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The kinematics of carbon stars in the large and small magellanic clouds Joslin, Garry Douglas 1983

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THE KINEMATICS OF CARBON STARS IN THE LARGE AND SMALL MAGELLANIC CLOUDS by GARRY DOUGLAS JOSLIN B . S c , Queen's U n i v e r s i t y , 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES (Department of G e o p h y s i c s and As t r o n o m y ) We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA J u l y 1983 © G a r r y D o u g l a s J o s l i n , 1983 i i 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 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 h e a d o f my d e p a r t m e n t o r by h i s o r h e r 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 c o p y i n g o r p u b l i c a t i o n 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 . D e p a r t m e n t o f C eo p <; t'cs a»\C^ / s y V o n o ^ 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 1956 Main M a l l V a n c o u v e r , Canada V6T 1Y3 DE-6 (3/81) A b s t r a c t R a d i a l v e l o c i t i e s were d e t e r m i n e d f o r a sample o f c a r b o n s t a r s p e c t r a . These c a r b o n s t a r s were l o c a t e d i n t h e L a r g e and S m a l l M a g e l l a n i c C l o u d s and were l i s t e d i n t h e c a t a l o g u e s o f W e s t e r l u n d , O l a n d e r , R i c h e r and C r a b t r e e ( 1 9 7 8 ) , B l a n c o , M c C a r t h y and B l a n c o (1980) and S a n d u l e a k and P h i l i p ( 1 9 7 7 ) . R o t a t i o n c u r v e s b a s e d upon t h e d e r i v e d r a d i a l v e l o c i t i e s i n d i c a t e d t h a t t h e LMC r o t a t e s l i k e a s o l i d body. The v e l o c i t y g r a d i e n t o f t h e c a r b o n s t a r s a l o n g t h e major a x i s was f o u n d t o be 10.4 km/sec/deg. F o r c i r c u l a r o r b i t s , t h i s p l a c e d a l o w e r mass l i m i t o f 6.0X10 9 Mo when e m p l o y i n g a d i s k - s h a p e d model o f t h e LMC. When c a r b o n s t a r s were compared w i t h o t h e r s t e l l a r p o p u l a t i o n s i n t h e LMC t a k e n f r o m t h e c a t a l o g u e o f F e i t z i n g e r and W e i s s (1979) an i n c r e a s e o f v e l o c i t y g r a d i e n t a l o n g t h e major a x i s was f o u n d t o c o r r e l a t e w i t h a d e c r e a s e i n o b j e c t age. T h i s was i n t e r p r e t e d a s an i n c r e a s e i n o r b i t a l e c c e n t r i c i t y w i t h o b j e c t age w h i c h a g r e e d w e l l w i t h p r e v i o u s s i m i l a r f i n d i n g s f o r our own g a l a x y (Eggen, L y n d e n - B e l l , Sandage 1962). The v e l o c i t y d i s p e r s i o n s o f t h e B l a n c o , M c C a r t h y and B l a n c o Bar f i e l d i n the SMC and t h e Bar West f i e l d i n t h e LMC were a l s o compared. The h i g h e r r a d i a l v e l o c i t y d i s p e r s i o n of t h e SMC f i e l d was a t t r i b u t e d t o t h e h i g h e r i n c l i n a t i o n a n g l e o f t h e SMC and t h e p o s s i b i l i t y t h a t t h e SMC f i e l d c o n t a i n e d s t a r s w h i c h were p a r t i c i p a t i n g i n s t r e a m i n g m o t i o n s f r o m o p p o s i t e ends o f t h e B a r . i v T a b l e o f C o n t e n t s A b s t r a c t i i i T a b l e o f C o n t e n t s . i v L i s t of T a b l e s v L i s t of F i g u r e s v i Acknowledgements v i i i C h a p t e r I I n t r o d u c t i o n and M o t i v a t i o n 1 C h a p t e r II D a t a R e d u c t i o n , M e t h o d o l o g y and t h e R a d i a l V e l o c i t y C a t a l o g u e 5 C h a p t e r I I I R o t a t i o n o f t h e LMC as D e s c r i b e d by C a r b o n S t a r s 17 C h a p t e r IV R a d i a l V e l o c i t y D i s p e r s i o n s i n t h e L a r g e and S m a l l M a g e l l a n i c C l o u d s 58 C h a p t e r V C o n c l u s i o n s 71 A p p e n d i x I : G e o m e t r i c a l P r e d i c t i o n s o f R a d i a l V e l o c i t y .... 75 B i b l i o g r a p h y 79 V L i s t o f T a b l e s T a b l e I S p e c t r a P a r a m e t e r s ; . . 5 T a b l e I I R a d i a l V e l o c i t i e s o f M a g e l l a n i c C l o u d C a r b o n S t a r s .... 