d e r a t i o n . N e x t i s t h e e f f e c t o f b o t h t h e m a g n i t u d e and d i r e c t i o n o f r a d i a t i o n on t h e o r b i t o f a s a t e l l i t e . A r e v i e w o f t h e c o n t r i b u t i o n s by P a r k i n s o n e t a l . 3 2 ( 1 9 6 0 ) , M u s e n 3 3 ' 3 4 ( 1 9 6 0 ) , B r y a n t 3 5 ( 1 9 6 1 ) , K o z a i 3 6 ' 3 7 (1961, 1 9 6 2 ) , S w a l l e y 3 8 ( 1 9 6 2 ) , W y a t t 3 9 ' 4 0 ( 1 9 6 3 ) , 41 42 43 Shapiro (1963), Hodge (1965), and Baker (1965) , emphasizes the importance of r a d i a t i o n f o r c e s on o r b i t a l p e r t u r b a t i o n s . The remaining areas are more p e r t i n e n t to the p r e s e n t a n a l y s i s and hence m e r i t a more d e t a i l e d review. However, the l i t e r a t u r e survey suggests t h a t they have r e c e i v e d l i t t l e a t t e n t i o n . The u t i l i z a t i o n of r a d i a t i o n p r essure to achieve 44 s a t e l l i t e a t t i t u d e s t a b i l i t y has been i n v e s t i g a t e d by Sohn (1959), Newton 4 5 (1960), V i l l e r s e t a l . 4 6 (1961), I v e s 4 7 (1961), 4 8 49 Windeknecht. (1961), and Acord e t a l . (1963), but t h e i r work has concentrated on c o n f i g u r a t i o n s designed to o r i e n t a t e the 50 s a t e l l i t e with r e s p e c t t o the sun. The paper by Ule (1963) c o n s i d e r s the use of r a d i a t i o n p r essure to s p i n an a r r a y of 51 m i r r o r s t o achieve a t t i t u d e s t a b i l i t y . M a l l a c h (1966) pro-posed a system f o r s o l a r damping of a g r a v i t y o r i e n t e d s a t e l l i t e , His phase-plane a n a l y s i s , though i n t e r e s t i n g , i s o v e r s i m p l i f i e d 5 through l i n e a r i z a t i o n and the use of average torques. The f i n a l area of study has been the i n f l u e n c e of r a d i a t i o n , both d i r e c t and r e f l e c t e d , on the a t t i t u d e dynamics 52 of a s a t e l l i t e . H a l l (1961) d e r i v e d an e m p i r i c a l f o r c e ex-p r e s s i o n f o r d i r e c t s o l a r r a d i a t i o n of the form F = P, AC,. ^ ds f where i s the f o r c e c o e f f i c i e n t found to be l e s s than two f o r 53 s e v e r a l convex shapes. M c E l v a i n (1961) obtained an a n a l y t i c a l e x p r e s s i o n f o r d i r e c t s o l a r r a d i a t i o n torque and determined, f o r two geometries, the change i n the s a t e l l i t e ' s angular momentum necessary to maintain a s p e c i f i e d o r i e n t a t i o n . Flook 54 and Warren (1963) reviewed the problem.of s o l a r torque e f f e c t s caused by the unequal thermal bending of g r a v i t y g r a d i e n t booms 8 i n synchronous o r b i t , and recommended the use of compensating d i s c s at the boom e x t r e m i t i e s . However, they d i d not analyse the problem dynamically and neglected the inherent v a r i a t i o n s i n the s a t e l l i t e i n e r t i a s . A more complete a n a l y s i s account-in g f o r the three major sources of r a d i a t i o n forces on a 55 s a t e l l i t e has been reported by Clancy and M i t c h e l l (1964) . They found the dynamical behavior of the system as the motion of and about the angular momentum ve c t o r . In a d d i t i o n to the inherent l i m i t a t i o n s of the a n a l y s i s , the r e s u l t i n g f o r c e expressions, given i n i n t e g r a l form, have to be evaluated n u m e r i c a l l y . In a t e s t on an IBM 7044 d i g i t a l computer, i t was found t h a t the time r e q u i r e d to evaluate these i n t e g r a l s renders a comprehensive study of s a t e l l i t e a t t i t u d e dynamics v i r t u a l l y i mpossible. The e f f e c t of small aerodynamic and g r a v i t a t i o n a l torques were t r e a t e d by B e l e t s k i i 5 6 (1960) as independent p e r t u r b a t i o n s of the motion of r a p i d l y spinning s a t e l l i t e s . Aerodynamic and r a d i a t i o n disturbances were presented i n the fundamental form 57 5 8 of pressure and shear s t r e s s by Evans (1962). Garber (1963) t r e a t e d the e f f e c t of constant d i s t u r b i n g torques on the l i b r a -t i o n a l motion of a r i g i d , g r a v i t y o r i e n t e d system i n a c i r c u l a r o r b i t by means of an i n f i n i t e s i m a l a n a l y s i s . More d i r e c t l y , 59 M e i r o v i t c h and Wallace (1966) i n v e s t i g a t e d the combined e f f e c t s of aerodynamic and g r a v i t y torques on the s t a b i l i t y of passive s a t e l l i t e s u sing Liapounov's d i r e c t method. For two s a t e l l i t e c o n f i g u r a t i o n s , e q u i l i b r i u m p o s i t i o n s were t e s t e d f o r s t a b i l i t y i n the s m a l l . Obviously, t h i s work i s r e s t r i c t e d to c i r c u l a r o r b i t s . 9 1.3 Purpose and Scope of the Investigation From the l i t e r a t u r e review, i t i s apparent that the dynamics of a gravity s t a b i l i z e d s a t e l l i t e accounting for the predominant environmental torques has received only cursory treatment. The main purpose of the thesis i s to analyse the effects of r a d i a t i o n and aerodynamic perturbing forces on this class of s a t e l l i t e s . Both the general response of the system and the l i m i t i n g i n i t i a l conditions for stable motion are explored, for a r b i t r a r y o r b i t s , as a function of the design parameters. The dependence of the environmental forces on a l t i t u d e , as outlined i n section 1.1, provides a r a t i o n a l approach to the problem, i n stages of increasing complexity. Chapter 2 deals with the motion of a s a t e l l i t e at high altitudes (approximately greater than 6000 miles) where gravity gradient and d i r e c t solar radiation constitute the predominant torques. In addition to the general response and s t a b i l i t y bounds, motion in the small i s analysed using a n a l y t i c a l techniques. Further, radiation damping i s investigated and an i n t e r e s t i n g concept of radi a t i o n attitude control i s discussed. In addition to the above forces, Chapter 3 includes the e f f e c t of d i r e c t earth radiation and i t s albedo, thus making the analysis applicable down to altitudes of - 500 miles. Closed form expressions are found for the additional forces which permit a comprehensive investigation. For lower a l t i t u d e s , aerodynamic forces must also be included. Chapters 4 and 5 not only investigate the motion i n 1 0 t h i s r e g i o n , b u t a l s o e x e m p l i f y t h e p e r t u r b i n g e f f e c t s t h r o u g h -3 o u t t h e e n t i r e a l t i t u d e r a n g e by a s t u d y o f t h e GEOS-A g r a v i t y s t a b i l i z e d s a t e l l i t e . F i g u r e 1 . 2 s c h e m a t i c a l l y i l l u s t r a t e s t h e v a r i o u s s t a g e s i n v o l v e d i n t h e p r o p o s e d p l a n o f s t u d y . I t i s f e l t t h a t t h i s a p p r o a c h p r o v i d e s a c o h e r e n t p r o g r a m t o e x p l o r e t h e s u b j e c t . Effect of Environmental Forces on the Attitude Dynamics of Gravity Oriented Satel l i tes Phase I Above 6000 Mi les Forces; Gravity Gradient Direct Solar Radiation Phase II Between 500 and 6000 Miles Forces: Gravi ty Gradient Direct Solar Radiation Direct Earth Radiation Earth Albedo Shadow Phase III Below 500 Miles Forces-. Grav i ty Gradient Direct Solar Radiation Direct Earth Radiation Earth Albedo Aerodynamic Shadow System Response Analyt ical Numerical Librational Stability Numerical Radiation Damping Numerical System Response Numerica I Librational Stability Numerical System Response Numerical Librational Stability Numerical Case Study G E O S - A Numerical Ci rcular Orb i t Elliptic Orbi t Circular Orbi t El l ipt ic Orb i t C ircular Orb i t E l l i p t i c Orb i t F i g u r e 1.2 S c h e m a t i c d i a g r a m o f t h e p r o p o s e d p l a n o f s t u d y 2. EFFECT OF DIRECT SOLAR RADIATION ON ATTITUDE \ DYNAMICS OF A SATELLITE This chapter deals w i t h planar l i b r a t i o n a l motion of a g r a v i t y o r i e n t e d system at high a l t i t u d e s where d i r e c t s o l a r r a d i a t i o n i s the predominant environmental disturbance. The a t t i t u d e dynamics of the system i s analysed, f o r a r b i t r a r y e c c e n t r i c i t y , i n terms of i t s design parameters, c and . P a r t i c u l a r a t t e n t i o n i s d i r e c t e d towards the e f f e c t of the s o l a r parameter to evaluate i t s importance i n a t t i t u d e dynamic s t u d i e s . In p a r t i c u l a r , the system response i s analysed using the a n a l y t i c a l WKBJ and Equi v a l e n t R i t z methods. The t y p i c a l r e -sponse and system p l o t s , obtained f o r r e p r e s e n t a t i v e values of the parameters, are compared wi t h those given by numerical i n t e g r a t i o n to assess the v a l i d i t y of the a n a l y t i c a l approach. Fu r t h e r , the bounds that must be placed on the i n i t i a l c o n d i t i o n s to ensure s t a b l e motion are i n v e s t i g a t e d as a f u n c t i o n of the e c c e n t r i c , i n e r t i a , and s o l a r parameters. These s t a b i l i t y c h a r t s , together w i t h the system p l o t s , provide an i n s i g h t i n t o the e n t i r e character of the l i b r a t i o n a l motion and should prove valuable during p r e l i m i n a r y design. The concept of s o l a r damping i s developed, and optimized p l o t s are obtained f o r a wide range of the design parameters. The r e s u l t s are compared 6 0 with those of the spring-mass-damper system proposed by Paul 61 62 and i t s g e n e r a l i z a t i o n i n v e s t i g a t e d by Tschann et a l . ' . The a n a l y s i s c l e a r l y p o i n t s out the p o t e n t i a l of s o l a r r a d i a t i o n pressure i n damping and c o n t r o l of s a t e l l i t e l i b r a t i o n s . 2.1 F o r m u l a t i o n o f t h e P r o b l e m 2.1.1 E q u a t i o n s o f M o t i o n f o r an A r b i t r a r i l y Shaped S a t e l l i t e C o n s i d e r a s a t e l l i t e o r b i t i n g i n t h e e c l i p t i c p l a n e a b o u t t h e c e n t e r o f f o r c e 0 and e x e c u t i n g p l a n a r l i b r a t i o n a l m o t i o n ( F i g u r e 2 . 1 ) . The c o o r d i n a t e s R and G d e f i n e t h e p o s i t i o n o f t h e s a t e l l i t e c e n t e r o f mass S q w i t h r e s p e c t t o p e r i g e e and t h e c e n t e r o f f o r c e . L e t X g , Y g , Z g r e p r e s e n t t h e p r i n c i p a l body c o o r d i n a t e s w i t h o r i g i n a t S q s u c h t h a t t h e Y g - a x i s i s n o r m a l t o t h e o r b i t a l p l a n e . The a n g l e t h a t t h e Z - a x i s makes w i t h t h e l o c a l v e r t i c a l d e f i n e s t h e l i b r a t i o n a l s a n g l e , ^ . W i t h r e f e r e n c e t o t h e p r i n c i p a l body c o o r d i n a t e s , t h e k i n e t i c and p o t e n t i a l e n e r g i e s o f t h e s y s t e m c a n be w r i t t e n as : (2.1) U = -dm 77 r (Z.Z) R e c o g n i z i n g t h a t r R Z w 2 _ , E (2.3) and u s i n g t h e b i n o m i a l e x p a n s i o n g i v e s 14 F i g u r e 2.1 Geometry o f s a t e l l i t e m o t i o n 15 (2.4-) 4-S i n c e t h e c o o r d i n a t e s a r e t a k e n t o be a p r i n c i p a l s e t l o c a t e d a t t h e c e n t e r o f m a s s , t h e f o l l o w i n g r e l a t i o n s h o l d : y * s d m = dm = / Z s d m =0 a n d (2.5) / X,2dm = -J- ( I + 1 - T } AY'dm = i - ( I + I - T ) s 2. x x yy ^ z ' X s Z 5 dm= 0 inn H e n c e , t h e e x p r e s s i o n f o r p o t e n t i a l e n e r g y ( e q u a t i o n 2 . 2 ) c a n b e w r i t t e n a s : 16 (2.7) The L a g r a n g i a n f o r m u l a t i o n g i v e s t h e e q u a t i o n s o f m o t i o n i n R, 6, and \> d e g r e e s o f f r e e d o m a s : no R - r o R O * t- ^ - 4 £ R' 2 R 4 2 I x x - Iyy -1-ZZ. -3 (X x > c - I E Z)Cos 2\|J (2.8) jL (m R 2 6 ) + i v y (e + <f) = Q dt '0 (2.9) (2.10) where Q^(i=R,9,^) a r e t h e g e n e r a l i z e d f o r c e s due t o s o l a r r a d i a t i o n . As t h e o r b i t a l p e r t u r b a t i o n s due t o l i b r a t i o n a l m o t i o n o f a s a t e l l i t e a r e s m a l l 6 3 ' 6 4 ' 5 6 ' 2 2 , t h e s o l u t i o n o f (2.8) and (2.9) l e a d s t o t h e c l a s s i c a l K e p l e r i a n r e l a t i o n s R 2 0 = h R = (2.11) R e c o g n i z i n g t h a t R e S i n e l +• e Cos e (2.12a) 17 R* de 2 R 4 e S i n e 1 + e C o s e d ^ de (a.iab). the equation of motion i n the l i b r a t i o n a l degree of freedom becomes 1 (l + e C o s e ) y - 2 e + 1) S i n 6 +-3K-, Sin ij> Cos if = -7-^ — Q (2.13) where . _ ^ - * x "~ Z Z I 2.1.2 Direct Solar Radiation Torque and General Equation of Motion The physical s a t e l l i t e configuration, consisting of a central body, solar panels and other appendages can be repre-sented, quite e f f e c t i v e l y so far as the radiation effects are concerned, by an a r b i t r a r i l y shaped f l a t surface with i t s geometric center separated from the center of mass of the system. As most s a t e l l i t e s do have a geometric symmetry about an axis, the center of pressure i s taken to be separated from the cen-ter of mass along the Z g-axis. In accordance with the quantum theory of electromagnetic radiation, the pressure exerted by the radiation flux incident on a surface i s the rate of change of momentum of the photons. 18 T h u s , f o r a s u r f a c e o f u n i t a b s o r p t i v i t y p l a c e d n o r m a l t o t h e i n c i d e n t f l u x P d s = n n p C / . (2.14) From E i n s t e i n ' s e n e r g y r e l a t i o n s h i p , e q u a t i o n (2.14) t a k e s t h e f o r m P d s = 4r (2.15) C . R a d i a t i o n p r e s s u r e on a s u r f a c e i s a f u n c t i o n o f i t s r e f l e c t i v i t y , a b s o r p t i v i t y , and t r a n s m i s s i b i l i t y , as w e l l as i t s p o s i t i o n r e l a t i v e t o t h e e m i t t i n g body. F u r t h e r , p, a, and x t h e m s e l v e s v a r y w i t h s u r f a c e t e m p e r a t u r e , t h e w a v e l e n g t h o f t h e i n c i d e n t r a d i a t i o n , and t h e a n g l e o f i n c i d e n c e . However, s u c h v a r i a t i o n s a r e g e n e r a l l y s m a l l and w i l l be n e g l e c t e d . F o r a f l a t p l a t e o r b i t i n g i n t h e e c l i p t i c p l a n e , u n d e r -g o i n g p l a n a r l i b r a t i o n a l m o t i o n ( F i g u r e 2 . 2 ) , t h e f o r c e on an e l e m e n t o f a r e a due t o d i r e c t s o l a r r a d i a t i o n , t a k e n p a r a l l e l t o t h e e a r t h - s u n l i n e and c o n s t a n t o v e r t h e s a t e l l i t e o r b i t , c a n be w r i t t e n as -^ T (1 + - T ) COS OL T + —j ( 1 - p - T ) S i n cL (2.16) d r \ Here t h e r e f l e c t i o n i s t a k e n t o be s p e c u l a r . R e c o g n i z i n g t h a t S 1 = s|Cosa| and a = c o n s t a n t f o r a p l a n e s u r f a c e , t h e e x p r e s s -i o n f o r t h e g e n e r a l i z e d f o r c e becomes Figu re 2.2 D i r e c t s o l a r r a d i a t i o n and s a t e l l i t e geometry Q v = f ( r s x d F ) • j 20 (2.17) = -^7 (1 +• p - T) Cos OL |Cos oL| Ayy . S u b s t i t u t i n g (2.17) i n t o (2.13) and r e c o g n i z i n g t h a t : h 2 - = yu R (1 + e Cos 6 ) = yU T p ( 1 + e ) (2.18) C o s OL - S i n (6 + 1 P - 4 M t h e e q u a t i o n o f m o t i o n t a k e s t h e f o r m : (1 + eCosO) y" - a e t y u ) S i n 8 + 3 K j Sin \|> Cos ip - ( U e f o s ^ S i n ( 6 ^ - 4 ) ) l S i n ( 8 ^ ^ ) l where y u . c ' J- yy 2.2 S y s t e m R e s p o n s e 2.2.1 A n a l y t i c a l and N u m e r i c a l A p p r o a c h The n o n - l i n e a r , non-autonomous d i f f e r e n t i a l e q u a t i o n ( 2 . 1 9 ) , f u r t h e r c o m p l i c a t e d by t h e p r e s e n c e o f t h e a b s o l u t e v a l u e t e r m , does n o t p o s s e s s any known c l o s e d f o r m s o l u t i o n . A l t h o u g h n u m e r i c a l i n t e g r a t i o n o f t h e e q u a t i o n c a n be p e r f o r m e d d i r e c t l y , any a t t e m p t f o r an a n a l y t i c a l s o l u t i o n r e q u i r e s p r e s e n t i n g t h e e q u a t i o n as (1+ eCose)ip" - ze ( l / +-D 5 i n 9 + 3 K; S i n i f Cos "UJ •j ~ \ Cos2V Cbs(29-Z$) (2.2o) + c ( i + e )3 ( i + e C o s e ) 3 . + | - S inEi | ) Sin(ae-a4>) w i t h m a t c h i n g o f i n i t i a l c o n d i t i o n s a t (e + \|) - <l>) = nir , (n- I , e, ) A t t i m e s u s e f u l i n f o r m a t i o n c o n c e r n i n g t h e b e h a v i o r o f a complex s y s t e m , n o t amendable t o a c l o s e d f o r m s o l u t i o n , c a n be o b t a i n e d t h r o u g h a l i n e a r i z e d a n a l y s i s . As s a t e l l i t e s r e -q u i r i n g a h i g h d e g r e e o f p o i n t i n g a c c u r a c y n o r m a l l y p e r m i t o n l y s m a l l a m p l i t u d e l i b r a t i o n , a l i n e a r i z e d a p p r o a c h c a n be 65 u s e d t o a d v a n t a g e . Hence t h e WKBJ a n a l y s i s o f t h e s y s t e m was u n d e r t a k e n . To a p p l y t h e WKBJ method i t i s n e c e s s a r y t o l i n e a r i z e t h e e q u a t i o n o f m o t i o n and remove t h e f i r s t d e r i v a t i v e t e r m . L i n e a r i z i n g t h e e q u a t i o n g o v e r n i n g l i b r a t i o n a l m o t i o n (2.20) w i t h t h e p o s i t i v e s i g n on t h e r i g h t hand s i d e y i e l d s 22 (2.21) where l + e C o s 6 R a { 6 ) = 3K, C ( l + e r Sin(26-2CD) l + eCose ( i t e C o s e ) 4 ^ ae S i n 6 , C( 1 + e ) 3 r -] Tie) = + — r r r 1~ Cos (26- 2$) A p p l y i n g t h e t r a n s f o r m a t i o n y = exp ( /P(e)de) and n e g l e c t i n g s e c o n d and h i g h e r o r d e r terms i n e and c , e q u a t i o n (2.21) r e -d u c e s t o Y + 3 K\ l + (-^ 77 - e)Cose - sinlee-^) = e e S i n e +• Cll+3e) 1 - Cbs(2e -z$) (2.22) w h i c h i s o f t h e f o r m ID" 4- G 2 ( e ) 1}J =• Q (6) . The f o r m o f e q u a t i o n (2.22) s u g g e s t s s e v e r a l a n a l y t i c a l a p p r o a c h e s s u c h as V a r i a t i o n o f P a r a m e t e r s , R i t z Method, WKBJ, Harmonic B a l a n c e , e t c . However, WKBJ i n c o n j u n c t i o n w i t h t h e Harmonic B a l a n c e method p r o v e d t o be more a c c u r a t e and hence i s d e s c r i b e d i n some d e t a i l . F o r t h e WKBJ method t o be a p p l i c a b l e , t h e i n e q u a l i t y F G " ( 0 ) 2 G(6) 3^ (G'(d) ]2 4 \ 6(0) / « 1 (2.23) must be s a t i s f i e d . The c o m p l i m e n t a r y s o l u t i o n c a n t h e n be w r i t t e n as - A, X,(e) + A £ x 2(e) (2.24) where X t(6) =G" §(8) C o s [ / 6 G ( 6 ) d e } X 2 (6) = G"^(e» S in ( / B G ( e ) d a } The c o m p l i c a t e d n a t u r e o f t h e f u n c t i o n G (6) w o u l d n e c e s s i t a t e e v a l u a t i o n o f X -^ (e ) and x 2 ( e ) n u m e r i c a l l y . T h i s w o u l d d e f e a t t h e o r i g i n a l a i m o f f i n d i n g a c l o s e d f o r m s o l u t i o n . -1 /2 However, f o r s m a l l e and c t h e f u n c t i o n G (9) and t h e i n -t e g r a l c a n be a p p r o x i m a t e d as G 2 (6) ( 3 K i ) ' * L_ ~ \ r n . c(U3e) r . . .. (2.25) 6 yGie)dBM3K0,fee + ^ ( ^ r - e ) s . n e + c ( 1 1 2 ^ e ) cba(2e-a<E) 24 p rov ided the f o l l o w i n g i n e q u a l i t y i s s a t i s f i e d : « 1 (2.26) As F and J depend on a l a rge number of parameters , the de t e rmina t ion of the range of a p p l i c a b i l i t y of equa t ion (2.24) would i n v o l v e a cons ide r ab l e amount of computat ion. F o r t u n a t e l y , fo r > 1/3, which i s normal ly the case fo r a g r a v i t y o r i e n t e d system, F and J a t t a i n t h e i r maximum values fo r 0 = 0° and 4> = J ' c c 4 5 ° , when the p o s i t i v e s i g n i s used i n equat ion (2 .20 ) . F i g u r e 2.3 shows the c r i t i c a l va lues of F and J f o r a wide range of e c c e n t r i c i t y , i n e r t i a and s o l a r parameters . For a synchronous s a t e l l i t e w i t h a r e p r e s e n t a t i v e va lue of c = 0.2 and = 0 . 6 , i t i s apparent t ha t the maximum value of the func t ions i s approximate ly 0 . 1 , thus s u b s t a n t i a t i n g the a p p l i c a b i l i t y of the method. A knowledge of the c r i t i c a l va lues of 9 and <J> f o r J to be maximum, provides a more d i r e c t method to determine the range of a p p l i c a b i l i t y . Equa t ion (2.26) y i e l d s m a x, c ( i + 3e) 3K-, ^ 1 (2.27) e=ec,<j><4>c A p p l y i n g (2.27) the f o l l o w i n g r e l a t i o n s are r e a d i l y obta ined 2 8 6(6) G ' l e ) <t>C e--ec e--&c 3K; 3K 3 K-, 3e) = 0 (2.28) G"(e) S u b s t i t u t i n g (2.28) i n t o (2.23) g i v e s t h e i n e q u a l i t y e n \ . c(i 3K, { l / 3K; J 3K; » _i.(_e__e] 20(1 + 3e) 3Kj 2 ^ 3 t l - e ) + 3 K; (2.2^ 1 R e c o g n i z i n g t h a t 3 K \ e + 3 K; « 1 (2.30) (2.29) r e d u c e s t o 6 K ; » i / e a l3K t ac (u 3e) 3 (2.31) 29 A p p a r e n t l y , f o r r e p r e s e n t a t i v e v a l u e s o f , e q u a t i o n (2.31) i s a l w a y s s a t i s f i e d . T h u s , t h e i n e q u a l i t y i n (2.27) r e p r e s e n t s t h e s u f f i c i e n t c o n d i t i o n f o r a p p l i c a t i o n o f t h e WKBJ a n a l y s i s . Now J c a n be p l o t t e d as a f u n c t i o n o f t h e p a r a m e t e r s max c ^ i n v o l v e d . However, f r o m p r a c t i c a l d e s i g n c o n s i d e r a t i o n s i t w o u l d be p r o f i t a b l e t o f i x an a c c e p t a b l e v a l u e o f J , t h u s r r max g i v i n g t h e bounds on t h e d e s i g n p a r a m e t e r s . T h i s i s shown i n F i g u r e 2.4 f o r J m a x = 1/3. R e c a l l i n g t h a t t h e a n a l y s i s r e q u i r e s b o t h e and c t o be s m a l l ( e q u a t i o n 2 . 2 2 ) , o n l y f o r a l i m i t e d r e g i o n u n d e r t h e s e c u r v e s w o u l d t h e s o l u t i o n c o r r e s p o n d t o t h e o r i g i n a l e q u a t i o n ( 2 . 2 1 ) . T h i s i s shown as a s h a d e d a r e a i n F i g u r e 2.4 w i t h t h e same l i m i t o f s m a l l n e s s on e and c. C o n s i d e r a p a r t i c u l a r s o l u t i o n o f t h e f o r m !Vp+ = V + B + 5in6 + C^CosB + Cos 26 +E^Sin20 (2.32) A p p l y i n g t h e H a r m o n i c B a l a n c e t e c h n i q u e , t h e c o n s t a n t s c a n be o b t a i n e d f r o m (2.33) 30 31 where C 3= -|- (1 +36) Sir, 2$ C 4 = | - ( l t 3 e ) Cos 2<J> C 5 - C, - C 3 - 1 C b = C t +C 3 - 1 C 7 « C t - 4 - . A r b i t r a r y c o n s t a n t s , A 1 and A , a r e d e t e r m i n e d f r o m t h e i n i t i a l c o n d i t i o n s as (2.34-) A - X,(6) X>(8) X[ie) X((9) X,[e) X'Ad) 1 " x tte) x,(6) ^ 2 l x t (0 ) J . S i m i l a r l y an a n a l y t i c a l s o l u t i o n t o e q u a t i o n (2.20) f o r t h e n e g a t i v e s i g n c a n be o b t a i n e d . The s o l u t i o n i s g i v e n as f o l l o w s : ^ c - = A s X j C O ) + A 4 X 4 ( 0 ) (2.35) 32 where X 3 (6 ) =* G"^ (©) Co5{ | G(6)de } o i e X 4 (6 ) = G " 2 ( 6 ) S in [ | G ( 6 ) d e | o G~E(e) = (3K;)'/4 i - i - ( ^ - e ) a » e - = ^ s i n ( 2 e - 2 « > ) '/a and Y P = A_ -h B_S in6 + C_Cos9 + D_ Cos ae+E_SinZe (E.3fe) where r 0 ~ C3 CA o c b C4 O 0 C 2 0 O J \-2C 3 O c 7 ol B. C. 0. e. ai+3e)^ o - c 3 33 M a t c h i n g t h e r e s p o n s e a t (6+ij;-<f>) = TT , 2TT, . . . y i e l d s an a p p r o x i m a t e a n a l y t i c a l s o l u t i o n t o t h e g e n e r a l e q u a t i o n o f m o t i o n (2.20). To c h e c k t h e v a l i d i t y o f t h i s a p p r o x i m a t e c l o s e d f o r m s o l u t i o n , t h e e q u a t i o n o f m o t i o n (2.19) was i n t e g r a t e d n u m e r i c -a l l y u s i n g an IBM 360-67 d i g i t a l c o m p u t e r . The A d a m s - B a s h f o r t h p r e d i c t o r - c o r r e c t o r p r o c e d u r e w i t h a f o u r t h o r d e r R u n g e - K u t t a s t a r t e r was u s e d i n c o n j u n c t i o n w i t h a s t e p s i z e o f 5° f o r r e s p o n s e p l o t s and 3 ° f o r t h e i n t e g r a l m a n i f o l d c r o s s - s e c t i o n s . The s t e p s i z e s u s e d g i v e r e s u l t s o f s u f f i c i e n t a c c u r a c y w i t h o u t i n v o l v i n g e x c e s s i v e c o m p u t a t i o n a l e f f o r t . 2 .2 .2 R e s p o n s e P l o t s and D i s c u s s i o n o f R e s u l t s F i g u r e 2.5 c o m p a r e s , f o r s e v e r a l r e p r e s e n t a t i v e v a l u e s o f t h e s y s t e m p a r a m e t e r s , t h e l i b r a t i o n a l r e s p o n s e o f a s a t e l l i t e as d e t e r m i n e d by t h e two m e t h o d s . S y s t e m b e h a v i o r i n a b s e n c e o f r a d i a t i o n p r e s s u r e (c = 0 ) , o b t a i n e d n u m e r i c a l l y , i s a l s o i n d i c a t e d f o r c o m p a r i s o n . ; I t i s a p p a r e n t t h a t f o r r e l a t i v e l y s m a l l d i s t u r b a n c e s , t h e two r e s p o n s e s a g r e e q u i t e w e l l . As c a n be e x p e c t e d , t h e n o n l i n e a r e f f e c t s m a n i f e s t t h e m s e l v e s a t l a r g e r a m p l i t u d e s l e a d -i n g t o d i s c r e p a n c i e s i n t h e two s o l u t i o n s . E v e n h e r e , t h e method c o n t i n u e s t o p r e d i c t t h e maximum a m p l i t u d e and p e r i o d o f t h e m o t i o n q u i t e a c c u r a t e l y , p a r t i c u l a r l y f o r n e a r c i r c u l a r o r b i t s ( F i g u r e 2 . 7 ) . Due t o t h e p e r i o d i c n a t u r e o f t h e c o e f f i c i e n t s ( e q u a t i o n 2 . 1 9 ) , t h e s y s t e m r e s p o n s e when r e p r e s e n t e d i n a t h r e e d i m e n s i o n a l p h a s e s p a c e (\\>, ty* , 6) r e s u l t s i n an i n t e g r a l m a n i f o l d o r an e= o.o c »o.i K| = 1.0 4>s0.0 l|i (0) =0.0 l|>'(0) =0.1 Numerical Analytical C = 0.0 F i g u r e 2.5 8 - orbi ts E f f e c t o f r a d i a t i o n p r e s s u r e on l i b r a t i o n a l r e s p o n s e o f a s a t e l l i t e as o b t a i n e d u s i n g a n a l y t i c a l and n u m e r i c a l methods; (a) s m a l l a m p l i t u d e i m p u l s i v e d i s t u r b a n c e ; ' (0) = 0.1 7.5 5.0 -2.5 -5.0 -7 .5 6-0.0 +(01=2.9° Numerical t • Analytical Kj=I.O lL(0) = 0.05 . (j, = 0.0 * c = 0-° 1 1 A A la A \ A / \ / \ • / ' i i I 1 • * • \ / i \ \ I1 \ U 1 A / * \ ' f / i // \ \ i / \ r* / VX /' \\ /' \» / /' V / \ / 11 f 1 \ A » /•' \ » w\\/ \/\\ - 1/ \ y V V i / •' v» / V ' 1 v W w r J \ v l f / \ 7 v \ y v 0 I 1 3 4 5 6 8 - o rb i ts F i g u r e 2 . 5 E f f e c t of r a d i a t i o n pressure on l i b r a t i o n a l response of a s a t e l l i t e as ob ta ined u s i n g a n a l y t i c a l and numer ica l methods: (b) s m a l l ampli tude a r b i t r a r y d i s t u r b a n c e ; <JJ(0) = 2 . 9 ° , i } > , ( 0 ) = 0 . 0 5 oo jure 2.5 9 - o r b i t s  E f f e c t o f r a d i a t i o n pressure on l i b r a t i o n a l response of a s a t e l l i t e as ob ta ined u s ing a n a l y t i c a l and numer ica l methods: (c) in te rmedia te response i n e c c e n t r i c o r b i t ; e = 0.1 u> 40 30 20 10 -10 -20 •30 e = 0.1 . Numerical C - 0 . 3 + , 0 ) ° a 0 . i t - i —————— Analy t ica l j / \ I" i / M ff A J i i~\ I \ / * / \ > — » \\ A 1 1 \ > 1 \ i \ V - y i i \\ /'/ \ r / \\// I v . / i i i / w _ 1 0 I 2 3 4 5 6 0 - o r b i t s F i g u r e 2.5 E f f e c t of r a d i a t i o n pressure on l i b r a t i o n a l response of a s a t e l l i t e as ob ta ined us ing a n a l y t i c a l and numer ica l methods: (d) l a rge ampli tude response showing the i n f l u e n c e of inc rease i n s o l a r parameter; c = 0.3 -o . 38 i n v a r i a n t s u r f a c e 6 6 , 6 7 ' 6 8 . The l i m i t i n g m a n i f o l d d e f i n e s t h e bound o f s t a b l e m o t i o n and i s d i s c u s s e d i n d e t a i l i n S e c t i o n 2 . 3 . 1 . The c r o s s - s e c t i o n s o f t h e s u r f a c e as g e n e r a t e d by t h e two methods s h o u l d s e r v e as an e f f e c t i v e c h e c k on t h e a c c u r a c y o f t h e a n a l y t i c a l a p p r o a c h . F i g u r e 2.6 compares t h e ph a s e s p a c e c r o s s - s e c t i o n s a t 0 = 0 . I t i s i n t e r e s t i n g t o n o t e t h a t t h e d i f f e r e n c e i n t h e r e s p o n s e t r a j e c t o r y r e s u l t s i n c i r c u m f e r -e n t i a l movement o f i t s i n t e r s e c t i o n w i t h t h e p h a s e p l a n e w i t h o u t s u b s t a n t i a l l y a l t e r i n g t h e i n v a r i a n t s u r f a c e c r o s s - s e c t i o n . To b e t t e r u n d e r s t a n d t h e g e n e r a l b e h a v i o r o f t h e s a t e l -l i t e , i . e . , t o p r e d i c t t h e i n f l u e n c e o f s y s t e m p a r a m e t e r s and i n i t i a l c o n d i t i o n s on i t s l i b r a t i o n a l a m p l i t u d e and p e r i o d , w o u l d r e q u i r e numerous r e s p o n s e p l o t s . The s y s t e m p a r a m e t e r s and i n i t i a l c o n d i t i o n s were v a r i e d s y s t e m a t i c a l l y o v e r a wide r a n g e . The r e s u l t i n g i n f o r m a t i o n i s c o n d e n s e d i n t h e f o r m o f s y s t e m p l o t s shown i n F i g u r e 2.7. F i g u r e 2.7(a) shows t h e e f f e c t o f s o l a r and i n e r t i a p a r a m e t e r s on maximum a m p l i t u d e and p e r i o d o f t h e m o t i o n as g i v e n by t h e two methods. I t r e v e a l s t h e a r e a d i s t r i b u t i o n , w h i c h a f f e c t s c , t o be as i m p o r t a n t as t h e i n e r t i a p a r a m e t e r , n o r m a l l y a c o n t r o l l i n g f a c t o r i n t h e d e s i g n o f a g r a v i t y o r i e n t e d s y s t e m . The WKBJ method p r e d i c t s maximum a m p l i t u d e and p e r i o d o f t h e m o t i o n q u i t e a c c u r a t e l y f o r c i r c u l a r o r b i t s , however t h e c o r r e l a t i o n d e t e r i o r a t e s w i t h i n c r e a s e i n e c c e n r t r i c i t y and f o r n e g a t i v e c . I n g e n e r a l , n e g a t i v e c l e a d s t o h i g h f r e q u e n c y l i b r a t i o n w h i l e t h e p o s i t i v e c and K^ have v e r y l i t t l e e f f e c t on l i b r a t i o n a l p e r i o d . F i g u r e 2.6 C o m p a r i s o n o f i n t e g r a l m a n i f o l d c r o s s - s e c t i o n s a t 6 = 0 f o r c o n d i t i o n s c o r r e s p o n d i n g t o t h o s e i n F i g u r e s 2.5 ( a ) , (b) Figu re 2.6 Comparison of i n t e g r a l man i fo ld c r o s s - s e c t i o n s at e = 0 f o r c o n d i t i o n s cor responding to those i n F igu res 2.5 ( c ) , (d) -0.5 - 0 3 -0.1 0.1 0.3 0.5 1.0 0.9 0.8 0.7 0.6 0.5 C Kj F i g u r e 2.7 S y s t e m p l o t s s h o w i n g t h e maximum l i b r a t i o n a l a m p l i t u d e and a v e r a g e p e r i o d f o r a r a n g e o f e c c e n t r i c i t y as a f f e c t e d by: (a) s o l a r and i n e r t i a p a r a m e t e r s F i g u r e 2.7 S y s t e m p l o t s s h o w i n g t h e maximum l i b r a t i o n a l a m p l i t u d e and a v e r a g e p e r i o d f o r a r a n g e o f e c c e n t r i c i t y as a f f e c t e d b y : (b) s o l a r a s p e c t a n g l e and i n i t i a l c o n d i t i o n 6 0.4 0.0 0.4 0.8 1.2 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2 l|)(0) -radians (0) System plots showing the maximum l i b r a t i o n a l amplitude and average period for a range of e c c e n t r i c i t y as affected by: (c) i n i t i a l disturbance ij) (0) and \\i ' (0) The s o l a r a s p e c t a n g l e does n o t s i g n i f i c a n t l y a f f e c t t h e maximum r e s p o n s e f o r a s a t e l l i t e i n a c i r c u l a r o r b i t , w h i c h i s u s u a l l y t h e c a s e o f i n t e r e s t . On t h e o t h e r hand, <j> c a n a f f e c t ib c o n s i d e r a b l y f o r s a t e l l i t e s i n e l l i p t i c t r a j e c t o r i e s rmax J ( F i g u r e 2 . 7 ( b ) ) . The same i s t r u e f o r t h e p o i n t o f a p p l i c a t i o n (9) o f t h e d i s t u r b a n c e . So f a r as t h e i n f l u e n c e o f t h e n a t u r e o f d i s t u r b a n c e i s c o n c e r n e d , F i g u r e 2.7(c) c l e a r l y p o i n t s o u t t h e l i m i t a t i o n o f g r a v i t y g r a d i e n t s t a b i l i z a t i o n i n t h e p r e s e n c e o f s o l a r r a d i -a t i o n . I n g e n e r a l , an e x t e r n a l d i s t u r b a n c e t e n d s t o r e d u c e t h e p e r i o d o f l i b r a t i o n , however i t does n o t f o l l o w any d e f i n i t e t r e n d . I t i s i n t e r e s t i n g t o n o t e t h a t t h e minimum l i b r a t i o n a l r e s p o n s e does n o t c o r r e s p o n d t o z e r o i n i t i a l c o n d i t i o n s . I n f a c t , i t i s d e f i n e d by t h e i n i t i a l c o n d i t i o n s w h i c h l e a d t o m o t i o n p e r i o d i c i n 2ir. 2.3 L i b r a t i o n a l S t a b i l i t y The a n a l y s i s o f m o t i o n and s t a b i l i t y i n t h e l a r g e p r o -v i d e s v a l u a b l e d e s i g n c r i t e r i o n . Not o n l y does i t g i v e t h e l i m i t and m a r g i n o f s t a b i l i t y f o r a s a t e l l i t e , b u t a l s o p r o -v i d e s t h e r e g i o n o f i t s c a p t u r e . F u r t h e r , s t a b i l i t y c h a r t s p r o v e u s e f u l i n d e t e r m i n i n g t h e e f f e c t o f t h e p h y s i c a l param-e t e r s w h i c h i s t h e p r i m e o b j e c t i v e o f t h i s a n a l y s i s . 2.3.1 N u m e r i c a l A n a l y s i s An a n a l y t i c a l s o l u t i o n t o t h e g e n e r a l e q u a t i o n g o v e r n i n g l i b r a t i o n a l m o t i o n (2.19) i s v i r t u a l l y i m p o s s i b l e t o f i n d . On t h e o t h e r hand, t h e n u m e r i c a l s o l u t i o n o f any o r d i n a r y d i f f e r -45 e n t i a l e q u a t i o n , r e g a r d l e s s o f i t s c o m p l e x i t y , p r e s e n t s no p r o b l e m as s e v e r a l w e l l e s t a b l i s h e d i n t e g r a t i o n t e c h n i q u e s a r e 69 a v a i l a b l e . The A d a m s - B a s h f o r t h p r e d i c t o r - c o r r e c t o r p r o -c e d u r e w i t h a f o u r t h o r d e r R u n g e - K u t t a s t a r t e r was u s e d t h r o u g h -o u t t h i s a n a l y s i s , i n c o n j u n c t i o n w i t h a p p r o p r i a t e s t e p s i z e s r a n g i n g f r o m 1 - 5 ° . A l t h o u g h t h e t r a j e c t o r i e s may be e a s i l y g e n e r a t e d , t h e i r i n t e r p r e t a t i o n , and c o n s e q u e n t l y i n s i g h t i n t o t h e p r o b l e m , i s o f t e n d i f f i c u l t . I t i s h e r e where t h e i n t e g r a l -m a n i f o l d i n t h e s t r o b o s c o p i c p h a s e s p a c e p r o v e s t o be v e r y v a l u a b l e . The c o n c e p t o f i n v a r i a n t s u r f a c e s was i n t r o d u c e d by Moser e t a l . ^ ' ^ ' ^ . B o t h B r e r e t o n ^ and N e i l s o n ^ demon-s t r a t e d , n u m e r i c a l l y , t h a t t h e s e s u r f a c e s c a n be g e n e r a t e d f o r t h e n o n - l i n e a r , non-autonomous s y s t e m s w i t h p e r i o d i c c o e f f i c -i e n t s t h a t t h e y i n v e s t i g a t e d . As s u c h , a d e t a i l e d d e v e l o p m e n t o f t h i s c o n c e p t i s n o t j u s t i f i e d , h e nce o n l y t h e p r i n c i p l e s u s e d i n t h e a n a l y s i s w i l l be r e v i e w e d h e r e . S i n c e t h e g o v e r n i n g e q u a t i o n (2.19) s a t i s f i e s t h e 70 L i p s h i t z c o n d i t i o n f o r u n i q u e n e s s , and s i n c e t h e c o e f f i c i e n t s o f (2.19) a r e p e r i o d i c i n 2TT , t r a j e c t o r i e s e m a n a t i n g f r o m e" = 9 q + 2niT (n = 0, 1, ...) w i t h t h e same i n i t i a l c o n d i t i o n s i n i> and i|>1 w i l l be i n v a r i a n t . C o n s e q u e n t l y , a c o m p l e t e t r a -j e c t o r y c a n be r e p r e s e n t e d i n t h e phase s p a c e by a s u c c e s s i o n o f t r a j e c t o r i e s e m a n a t i n g f o r 8 = 0 and t e r m i n a t i n g a t 6 = 2TT I where, l o g i c a l l y , ^ n ( 0 ) and ^ n ( 0 ) t a k e t h e v a l u e s o f i p n _ 1 (2TT) and ^ ' n (2TT). Such a s e r i e s o f t r a j e c t o r i e s d e f i n e an i n v a r i a n t 46 s u r f a c e o r i n t e g r a l m a n i f o l d . I t s h o u l d be n o t e d t h a t any p o i n t on t h e s u r f a c e i s s u f f i c i e n t t o g e n e r a t e and t h u s d e f i n e t h e e n t i r e s u r f a c e . L o g i c a l l y , t h e c o n c e p t o f an i n t e g r a l m a n i f o l d b r e a k s down f o r u n s t a b l e t u m b l i n g m o t i o n . T h u s , t h e l i m i t o f s t a b i l i t y i s d e f i n e d by t h e l a r g e s t i n v a r i a n t s u r f a c e t h a t c a n be f o u n d . Any s t a t e o f m o t i o n t h a t l i e s w i t h i n t h i s s u r f a c e g e n e r a t e s a t r a j e c t o r y , and hence a new s u r f a c e , w h i c h r e m a i n s w i t h i n t h e l i m i t i n g m a n i f o l d . T y p i c a l l i m i t i n g s u r f a c e s a r e shown i n F i g u r e 2.8. S i n c e m a n i f o l d s a r e u n i q u e , i t f o l l o w s t h a t a s u c c e s s i o n o f n e s t e d s u r f a c e s w i l l be g e n e r a t e d by c h o o s i n g i n i t i a l c o n d i t i o n s i n t e r i o r t o t h e p r e c e d i n g s u r f a c e . I n t h e l i m i t , t h e s u r f a c e w i l l s h r i n k t o a l i n e o r a s e t o f l i n e s r e p r e s e n t i n g p e r i o d i c s o l u t i o n s . O b v i o u s l y , a s e t of. m - l i n e s r e p r e s e n t s a m o t i o n p e r i o d i c i n m - o r b i t s . The n o t a t i o n n/m r e p r e s e n t s a p e r i o d i c s o l u t i o n e x e c u t i n g n - o s c i l l a t i o n s i n m - o r b i t s . N ote t h a t F i g u r e 2 . 8 ( c ) shows t h r e e l i m i t i n g s u r f a c e s , a l a r g e m a i n l a n d s u r r o u n d e d by two s m a l l e r s e c o n d a r y r e g i o n s ( i s l a n d s ) o f s t a b l e m o t i o n . F o r c l a r i t y , t h e s e s e c o n d a r y s u r f a c e s a s s o c i a t e d w i t h t h e 3/2 p e r i o d i c s o l u t i o n a r e shown s e p a r a t e l y i n F i g u r e 2.8 ( C 2 ) . From p r a c t i c a l d e s i g n c o n s i d e r a t i o n s , t h e y a r e o f l i t t l e s i g n i f i c a n c e s i n c e t h e o p e r a t i o n o f a s a t e l l i t e i n t h i s r e g i o n w o u l d be u n d e s i r a b l e . Hence', t h e m a j o r e n v e l o p e o f m o t i o n ( m a i n l a n d ) , w h i c h i s i n v a r i a b l y a s s o c i a t e d w i t h t h e f u n d a m e n t a l p e r i o d i c s o l u t i o n 1 / 1 , r e p r e s e n t s t h e main r e g i o n o f i n t e r e s t . F i g u r e 2.8 T y p i c a l l i m i t i n g i n v a r i a n t s u r f a c e s : (a) e=0 , c=0.5, K.=1.0 360° 51 The i m p o r t a n c e o f t h e i n t e g r a l m a n i f o l d c a n n o t be o v e r -e m p h a s i z e d , as i t p r o v i d e s a l l p o s s i b l e c o m b i n a t i o n s o f t h e d i s t u r b a n c e s t o w h i c h a s a t e l l i t e c a n be s u b j e c t e d , a t any p o i n t i n t h e o r b i t , w i t h o u t c a u s i n g i t t o t u m b l e . However, f o r a p a r a m e t r i c a n a l y s i s , t h e c h a r a c t e r i s t i c s o f t h e s y s t e m c a n be b e t t e r r e p r e s e n t e d i n t h e s t r o b o s c o p i c p h a s e p l a n e w h i c h r e p r e s e n t s a c r o s s - s e c t i o n o f t h e i n v a r i a n t s u r f a c e . The WKBJ a n a l y s i s ( S e c t i o n 2.2) u s e d t h e p h a s e - p l a n e c r o s s - s e c t i o n s ( F i g u r e 2.6) t o s u b s t a n t i a t e t h e v a l i d i t y o f t h e a n a l y t i c a l s o l u t i o n . F i g u r e 2.9 shows t h e c r o s s - s e c t i o n s o f t h e l i m i t -i n g m a n i f o l d s a t 6 = 0 f o r a r a n g e o f e c c e n t r i c i t y . Such a c o n d e n s a t i o n o f d a t a p r o v i d e s c o n s i d e r a b l e i n s i g h t i n t o t h e i n f l u e n c e o f t h e c o n t r o l l i n g p a r a m e t e r s on t h e s t a b i l i t y bounds o f a s y s t e m . Y e t a n o t h e r e q u a l l y p o w e r f u l e x t e n s i o n o f t h e i n v a r i a n t s u r f a c e c o n c e p t i s t h e d e v e l o p m e n t o f ' s t a b i l i t y c h a r t s ' . Such c h a r t s a r e c o n s t r u c t e d u s i n g some s p e c i f i c i n t e r c e p t o f t h e l i m i t i n g s u r f a c e as a measure o f s t a b i l i t y p l o t t e d a g a i n s t a c o n t r o l l i n g p a r a m e t e r . T h i s a n a l y s i s p l o t s t h e l i m i t i n g v a l u e o f f a t 9 = - 0 a g a i n s t e c c e n t r i c i t y , t h u s r e a l i z i n g a f u r t h e r c o n d e n s a t i o n o f d a t a . F i g u r e 2.10 shows a t y p i c a l f a m i l y o f s t a b i l i t y c h a r t s f o r v a r i a t i o n s i n t h e s o l a r p a r a m e t e r . The r u g g e d n a t u r e o f t h e s t a b i l i t y s u r f a c e i s a s s o c i a t e d w i t h t h e emergence o f v a r i a t i o n a l l y s t a b l e p e r i o d i c s o l u t i o n s . I t s h o u l d be n o t e d t h a t u n s t a b l e p e r i o d i c s o l u -t i o n s do n o t a f f e c t t h e s t a b i l i t y b o u n d a r y . F i g u r e 2.9 F a m i l y o f l i m i t i n g i n t e g r a l m a n i f o l d c r o s s s e c t i o n s a t 9 = 0 f o r a r a n g e o f e a t K. = 1.0: (a) c=0, 0.1 m 1 ro stable regions stable periodic motion unstable periodic motion 2.0 1.5 1.0 0.5 o it CD -9- CD 0.0 •0.5 - 1.0 1.5 -2 .0 stable regions stable periodic motion unstable periodic motion C = 0.3 Kj = 1.0 <f)=0.0 1 ^ 3 / 2 1 1 -1 1 1 o II' CD -9-CD 2.0 1.5 1.0 0.5 0.0 -0.5 -1.0 -•1.5 -0.0 0.1 0.2 0.3 0.4 •2-0 C=0.5 Kj =1.0 <j) = 0.0 J L 2.0 9 -2.0 C=0.75 K=1.0 rj>=0.0 1.5 -'•0 r.3/2 0.5 0.0 •0.5 - 1.0 --1.5 0.0 0.1 0.2 0.3 0.0 0.1 e 0.2 F i g u r e 2.10 - ^ S t a b i l i t y c h a r t s s h o w i n g t h e e f f e c t o f s o l a r p a r a m e t e r a t ~ K . = l 0 (b) c=0.3, 0.5, 0.75 i 56 2.3.2 V a r i a t i o n a l A n a l y s i s of P e r i o d i c S o l u t i o n s The concept th a t s t a b i l i t y regions are c l o s e l y a s s o c i a t e d w i t h s t a b l e p e r i o d i c s o l u t i o n s was introduced i n the preceding development. Indeed, p e r i o d i c s o l u t i o n s provide a powerful t o o l i n the c o n s t r u c t i o n of s t a b i l i t y c h a r t s , forming the 'frame' about which they are b u i l t . Moreover, by applying 70 Floquet theory, the v a r i a t i o n a l s t a b i l i t y of the p e r i o d i c s o l u t i o n can be determined, thus l o c a t i n g the t e r m i n a t i o n of an i s l a n d ' s p i k e 1 or mainland t i p . I t a l s o helps i n d i s c o v e r i n g i s l a n d s of s t a b i l i t y which would be extremely d i f f i c u l t f o f i n d otherwise. As p e r i o d i c s o l u t i o n s form the b a s i s of the s t a b i l i t y c h a r t s , a note concerning t h e i r determination would be appro-p r i a t e . By nu m e r i c a l l y i n t e g r a t i n g the equation of motion over the r e q u i r e d p e r i o d and s y s t e m a t i c a l l y v a r y i n g ^ ' ( 0 ) , the s o l u t i o n of i n t e r e s t was bracketed. Comparing (2m\) w i t h i t s i n i t i a l value and applying a c o r r e c t i o n i n accordance wi t h the v a r i a b l e secant method, the s o l u t i o n can be determined to any d e s i r e d degree of accuracy. For the c o n s t r u c t i o n of the s t a b i l i t y c h a r t s , the p e r i o d i c s o l u t i o n s 1/1, 3/2, 4/3, and 6/4 were i n v a r i a b l y found to be the most s i g n i f i c a n t . As mentioned before, the mainland i s c l o s e l y a s s o c i a t e d w i t h the fundamental s o l u t i o n 1/1, whi l e the others c o n t r o l the spikes emerging from the mainland. , The v a r i a t i o n a l a n a l y s i s of the p e r i o d i c s o l u t i o n ' was accomplished as f o l l o w s . S u b s t i t u t i n g ^ = + e i n t o the 57 equation of motion (2.19) and l i n e a r i z i n g with respect to yields the v a r i a t i o n a l equation (2.37) where f ( t ) 2 e Sin6 1 + e Cos 6 0(1)- 3KiCos2tp P c ( i . e ) 3 y l + eCos6 ( l + ecosfc)4-^5in(© + typ-$) Sih(©+ ^p-^H-^)-5in(e*-Vp-4>) Simultaneously integrating the periodic solution and the va r i a t i o n a l equation (2.37) over the period 2mr with the i n i t i a l conditions £ p ( o )» i £p (o) - o £ p ( o ) = o £p (0) - 1 (2.38) yields 58 £ P i(2nTr) + £p z (2 i rYTr) _ The p e r i o d i c s o l u t i o n s and t h e r e g i o n s o f t h e i r s t a b i l i t y a r e i n d i c a t e d i n F i g u r e s 2.10, 2.11, 2.14. 2.3.3 D i s c u s s i o n o f R e s u l t s The t y p i c a l i n t e g r a l m a n i f o l d s i n F i g u r e 2.8 show t h e e f f e c t o f t h e s o l a r p a r a m e t e r q u i t e c l e a r l y . As o p p o s e d t o p u r e g r a v i t y g r a d i e n t s a t e l l i t e s where t h e i n t e g r a l m a n i f o l d s a r e g r i g h t c y l i n d e r s f o r z e r o e c c e n t r i c i t y , i t i s i n t e r e s t i n g t o n o t e t h e s p i r a l i n g e f f e c t i n t r o d u c e d by t h e r a d i a t i o n p r e s s u r e as shown i n F i g u r e 2 . 8 ( a ) . The l i m i t i n g i n t e g r a l m a n i f o l d , f o r a s a t e l l i t e i n an e l l i p t i c o r b i t , as shown i n F i g u r e s 2.8(b) and 2 . 8 ( c ) , i n d i c a t e s a c o n s i d e r a b l e r e d u c t i o n i n t h e s t a b i l i t y r e g i o n s w i t h i n c r e a s e i n c. A s i m i l a r e f f e c t i s e v i d e n t f o r i n c r e a s e i n e c c e n t r i c i t y by c o m p a r i n g F i g u r e s 2.8(a) and 2 . 8 ( c ) . I t s h o u l d be e m p h a s i z e d t h a t t h i s r e d u c t i o n w i t h i n c r e a s e i n c o c c u r s f o r a l l v a l u e s o f 0 and, i n g e n e r a l , l e a d s t o n e s t e d m a n i f o l d s o f d i m i n i s h i n g s i z e . As t h i s r e d u c t i o n i s r e l a t i v e l y i n d e p e n d e n t o f 0, t h e s t r o b o s c o p i c p h a s e p l a n e p l o t s ( F i g u r e 2.9) a t 6 = 0 and t h e s t a b i l i t y c h a r t s ( F i g u r e s 2.10, 2.11, 2.14) a t 0 = IJJ = 0 c a n e f f e c t i v e l y s e r v e as a measure o f s t a b i l i t y o f t h e s y s t e m . Note t h a t t h e y a r e a r e l a t e d s e t o f p l o t s and s h o u l d be s t u d i e d t o g e t h e r . They v i v i d l y show t h e > 1 , unstable ^ 1 , stable (2.3q) e f f e c t o f e c c e n t r i c i t y and t h e s o l a r p a r a m e t e r on t h e s t a b i l i t y b o u nds. The r e d u c t i o n i n t h e l i m i t i n g e c c e n t r i c i t y ( m a i n l a n d ) f o r s t a b l e m o t i o n , w h i c h c o r r e s p o n d s t o t h e d e g e n e r a t i o n o f t h e i n t e g r a l m a n i f o l d t o t h e f u n d a m e n t a l p e r i o d i c s o l u t i o n , i s s i g n i f i c a n t . T h i s i n f o r m a t i o n d e m o n s t r a t e s t h e i m p o r t a n c e o f a r e a d i s t r i b u t i o n i n t h e d e s i g n o f a s a t e l l i t e . F o r g i v e n c , F i g u r e 2.11 shows t h e i n f l u e n c e o f t h e i n e r t i a p a r a m e t e r on t h e m a r g i n o f s t a b i l i t y f o r a g r a v i t y g r a d i e n t s a t e l l i t e . The e f f e c t i s s i m i l a r t o t h a t o b s e r v e d by 9 B r e r e t o n and Modi i n a s t u d y n e g l e c t i n g t h e r a d i a t i o n e f f e c t . As s t a t e d b e f o r e , t h e c r o s s - s e c t i o n s o f t h e i n t e g r a l m a n i f o l d s ( F i g u r e 2.9) a r e w e l l s u i t e d f o r p a r a m e t r i c s t u d i e s . A l t h o u g h t h e c r o s s - s e c t i o n a l a r e a s do n o t a p p r e c i a b l y change w i t h 0, t h e s p i r a l i n g n a t u r e o f t h e s u r f a c e d i s p l a c e s them s u f f i c i e n t l y t o r e d u c e t h e s t a b i l i t y r e g i o n a v a i l a b l e o v e r t h e e n t i r e o r b i t . As t h e s e 6 - i n d e p e n d e n t s t a b i l i t y r e g i o n s w o u l d be o f i n t e r e s t t o t h e d e s i g n e n g i n e e r , t h e y a r e p l o t t e d as a f u n c t i o n o f e c c e n t r i c i t y i n F i g u r e 2.12. N o t e t h a t t h e p l o t s show c o n s i d e r a b l e r e d u c t i o n n o t o n l y i n t h e s t a b i l i t y bounds b u t a l s o i n t h e c r i t i c a l e c c e n t r i c i t y . As shown e a r l i e r i n F i g u r e 2 . 7 ( a ) , f o r a s a t e l l i t e i n an e l l i p t i c o r b i t , t h e r e e x i s t s a n o n - z e r o v a l u e o f c f o r w h i c h l i b r a t i o n a l m o t i o n i s a minimum. As s t a b i l i t y c h a r t s a r e r e -l a t e d t o t h e l i b r a t i o n a l r e s p o n s e , i t s u g g e s t s t h a t t h e r e s h o u l d a l s o be a v a l u e o f c f o r t h e maximum c r i t i c a l e c c e n t r i c i t y . To s u b s t a n t i a t e t h i s , ^ m a x was p l o t t e d a g a i n s t t h e s o l a r param-e t e r f o r a r a n g e o f v a l u e s o f ( F i g u r e 2 . 1 3 ) . I t i s a p p a r e n t t h a t f o r a g i v e n i n e r t i a p a r a m e t e r t h e r e i s a v a l u e o f c f o r 6 1 which c r i t i c a l e c c e n t r i c i t y i s a maximum. The o v e r a l l maximum mainland e c c e n t r i c i t y i s giv e n by a system w i t h = 0.7 and c = -0.25. For these c o n d i t i o n s , e was found t o be n e a r l y max J double t h a t p r e d i c t e d by g r a v i t y f o r c e s alone (c = 0). The s t a b i l i t y c h a r t s corresponding t o = 1.0, 0.7 u s i n g the optimum values of c, are p l o t t e d i n F i g u r e 2.14. Note t h a t the c h a r a c t e r of the s t a b i l i t y r e g i o n i s changed c o n s i d e r a b l y , however, the net area appears t o be r e l a t i v e l y u n a f f e c t e d . I t should be emphasized t h a t such s t a b i l i t y c h a r t s are a p p l i c a b l e to s h o r t - l i f e s a t e l l i t e s only s i n c e the s o l a r aspect angle changes wi t h time. Hence, f o r long term p r e d i c t i o n , i t i s necessary t o vary <j> over the e n t i r e range t o o b t a i n the absolute r e g i o n of s t a b i l i t y . T h i s would e n t a i l an enormous amount of computation. F o r t u n a t e l y , some r e d u c t i o n i n e f f o r t can be r e a l i z e d s i n c e the governing equation of motion remains i n v a r i a n t when <J> i s r e p l a c e d by <j> + 180° and c by - c . However, a q u e s t i o n a r i s e s as t o the value of <J> t h a t would s a t i s f a c t o r i l y p r e d i c t long term s t a b i l i t y . C o n s i d e r i n g e t o be r e p r e s e n t a t i v e of the s t a b i l i t y ^ max c • 2 area as i n d i c a t e d by F i g u r e s 2.10 and 2.11, i t s v a r i a t i o n with <f> was p l o t t e d f o r a range of values of the s o l a r parameter (Figure 2.15). I t i s i n t e r e s t i n g t o note t h a t , i n g e n e r a l , the minimum value of c r i t i c a l e c c e n t r i c i t y occurs at <j> = 0, which e x p l a i n s i t s use i n the f o r e g o i n g a n a l y s i s . 66 2.4 R a d i a t i o n Damping and A t t i t u d e C o n t r o l 2.4.1 Damping C o n c e p t The s a t e l l i t e r e s p o n s e , o b t a i n e d o v e r a wide r a n g e o f i n i t i a l c o n d i t i o n s and p h y s i c a l p a r a m e t e r s ( S e c t i o n s 2.2 and 2. 3 ) , showed a s u b s t a n t i a l i n f l u e n c e o f s o l a r r a d i a t i o n p r e s -s u r e . Under t h e n o r m a l c i r c u m s t a n c e s i n v e s t i g a t e d , r a d i a t i o n a f f e c t s t h e s y s t e m p e r f o r m a n c e a d v e r s e l y . However, t h e r e s u l t s s u g g e s t t h a t by j u d i c i o u s c o n t r o l o f t h e s o l a r param-e t e r / r a d i a t i o n c a n p r o v i d e an e f f e c t i v e damping t o r q u e t o m a i n t a i n a d e s i r e d a t t i t u d e . Hence, t h e u s e o f s o l a r p r e s s u r e i n i m p r o v i n g t h e a t t i t u d e c o n t r o l o f a g r a v i t y o r i e n t e d s a t e l l i t e m e r i t s a d e t a i l e d i n v e s t i g a t i o n . The g e n e r a l e q u a t i o n o f m o t i o n g o v e r n i n g l i b r a t i o n a l r e s p o n s e o f a g r a v i t y o r i e n t e d s a t e l l i t e i n t h e p r e s e n c e o f s o l a r r a d i a t i o n was g i v e n by e q u a t i o n (2.19) a s : (l + e C o s 6 ) ^ " - ae(i|J'M)Sih 6 + 3K ; S i n ^ Cosy (s.iq) C(n-e) 3 (i+-eCose)3 Sin(e + -^4>) 5in(e+ -4>) I t i s a p p a r e n t t h a t t h e r a d i a t i o n e f f e c t on t h e s a t e l l i t e d y n a m i c s i s g o v e r n e d by t h e s o l a r p a r a m e t e r c w h i c h i s a f u n c -t i o n o f p e r i g e e d i s t a n c e , g e o m e t r y , and i n e r t i a o f t h e s a t e l l i t e . Hence, f o r g i v e n o r b i t a l m o t i o n , c c a n be c o n t r o l l e d i n m a g n i t u d e and s e n s e by m a n e u v e r i n g t h e a r e a A so as t o y y a p p r o p r i a t e l y p o s i t i o n t h e c e n t e r o f p r e s s u r e r e l a t i v e t o t h e c e n t e r o f mass. Thus t h e s o l a r t o r q u e c a n be made t o o p p o se t h e l i b r a t i o n a l m o t i o n . I n p r a c t i c e t h i s w o u l d be a c c o m p l i s h e d by a c o n t r o l l e r o f s u i t a b l e c h a r a c t e r i s t i c s r e g u l a t i n g t h e s o l a r p a r a m e t e r i n a d e s i r e d f a s h i o n . F o r e x ample, c o n s i d e r a bank o f s l a t s a t t a c h e d t o t h e s a t e l l i t e a l o n g t h e Z - a x i s . ^ s I n a c c o r d a n c e w i t h t h e demand f r o m t h e c o n t r o l l e r , t h e s e s l a t s c o u l d be opened o r c l o s e d t h u s r e g u l a t i n g A , and h e n c e c as d e s i r e d . I t s h o u l d be n o t e d t h a t s u c h c o n t r o l w i l l have, n e g l i g i b l e e f f e c t on t h e i n e r t i a o f t h e s y s t e m . I n t h i s a n a l y s i s , t h e s o l a r p a r a m e t e r i s t a k e n t o be c o n t r o l l e d a c c o r d i n g t o t h e r e l a t i o n (2.40) | C | 4 C max r> where c r e p r e s e n t s t h e l i m i t on t h e s o l a r p a r a m e t e r as im-max c c p o s e d by t h e s a t e l l i t e d e s i g n . N ote t h a t b o t h p o s i t i v e and n e g a t i v e s i g n s a r e i n c l u d e d i n t h e e x p r e s s i o n t o p r o v i d e t h e damping t o r q u e t h r o u g h o u t t h e o r b i t . Thus (2.41) C = 4 - y U c t p / ; TT < (6 + - <b) ^ 2.TT . 68 2.4.2 A n a l y s i s and D i s c u s s i o n o f R e s u l t s The g o v e r n i n g n o n - l i n e a r , non-autonomous d i f f e r e n t i a l e q u a t i o n (2.19) w i t h c d e f i n e d i n (2.41) does n o t a d m i t o f a known c l o s e d f o r m s o l u t i o n . A n u m e r i c a l a p p r o a c h u n d e r s u c h a s i t u a t i o n c a n be u s e d t o an a d v a n t a g e . The e q u a t i o n o f m o t i o n was i n t e g r a t e d ' u s i n g t h e A d a m s - B a s h f o r t h p r e d i c t o r -c o r r e c t o r p r o c e d u r e i n c o n j u n c t i o n w i t h a s t e p s i z e o f 5 ° . F i g u r e s 2.16, 2.17, and 2.18 show t h e r e s p o n s e o f t h e s y s t e m t o v a r i o u s i n i t i a l c o n d i t i o n s w h i c h i l l u s t r a t e s e v e r a l i n t e r e s t i n g c h a r a c t e r i s t i c s o f t h e c o n t r o l l e d p e r f o r m a n c e . I t i s a p p a r e n t t h a t t h e c o n t r o l l e r p r o v i d e s a r a p i d d e c a y o f l i -b r a t i o n , l e a d i n g t o s t a b l e , s m a l l a m p l i t u d e m o t i o n i n t h r e e t o f o u r o r b i t s . F o r c i r c u l a r o r b i t s , F i g u r e 2.16 shows t h e l i b r a t i o n s t o be c o m p l e t e l y damped. Note t h a t f o r t h e same i n i t i a l c o n d i t i o n s i n a b s e n c e o f t h e r a d i a t i o n damping, t h e s y s t e m u n d e r g o e s l a r g e a m p l i t u d e ( F i g u r e 2.16) o r e v e n t u m b l i n g ( F i g u r e 2.17) m o t i o n . F i g u r e 2.18 i s o f p a r t i c u l a r i n t e r e s t and c l e a r l y shows t h e t e n d e n c y o f t h e s y s t e m t o o r i e n t i t s e l f i n t h e ' r i g h t s i d e up' p o s i t i o n o n l y . T h u s , t h e p o s s i b i l i t y o f s t a b i l i z i n g a g r a v i t y g r a d i e n t s a t e l l i t e i n an u p s i d e down p o s i t i o n , as c a n e a s i l y happen w i t h a c o n v e n t i o n a l s y s t e m , no l o n g e r e x i s t s . F u r t h e r m o r e , i t i s o f i n t e r e s t t o n o t e t h a t t h e s y s t e m r e s p o n s e a t t a i n s a s t a b l e , s m a l l a m p l i t u d e , p e r i o d i c m o t i o n i n e c c e n t r i c o r b i t s i n d e p e n d e n t o f i n i t i a l c o n d i t i o n s ( F i g u r e 2 . 1 9 ) . The a m p l i t u d e o f t h e p e r i o d i c m o t i o n i s 0 - orbits Figure 2.16 Typical librational decay in presence of solar damping 70 Figure 2.17 S t a b i l i z i n g i n f l u e n c e of s o l a r damping 440k e = o.i Kj = 1.0 (|> =0.0 l|)(0) = 2.0 l|/(0) =0.0 = 7.0 Cmax = 0.2 0 - orbits F i g u r e 2.18 S y s t e m r e s p o n s e s h o w i n g u p r i g h t c o n t r o l due t o s o l a r damping 0 1 2 3 4 5 0 1 2 3 4 5 Q - orbi ts 8 - o r b i t s F i g u r e 2.19 System response showing independence of l i m i t c y c l e from i n i t i a l c o n d i t i o n s g o v e r n e d by c and t h e p r o p o r t i o n a l i t y c o n s t a n t , t h e minimum max J v a l u e b e i n g g i v e n by t h e optimum s i z e o f t h e p a r a m e t e r s . The s e a r c h f o r t h e s e optimum v a l u e s i s g r e a t l y f a c i l i t a t e d by t h e c h o i c e o f a s m a l l i n i t i a l d i s t u r b a n c e w h i c h q u i c k l y l e a d s t o t h e s t a b l e l i m i t c y c l e . F i g u r e 2.20 shows t h e v a r i a t i o n o f IJJ and optimum y^ as a f u n c t i o n o f c o v e r a r a n g e o f o r b i t e c c e n t r i c i t y and max ^ J f o r r e p r e s e n t a t i v e v a l u e s o f t h e i n e r t i a p a r a m e t e r . The p l o t s were o b t a i n e d by s y s t e m a t i c a l l y v a r y i n g y f o r g i v e n c and c max e a c c o r d i n g t o t h e p r o c e d u r e i n d i c a t e d a b o v e . N o t e t h a t t h e c u r v e s t e n d t o a u n i f o r m v a l u e q u i t e r a p i d l y w i t h i n c r e a s e i n c , p a r t i c u l a r l y f o r s m a l l v a l u e s o f e. The i n f o r m a t i o n max c u s h o u l d p r o v e u s e f u l i n t h e d e s i g n o f an a t t i t u d e c o n t r o l s y s t e m . F o r a g i v e n s a t e l l i t e and i t s t r a j e c t o r y , t h e v a l u e s o f and e a r e known. D e p e n d i n g on t h e n a t u r e o f t h e m i s s i o n , p e r m i s s i b l e l i b r a t i o n a l m o t i o n i s f i x e d and h e n c e ip i s ^ rmax s p e c i f i e d . U s i n g t h i s i n f o r m a t i o n , t h e d e s i g n c h a r t s d i r e c t l y g i v e t h e optimum c and y r e q u i r e d t o a c h i e v e t h e d e s i r e d max c a t t i t u d e c o n t r o l . The l i b r a t i o n a l c o n t r o l t h r o u g h s o l a r r a d i a t i o n p r e s s u r e a p p e a r s t o be c o n s i d e r a b l y more e f f e c t i v e t h a n t h e s p r i n g - m a s s -damper a r r a n g e m e n t p r o p o s e d by P a u l ^ and i t s g e n e r a l i z a t i o n 6 X 6 2 i n v e s t i g a t e d by T s c h a n n e t a l . ' . They c o n c l u d e d t h a t t h e damped m o t i o n a p p r o a c h e s , i n t h e l i m i t , t h e f u n d a m e n t a l p e r i o d i c s o l u t i o n o f an undamped s a t e l l i t e . W i t h t h i s i n m i n d , t h e optimum v a l u e o f t h e l i b r a t i o n a l m o t i o n t h a t c a n be a t t a i n e d w i t h a v i s c o u s damper was d e t e r m i n e d as a f u n c t i o n o f e c c e n -76 t r i c i t y and i n e r t i a parameters. This i s shown i n Figure 2.21. I t i s apparent, through a comparison of F i g u r e s 2.20 and 2.21 th a t the proposed system can achieve much f i n e r a t t i t u d e con-t r o l . For example, w i t h = 0.8 and e = 0.2, the viscous damper can provide the minimum response of 18° as opposed to the c o n t r o l of 2-3° given by the s o l a r r a d i a t i o n damper. A comment concerning the r e l a t i v e r a t e of damping would be ap p r o p r i a t e . As against the viscous damper, which normally r e q u i r e s 10-15 o r b i t s , the present system takes only 3-4 o r b i t s to a t t a i n l i m i t i n g p e r i o d i c motion (Figure 2.19). I t should be pointed out t h a t the s o l a r r a d i a t i o n torque fun c t i o n s independent of the g r a v i t a t i o n a l e f f e c t . A consider-a t i o n of the system response f o r la r g e e c c e n t r i c i t i e s (Figure 2.20) c l e a r l y i n d i c a t e s t h a t the l i b r a t i o n a l motion of a r a d i a t i o n damped s a t e l l i t e i s considerably l e s s than that of a pure g r a v i t y o r i e n t e d system (Figure 2.21). Apparently, under these circumstances, the g r a v i t y gradient torques have an adverse e f f e c t and appear as a p e r t u r b i n g force to the s o l a r s t a b i l i z e d motion. Under these c o n d i t i o n s , i t would be d e s i r a b l to s t a b i l i z e a s a t e l l i t e using s o l a r r a d i a t i o n only. H e r e , - i t i s necessary to modify the c o n t r o l l e r such t h a t c a l s o becomes a f u n c t i o n of the l i b r a t i o n a l angle . Such a system has s e v e r a l d i s t i n c t advantages: f i r s t , the l i b r a t i o n a l motion w i l l be l a r g e l y independent of e c c e n t r i c i t y , thus improving upon the performance of a g r a v i t y o r i e n t e d system; the s a t e l l i t e could a l s o be s t a b i l i z e d about any reference angle, thus, multi-purpos missions, r e q u i r i n g d i f f e r e n t o r i e n t a t i o n s , could be performed wi t h the same s a t e l l i t e ; furthermore, the system acts'as i t s own 77 78 damper a l w a y s t e n d i n g t o t h e p r e f e r r e d o r i e n t a t i o n . 2.5 C o n c l u d i n g Remarks The s a l i e n t f e a t u r e s o f t h e a n a l y s i s and r e l e v a n t c o n -c l u s i o n s may be summarized as f o l l o w s : i ) F o r s m a l l a m p l i t u d e m o t i o n t h e WKBJ t o g e t h e r w i t h t h e Harmonic B a l a n c e method c a n p r e d i c t l i b r a t i o n a l r e s p o n s e o f a s a t e l l i t e , s u b j e c t e d t o g r a v i t y g r a d i e n t and s o l a r r a d i a t i o n f o r c e s , w i t h c o n s i d e r a b l e a c c u r a c y . The d i s c r e p a n c y due t o n o n l i n e a r i t i e s , w h i c h become s i g n i f i c a n t d u r i n g s e v e r e d i s t u r b a n c e s , p r i m a r i l y a f f e c t s t h e d e t a i l e d c h a r a c t e r i s t i c s o f t h e r e s p o n s e w i t h o u t s u b s t a n t i a l l y a l t e r i n g t h e maximum a m p l i t u d e o r f r e q u e n c y o f t h e m o t i o n . As most o f t h e communica-t i o n , w e a t h e r , and e a r t h r e s o u r c e s s a t e l l i t e s f o l l o w n e a r c i r c u l a r t r a j e c t o r i e s and p e r m i t o n l y s m a l l a m p l i -t u d e l i b r a t i o n s , t h e r e s u l t s g i v e n by t h e WKBJ method a r e o f s u f f i c i e n t a c c u r a c y t o be u s e f u l d u r i n g a p r e -l i m i n a r y d e s i g n s t a g e , i i ) The d i r e c t s o l a r r a d i a t i o n c o n s i d e r a b l y a f f e c t s t h e l i b r a t i o n a l d y n a m i c s o f a s a t e l l i t e . F o r n o n - z e r o c t h e s a t e l l i t e a l w a y s e x e c u t e s l i b r a t i o n a l m o t i o n . Hence n e g l e c t i n g t h e r a d i a t i o n f o r c e s when p r e d i c t i n g t h e a t t i t u d e b e h a v i o r o f t h e g r a v i t y g r a d i e n t s a t e l l i t e s c o u l d r e s u l t i n g r o s s e r r o r s . However, t h r o u g h p r o p e r d e s i g n , r a d i a t i o n p r e s s u r e c a n s u b s t a n t i a l l y r e d u c e t h e l i b r a t i o n a l m o t i o n o f a g r a v i t y g r a d i e n t s a t e l l i t e i n an e l l i p t i c o r b i t . 79 i i i ) F o r g i v e n c , t h e minimum l i b r a t i o n a l m o t i o n does n o t c o r r e s p o n d t o z e r o i n i t i a l c o n d i t i o n s e v e n f o r c i r c u l a r o r b i t . I n f a c t , c r i t i c a l i n i t i a l c o n d i t i o n s a r e i d e n -t i c a l t o t h o s e w h i c h r e s u l t i n p e r i o d i c s o l u t i o n s . On t h e o t h e r hand, f o r g i v e n i n i t i a l c o n d i t i o n s , t h e r e , e x i s t s a v a l u e o f c f o r w h i c h l i b r a t i o n a l m o t i o n i s a minimum. i v ) The c o n c e p t o f i n t e g r a l m a n i f o l d s i n a t h r e e d i m e n s i o n a l p h a s e s p a c e c a n be e f f e c t i v e l y u s e d t o s t u d y s t a b i l i t y o f t h e s y s t e m . The l i m i t i n g i n v a r i a n t s u r f a c e p r o v i d e s a l l p o s s i b l e c o m b i n a t i o n s o f d i s t u r b a n c e s w h i c h t h e s a t e l l i t e c a n w i t h s t a n d w i t h o u t t u m b l i n g , v) The i m p o r t a n c e o f p e r i o d i c s o l u t i o n s i n s u c h s t u d i e s c a n n o t be o v e r e m p h a s i z e d , as t h e y f o r m t h e f r a m e . a b o u t w h i c h s t a b i l i t y c h a r t s a r e b u i l t . A t c r i t i c a l e c c e n -t r i c i t y t h e o n l y a v a i l a b l e bounded m o t i o n c o r r e s p o n d s t o t h e f u n d a m e n t a l p e r i o d i c s o l u t i o n . ! v i ) The s o l a r p a r a m e t e r c s u b s t a n t i a l l y a f f e c t s t h e s t a b i l i t y o f t h e m o t i o n and h e n c e m e r i t s e q u a l c o n s i d e r a t i o n w i t h e c c e n t r i c i t y and i n e r t i a p a r a m e t e r s . The i n t e g r a l • m a n i f o l d s , m a r g i n o f s t a b i l i t y , and c r i t i c a l e c c e n t r i c -i t y r e d u c e w i t h i n c r e a s i n g c . Lo n g t e r m s t a b i l i t y , bounds c a n b e s t be p r e d i c t e d by c o n d u c t i n g t h e s t u d y a t <j) = 0. v i i ) S o l a r r a d i a t i o n p r e s s u r e c a n be u s e d q u i t e e f f e c t i v e l y t o c o n t r o l t h e l i b r a t i o n a l m o t i o n o f a s a t e l l i t e . N ot o n l y c a n i t a v o i d t u m b l i n g m o t i o n b u t c a n a l s o 80 s t a b i l i z e the s a t e l l i t e , i n i t s upright p o s i t i o n , even when subjected to a large disturbance. In general, the system response attains a stable, small amplitude, periodic motion independent of i n i t i a l conditions. v i i i ) The rate and amplitude of l i b r a t i o n a l control by solar radiation i s considerably more e f f e c t i v e than the viscous damper arrangement proposed by other authors. With the optimum values of c and y , a high degree ^ max c ' ' ' of pointing accuracy can be attained making the system suitable for communication, weather, m i l i t a r y and earth resources s a t e l l i t e s , ix) The system not only improves but also extends the effectiveness of gravity gradient s a t e l l i t e s to higher e c c e n t r i c i t y o r b i t s . 3. ATTITUDE DYNAMICS OF A SATELLITE ACCOUNTING FOR EARTH RADIATIONS In a d d i t i o n to d i r e c t s o l a r r a d i a t i o n , t h i s chapter i n c l u d e s the e f f e c t of d i r e c t e a r t h r a d i a t i o n and i t s albedo on the a t t i t u d e dynamics of g r a v i t y o r i e n t e d systems, thus extending the a n a l y s i s of Chapter 2 down t o the ear t h ' s e f f e c t i v e atmosphere. The importance of these r a d i a t i o n s was i n d i c a t e d by a p r e l i m i n a r y f o r c e a n a l y s i s (Chapter 1), which r e v e a l e d t h a t they can e x e r t a f o r c e comparable to t h a t of d i r e c t s o l a r r a d i a t i o n f o r c l o s e e a r t h s a t e l l i t e s . Thus, the need f o r a r i g o r o u s i n v e s t i g a t i o n of e a r t h r a d i a t i o n s i s e v i d e n t . E a r t h r a d i a t i o n f o r c e s were obtained i n i n t e g r a l form 55 . . . by Clancy and M i t c h e l l u s i n g elementary r a d i a t i o n p r i n c i p l e s . However, i n a t e s t on an IBM-7044 d i g i t a l computer, i t was found t h a t the time r e q u i r e d to eva l u a t e these i n t e g r a l s renders a study of s a t e l l i t e a t t i t u d e dynamics v i r t u a l l y i m p o s s i b l e . Thus, one of the major outcomes of t h i s a n a l y s i s i s the accurate e x p r e s s i o n of these f o r c e s i n c l o s e d form accomplished through the concept of c u t t i n g plane d i s t a n c e r a t i o s . Although the r e -s u l t i n g f o r c e e x p r e s s i o n s are lengthy, they are i d e a l l y s u i t e d to a numerical a n a l y s i s of e a r t h o r b i t i n g s a t e l l i t e s . The, c l o s e d form nature of the expr e s s i o n s reduces the computational time t o approximately 1/10Oth of t h a t r e q u i r e d w i t h the i n t e g r a l equations, thus p e r m i t t i n g the a n a l y s i s of a s a t e l l i t e at lower a l t i t u d e s . 82 The a t t i t u d e d y n a m i c s o f t h e s y s t e m i s a n a l y s e d , f o r a r b i t r a r y e c c e n t r i c i t y , i n t e r m s o f i t s d e s i g n p a r a m e t e r s , c and K.. P a r t i c u l a r a t t e n t i o n i s d i r e c t e d t o w a r d s t h e e f f e c t x o f t h e e a r t h r a d i a t i o n s t o e s t a b l i s h t h e i r r e l a t i v e i m p o r t a n c e i n a t t i t u d e dynamic s t u d i e s . The s y s t e m r e s p o n s e p l o t s and s t a b i l i t y c h a r t s a r e p r e s e n t e d w h i c h p r o v i d e c o m p l e t e i n f o r -m a t i o n c o n c e r n i n g t h e b e h a v i o r o f a s a t e l l i t e . 3.1 F o r m u l a t i o n o f t h e P r o b l e m The g e n e r a l e q u a t i o n g o v e r n i n g t h e l i b r a t i o n a l m o t i o n o f an a r b i t r a r i l y s h a p e d s a t e l l i t e was g i v e n by e q u a t i o n (2.13) a s : where r e p r e s e n t s t h e g e n e r a l i z e d f o r c e due t o d i r e c t s o l a r , and e a r t h r a d i a t i o n s . T h i s s e c t i o n i s p r i m a r i l y c o n c e r n e d w i t h t h e e v a l u a t i o n o f Q . The s a t e l l i t e i s c o n s i d e r e d t o be a s p e c u l a r l y r e f l e c t i n g body moving a l o n g an a r b i t r a r y t r a j e c -t o r y i n t h e p l a n e o f t h e e c l i p t i c and e x e c u t i n g p l a n a r l i b r a -t i o n a l m o t i o n , w h i l e t h e e a r t h , t a k e n s p h e r i c a l l y , i s assumed t o r e f l e c t and r a d i a t e d i f f u s e l y . I n a c c o r d a n c e w i t h t h e model c h o s e n , t h e r a d i a t i o n f o r c e s a r e c a l c u l a t e d on an a r b i t r a r i l y s h a p e d f l a t s u r f a c e o r p l a t e . 3.1.1 F o r c e Due t o D i r e c t S o l a r R a d i a t i o n From e q u a t i o n ( 2 . 1 6 ) , t h e d i r e c t s o l a r r a d i a t i o n f o r c e on an e l e m e n t o f a r e a c a n be w r i t t e n a s : GL (2.13) 83 ^ d a -s' 7(1+p - T ) Cos CL i (2.1W + -g- (1 - p - T ) S in oL d A Recognizing that S 1 = s|cosa|, cosa = Sin(e+^-<j>) and a = constant f o r a plane s u r f a c e , the x-component of force on a f l a t p l a t e i s given by: ^ - ( l t p - T ) M c ^ 13.1) 5 i n ( e + t|i-4>) S i n ( e «-vf,-<H 1 • The v a r i a t i o n of t h i s force w i t h angle of incidence i s shown i n Figure 3.1. 3.1.2 Force Due to D i r e c t Earth R a d i a t i o n The determination of the force due to d i r e c t e a rth r a d i a t i o n i s r e l a t i v e l y more d i f f i c u l t . From Figure 3.2 i t i s apparent t h a t the problem can be d i v i d e d i n t o two general cases. Case I can be simply defined by | ij; | > B m where the s a t e l l i t e plane l i e s on or outside the tangent cone and does not cut the s p h e r i c a l cap. Case I I adds f u r t h e r complexity to the problem and i s defined by |^ | < 3 . Here, o b v i o u s l y , the s a t e l l i t e plane l i e s where M c , = ds Figu re 3.1 V a r i a t i o n of d i r e c t s o l a r r a d i a t i o n pressure w i t h angle of i nc idence F i g u r e 3.2 D i r e c t e a r t h r a d i a t i o n and o r b i t a l g e o m e t r y 86 w i t h i n the tangent cone and consequently cuts the s p h e r i c a l cap. Thus, both s ides of the p l a t e exper ience force due to d i r e c t ea r th r a d i a t i o n . Cons ider f i r s t the genera l equa t ion . I r r a d i a t i o n on the p l a t e due to the e lementa l ea r th a r ea , dA , can be w r i t t e n as 4 Cos £2. <d A e (3.2) Thus, the x-component of the force becomes Recogn iz ing tha t d A e = (LS in(3 d T ) (3.4) C b s T the force expres s ion takes the form £0- ' T e 4 ( H - p - t ) A TT C ' 87 + S i n 2(3 3 Sm 3(3 Cos 2 T (3.5) - ^ C o s ^ 5 - . n p 3 Cos(3 S i n 2 ( 3 C o s T d Y d £ ^ or K / f((3 3, (3, T) d T d ^ ' (3.G) where K _ £ ( T ' T £ 4 ( u p - T ) A C ' A comment concerning the v a l i d i t y of t h i s expression f o r a l l angles would be appropriate. The changes i n s i g n of c e r t a i n t r i g o n o m e t r i c f u n c t i o n s n e c e s s i t a t e s the i n t r o d u c t i o n of the corresponding absolute terms. T h i s , however, renders i n t e g r a t i o n of the expression impossible. On the other hand, one can care-f u l l y study the i n t e g r a t e d expression and a p p r o p r i a t e l y c o n t r o l the s i g n of the req u i r e d terms to ob t a i n proper magnitude and sense of the r e s u l t a n t f o r c e . Such c o n t r o l i s not always obvious and may e n t a i l r e f o r m u l a t i o n of the equation using a new o r i e n t a t i o n . Case I : Introducing appropriate l i m i t s of i n t e g r a t i o n , equa-t i o n (3.6) becomes c -I 1 r e 88 ( 3 . 7 ) O which on i n t e g r a t i o n y i e l d s F = j a L 3 C I r K R 3 ' (3.8) = c T O - ' T ^ 4 ( l + p - ^ ) A n < ' C re where Case I I : Here both sides of the p l a t e are exposed to earth r a d i a t i o n , hence the force expression modifies to Sin ' 1/R 2TT F = 1 re O O S i n - ' >k Y 4 K TT 89 (3.9) where v i s a v a r i a b l e l i m i t d e f i n e d by X - Cos" 1 r 5 i n V Cos ^ ° L Cos -q) S in p E v a l u a t i n g t h e s e c o n d t e r m i n e q u a t i o n (3.9) and making u s e o f (3.8) g i v e s F = .1 —4 £0- Te (i + p-T)A 3 TT C • 2 1 T 5 i n ^ ( l - i B L l l ! ? ) -I- Tf C o s 2 i|) ( 3 ( R £ - n ' / a R + 6 Sin tfi| ^  - S i n E tp) 3 / a - 4 C o s " ' | R 5 i n u) -2 ( l - 3 S i n 2 t p) (-^i^- GDs- l((RM)' /a |Tan^)|) '  R (3.10) 2 ( 1 - 3 S i n s -qj) |S in ^\(-kz - S i n £ u j ) + 6 ( C o 3 2 ip) R C o s - , ( ( R a - l V ' f e |Tan - £ cr' r e 4 (ltj) - r) A ^ C r e 90 where Note t h a t e q u a t i o n s (3.8) and (3.10) g i v e t h e f o r c e due t o d i r e c t e a r t h r a d i a t i o n f o r any a l t i t u d e and o r i e n t a t i o n o f t h e p l a t e . A r e p r e s e n t a t i v e v a r i a t i o n o f t h e f o r c e f o r t h e s e v a r i a b l e s i s shown i n F i g u r e 3.3, where t h e e a r t h i s c o n s i d e r e d 71 t o be a b l a c k body r a d i a t i n g u n i f o r m l y a t 250°K. Here t h e f o r c e i s c a l c u l a t e d f o r u n i t a r e a and u n i t a b s o r p t i v i t y . 3.1.3 F o r c e Due t o E a r t h A l b e d o The d e t e r m i n a t i o n o f a g e n e r a l c l o s e d f o r m e x p r e s s i o n f o r t h e f o r c e on a f l a t p l a t e due t o e a r t h a l b e d o i s i n d e e d a com-p l e x p r o b l e m . The d i f f i c u l t y a r i s e s n o t p r i m a r i l y f r o m t h e l a r g e number o f v a r i a t i o n s t h a t e x i s t b u t r a t h e r f r o m t h e f a c t t h a t t h e e x p r e s s i o n c a n n o t be i n t e g r a t e d e x c e p t i n t h e s i m p l e s t o f c a s e s . From F i g u r e 3.4 i t i s a p p a r e n t t h a t t h e p r o b l e m c a n be d i v i d e d i n t o f i v e g e n e r a l c a s e s w i t h c a s e s IV and V h a v i n g two o r more v a r i a t i o n s d e p e n d e n t on t h e s a t e l l i t e a t t i t u d e and t h e s o l a r a s p e c t a n g l e . The c a s e s c a n be d e f i n e d as f o l l o w s : i ) C a se I e x i s t s when t h e t a n g e n t cone l i e s w h o l l y o u t -s i d e t h e s u n l i t h e m i s p h e r e , t h u s , I A I > + A . N o t e c 1 s 1 2 m t h a t t h i s c a s e r e s u l t s i n z e r o f o r c e , i i ) C a s e I I c o r r e s p o n d s t o t h e c o n f i g u r a t i o n i n w h i c h t h e t a n g e n t cone l i e s w h o l l y w i t h i n t h e s u n l i t h e m i s p h e r e and i s n o t c u t by t h e s a t e l l i t e p l a n e , t h u s , Usl< (f - A m ) , and M>B m. 91 Figure 3.3 V a r i a t i o n of d i r e c t ea r th r a d i a t i o n pressure w i t h a l t i t u d e and o r i e n t a t i o n F i g u r e 3 . 4 E a r t h albedo and o r b i t a l geometry i i i ) A s l i g h t v a r i a t i o n o f t h e above c a s e i s t h e s i t u a t i o n where t h e t a n g e n t cone l i e s w h o l l y w i t h i n t h e s u n l i t h e m i s p h e r e and i s i n t e r c e p t e d by t h e s a t e l l i t e p l a n e , i . e . , ]X S I « (J ~ X m) , and | T\> | < g m . i v ) I n C a s e IV t h e t a n g e n t cone l i e s p a r t i a l l y w i t h i n t h e s u n l i t h e m i s p h e r e and i s n o t c u t by t h e s a t e l l i t e p l a n e , t h u s , + A ) > I X I > (5- - A ) , and U I > B • ' 2 m 1 s 1 2 m ' 1 y 1 m v) C a s e V e x i s t s when t h e t a n g e n t cone l i e s p a r t i a l l y w i t h i n t h e s u n l i t h e m i s p h e r e and i s i n t e r c e p t e d by t h e s a t e l l i t e p l a n e , i . e . , (5- + A ) > I A I > (5- - A ) , and I \j> I < B . ^ ' ' 2 m 1 s 1 2 m ' 1 1 m C o n s i d e r f i r s t t h e g e n e r a l e q u a t i o n a p p l i c a b l e t o a l l t h e s e c a s e s . From F i g u r e 3.4, i r r a d i a t i o n on t h e p l a t e due t o t h e e l e m e n t o f e a r t h a r e a , dA , i s ' e dG = a * \ A Coscr Cosp 1 C o s ^ d A e o.irt > 1 L-t g i v i n g t h e x-component o f t h e f o r c e as S u b s t i t u t i n g t h e i d e n t i t i e s 94 Cos(3 2 = Cos (33 Cosp - S i n p 3 S i n (3 Cos r Cos CT = Cos Ao Cos X -r S in A<- S'mX Cos V -P- . S i n A p R - C o s A C o s X = R S i n 2 ( 3 +• C o s p ( 1 - R 2 S i n 2 (3 ) V a R - C o s \ = R Cos 2 p - Cosp ( 1 - R 2 S i n ^ p ) 1 ' 2 -(3.13) into (3.12) leads to the general expression for the earth albedo: + C o s 3 p Sinp ( 1 - R 2 S i n 2 p ) V 2 - } -f Sin A s C o s * p 3 ( R C o s 3 j 3 S i n a p C o s Y - (1-R* S . n 2 p ) V a Cos l p S i n 2 p Cos V ) + CosX s S i n 2 ^ 3 { R S \ n s p C o s 2 V + ( 1 - S i n 2 p V/z- Cos p S i r \ 3 p Cos^r} +- S i n A s S i n 2 p 3 (RGosp S i n * p C o s 3 r (3-14) - ( 1 - R £ S m 2 p ) ' / 2 5 i n * p Cos 3 r} - 2 C o s \ s Sin(3 3 Cosp 3 {RCosp S i n 4 p C o s r +- C o s 2 p S i n 2 p Cos T (1-Rz S i n 2 ^ ' 2 } - a S \ v \ X s S i n p 3 C o s p 3 ( R C o s 2 p S i n 3 p C o s 2 V - (1-R 2 S i n 2 p ) v * C o s p S i n 3 p C o s 2 r} c J T d p 95 where y, a e S (1 f p- T) A — ; o c / Case I I : D e t e r m i n i n g t h e l i m i t s o f i n t e g r a t i o n f o r t h i s c a s e f r o m F i g u r e 3.4, t h e f o r c e e x p r e s s i o n ( e q u a t i o n 3.14) becomes rr _ K Q r r a~ u Sin yR eir j " f 0 ( R > s , P s , ( 3 , r ) d r d p o o g e s (n-p- x) A c 2 Cos X s 5 in u> 3 ( R z - U ^ -_2_ + a R 15 R \5 ( R 2 - l ) 3 / £ -I") 3 R 2 + C o s X s C o s a ^ f 2 - 3 l R 2 - i ) 5 / £ 2 ( R 2 - l ) 3 / £ + -IS R' 3 R z (3.15) - ( R z - l ) ' / £ + + 2. S i n X 5 Cos\J> S i n tp f (R 2-l) 3 / 2_ _a- 3 ( R £ - l ) 5 / a 3R' IS" R' - Q.e S (1 *p - T) A rvi C ' r a Case I I I : F i g u r e 3.5 r e p r e s e n t s t h e g e o m e t r i c a l d e t a i l s c o r r e -s p o n d i n g to t h e s a t e l l i t e a t t i t u d e i n C a s e I I I . S u b s t i t u t i n g t h e a p p r o p r i a t e l i m i t s , t h e f o r c e e q u a t i o n c a n be w r i t t e n as where X. C u OS 5m IjJ Cos ft Cos t|) Si n p The f i r s t i n t e g r a l i n t h e above e q u a t i o n i s t h a t o f C a se I I e v a l u a t e d i n 3.15. However, e v a l u a t i o n o f t h e s e c o n d i n t e g r a l p o s e s a m a j o r o b s t a c l e . The n a t u r e o f t h e i n t e g r a n d t o g e t h e r w i t h t h e v a r i a b l e l i m i t i n y r e n d e r s a n a -l y t i c a l i n t e g r a t i o n o f t h e e x p r e s s i o n i m p o s s i b l e . Thus i t was n e c e s s a r y t o o b t a i n a s a t i s f a c t o r y a p p r o x i m a t e s o l u t i o n . More o b v i o u s a p p r o a c h e s s u c h as t h e n u m e r i c a l c o n s t r u c t i o n o f a t h r e e d i m e n s i o n a l g r i d p r o v e d u n d e s i r a b l e . As t h e main s o u r c e F i g u r e 3 . 5 Geometry o f p h y s i c a l s y s t e m f o r e a r t h a l b e d o s t u d y i n C a s e I I I 98 of d i f f i c u l t y was due to the v a r i a b l e l i m i t y , i t was decided to r e p r e s e n t i t by a constant value y . T h i s permits the a n a l y t i c a l i n t e g r a t i o n of the e x p r e s s i o n with the ex c e p t i o n o f * two terms, which can e a s i l y be approximated. Thus I * = i * . ( R , x s , p 3 , p, rQ) . ( 3 . i 7 ) The problem thus reduces to the d e t e r m i n a t i o n of Y q f o r a gi v e n s e t of parameters A , R, . The form of equation (3.17) suggests y to be s u b s t a n t i a l l y independent of A . Thus o s X " Y„(R , t |> ) . F i g u r e 3.6 shows the e f f e c t of changing the l i m i t of i n t e g r a t i o n from Y to Y . In e f f e c t t h i s i n v o l v e s i n t e g r a t i o n over the u o r s e c t o r r e p r e s e n t e d by 2 y Q i n s t e a d of the segment d e f i n e d by the s a t e l l i t e c u t t i n g plane. Thus Y should be c l o s e l y r e l a t e d to the r a t i o x / x Q r i . e . , To s u b s t a n t i a t e the v a l i d i t y of t h i s procedure, I 2 i n (3.16) was ev a l u a t e d n u m e r i c a l l y u s i n g an IBM-70 44 computer over a wide range of parameters, equated t o (3.17), and s o l v e d f o r Y - Table 2 shows a t y p i c a l s e t of data, o * These terms are i n d i c a t e d by t . r 9 9 F i g u r e 3.6 C u t t i n g plane geometry and i t s e f f e c t on l i m i t s of i n t e g r a t i o n i n Case I I I 100 TABLE 2 VARIATION OF y WITH SYSTEM PARAMETERS (a) X = 15.0° \ x / x 0.0 0.1 0 . 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2000 0.0 13.1 24.9 32.5 39 .1 45.6 52.3 59 . 6 67.8 77.4 89.6 5000 0.0 13.1 24.5 32.6 39.5 46. 3 53.2 6 0.6 68.8 77.9 89 .5 8000 0.0 12.9 25.0 32.7 39.8 46.3 53.4 60.8 68.9 78.1 89.5 (b) X = -15.0 0.0 0.1 0 . 2 0 . 3 0.4 0 . 5 0.6 0.7 0 . 8 0.9 1.0 2000 0 . 0 13. 2 24.8 32. 3 38.8 45.3 52.0 59 . 3 67.7 77.4 89.6 5000 0 . 0 10 . 8 24.0 31.8 38.9 45.6 52 . 7 60.2 68 . 6 77.8 89.5 8000 0.0 10 .0 23.0 31.4 38.7 45.6 52.7 60 . 4 68.6 78.0 89.5 The r e s u l t s c o n f i r m t h e i n d e p e n d e n c e o f Y q f r o m A G and i n d i c a t e t h a t i t can be e x p r e s s e d as a f u n c t i o n o f X/XQ. F u r t h e r m o r e , l^ ( Y Q ) w a s o b s e r v e d t o be r e l a t i v e l y i n s e n s i t i v e t o t h e v a l u e s o f y n e a r t h e n u m e r i c a l l y d e t e r m i n e d v a l u e o f o I 2 > I n f a c t , e r r o r s as l a r g e as ±5° i n d e t e r m i n i n g Y q have n e g l i -g i b l e e f f e c t on t h e r e s u l t s . F o r s m a l l v a l u e s o f (<0.5), t h e c o n t r i b u t i o n o f t h e s e c o n d i n t e g r a l t o t h e r e s u l t a n t f o r c e i s v e r y s m a l l . 101 The d e p e n d e n c e o f Y o on X/Xq as g i v e n by t h e t a b l e was e x p r e s s e d as a c u b i c u s i n g a l e a s t s q u a r e s c u r v e f i t : X = -0.15 259 + 3.055 66 (-£-) - 3. 2 £ S 63 ( ~ f (3.18) + \ . 9 2 9 6 9 (4" )3 Ao F o r a s a t e l l i t e i n a g i v e n o r b i t and h a v i n g s p e c i f i e d o r i e n -t a t i o n , R and \b a r e known, h e n c e X/Xq c a n be d e t e r m i n e d u s i n g X 0 = ( R 2 - n l / a R Thus t h e f i n a l e x p r e s s i o n f o r f o r c e i n C a s e I I I becomes 2TTCosX s S m 2i};(B 3--|B i-B 4+B 1 t ^ | ) (3.iq) 4-T Cos>Xs Cos atp (-| B 2 - B 3 +• 2 B 4 - B 5 + ~ | - ) +- 2TT S i n X s CosifJ Sin y ( - | B a - B 3 * B 4 - ) 4% Co<=,Xs S i n 2 tp ( B 3 - B 4 - B f c + B 7 + Be, * B„ - B l o ) -4 Sin X 5inX s Sinip |5inif>| ( B^- B 2 - B l 5 + B ^ - B l 2 ) - a(T 0 t 5inT 0 CosT 0) Cos \ Cos £ tp (-B3 t E B 4 -B s (3.20) + B f c-2B 7 + B 8 + B , 0 - B U ) - 4 ( S i n Y;- S i n 3 ^ ) SinX s C o 3 > ( B a - B l 4.- B 1 3 ) -8 Sin r. Cos A s |CosUj||SinUJ|(B|4 - B 2 -B 1 H) -4 - (X + S i n X Cos r o } S i n X s iCoslpllSintpl ( - B 3 * B 4 r B f o - B 7 - B„ + B l o ) - Q.eS(lf p - T ) A , M C ' r a where B e = R|Cos if>| 1 _ 3R ; B,= 3 R 2 3/2 B p = 5R' Bio = 3 R ' Q (R £-D 5 / A  B 3 = 5R« By, = 3R1 B 5 = ( R * - 1 ) V * b,| - — 5 R 4 ^z=fx-l\-X-)i/z ( 1 - R 2 X M / a d x I5intp| B . a ^ X ^ l - X ^ ' ^ d - R ' X ^ ^ d X 1 , |Sin^| 103 R R | C o s s i p l o R | 5 i n 5 V l bfc = b 1 4 _ = — — _ R l C o s 3 t p | R |S i r » 3 VI D 7 - - - b | 5 = I t may be p o i n t e d o u t t h a t f o r a r a n g e o f p a r a m e t e r s p i c k e d a t random, t h e f o r c e g i v e n by e q u a t i o n (3.20) had t h e a c c u r a c y o f n u m e r i c a l i n t e g r a t i o n . C a se IV: F o r c o n v e n i e n c e , t h e s t u d y i n Case IV ( F i g u r e 3.7) may be s u b d i v i d e d a s : 2 ' s 1 v2 m I V a . ' m ( T + A ) > I A I > TT/2 2 m 1 s 1 ' IVb. U I > B m The f o r c e f o r C a s e I V a c a n be w r i t t e n a s : s i n H yR zn F K o o IT Pu O V 104 F i g u r e 3 . 7 Geometry o f p h y s i c a l s y s t e m f o r e a r t h a l b e d o s t u d y i n C a se IV 105 where f3- - S i n -1 S i n x . r y = C o s - 1 Cos X<= Cos X S i n Xs S i n X Cos X = R Sin^jB + C o s ( 3 ( l - R a 5inapV / 2" The s o l u t i o n o f e q u a t i o n (3.21) p a r a l l e l s t h a t o f (3.16) w i t h c e r t a i n m o d i f i c a t i o n s . As b e f o r e , t h e s e c o n d i n t e g r a l has a v a r i a b l e l i m i t Y u w h i c h i s a p p r o x i m a t e d by a c o n s t a n t y ^ . However, h e r e y^ i s i n d e p e n d e n t o f \b and a f u n c t i o n o f R and X g ( T a b l e 3 ) . T h u s , y c a n be e x p r e s s e d as a f u n c t i o n o f t h e r a t i o X/Xq where x i s now t h e d i s t a n c e measured t o t h e p o i n t where t h e s u n l i t h e m i s p h e r e c u t s t h e b a s e o f t h e s p h e r i c a l c a p ( F i g u r e 3 . 8 ) . F i g u r e 3.8 C u t t i n g p l a n e geometry and i t s e f f e c t on l i m i t s o f i n t e g r a t i o n i n Case IV 107 TABLE 3 VARIATION OF yn WITH SYSTEM PARAMETERS (a) ib = 60° \ x / x A 1 ? X > 0.0 0.1 0 . 2 0.3 0.4 0.5 0.6 0.7 0 . 8 0.9 1.0 2000 0 . 0 15.0 21.6 27.0 32.0 37.2 42.9 49.7 58.8 71.7 90.0 5000 0.0 15.0 21.3 26 .5 31.2 35 . 8 41.0 47 . 3 56 . 3 70.0 90 .0 8000 0 . 0 15 . 0 21.2 26 . 2 30.7 35.0 39.6 45 . 3 53.5 67.3 90.0 (b) $ = -60° ^ s X / x A I I X ^ 0.0 0.1 0.2 0.3 0.4 0.5 0 . 6 0.7 0.8 0.9 1.0 2000 0.0 14.9 21.5 26.8 31.6 36.3 41.4 47.6 56.4 70.4 90.0 5000 0.0 15.1 21.2 26.4 31.0 35 . 6 40.5 46.6 55.2 69 .2 90.0 8000 0.0 13 . 3 21.1 26.2 30.6 34.9 39 .4 44.9 52.8 66.7 90.0 The r e m a i n d e r o f t h e f o r m u l a t i o n p a r a l l e l s t h a t o f Case I I I w i t h t h e c o r r e s p o n d i n g e x p r e s s i o n s l i s t e d b e l o w : T t = 2.0466 _ 4 . 7 q 6 3 q ( - A - ) + 4. 3 1 3 2 q ( _ ^ _ j 2 ' [ 3 Z 2 ) f = 1 _ T a n r / a - I X 5 l ) ( 3 £ 3 ) 108 2TrGosXs S i n £ ^ ( B 3 - f 8 £ - B 4 + B ! +• ^ ) r T T C o a X c , COS2H)(-| B a - B 3 + - Z B 4 - B 5 * ^ | ) . + ETT S m X s C o s u ) 5 i n t v ( J - B 2 - B 3 4- B 4 - | | - ) C o s X s S i n £ ^ ( B 3 - B 4 - B f a -t- B 7 + Bq t B „ - B l o ) - r a S i n V ; | 5 i n X s | S i n 2 V (B»- B 2 - B , s t B , 4 - B^, ) (3.24) -(Yj + S in Y, C o s Y j C o s X s C o s £ t p ( - B 3 + 2 B 4 - B s 4- B 6 - 2 B T v B 8 - B u + B I 0 ) -H2(Sin7; - ± 5 \ n 3 ^ ) | 5 - . n A s | C o s e i p ( B £ - B w - B l 3 ) + 4 5 i n Y J CoaX s S ' m X s Sjnij) Cos ^ ( B L 4 - B A - B 1 2 L ) -2(Y t 4- SmYj CosY^ SinX 5 Cos if Sintp(-B3 * B 4 + B f c - B 7 - B« + B, 0 ) ] a e S ( l t p - T ) A ^ C rex where l ^ . t t.SMy - T I A B 8 = R C b 5 8 , 5 R ' 109 B to - ( 1 - R Z S i n * p u ) 3 / i 3R' ( R ^ - l ) 5 / Z 5 R A Q ( l - R * Sin* ft J B^ = ( R ^ - l )3 / a 3 R 2 -o _ R C o s 5 f i L  b- -B»= / x 2 ( i - X 2 ) , / a ( l - R 2 X 2 ) V £ a x I t S i n £ L B,, = / x 4 , ( i - x 2 ) " , ' f e ( i - R t x a V / a c i x '13 Sin ^ u a _ R 5 \ n s g u • . R C o s 3 & " 3 n R 5 m 3 p L b, 5 5 — The f o r c e e x p r e s s i o n f o r Case IVb, being e s s e n t i a l l y the same as the second i n t e g r a l of (3.21), does not warrant any f u r t h e r d i s c u s s i o n and can be w r i t t e n as: F, ra 2Y ; C o s X s S i n 2 - < i ) ( B 3 - B 4 - B f o + - B 7 + Bc, + B „ - B , 0 ) + 2 S i n T j | 5 i n X s | S i n 2 i ) J ( B, - B 2 - B l 5 + B ,4 - B,a) + lYi+-Smr 1Cosr 1)CDsX s Coazy(-&3 ^ B 4 - B 5 + B f o - 2 B T + B 8 - B u + B , 0 )+ a f S i n ^ - ± 5 i n 3 \ q ) | 5 i n \ s | G>s2tj> (Bz. - B l 4 - B l 3 ) 110 + - 4 S i n T 1 C o s X S | 5 i n X s | S i n t p C o s v p ( B I A - B £ - B i a ) (3.25) ISin X s \ 4- 2(Y[ * S m ^ C o s T j ) 5 i n X s Cost}) S in ip ( -B 3 +B 4 v B f c - B T - B U * B 1 0 ) where the values of B^(i=l,...15) are g i v e n by (3.24) with y± d e f i n e d by equations (3.22) and (3.23). Case V: Here the p h y s i c a l system presents seven d i s t i n c t con-f i g u r a t i o n s as shown i n F i g u r e 3.9. F o r t u n a t e l y , the v a r i a t i o n s r e p r e s e n t combinations of Cases I I I and IV except when the s a t e l l i t e plane cuts the t e r m i n a t o r w i t h i n the s p h e r i c a l cap (Figure 3.9b). However, the f o r c e c o n t r i b u t i o n from areas near the c u t t i n g planes i s n e g l i g a b l y s m a l l , thus the s o l u t i o n i n terms of Cases I I I and IV i s s t i l l v a l i d . With r e f e r e n c e to F i g u r e 3.9, the f o r c e equations can be expressed i n terms of the d e r i v e d equations as: A F r a (3. 20) + (3.24) - (3.15) B F r a (3. 20) - (3.24) + (3.15) C F r a (3. 24) D F r a (3. 25) + (3.20) • - (3.15) E F r a (3. 25) ' 111 F i g u r e 3.9 Geometry of p h y s i c a l system f o r e a r t h albedo study i n Case V: (a) S a t e l l i t e and t e r m i n a t o r planes not i n t e r s e c t i n g i n the s p h e r i c a l cap G Figure 3.9 Geometry of p h y s i c a l system f o r e a r t h albedo study i n Case V: (b) S a t e l l i t e and terminator planes i n t e r s e c t i n g i n the s p e r i c a l cap 113 F G I f : D T / D A » 0.5 ; F = ( 3 . 2 4 ) D,/D0 < 0.5 ; F = ( 3 . 2 0 ) - ( 3 . 2 4 ) + ( 3 . 1 5 ) 1 2 r a I f : D./Dn > 0.5 ; F = ( 3 . 2 5 ) + ( 3 . 2 0 ) - ( 3 . 1 5 ) 1 2 r a D,/D„ < 0.5 ; F = ( 3 . 2 5 ) V 2 ra where: _ R S i n ^ P _ Cos A m S i n co 0 | Cos | 5 D £ = 1 - (3.24.) I t may be emphasized t h a t the f o r c e e x p r e s s i o n s developed here h o l d f o r any a l t i t u d e (R), a t t i t u d e (tip) and s o l a r aspect angle ( ± A g ) . The r e p r e s e n t a t i v e p l o t s showing the v a r i a t i o n of e a r t h albedo f o r c e with the s a t e l l i t e a l t i t u d e and o r i e n t a t i o n are shown i n F i g u r e 3.10. In these c a l c u l a t i o n s the v a l u e of a i s taken to be 0 . 3 9 . As compared to the numerical e v a l u a t i o n , the maximum e r r o r i n the f o r c e was found to be l e s s than 2.5%. I t i s i n t e r e s t i n g t o note t h a t , depending on the s o l a r aspect angle, the r a d i a t i o n f o r c e may m o n o t o n i c a l l y decrease or show temporary i n c r e a s e with a l t i t u d e . In g e n e r a l the f o r c e i n c r e a s e s with the l i b r a t i o n a l angle and, depending on the value of the s o l a r angle, may a t t a i n a n egative v a l u e even f o r p o s i t i v e \|> . c i r c u l a r or near c i r c u l a r o r b i t s , the p l o t s given i n F i g u r e 3.11 should be of i n t e r e s t . I t i s easy to see t h a t a s a t e l l i t e would experience r a d i a t i o n torque, which can be p o s i t i v e and/or negative depending on i t s a t t i t u d e i n an o r b i t . e Since most of the s a t e l l i t e s launched to date are i n Figure 3 .10 V a r i a t i o n of ea r th albedo force w i t h s a t e l l i t e a l t i t u d e and a t t i t u d e : (a) X = 0 115 F igure 3.10 V a r i a t i o n of ea r th albedo force w i t h s a t e l l i t e a l t i t u d e and a t t i t u d e : (b) X = 30° 116 117 F i g u r e 3.10 V a r i a t i o n of e a r t h albedo f o r c e w i t h s a t e l l i t e a l t i t u d e and a t t i t u d e : (d) X = 90° 118 A s - deg. F i g u r e 3.11 V a r i a t i o n o f e a r t h a l b e d o f o r c e w i t h s a t e l l i t e a t t i t u d e and s o l a r a s p e c t a n g l e : (a) A l t i t u d e = 500 m i l e s 119 F i g u r e 3.11 V a r i a t i o n of ea r th albedo force w i t h s a t e l l i t e a t t i t u d e and s o l a r aspect ang le : (b) A l t i t u d e = 3000 mi le s 120 The method d i s c u s s e d h e r e c a n be e x t e n d e d q u i t e r e a d i l y t o o b t a i n t h e f o r c e on s a t e l l i t e s o f d i f f e r e n t g e o m e t r i e s . F o r e x ample, c o n s i d e r a r e c t a n g u l a r c y l i n d e r o r i e n t e d s u c h t h a t one o f i t s f a c e s i s p a r a l l e l t o t h e o r b i t a l p l a n e . A p p a r e n t l y t h e f o r c e e x p r e s s i o n s d e v e l o p e d e a r l i e r a r e d i r e c t l y a p p l i c a b l e t o t h e s i d e s n o r m a l t o t h e o r b i t a l p l a n e , however, t h e p a r a l l e l s i d e s may o r may n o t r e c e i v e a c o n t r i b u t i n g t o r q u e . F o r d i r e c t s o l a r r a d i a t i o n and d i r e c t e a r t h r a d i a t i o n no t o r q u e w i l l be e x p e r i e n c e d by t h e s i d e p a r a l l e l t o t h e o r b i t a l p l a n e ; on t h e o t h e r hand, e a r t h a l b e d o w i l l c o n t r i b u t e a s m a l l t o r q u e w h i c h goes t o z e r o as p goes t o u n i t y . T h u s , t h e f o r c e e x p r e s s i o n s c a n be d i r e c t l y a p p l i e d t o g i v e a good f i r s t a p p r o x i m a t i o n f o r a r e c t a n g u l a r c y l i n d r i c a l s a t e l l i t e r e p r e s e n t e d as two f l a t p l a t e s o f f s e t by 90 d e g r e e s . U n d e r s t a n d a b l y , t h e f o r c e e x p r e s s i o n s a r e n o t d i r e c t l y a p p l i c a b l e t o a c i r c u l a r c y l i n d e r e x c e p t f o r i t s e n d s . The c y l i n d r i c a l p o r t i o n o f t h e s a t e l l i t e w i l l y i e l d a m o d i f i e d i n t e g r a l e x p r e s s i o n f o r t h e x-component o f f o r c e . As b e f o r e , however, o n l y t h i s component w i l l c o n t r i b u t e t o a d i s t u r b i n g t o r q u e . T h u s , a l t h o u g h t h e b a s i c f o r c e e x p r e s s i o n s a r e m o d i f i e d , t h e method d e v e l o p e d h e r e c a n be u s e d f o r t h e i r e v a l u a t i o n . I n t h e c a s e o f a s p h e r i c a l s a t e l l i t e , t h e i n t e g r a l f o r c e e q u a t i o n s must a g a i n be m o d i f i e d . T h i s s i t u a t i o n , however,, i s r e l a t i v e l y more e a s y t o a n a l y s e due t o t h e f a c t t h a t a s p h e r i c a l body does n o t p r o d u c e a c u t t i n g p l a n e . Of c o u r s e , t h e c o m p l i -c a t i o n due t o t h e t e r m i n a t o r p l a n e i s s t i l l p r e s e n t b u t i t c a n be h a n d l e d q u i t e r e a d i l y u s i n g t h e p r o c e d u r e p r e v i o u s l y d e v e l o p e d . 3.1.4 E f f e c t of Earth's Shadow and the General Equation of Motion The force expressions being known, t h e i r moment about the Y -axis can be determined giving the generalized force expression as: Q = -5-, U + p - t ) k„ M c d & + ( 1 + p - T ) C ( 3 . E l ) a e s re c r a Substituting (3.27) into (2.13), using the r e l a t i o n (2.18), and introducing the values of T g and a g as previously indicated, the equation of motion takes the form (1 + e Cbs6) u)" - £e ( i | ) ' f 1) 5 i n 9 + 3 K ; 5inXf> Cos tp be appropriate to comment on the e f f e c t of the earth's shadow, which merits consideration i n the analysis of close earth s a t e l l i t e s . Obviously, t h i s e f f e c t should be a function of the portion of the o r b i t a l period spent i n the shadow. Consequently, i t i s l i k e l y to be a maximum for close earth s a t e l l i t e s i n c i r c u l a r o r b i t s . It should be noted that the governing equation of motion, being d i f f e r e n t i n the shadow, requires, upon (i + e) 3 C r^ c +o. it> M c r e o. 3 9 M Before proceeding to integrate equation (3.28), i t would entrance and e x i t , the matching of end c o n d i t i o n s d u r i n g i n t e g r a t i o n . To determine the a l t i t u d e a t which the shadow e f f e c t becomes s i g n i f i c a n t , the p l o t s o f the l i m i t i n g v a l u e of i>' f o r s t a b l e motion were obtained u s i n g equation (2.19) and are shown i n F i g u r e 3.12. I t i s apparent t h a t the e f f e c t i s s m a l l becoming n e g l i g i b l e above 4000 m i l e s . In the f o l l o w i n g study of c l o s e e a r t h s a t e l l i t e s , the shadow e f f e c t was taken i n t o account by w r i t i n g equation (3.28) as (l + eCose)iy" - 2e(nV+ D S m e +- 3K\ Simp Cos tp (3.23) ( i + e c o s e y C S T l c ^ +0.lfc M c r e ^ 0 .33 5 M c r a where O. 9 in shadow 1 . ^  outside sho-dow The response and s t a b i l i t y of the system, obtained u s i n g t h i s e q u a t i o n , are d i s c u s s e d i n the f o l l o w i n g s e c t i o n s . 3.2 System Response and S t a b i l i t y A n a l y s i s The governing equation of motion (3.29) was analysed n u m e r i c a l l y over a wide range of system parameters and i n i t i a l c o n d i t i o n s to o b t a i n response and s t a b i l i t y c h a r a c t e r i s t i c s of the system. However, f o r c o n c i s e n e s s , only r e p r e s e n t a t i v e 1.6 1.2 0.8 -6 O — - ~ ° 6 o e e e e e e Q.5 o II CD II •el o 0.4 0.0 -0.4 -0.8 1.2 -- 1.6 e = o.o c = o.i Kj * K i O: O-without shadow with shadow T T o — o o o • o u o o o o K; = 1.0 O 'O 8 C O O O O O O Q 0.7 .u o * © © © © © © © 0.5 -O o o o o- • u- © e e 0.7 o o u u-=—o o o o o o o K, = I.O 6 8 Al t i t u d e -miles x I 0 3 10 12 F i g u r e 3.12 E f f e c t o f shadow on l i m i t i n g v a l u e o f (0) f o r s t a b i l i t y i n c i r c u l a r o r b i t : (a) c=0.1; K.=1.0, 0.7, 0.5 124 1.6 0 . 8 o II CD •i 0.4 0 . 0 -0.4 0 . 8 - 1 . 2 1.6 e = 0 . 0 C = C K J S 1.0 without shadow • • • with shadow T ^ • • 6 8 Alt i tude - miles x I 0 3 0.1 =» 0 . 3 - • 0 . 5 - * • 0 . 5 * • 0 . 3 C * 0.1 10 12 Figure 3.12 E f f e c t of shadow on l i m i t i n g va lue of ^ ' (0 ) f o r s t a b i l i t y i n c i r c u l a r o r b i t : (b) K .=1 .0 ; c=0.1 , 0 . 3 , 0.5 1 125 p l o t s e m p h a s i z i n g t h e s y s t e m b e h a v i o r a r e p r e s e n t e d h e r e . F i g u r e 3.13 shows t h e l i b r a t i o n a l r e s p o n s e o f a s a t e l l i t e u n d e r t h e i n f l u e n c e o f : a) g r a v i t y g r a d i e n t f o r c e s a l o n e ; b) g r a v i t y g r a d i e n t and d i r e c t s o l a r r a d i a t i o n f o r c e s ; c) g r a v i t y g r a d i e n t , d i r e c t s o l a r and e a r t h r a d i a t i o n s i n c l u d i n g t h e shadow e f f e c t . The p l o t s show t h e i m p o r t a n c e o f r a d i a t i o n i n a t t i t u d e d y namic s t u d i e s , t h e t r a j e c t o r i e s b e i n g v a s t l y d i f f e r e n t f r o m t h e p u r e g r a v i t y g r a d i e n t r e s p o n s e . N o t e t h a t t h e e f f e c t o f e a r t h r a d i a t i o n s c o u l d be s u b s t a n t i a l , s o much so as t o change t h e s t a b i l i t y c h a r a c t e r i s t i c s o f t h e s y s t e m ( F i g u r e s 3.13 b, c ) . A c o m p a r i s o n o f F i g u r e s 3.13 (a) and (b) i n d i c a t e t h e e f f e c t o f on t h e r e s p o n s e o f t h e s y s t e m , whereas 3.13 (a) and (c) show t h e i m p o r t a n c e o f e c c e n t r i c i t y . As c a n be e x p e c t e d , t h e p e r f o r m a n c e o f t h e s y s t e m d e t e r i o r a t e s w i t h i n c r e a s i n g e and d e c r e a s i n g K^. As shown i n C h a p t e r 2, an i n c r e a s e i n t h e s o l a r p a r a m e t e r r e d u c e s t h e s t a b i l i t y bound o f t h e m o t i o n . However, t h e t r a j e c t o r i e s i n 3.13 (c) and (d) i n d i c a t e t h e r e v e r s e t r e n d . T h i s u n e x p e c t e d b e h a v i o r c a n be e x p l a i n e d t h r o u g h t h e s t a b i l i t y c h a r t s . F o r t h e f i r s t c a s e , t h e c o n d i t i o n s a r e s u c h t h a t t h e t r a j e c t o r i e s emanate f r o m t h e u n s t a b l e r e g i o n between t h e m a i n l a n d and t h e i s l a n d a s s o c i a t e d w i t h t h e 3/2 p e r i o d i c s o l u t i o n ( F i g u r e s 2.10, 3.15). On t h e o t h e r h a n d , by i n c r e a s i n g c , t h e s t a b i l i t y bound s h r i n k s s h i f t i n g t h e 3/2 i s l a n d 40 30 20 10 - 10 - 20 -40 e " ° * 1 0 ib«»=0.0 GG + DSR + DER+ EA+shadow, 600 miles C =0.30 ™ , GG + DSR Ki = 1.00 d)(0) = 0.0 <|, =0.00 GG,|C = 0) 1 1 C\ A if \ /I k \ K '/A '/ \ '/ 1 '/ 1 '/ 1 r \ 1 ' '"A h \ \ 7 X * 1 / \ / / I ft V/' 1 1 9 - orbits F i g u r e 3.13 E f f e c t o f e a r t h r a d i a t i o n s on l i b r a t i o n a l response: (a) e=0.1, c=0.3, K.=1.0 1 F i g u r e 3.13 E f f e c t o f e a r t h r a d i a t i o n s on l i b r a t i o n a l r e s p o n s e : (b) e=0.1, c=0.1, K.=0.65 l H1 to G G • DSR +DER + EA + shadow , 600 miles Q - orbits F i g u r e 3.13 E f f e c t o f ea r th r a d i a t i o n s on l i b r a t i o n a l response: (c) e=0.2, c=0.3, K.=1.0 Figu re 3.13 0 - orbits E f f e c t of e a r t h r a d i a t i o n s on l i b r a t i o n a l response: K.=1.0 1 (d) e=0.2, c=0.4, F i g u r e 3.13 0-orbits E f f e c t o f ea r th r a d i a t i o n s on l i b r a t i o n a l response K.=1.0 (e) e=0.2, c=-0 .3 , 131 i n w a r d s u c h t h a t t h e i n i t i a l c o n d i t i o n s c o r r e s p o n d t o s t a b l e m o t i o n . F i g u r e 3.13 (e) shows t h e h i g h f r e q u e n c y m o t i o n r e s u l t i n g f r o m n e g a t i v e c. A s y s t e m a t i c s t u d y showed t h e e f f e c t o f e a r t h r a d i a t i o n s t o d i m i n i s h w i t h a l t i t u d e , t h e t r a j e c t o r i e s b e c o m i n g a l m o s t i d e n t i c a l t o t h o s e o b t a i n e d i n C h a p t e r 2 f o r a l t i t u d e s o f 4000-6000 m i l e s . T h i s i s f u r t h e r s u b s t a n t i a t e d by t h e s y s t e m p l o t s and s t a b i l i t y c h a r t s d i s c u s s e d b e l o w . F i g u r e 3.14 shows t h e s y s t e m p l o t s f o r a w i de r a n g e o f t h e d e s i g n p a r a m e t e r s and i n i t i a l c o n d i t i o n s . To b e t t e r a p p r e c i a t e t h e e f f e c t o f e a r t h r a d i a t i o n s , t h e s e p l o t s s h o u l d be s t u d i e d i n c o n j u n c t i o n w i t h t h e c o r r e s p o n d i n g p l o t s f o r d i r e c t s o l a r r a d i a t i o n g i v e n i n F i g u r e 2.7. The c o r r e s p o n d i n g d i s c u s s i o n i n S e c t i o n 2.2.2 c a n be e q u a l l y w e l l a p p l i e d t o F i g u r e s 3.14 ( a ) , ( b ) , and ( c ) . N o t e t h a t t h e b r o k e n n a t u r e o f t h e c u r v e f o r e = 0.2 i n F i g u r e 3.14 (c) c a n be a t t r i b u t e d t o t h e 3/2 i s l a n d ( F i g u r e 3.15 a ) . F o r a g i v e n s a t e l l i t e , F i g u r e 3.14 (d) shows a s t r o n g d e p e n d e n c e o f s y s t e m p e r f o r m a n c e on a l t i t u d e , w h i c h i s p r i m a r i l y a r e s u l t o f d i r e c t s o l a r r a d i a t i o n ; as a g a i n s t a p u r e g r a v i t y o r i e n t e d s a t e l l i t e . F rom a c o m p a r i s o n w i t h F i g u r e 2.7, t h e s y s t e m p l o t s show v e r y l i t t l e d e p e n d e n c e on e a r t h r a d i a t i o n s , t h e l i b r a t i o n a l a m p l i t u d e s b e i n g n e a r l y i d e n t i c a l . S m a l l d i s c r e p a n c i e s c a n be n o t e d i n t h e a v e r a g e p e r i o d a l t h o u g h t h e t r e n d i s m a i n t a i n e d . C _Ki Figu re 3.14 System p l o t s showing the maximum l i b r a t i o n a l ampli tude and average p e r i o d f o r a range of e c c e n t r i c i t y as a f f ec t ed by: (a) s o l a r and i n e r t i a parameters i—• 1 1 1 o I I I I TT 3TT/2 2TT 0 TT/2 TT 3TT/2 2TT (}) - radians 9 ~ rod ions System plots showing the maximum l i b r a t i o n a l amplitude and average period for a range of e c c e n t r i c i t y as affected by: (b) solar aspect angle and i n i t i a l condition 0 H-1 U) OS -0.7 .05 a9 I 0.7 0.5 0.9 l|o.7 . 0.5 80 60 40 20 e = e Kj = 1.0 C =0.1 <j) =0.0 Alt = 1000 mi. e°Q-2 -1.2 -0.8 F i g u r e 3.14 l|l(0) = l|KO} l|/(0> = 0.0 0.9 07 05 0.9 07 0.5 09 > H0.7 05 80 60 40 20 e=e K i = 1.0 C =0.1 (j) =0.0 Alt = l000 mi. ||M0) = 0.0 t|/(0) = l|/(0) e=o.i e=o.o t|jf0) - radians System p l o t s showing .the maximum l i b r a t i o n a l ampli tude and average p e r i o d for a range of e c c e n t r i c i t y as a f f ec t ed by: (c) i n i t i a l d i s tu rbances t (0) and ij; 1 (0) OJ 4^. e = e (b =o.o . Y ib (0) =o.o Ki -1.0 Alt = Alt T y C =0.05 at r p=R e <J/(0) =0.0 e = e <b =o.o . Y J) (0) =0.0 K; = 1.0 Alt =• Alt , C =0.10 at r p = R e l|/(0)=0.0 0.9 3|o.7 0.-5 I I i i i e=o.2 0.9 3|0.7 0.5 1 1 1 1 1 \ e =0.2 0.9 •4.7 0.5 0.9 H°|O.7 0.5 e-o.i e=o.i 0.9 4.7 0.5 e=o.o 0.9 3|o.7 0.5 e=o.o " I I I i i 80 60 6) <u 1 | 40 20 i i i 1 1 ' ^ ^ ^ ^ — e=aft-^"^^ 1 1 1 1 1 80 60 0) T3 1 X 1 40 20 n I i l i i e=0-2 / / / / e=o.l / / / / e=0.0 s i i i 1 i Altitude - miles " IP 3 Altitude - miles x 10 3 F i g u r e 3.14 System p l o t s showing the maximum l i b r a t i o n a l ampli tude and average p e r i o d f o r a range of e c c e n t r i c i t y as a f f ec t ed by : (d) a l t i t u d e fo r a g iven s a t e l l i t e geometry . u> U1 136 F i g u r e 3.15 shows t h e e f f e c t o f t o t a l r a d i a t i o n on t h e s t a b i l i t y bounds o f a s a t e l l i t e . The e f f e c t o f t h e i n e r t i a i s a l s o i n d i c a t e d . To d e t e r m i n e t h e i m p o r t a n c e o f e a r t h r a d i a t i o n s on s t a b i l i t y , F i g u r e 3.15 s h o u l d be s t u d i e d i n c o n j u n c t i o n w i t h t h e c o r r e s p o n d i n g p l o t s f o r d i r e c t s o l a r r a d i a t i o n g i v e n i n F i g u r e s 2.10 and 2.11. I t i s a p p a r e n t t h a t t h e c h a r a c t e r o f t h e s t a b i l i t y bounds has c h a n g e d c o n s i d e r a b l y . The c h a n g e s a r e p r i m a r i l y due t o t h e s e p a r a t i o n and g r o w t h o f t h e i s l a n d a s s o c i a t e d w i t h t h e 3/2 p e r i o d i c s o l u t i o n . However, t h e m a i n l a n d , w h i c h i s t h e a r e a o f p r i m a r y i m p o r t a n c e i n t h e o p e r a t i o n and d e s i g n o f a s p a c e c r a f t ( S e c t i o n 2.3.1, p.46 ) , i s o n l y s l i g h t l y a f f e c t e d by t h e i n c l u s i o n o f e a r t h r a d i a t i o n f o r c e s . T h u s , t h e i n f l u e n c e o f e a r t h r a d i a t i o n s t e n d s t o a f f e c t t h e r e s p o n s e t r a j e c t o r y and t h e i s l a n d s t r u c t u r e o f t h e s t a b i l i t y c h a r t s w i t h o u t s u b s t a n t i a l l y a f f e c t i n g t h e maximum l i b r a t i o n a l a m p l i t u d e o r t h e m a i n l a n d s t a b i l i t y a r e a . A comment c o n c e r n i n g p e r i o d i c s o l u t i o n s o f t h i s s y s t e m w o u l d be a p p r o p r i a t e . I t s h o u l d be p o i n t e d o u t t h a t h e r e , i n g e n e r a l , t h e i n i t i a l c o n d i t i o n s f o r p e r i o d i c m o t i o n i n v o l v e n o n - z e r o v a l u e s o f \b as w e l l as tp . T h u s , t h e p e r i o d i c s o l u t i o n s l i e s l i g h t l y o u t s i d e t h e p l a n e o f t h e s t a b i l i t y c h a r t . T h i s e x p l a i n s why t h e y , a t t i m e s , do n o t p a s s t h r o u g h s p i k e s and i s l a n d t i p s i n F i g u r e 3.15. F u r t h e r m o r e , t h e c o m p l e x i t y o f t h e e q u a t i o n o f m o t i o n r e n d e r e d a v a r i a t i o n a l a n a l y s i s o f t h e p e r i o d i c s o l u t i o n s v i r t u a l l y i m p o s s i b l e . C =0.1 K; =1.0 <J> =0.0 Alt =600 mi. 1.5[ 0.0 ^^^SSSxt stable regions periodic motion 1 0.2 0.3 0.4 1-5 -I.Oh l.5l 0.0 C =0.3 (|> = 0.0 WW* stable regions Kj = 1.0 • Alt=600mi. periodic motion -.4/3 0.1 _L_ 0.2 e 0.3 0.4 F i g u r e 3.15 S t a b i l i t y cha r t s showing the e f f e c t of s o l a r and i n e r t i a parameters: (a) K i = 1 . 0 ; c=0.1 , 0.3 OJ Figu re 3.15 S t a b i l i t y char t s showing the e f f e c t of s o l a r and i n e r t i a parameters : (b) K i = 1 . 0 ; c=0.5, 0.75 w K i = 0 . 7 ; c=0.1 oo 139 3.3 Concluding Remarks The important c o n c l u s i o n s based on the a n a l y s i s may be summarized as f o l l o w s : i ) The a n a l y s i s g i v e s g e n e r a l c l o s e d form s o l u t i o n s t o the r a d i a t i o n f o r c e s on a f l a t p l a t e i n e a r t h o r b i t s thus adding t o our presen t knowledge which i s r e s t r i c t e d t o i n t e g r a l e x p r e s s i o n s . i i ) The f o r c e e x p r e s s i o n s are i d e a l l y s u i t e d f o r the dynamical study of an e a r t h o r b i t i n g s a t e l l i t e under the i n f l u e n c e of r a d i a t i o n p r e s s u r e s . The c l o s e d form nature of the expre s s i o n s reduces the compu-t a t i o n a l time to approximately l/100th of t h a t r e q u i r e d f o r the i n t e g r a l e x p r e s s i o n s . T h i s p e r m i t t e d the e x t e n s i o n of the e x i s t i n g s t u d i e s of g r a v i t y o r i e n t e d s a t e l l i t e s , which ignore the e f f e c t s of r a d i a t i o n f o r c e s . The d i r e c t s o l a r r a d i a t i o n g i v e s the maximum f o r c e — 8 2 of 9.72 x 10 l b . / f t . on a s u r f a c e of u n i t a b s o r p t i v i t y (a = 1), d e c r e a s i n g m o n o t o n i c a l l y to zero as the angle of i n c i d e n c e (a) goes t o 90° . The maximum f o r c e f o r a = 1 due to the d i r e c t e a r t h r a d i a t i o n i s 1.04 x 10~ 8 l b . / f t 2 , a t R = 1 and \b = TT/2 The f o r c e decreases m o n o t o n i c a l l y both w i t h i n c r e a s -i n g a l t i t u d e and d e c r e a s i n g ip . i i i ) i v ) 140 v) The f o r c e due t o e a r t h a l b e d o gave t h e maximum v a l u e o f 2.52 x 1 0 ~ 8 l b . / f t . 2 f o r a p l a t e o f a = 1 a t X G = 0, ip = TT/2 , and R = 1. - I n g e n e r a l t h e f o r c e d e c r e a s e s w i t h i n c r e a s i n g a l t i t u d e and X G and d e c r e a s i n g VJJ v i ) The i m p o r t a n c e o f d i r e c t s o l a r r a d i a t i o n i n a t t i t u d e d ynamic s t u d i e s was p o i n t e d o u t i n C h a p t e r 2. On t h e o t h e r hand, t h i s a n a l y s i s shows v e r y l i t t l e e f f e c t o f e a r t h r a d i a t i o n s on t h e maximum l i b r a t i o n a l a m p l i -t u d e and t h e m a i n l a n d s t a b i l i t y a r e a . Hence, f o r a l l p r a c t i c a l p u r p o s e s , d i r e c t e a r t h r a d i a t i o n , i t s a l b e d o and shadow e f f e c t s c a n be n e g l e c t e d i n s u c h s t u d i e s . 4. ATTITUDE DYNAMICS OF CLOSE EARTH SATELLITES F o r s a t e l l i t e s t h a t have p a r t - o r a l l o f t h e i r o r b i t s e x t e n d i n g i n t o t h e e a r t h ' s e f f e c t i v e a t m o s p h e r e , i . e . , l e s s t h a n 500 m i l e s , i t i s n e c e s s a r y t o c o n s i d e r t h e c o n t r i b u t i o n o f t h e - a e r o d y n a m i c f o r c e i n a t t i t u d e dynamic s t u d i e s . The i m p o r t a n c e o f t h e a e r o d y n a m i c t o r q u e i s r e v e a l e d by t h e f a c t t h a t i t i s c o m p a r a b l e t o e a r t h r a d i a t i o n a t 500 m i l e s , d i r e c t s o l a r r a d i a t i o n a t 400 m i l e s , and d e p e n d i n g on s a t e l l i t e geom-e t r y , c a n be as l a r g e as g r a v i t y g r a d i e n t t o r q u e a t 200-300 m i l e s . Hence, i n a d d i t i o n t o s o l a r and e a r t h r a d i a t i o n t o r q u e s , t h i s c h a p t e r i n c l u d e s t h e e f f e c t o f t h e r a r e f i e d a t m o s p h e r e on t h e r e s p o n s e o f g r a v i t y g r a d i e n t s a t e l l i t e s , t h u s c o v e r i n g t h e e n t i r e a l t i t u d e r a n g e . A comment on t h e s a t e l l i t e model i s a p p r o p r i a t e here-. I n t h e p r e v i o u s a n a l y s i s , t h e g e n e r a l s a t e l l i t e c o n f i g u r a t i o n was r e p r e s e n t e d by an a r b i t r a r i l y s h a p e d f l a t s u r f a c e w i t h i t s c e n t e r o f p r e s s u r e d i s p l a c e d f r o m t h e c e n t e r o f mass o f t h e s y s t e m . A q u e s t i o n a r i s e s as t o t h e a p p l i c a b i l i t y o f t h i s model i n t h e a e r o d y n a m i c s t u d y . The m a i n p o i n t o f c o n c e r n w o u l d be t h e movement o f t h e c e n t e r o f p r e s s u r e , w h i c h , i n g e n e r a l i s a p e r i o d i c f u n c t i o n o f t h e a n g l e o f a t t a c k . However, as p o i n t e d o u t by B e l e t s k i i , 5 ^ t h i s d e p endence on a n g l e o f a t t a c k i s i n -s i g n i f i c a n t compared t o t h e changes i n t h e o t h e r q u a n t i t i e s , s u c h as t h e d r a g c o e f f i c i e n t . F u r t h e r m o r e , M e i r o v i t c h and 59 W a l l a c e f o u n d , f o r drum-shaped and d u m b - b e l l s a t e l l i t e s , t h e a e r o d y n a m i c moment t o be e s s e n t i a l l y z e r o f o r a l l a t t i t u d e s , when t h e g e o m e t r i c c e n t e r was c o i n c i d e n t w i t h t h e c e n t e r o f mass. T h u s , t h e model i s a good r e p r e s e n t a t i o n o f t h e p h y s i c a l s y s t e m f o r t h e a n a l y s i s o f r a d i a t i o n and a e r o d y n a m i c t o r q u e s . t e r m s o f t h e d e s i g n p a r a m e t e r s c and , f o r a r a n g e o f a l t i -t u d e s i n t h e a e r o d y n a m i c r e g i m e . The c o n c e p t o f p h a s e s p a c e r e p r e s e n t a t i o n and t h e c o r r e s p o n d i n g s t a b i l i t y c h a r t s a r e u s e d t o s t u d y t h e b e h a v i o r o f t h e s y s t e m and t h e r e l a t i v e i n f l u e n c e A t t h e a l t i t u d e s u n d e r c o n s i d e r a t i o n , f r e e m o l e c u l e f l o w t h e o r y i s a p p l i c a b l e f o r p r e d i c t i o n o f a e r o d y n a m i c f o r c e s . T h e o r e t i c a l c a l c u l a t i o n s o f t h e s e f o r c e s c a n be c a r r i e d o u t by t r e a t i n g t h e f l o w s o f i n c i d e n t and r e - e m i t t e d o r r e f l e c t e d m o l e c u l e s s e p a r a t e l y , where i t i s assumed t h a t t h e a p p r o a c h i n g gas i s i n l o c a l M a x w e l l i a n e q u i l i b r i u m . U s i n g t h e method o f 72 S c h a a f and Charabre, t h e n o r m a l p r e s s u r e and s h e a r f o r c e on a u n i t a r e a o f t h e s a t e l l i t e c a n be e x p r e s s e d a s : The s y s t e m i s a n a l y s e d , f o r a r b i t r a r y e c c e n t r i c i t y , i n o f t h e f o r c e s c o n s i d e r e d . 4.1 F o r m u l a t i o n o f t h e P r o b l e m 4.1.1 A e r o d y n a m i c T o r q u e (4.1) and <Tr Po V * S i n tp , - ( 5 r Cos V)' (4.2) + (Tf)' / Z S r Cbstj){l + er-T ( 5 r Cosip)} with r \ = J r s x ( Pn * p s ) . ^ . 3 ) ' 59 Using equation (4 . 3) , M e i r o v i t c h and Wallace found t h a t the aerodynamic moment c o u l d be expressed as a f u n c t i o n of the drag c o e f f i c i e n t a t \p = 0, the o r i e n t a t i o n of the body, and the dynamic p r e s s u r e as = C d S' A Cos 2" j . (4.4) Equation (4.4) i s b a s i c a l l y t h a t which B e l e t s k i i ^ used and w i l l be m o d i f i e d s l i g h t l y f o r t h i s a n a l y s i s . Apparently (4.4) holds f o r 4> .< ± TT/2, however, i t must be a l t e r e d i f should exceed ±90°. F u r t h e r , r e c o g n i z i n g t h a t S'A = A as p r e v i o u s l y d e f i n e d , the equation f o r aerodynamic moment becomes ^ a - J r y 1 ^ C a C o s ^ ICos.pl A y v . (4.5) 144 The density of the upper atmosphere varies for a given a l t i t u d e primarily due to solar radiation (day-night, seasons, e t c . ) . However, s a t e l l i t e measurements have made possible a comparatively long time study of the a i r density r e s u l t i n g i n a s t a t i s t i c a l averaging of i t s short period fl u c t u a t i o n s . Thus, a sa t i s f a c t o r y atmospheric density model can be inf e r r e d . Although i t may be argued that short term density fluctuations would a f f e c t the attitude dynamics of a s a t e l l i t e , for pre-liminary design i t i s convenient to consider an average set of 73 atmospheric properties as given by the ARDC-1959 model. For a l t i -tudes greater than 300 miles, the atmospheric density can be approximated by ^ P. 'Po( -^ - ) «'6> where the reference values of p' = 10 s l u g s / f t . at h' = 460 o o miles are chosen. Further, from o r b i t a l r e l a t i o n s , the s a t e l l i t e v e l o c i t y can be expressed as: v 2 1 * a e Cose + e a l + e (4.7) where Mr = E. 6 > 10 + 4 ft/sec Substituting (4.6) and (4.7) into equation (4.5), the aerodynamic moment for a s a t e l l i t e i n an e l i p t i c o r b i t becomes 145 Hcx -(-T Rc V c a C d ) A y y Klc^ (4.8) where 4.1.2 General Equation of Motion The g e n e r a l equation governing the l i b r a t i o n a l motion of an a r b i t r a r i l y shaped, g r a v i t y g r a d i e n t s a t e l l i t e was gi v e n by equation (2.13) as: (1 + eCoseyV - a e {y' +• l) Sine + 3K\ Sintp Cos^ , (H.13) R 3 Q. T ^ Y A1 In t h i s a n a l y s i s , must be m o d i f i e d to i n c l u d e the aerodynamic e f f e c t . From equations (4.8) and (3.27), the g e n e r a l i z e d f o r c e becomes: . . 14.9) + ^ (1+p-T) M c r a - p ; V* C d ) M c a . 146 Substituting (4.9) into (2.13) and using the r e l a t i o n (2.18), the equation of motion takes the form ( 1 + eCos6) tp" - 2e ' ( V +1) S i n e f 3 Ki 5inu> Cos ^  (4.10) live)' ( l + e C o s 6 ) 3 C & M c ^ + Q .V6 M c r e + 0 . 3 9 S Mc r o _ c'pd v| Cd ^ 2S(i*p-r) o-In a comprehensive study, Cook 74 deduced the value of to be approximately 2.2. Further, introducing the rep-resentative values of T = 0 and p = 0.5, together with the previously indicated values of c' , , V"c and S, the general equation of motion for close earth s a t e l l i t e s becomes (1 + e QosQ)y" - ZQ(v)' +1) Sin 6 O K ; Sin-q) Cos t_p (1+ e ) -(1 - r -eGosO) 3 £ M c a s +• 0.16 M c r e + 0 . 3 c ? £ M c r a ~ 0 . 5 1 M c a (4.11) The analysis of gravity oriented s a t e l l i t e s under the influence of aerodynamic and radiation pressures, obtained using t h i s equation, i s presented i n the following sections. 147 4.2 Influence of Earth Radiations on S a t e l l i t e Response It was shown i n Chapter 3 that the e f f e c t of d i r e c t earth r a d i a t i o n , i t s albedo and shadow, on the maximum l i b r a -t i o n a l amplitudes and mainland s t a b i l i t y areas i s small, and hence could be neglected i n such studies. As this results i n a large reduction i n computational time, i t was considered desirable to investigate t h e i r r e l a t i v e importance i n thi s analysis. Figure 4.1 shows representative response plots of the system with and without the inclu s i o n of earth radiations. The maximum l i b r a t i o n a l angle, even for close earth o r b i t s (Figure 4(a)) and large values of the solar parameter (Figure 4(c)), remains e s s e n t i a l l y unchanged, although the response t r a j e c t o r i e s themselves are modified. A s i m i l a r e f f e c t was observed i n Chapter 3. This i s further substantiated by the s t a b i l i t y charts given i n Figure 4.2. As i n the previous analysis, the variations in the s t a b i l i t y l i m i t s are very s l i g h t when earth radiations are neglected. Hence, for the analysis of gravity oriented S a t e l l i t e s under the influence of aerodynamic and radiation forces, one i s j u s t i f i e d i n neglecting the e f f e c t of earth radiations and shadow. It should be emphasized that this sim-p l i f i c a t i o n i s not es s e n t i a l to the analysis of the problem, but is introduced only to reduce the computational e f f o r t without s i g n i f i c a n t l y a f f e c t i n g the accuracy. Hence, the equation of motion reduces to: 20 15 10 5 -5 -10 -15 c l o ' a A1U500 miles GG +DSR+AD L ~ 0 - 3 (J)(0)=0-0 K i = 1 , ° djJ0)=0.O GG + DSR + AD + DER + EA + shadow (j) = 0.0 y - • ........ , / \ /s / I / " \ * \ 1 / / J / A * 11 \ i *% / A i -sA /Al ' t_ A J—/ 1—I . 1 r v K y 1 ' 1 V \ 1 A \ '• V/ w \J \J -0 1 2 3 4 5 6 8 - o r b i t s F i g u r e . 4.1 Response p l o t s i n a e r o d y n a m i c r e g i m e s h o w i n g e f f e c t o f e a r t h r a d i a t i o n s : (a) e=0, c=0.3, K.=1.0, A l t = 500 m i l e s M 1 J*. oo 40 30 20 10 -10 -20 -30 -40 e = 0 , 0 Alt=400 miles C = 0-3 , G G + DSR + AD K - 1 0 l|>(0)=0.0 ' ' lb'(0l=0.0 G G + DSR + AD+DER+EA+shadow <p = 0-0 ™ 1 / \ 1 /x> 1 A \ I ' l l / \ //? A 1 \ i n / / \» - \ / V / \ \ \ i K / V / / V A * /' \ i / / n // V I ' i i? / / \i w \ / \l H V/' \ / Y * \ / i i / \/ i 0 1 2 3 4 5 6 9 - orb i ts F i g u r e 4.1 Respo n s e p l o t s i n a e r o d y n a m i c r e g i m e s h o w i n g e f f e c t o f e a r t h r a d i a t i o n s (b) e=0, c=0.3, K i=1.0, A l t = 400 m i l e s 80 60 40 20 -20 -40 -60 6 = 0 - 1 Alt= 500 miles „ „ _ C = 0.5 i GG+DSR+AD . -10 l(» 101=0-0 ^' J|'(0I = 0 0 GG+DSR + AD + DER+EA+shadow <p • 0.0 T 1 1 ^\ Ii \ 1 \ ft I \ \ I A r\ / \ -1 I \ -» I / h / 1 / \* 1 1 / \ v - 7 1 / -I 0 1 2 3 4 5 6 9 - orbits F i g u r e 4.1 Response p l o t s i n aerodynamic regime showing e f f e c t o f ea r th r a d i a t i o n s (c) e=0.1, c=0.5, K i = 1 . 0 , A l t = 500 mi l e s GG + DS + AD GG + DSR + AD + DER + EA + shadow 2.0 1.5 1.0 0.5 0.0 •0.5 •1.0 1.5 2.0 C=0.3 • 1.0 - r <|> = 0.0 Alt =400 miles 0.0 0.1 0.4 0.5 0.0 0.1 0.2 0.3 e Figure 4.2 Stability charts in aerodynamic regime showing effect of earth radiations 0-2 0.3 0.4 0.5 e H I—1 152 (i * eCose) V ' - ae ( . tv ' + n S i n e + 3K-, S i n ^ Cos^p (1 f e V d + ecos©) 3 c 0 . 5 1 M c , 4.3 S y s t e m R e s p o n s e and S t a b i l i t y A n a l y s i s The g o v e r n i n g e q u a t i o n o f m o t i o n (4.12) was i n t e g r a t e d n u m e r i c a l l y t o o b t a i n n o t o n l y t h e r e s p o n s e o f t h e s y s t e m , b u t a l s o t h e r e l a t i v e i m p o r t a n c e o f t h e d i r e c t s o l a r r a d i a t i o n and a e r o d y n a m i c f o r c e s . F o r a r a n g e o f p a r a m e t e r s , F i g u r e 4.3 shows t h e r e s p o n s e o f a s a t e l l i t e as p r e d i c t e d by: a) g r a v i t y g r a d i e n t f o r c e s a l o n e ; b) g r a v i t y g r a d i e n t and a e r o d y n a m i c f o r c e s ; c) g r a v i t y g r a d i e n t , a e r o d y n a m i c , and d i r e c t s o l a r r a d i a t i o n f o r c e s . The r e s u l t s c l e a r l y show t h e s i g n i f i c a n c e o f t h e f o r c e s i n v o l v e d . F u r t h e r , t h e y e m p h a s i z e t h e i m p o r t a n c e o f s o l a r r a d i a t i o n t h r o u g h v a s t l y d i f f e r e n t r e s p o n s e t r a j e c t o r i e s . I t i s i n t e r e s t i n g t o n o t e ( F i g u r e 4 .3(f)) t h a t t h e i n s t a b i l i t y o f t h e t r u e r e s p o n s e was n o t p r e d i c t e d i n t h e a b s e n c e o f s o l a r , r a d i a t i o n . Hence, t h e b e h a v i o r o f a g r a v i t y o r i e n t e d s a t e l l i t e i n n e a r e a r t h o r b i t s c a n o n l y be p r e d i c t e d by t h e i n c l u s i o n o f b o t h a e r o d y n a m i c and s o l a r r a d i a t i o n f o r c e s . 20 15 10 5 - 5 -10 -15 -20 e = ° - ° Alt = 500 miles G G + D S R + A D (bWo V[0]-°-0 - - - - G G I' 1 1 \./\ \J \ y N 9 - o rb i ts F i g u r e 4.3 S a t e l l i t e r e s p o n s e p l o t s s h o w i n g r e l a t i v e i m p o r t a n c e o f e n v i r o n m e n t a l f o r c e s : (a) e=0, c=0.3, K i=1.0, A l t = 500 m i l e s (JO 4 0 l e = o.o C = 0.3 Ki= 1.0 = 0-0 Alt = 400 miles IJJ 101 = 0-0 l|)'t0)=0.0 G G + DSR + AD G G + AD . G G 30 U 20 h - 3 0 h •401 e I 3 orb i ts F i g u r e 4.3 S a t e l l i t e r e s p o n s e p l o t s s h o w i n g r e l a t i v e i m p o r t a n c e o f e n v i r o n m e n t a l f o r c e s : (b) e=0, c=0.3, K i=1.0, A l t = 400 m i l e s 40 30 20 10 - 10 -20 - 30 e = 0 - 1 Alt = 500 miles G G + DSR + AD k " ° ' 3 ) l|l|0)=0.0 G G + A D d/io'-O +' ( 0 ) = 0 - ° — — — G G : 1 / ^ \ | / \ Jf \ f i 1 /A 1 r ''/ \"» K'/ \v/ 1 7 ^7 1 0 1 2 3 4 5 6 Q - orbits F i g u r e 4.3 S a t e l l i t e r e s p o n s e p l o t s s h o w i n g r e l a t i v e i m p o r t a n c e o f e n v i r o n m e n t a l f o r c e s : (c) e=0, c=0.3, K i=1.0, A l t = 300 m i l e s e = o.o c = 0-3 Kj= 1.0 (j> =0-0 Alt = 300 miles lb (0) =0-0 l|)'(0) = 0.0 G G + DSR + AD G G + AD G G 240 180L P O h 60 h •2401 F i g u r e 4.3 6 - orb i ts S a t e l l i t e response p l o t s showing r e l a t i v e importance of environmenta l f o r c e s : .(d) e=0.1, c=0.3, 1^=1.0, A l t = 500 mi le s 40 e • o.i C = 0.3 Kj= 1.0 (b =o.o T Alt = 400 miles lb (01 = 0.0 l|/(0)=0.0 G G + DSR + AD G G + AD G G 30 k Figure 4.3 S a t e l l i t e response plots showing r e l a t i v e importance of environmental forces: (e) e=0.1, c=0.3, K.=1.0, A l t = 400 miles £J 1 F i g u r e 4 . 3 S a t e l l i t e r e s p o n s e p l o t s s h o w i n g r e l a t i v e i m p o r t a n c e o f e n v i r o n m e n t a l f o r c e s : ( f ) e=0.1, c=0.5 , K i = 1 . 0 / A l t = 400 m i l e s 00 159 Figures 4.3 (a), (b) and (c) show the dominating e f f e c t of the r a r e f i e d atmosphere with decreasing a l t i t u d e . At 500 miles, solar radiation appears to be the most s i g n i f i c a n t force; at 400 miles aerodynamic and solar radiation effects are of comparable magnitude; however, at 300 miles, aerodynamic force / controls the system behavior. Note that the study of l i b r a t i o n a l motion i n excess of 90°, as shown i n Figure 4.3(c), i s made possible through the modified expression of the aerodynamic moment (equation 4.5). Furthermore, i t i s in t e r e s t i n g to rec-ognize that the s a t e l l i t e does not tumble i n spite of the large amplitude (>90°) motion, which would be the case for a pure gravity oriented system. The expected larger amplitude motion for increasing e c c e n t r i c i t y i s shown by a comparison of Figures 4.3(a) and (b) with (d) and (e) . Figure 4.4 further substantiates the r e l a t i v e influence of these forces through a comparison of mainland s t a b i l i t y bounds as predicted by a: a) pure gravity gradient system; b) gravity gradient system with aerodynamic forces; c) gravity gradient system with d i r e c t solar radiation forces; d) gravity gradient system with both aerodynamic and radiation forces. In general, the s t a b i l i t y area reduces with the inc l u s i o n of radiation and/or aerodynamic forces. Apparently, d i r e c t solar radiation has a detrimental e f f e c t for a l l e c c e n t r i c i t i e s . On Alt=400 G G 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Figure 4.4 Stability charts showing relative importance of environmental forces 161 t h e o t h e r hand, t h e a e r o d y n a m i c e f f e c t i s most p r e v a l e n t a t l o w e r e c c e n t r i c i t i e s and, f o r t h e p a r a m e t e r s c o n s i d e r e d , i s a l m o s t n e g l i g i b l e when e>0.18. However, i t c a n be s e e n t h a t t h e t r u e r e p r e s e n t a t i o n o f t h e s t a b i l i t y bound i s g i v e n o n l y when b o t h t h e s e e f f e c t s a r e i n c l u d e d . A s y s t e m p l o t s h o w i n g t h e v a r i a t i o n o f tp w i t h t h e m a j o r p a r a m e t e r s , e, c, and a l t i t u d e , i s g i v e n by F i g u r e 4.5. A l t h o u g h t h e l i b r a t i o n a l a m p l i t u d e i n c r e a s e s q u i t e r a p i d l y a t l o w e r a l t i t u d e s , t h e e f f e c t o f e c c e n t r i c i t y i n t h e same r e g i o n i s q u i t e s m a l l . As a g a i n s t t h e p r e v i o u s d i a g r a m , F i g u r e 4.6 e m p h a s i z e s t h e p e r f o r m a n c e o f a g i v e n s a t e l l i t e as a f u n c t i o n o f a l t i t u d e . I t i s i n t e r e s t i n g t o n o t e t h e t r a n s i t i o n f r o m a e r o d y n a m i c t o s o l a r r a d i a t i o n d o m i n a n c e . N o t e t h a t a p u r e g r a v i t y g r a d i e n t a n a l y s i s i s most a p p l i c a b l e i n t h e i n t e r m e d i a t e a l t i t u d e r a n g e . F i g u r e 4.7 shows t h e s u p e r i m p o s e d s t a b i l i t y c h a r t s , f o r a r a n g e o f a l t i t u d e s a t d i s c r e t e v a l u e s o f c. The s t a b i l i t y bound f o r a p u r e g r a v i t y g r a d i e n t s a t e l l i t e i s a l s o i n c l u d e d . The f i g u r e c l e a r l y shows t h e r e l a t i v e i m p o r t a n c e o f t h e e n v i r o n -m e n t a l f o r c e s . I n g e n e r a l , t h e a e r o d y n a m i c e f f e c t i s r e l a t i v e l y i n s i g n i f i c a n t above 500 m i l e s a l t i t u d e . I t may be p o i n t e d o u t t h a t , as i n C h a p t e r 3, t h e p e r i o d i c s o l u t i o n s a r e s h i f t e d o u t o f t h e s t a b i l i t y c h a r t p l a n e . However, t h i s s h i f t i s v e r y s m a l l , and h e n c e , t h e v a r i a t i o n i n t h e , s t a b i l i t y l i m i t i s n e g l i g i b l e . Figure 4.6 Variation of . with altitude for given sate l l i t e geometry 164 Figure 4.7 S t a b i l i t y char t s i n presence of environmenta l forces as a f fec ted by a l t i t u d e : (a) c=0.1 2.0 G G C = 0-3 Kj = 1.0 (|) = 0.0 G G + D S R + A D , 400 miles no stability at 300 miles 0.0 0.1 0.2 0.3 0.4 0.5 0.6 Figu re 4.7 S t a b i l i t y char t s i n presence of environmental forces as a f f ec t ed by a l t i t u d e : (b) c=0.3 166 G G C = 0.5 G G + DSR Kj = 1.0 • G G + DSR + AD , 500 miles <|) = 0.0 G G + DSR + A D , 400 miles no stability at 3 0 0 miles 0.1 0.2 0.3 0.4 0.5 0.6 e -2.0*-0.0 Figu re 4.7 S t a b i l i t y char t s i n presence of environmental forces as a f fec ted by a l t i t u d e : (c) c=0.5 1 6 7 4.4 Concluding Remarks The a n a l y s i s extends our understanding of the e n v i r o n -mental e f f e c t s on c l o s e - e a r t h , g r a v i t y g r a d i e n t s a t e l l i t e s and should prove u s e f u l d u r i n g t h e i r p r e l i m i n a r y d e s i g n . Based on the a n a l y s i s , the f o l l o w i n g o b s e r v a t i o n s can be made: i ) For a l l p r a c t i c a l purposes, the e f f e c t of e a r t h r a d i a t i o n s and shadow can be n e g l e c t e d i n the aero-dynamic regime as the maximum l i b r a t i o n a l amplitude and mainland s t a b i l i t y bounds are e s s e n t i a l l y u n a f f e c t e d . i i ) A p r e c i s e dynamic a n a l y s i s of such s a t e l l i t e s r e q u i r e s the c o n s i d e r a t i o n of both aerodynamic and d i r e c t s o l a r r a d i a t i o n f o r c e s , i i i ) In g e n e r a l , the environmental f o r c e s a d v e r s e l y a f f e c t the s t a b i l i t y areas. For the aerodynamic f o r c e / t h i s e f f e c t reduces w i t h i n c r e a s i n g e c c e n t r i c i t y ; on the o t h e r hand, the i n f l u e n c e of d i r e c t s o l a r r a d i a t i o n i s s u b s t a n t i a l at a l l e c c e n t r i c i t i e s . i v ) The pure g r a v i t y g r a d i e n t a n a l y s i s i s l i k e l y t o be more a p p l i c a b l e at i n t e r m e d i a t e a l t i t u d e s (approximately 400 - 4000 m i l e s ) . 5. CLOSING COMMENTS 5.1 Case Study, GEOS-A The a n a l y s i s showing the e f f e c t of the major environmental f o r c e s on the a t t i t u d e dynamics of a s a t e l l i t e having been developed, i t would be p e r t i n e n t to apply i t t o an e x i s t i n g con-f i g u r a t i o n . For t h i s purpose the Geodetic E a r t h O r b i t i n g S a t e l l i t e (GEOS-A), which i s s t a b i l i z e d u s i n g the g r a v i t y g r a d i e n t torque was s e l e c t e d . Designed and b u i l t f o r NASA by the Johns Hopkins U n i v e r s i t y ' s A p p l i e d P h y s i c s L a b o r a t o r y , the s a t e l l i t e weighing 386.6 l b . and c a r r y i n g a 40 f t . boom was launched i n an e c c e n t r i c o r b i t (perigee = 600 nm, apogee = 1230 nm) on November 6, 1965. The p h y s i c a l p r o p e r t i e s of the s a t e l l i t e are summarized i n Table 1. Accounting f o r the dominant environmental f o r c e s ( g r a v i t y g r a d i e n t and d i r e c t s o l a r r a d i a t i o n above 500 m i l e s , and i n a d d i t i o n , the aerodynamic f o r c e below 500 miles) the s t a b i l i t y c h a r t s f o r the s a t e l l i t e were obtained as a f u n c t i o n of a l t i t u d e as shown i n F i g u r e 5.1. The s t r o n g e f f e c t of d i r e c t s o l a r r a d i a t i o n a t high a l t i t u d e s as p r e d i c t e d by the parametric a n a l y s i s i s q u i t e apparent. The s t a b i l i t y r e g i o n decreases s u b s t a n t i a l l y w i t h i n c r e a s i n g a l t i t u d e and disap p e a r s completely around 14000 miles (Figure 5.1(a)). The e f f e c t of the aerodynamic f o r c e i s c l e a r l y shown i n F i g u r e 5.1(b). The d r a s t i c r e d u c t i o n o f the s t a b i l i t y r e g i o n w i t h d e c r e a s i n g a l t i t u d e confirms the c o n c l u s i o n s of the i n v e s t i -g a t i o n i n Chapter 4. The s a t e l l i t e appears t o have no s t a b i l i t y below 200 miles a l t i t u d e . 169 2000 miles — — 12000 miles 6000 miles 14000 miles 10000 miles G E O S - A 0-0 0.T 0.2 0.3 0.4 0.5 0.6 Figure 5.1 S t a b i l i t y charts f o r GEOS-A showing (a) d i r e c t s o l a r r a d i a t i o n e f f e c t at higher a l t i t u d e s 170 2.01 500 miles 300 miles 250 miles T 1— GEOS-A 225 miles 210 miles •2.0' 0-0 0.1 0.2 0.3 e 0-4 0.5 0.6 171 The i m p o r t a n t g e n e r a l o b s e r v a t i o n s b a s e d on t h e a n a l y s i s may be summarized as f o l l o w s : i ) The n o r m a l l y n e g l e c t e d d i r e c t s o l a r r a d i a t i o n i s an i m p o r t a n t p a r a m e t e r i n t h e a n a l y s i s o f a g r a v i t y o r i e n t e d s y s t e m . I n f a c t t h e s o l a r p a r a m e t e r c w h i c h i s a f u n c t i o n o f t h e s a t e l l i t e a r e a d i s t r i b u t i o n , i s as i m p o r t a n t as t h e i n e r t i a p a r a m e t e r . i i ) S o l a r r a d i a t i o n c a n be u s e d q u i t e e f f e c t i v e l y i n damp-i n g t h e l i b r a t i o n a l m o t i o n o f a s a t e l l i t e . The c o n c e p t c a n e a s i l y be e x t e n d e d t o a c h i e v e . s a t e l l i t e a t t i t u d e c o n t r o l . i i i ) The c l o s e d f o r m e x p r e s s i o n s f o r e a r t h r a d i a t i o n f o r c e s p e r m i t c o m p r e h e n s i v e s t u d y o f s a t e l l i t e d y n a m i c s . The c o n c e p t s h o u l d a l s o p r o v e u s e f u l i n an a c c u r a t e t h e r m a l b a l a n c e o f c l o s e e a r t h s y s t e m s . i v ) The maximum l i b r a t i o n a l a m p l i t u d e and m a i n l a n d s t a b i l i t y a r e a s o f g r a v i t y o r i e n t e d s a t e l l i t e s c a n be a c c u r a t e l y p r e d i c t e d , e v e n f o r c l o s e o r b i t s , w i t h o u t c o n s i d e r i n g d i r e c t e a r t h r a d i a t i o n , i t s a l b e d o and shadow. However, t h e t r u e r e s p o n s e o f s a t e l l i t e s i n t h e a e r o d y n a m i c r e g i m e down t o 300 m i l e s c a n o n l y be p r e d i c t e d by c o n -s i d e r i n g b o t h a e r o d y n a m i c and d i r e c t s o l a r r a d i a t i o n f o r c e s . 5.2 Recommendations f o r F u t u r e Work The t h e s i s p r e s e n t s numerous t o p i c s f o r p o s s i b l e e x t e n -s i o n , however, o n l y t h e m a j o r a r e a s w i l l be l i s t e d h e r e . I t w o u l d be u s e f u l t o e x t e n d t h e d i r e c t s o l a r r a d i a t i o n a n a l y s i s t o i n c l u d e t h e t h e r m a l b e n d i n g o f booms w h i c h s h o u l d have s u b s t a n t i a l e f f e c t on t h e s a t e l l i t e d y n a m i c s . " ^ Such an i n v e s t i g a t i o n s h o u l d t a k e i n t o c o n s i d e r a t i o n n o t o n l y t h e e a r t h ' s shadow b u t a l s o t h e d i f f e r e n t i a l t h e r m a l d i s t o r t i o n o f t h e booms 54 r e s u l t i n g i n a p e r i o d i c d i s p l a c e m e n t o f t h e c e n t e r o f p r e s s u r e . I t i s c o n c e i v a b l e t h a t t h e t h e r m a l e f f e c t o f t h e e a r t h r a d i a t i o n s i n s u c h a s t u d y may p r o v e t o be s i g n i f i c a n t f o r low a l t i t u d e o r b i t s . The c o n c e p t o f l i b r a t i o n a l damping u s i n g s o l a r r a d i a t i o n p r e s s u r e a p p e a r s t o be p r o m i s i n g and s h o u l d be e x p l o r e d f u r t h e r . The use o f a s a t e l l i t e o r b i t a l p l a n e o t h e r t h a n t h e e c l i p t i c and t h e e x t e n s i o n t o two d e g r e e s o f f r e e d o m s h o u l d be i n t e r e s t i n g . However, t h e l a t t e r w o u l d i n c r e a s e t h e number o f s t a t e v a r i a b l e s t o f i v e w i t h t h r e e t o f o u r d e s i g n p a r a m e t e r s . T h i s w o u l d r e n d e r t h e p r e s e n t a t i o n o f n u m e r i c a l d a t a e x t r e m e l y c o m p l e x and e v e n a l i m i t e d p a r a m e t r i c a n a l y s i s may i n v o l v e a m a s s i v e c o m p u t a t i o n a l e f f o r t . BIBLIOGRAPHY 1. K a t u c k i , R.J., " G r a v i t y - G r a d i e n t S t a b i l i z a t i o n , " Space/  A e r o n a u t i c s , Oct. 1964, pp. 42-47. Shapi r o , I . I . , and Jones, H.M., " L i f e t i m e s of O r b i t i n g D i p o l e s , " S c i e n c e , V o l . 134, No. 3484, Oct. 1961, pp. 973-979. P i s c a n e , V i n c e n t L., Pardoe, P e t e r P., and Hook, P. Joy. " S t a b i l i z a t i o n System A n a l y s i s and Performance of the Geos-A G r a v i t y - G r a d i e n t S a t e l l i t e ( E x p l o r e r XXIX)," Proceedings of the AIAA/JACC Guidance and C o n t r o l Con- ference , American I n s t i t u t e of A e r o n a u t i c s and A s t r o n -a u t i c s , 1966, pp. 226-237. 4. Klemperer, W.B., " S a t e l l i t e L i b r a t i o n s of Large Amplitude," ARS J o u r n a l , V o l . 30, No. 1, Jan. 1960, pp. 123-124. 5. Baker, Robert M.L., J r . , " L i b r a t i o n s on a S l i g h t l y E c c e n t r i c O r b i t , " ARS J o u r n a l , V o l . 30, No. 1, Jan. 1960, pp. 124-126. 6. Schechter, Hans B., "Dumbbell L i b r a t i o n s i n E l l i p t i c O r b i t s , " AIAA J o u r n a l , V o l . 2, No. 6, June 1964, pp. 1000-1003. 7. Zlatousov, V.A. , Okhotsimsky, D.E., Sarghev, V.A., and Torzhevsky, A.P., " I n v e s t i g a t i o n of a S a t e l l i t e O s c i l l a t i o n s i n the Plane of an E l l i p t i c O r b i t , " Proceedings of the  E l e v e n t h I n t e r n a t i o n a l Congress of A p p l i e d Mechanics, G o r t l e r , Henry, ed., S p r i n g e r - V e r l a g , B e r l i n , 1964, pp. 436-439. 8. B r e r e t o n , R.C, and Modi, V . J . , "On the S t a b i l i t y of P l a n a r L i b r a t i o n s of a Dumbbell S a t e l l i t e i n an E l l i p t i c O r b i t , " J o u r n a l of the Royal A e r o n a u t i c a l S o c i e t y , V o l . 70, No. 12, 1966, pp. 1098-1102. 9. B r e r e t o n , R.C, and Modi, V . 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Modi, V.J., and Brereton, R.C., " S t a b i l i t y of a Dumbbell S a t e l l i t e i n a C i r c u l a r Orbit During Coupled L i b r a t i o n a l Motions," Proceedings of the XVIIIth International Astro- nautical Congress, Pergamon Press, London, 196 8, pp. 109-120. 14. Brereton, R.C., "A S t a b i l i t y Study of Gravity Oriented S a t e l l i t e s , " Ph.D. d i s s e r t a t i o n , University of B r i t i s h Columbia, Nov. 1967. 15. Thompson, W.J. , "Spin S t a b i l i z a t i o n of Attitude Against Gravity Torques," The Journal of the Astronautical  Sciences, Vol. 9, No. 1, Jan. 1962, pp. 31-33. 16. Kane, T.R., Marsh, E.L., and Wilson, W.G., "Letter to the Editor," The Journal of the Astronautical Sciences, Vol. 9, No. 1, Jan. 1962, pp. 108-109. 17. Kane, T.R., and Barba, P.M., "Attitude S t a b i l i t y of a Spinning S a t e l l i t e i n an E l l i p t i c Orbit," Journal of  Applied Mechanics, June 1966, pp. 402-405. 18. 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N e i l s o n , J.E., "On the A t t i t u d e Dynamics of Slowly S p i n -ning Axisymmetric S a t e l l i t e s under the I n f l u e n c e of G r a v i t y - G r a d i e n t Torques," Ph.D. d i s s e r t a t i o n , U n i v e r s i t y of B r i t i s h Columbia, Nov. 1968. 23. Garwin, R.L., " S o l a r S a i l i n g - A P r a c t i c a l Method of Pro-p u l s i o n W i t h i n the S o l a r System," J e t P r o p u l s i o n , V o l . 28, No. 3, March 1958, pp. 188-190. 24. Roberson, R.E., " A t t i t u d e C o n t r o l of a S a t e l l i t e V e h i c l e - An O u t l i n e of the Problem," Proceedings of the V l l l t h  I n t e r n a t i o n a l A s t r o n a u t i c a l Congress, Wein-Springer-V e r l a g , B e r l i n , 1958, pp. 317-339. 25. Camac, W.G., and Edwards, K.K., " E f f e c t of Surface Thermal R a d i a t i o n C h a r a c t e r i s t i c s on the Temperature C o n t r o l Problem i n S a t e l l i t e s , " i n Surface E f f e c t s on Space C r a f t M a t e r i a l , ed. by F. J . C l a u s s , New York, John Wiley, 1960. 26. Cunningham, F.G., "Power Input to a Small F l a t P l a t e from a D i f f u s e l y R a d i a t i n g Sphere, with A p p l i c a t i o n to E a r t h S a t e l l i t e s , " NASA TN D-710, 1961. 27. Cunningham, F.G., "Earth R e f l e c t e d S o l a r R a d i a t i o n Input to S p h e r i c a l S a t e l l i t e s , " NASA TN D-1099, 1961. 28. K r e i t h , F., R a d i a t i o n Heat T r a n s f e r f o r Space C r a f t and  S o l a r Power P l a n t Design, I n t e r n a t i o n a l Textbook Comp., 1962. 29. Dennison, A . 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C , " T h e o r e t i c a l and P r a c t i c a l Aspects of S o l a r Pressure A t t i t u d e C o n t r o l f o r I n t e r p l a n e -t a r y S p a c e c r a f t , " J e t P r o p u l s i o n Laboratory T e c h n i c a l  Report No. 32-467, May 15, 1964. 50. U l e , L.A., " O r i e n t a t i o n of Spinning S a t e l l i t e s by R a d i a t i o n P r e s s u r e , " AIAA J o u r n a l , V o l . 1, No. 7, 1963. 51. M a l l a c h , E.G., " S o l a r Pressure Damping of the L i b r a t i o n s of a G r a v i t y - G r a d i e n t O r i e n t e d S a t e l l i t e , " AIAA Student  J o u r n a l , V o l . 4, No. 4, Dec. 1966, pp. 143-147. 52. H a l l , H.B., "The E f f e c t s of R a d i a t i o n Force on S a t e l l i t e s of Convex Shape," NASA TN D-604, 1961. 53. M c E l v a i n , R.J., " E f f e c t s of S o l a r R a d i a t i o n Pressure Upon S a t e l l i t e A t t i t u d e C o n t r o l , " ARS P r e p r i n t 1918-61, Aug. 1961. 54. Flook, L. , and Warren, H.R., " E f f e c t s of Thermal Bending and S o l a r Torques on G r a v i t y G r a d i e n t Booms," Hawker  S i d d e l e y Dynamics L i m i t e d , H.S.D. TN No. 92850, 1963. 55. Clancy, T.F., and M i t c h e l l , T.P., " E f f e c t s of R a d i a t i o n Forces on the A t t i t u d e of an A r t i f i c i a l E a r t h S a t e l l i t e , " AIAA J o u r n a l , . V o l . 2, No. 3, March 1964, pp. 517-524. 56. B e l e t s k i i , V.V., "Motion of an A r t i f i c i a l E a r t h S a t e l l i t e About i t s Center of Mass," A r t i f i c i a l E a r t h S a t e l l i t e s , V o l . 1, ed. by L.V. Kurnosova, Plenum P r e s s , New York, 1960, pp. 30-59. 57. Evans, W.J., "Aerodynamic and R a d i a t i o n Disturbance Torques on S a t e l l i t e s Having Complex Geometry," The J o u r n a l of the A s t r o n a u t i c a l S c i e n c e s , V o l . 9, 1962, pp. 93-99. 58. 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J . , and B r e r e t o n , R.C., "The P l a n a r Motion of a Damped G r a v i t y G r a d i e n t S t a b i l i z e d S a t e l l i t e , " T r a n s a c t i o n s of the Canadian A e r o n a u t i c s and Space I n s t i t u t e , V o l . 2, No. 1, March, 1969, pp. 44-48. 63. Moran, John P., " E f f e c t s of Plane L i b r a t i o n s on the O r b i t a l Motion of a Dumbbell S a t e l l i t e , " ARS J o u r n a l , V o l . 31, No. 8, Aug. 1961, pp. 1089-1096. 64. Yu, E.Y., "Long-term C o u p l i n g E f f e c t s Between the L i b r a t i o n a l ' and O r b i t a l Motions of a S a t e l l i t e , " AIAA J o u r n a l , V o l . ; 2 , No. 3, March 1964, pp. 553-555. 65. Cunningham, W.J., I n t r o d u c t i o n to N o n l i n e a r A n a l y s i s , McGraw-Hill, New York, 1958, pp. 250-257. 66. Moser, Jurgen. " P e r t u r b a t i o n Theory f o r Almost P e r i o d i c S o l u t i o n s f o r Undamped N o n l i n e a r D i f f e r e n t i a l E q u ations," I n t e r n a t i o n a l Symposium on N o n l i n e a r D i f f e r e n t i a l Equations  and N o n l i n e a r Mechanics, e d i t e d by La S a l l e , J.P., and L e f s c h e t z , S., Academic P r e s s , New York, 1963, pp. 71-79. 67. Henon, M., and H e i l e s , C , "The A p p l i c a b i l i t y of the T h i r d I n t e g r a l of Motion: Some Numerical Experiments," A s t r o -nomical J o u r n a l , V o l . 69, No. 1, Feb.: 1964 , pp. 73-79. 68. J e f f r e y s , W.H., "Some Dynamical Systems of Two Degrees of Freedom i n C e l e s t i a l Mechanics," A s t r o n o m i c a l J o u r n a l , V o l . 71, No. 5, June 1966, pp. 306-313. 69. Hamming, R.W. , Numerical Methods f o r S c i e n t i s t s and  En g i n e e r s , McGraw-Hill Book Company, Inc., New York, 1962, pp. 183-222. 70. Minorsky, N i c h o l a s , N o n l i n e a r O s c i l l a t i o n s , D. Van Nostrand Company, Inc., P r i n c e t o n , 1962, pp. 127-133, 390-415. 71. Johnson, J.C., P h y s i c a l Meteorology, John Wiley and Sons, Inc., New York, p. 171. 72. Schaaf, S.A., and Chambre, P.L., "The Flow of R a r e f i e d Gases," High Speed Aerodynamics and J e t P r o p u l s i o n , V o l . 3, S e c t i o n H, ed. by H.W. Emmons, P r i n c e t o n U n i v e r s i t y P r e s s , 1958, pp. 687-708. 179 73. Jensen, J . , Townsend, G., Kork, J . , and K r a f t , D. Design Guide t o O r b i t a l F l i g h t , McGraw-Hill, New York, 1962, pp. 179-264. 74. Cook, G.E., ARC C u r r e n t Paper No CP523, M i n i s t r y of A v i -a t i o n , H.M.S.O., London, 1960. PUBLICATIONS Flanagan, R . C , and Modi , V . J . , "Radiation Forces on a F l a t Plate i n Close Earth O r b i t s , " Proceedings of the Canadian Congress o f Appl ied Mechanics, edi ted by N . C . L i n d , U n i v e r s i t y of Waterloo, Waterloo, Ontar io , 1969, pp. 249-250. Flanagan, R . C , and Modi , V . J . , "Att i tude Dynamics of a Grav i ty Oriented S a t e l l i t e Under the Influence of Solar Radiat ion Pressure ," i n press , Journal of the Royal Aeronaut ica l Society . Flanagan, R . C , and Modi , V . J . , " L i b r a t i o n a l Damping of a Grav i ty Oriented System Using Solar Radiat ion Pressure", presented for p u b l i c a t i o n . Flanagan, R . C , and Modi , V . J . , "Effect of Environmental Forces on the A t t i t u d e Dynamics of Grav i ty Oriented S a t e l l i t e s : Part I - Radiat ion Forces", presented for p u b l i c a t i o n . Modi , V . J . , and Flanagan, R . C , "Effect of Environmental Forces on the A t t i t u d e Dynamics of Gravi ty Oriented S a t e l l i t e s : Part II - High A l t i t u d e O r b i t s " , presented for p u b l i c a t i o n . Modi , V . J . , and Flanagan, R . C , "Effect of Environmental Forces on the At t i tude Dynamics of Grav i ty Oriented S a t e l l i t e s : Part I I I - Intermediate A l t i t u d e Orb i t s Accounting for Earth Radiat ions" , presented for p u b l i c a t i o n . Modi, V . J . , and Flanagan, R . C , "Effect of Environmental Forces on the A t t i t u d e Dynamics of Grav i ty Oriented S a t e l l i t e s : Part IV - Close Earth O r b i t s Accounting for Aerodynamic Forces", presented for p u b l i c a t i o n . 

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