11 T a b l e I I I S o l i d Body R o t a t i o n C u r v e P a r a m e t e r s 28 T a b l e IV V e l o c i t y G r a d i e n t v s . Age o f t h e LMC O b j e c t 45 T a b l e V Mass o f t h e L a r g e M a g e l l a n i c C l o u d 48 T a b l e VI F l a t R o t a t i o n C u r v e P a r a m e t e r s f o r t h e LMC 56 v i L i s t o f F i g u r e s F i g . I Ca r b o n S t a r S p e c t r u m 7 F i g . I I X-Y Map f o r WORC Car b o n S t a r s 19 F i g . I l l X-Y Map f o r S-P Car b o n S t a r s 20 F i g . IV X-Y Map f o r HII R e g i o n s 21 F i g . V X-Y Map f o r S t a r s 22 F i g . VI X-Y Map f o r S t e l l a r C l u s t e r s 23 F i g . V I I X-Y Map f o r P l a n e t a r y N e b u l a e 24 F i g . V I I I S o l i d Body R o t a t i o n C u r v e : WORC C a r b o n S t a r s .. 29 F i g . IX S o l i d Body R o t a t i o n C u r v e : S-P C a r b o n S t a r s ... 30 F i g . X S o l i d Body R o t a t i o n C u r v e : HII R e g i o n s 31 F i g . XI S o l i d Body R o t a t i o n C u r v e : S t a r s 32 F i g . X I I S o l i d Body R o t a t i o n C u r v e : S t e l l a r C l u s t e r s ... 33 F i g . X I I I S o l i d Body R o t a t i o n C u r v e : P l a n e t a r y N e b u l a e .. 34 F i g . X I V V e l o c i t y Mean D i f f e r e n c e : WORC C a r b o n S t a r s 36 F i g . XV V e l o c i t y Mean D i f f e r e n c e : HII R e g i o n s 37 F i g . XVI V e l o c i t y Mean D i f f e r e n c e : S t a r s 38 F i g . XVII V e l o c i t y Mean D i f f e r e n c e : S t e l l a r C l u s t e r s .... 39 F i g . X V I I I V e l o c i t y Mean D i f f e r e n c e : P l a n e t a r y N e b u l a e ... 40 F i g . XIX S o l i d Body R o t a t i o n C u r v e : S-P S t a r s (0=60°) .. 41 F i g . XX S o l i d Body R o t a t i o n C u r v e : S-P S t a r s (9=120°) . 42 F i g . XXI F l a t R o t a t i o n C u r v e : WORC Carbon S t a r s 50 F i g . XXII F l a t R o t a t i o n C u r v e : S-P Car b o n S t a r s 51 F i g . X X I I I F l a t R o t a t i o n C u r v e : HII R e g i o n s 52 F i g . XXIV F l a t R o t a t i o n C u r v e : S t a r s 53 F i g . XXV F l a t R o t a t i o n C u r v e : S t e l l a r C l u s t e r s 54 F i g . XXVI F l a t R o t a t i o n C u r v e : P l a n e t a r y N e b u l a e ... 55 F i g . XXVII V e l o c i t y S t a n d a r d D e v i a t i o n : WORC S t a r s 60 v i i F i g . X X V I I I V e l o c i t y S t a n d a r d D e v i a t i o n : HII R e g i o n s 61 F i g . XXIX V e l o c i t y S t a n d a r d D e v i a t i o n : S t a r s 62 F i g . XXX V e l o c i t y S t a n d a r d D e v i a t i o n : S t e l l a r C l u s t e r s . 63 F i g . XXXI V e l o c i t y S t a n d a r d D e v i a t i o n : P l a n e t a r y N e b u l a e 64 F i g . XXXII Geometry of D i s k M o d e l s 77 v i i i A c knowledgements I w i s h t o thank D r . H. R i c h e r f o r a c t i n g as my g r a d u a t e s u p e r v i s o r and f o r s u g g e s t i n g t h i s p r o j e c t . H i s p a t i e n t e n c o u r a g e m e n t from b e g i n n i n g t o c o m p l e t i o n has made t h i s t h e s i s b o t h e d u c a t i o n a l and e n j o y a b l e . Many t h a n k s a l s o t o Dr. M. Ovenden whose i n s i g h t f u l d i s c u s s i o n s and h e l p f u l c r i t i c i s m s a i d e d g r e a t l y i n t h e p r e p a r a t i o n o f t h i s t h e s i s . F o r b e g i n n i n g my e d u c a t i o n i n t h e a r e a of a s t r o n o m y t h r o u g h c l a s s e s , d i s c u s s i o n s a n d i n f o r m a l i n t e r a c t i o n , I e x t e n d my g r a t i t u d e t o t h e P r o f e s s o r s and my f e l l o w g r a d u a t e s t u d e n t s i n a s t r o n o m y a t UBC. S p e c i a l t h a n k s go t o J o h n N i c o l , whose p l o t t i n g r o u t i n e p r o d u c e d t h e p l o t s f o r t h i s t h e s i s . L a s t but n o t l e a s t , I w i s h t o t h a n k my p a r e n t s , M a u r i c e and L o i s , f o r e n c o u r a g i n g , me t o f o l l o w my dreams and I w i s h t o thank my f i a n c e , J a n Moore, f o r f o l l o w i n g me t o V a n c o u v e r and c o n s t a n t l y e n c o u r a g i n g and s u p p o r t i n g me a s I made t h a t dream a r e a l i t y . 1 CHAPTER I INTRODUCTION AND MOTIVATION The p u r p o s e o f t h i s t h e s i s was t o d e r i v e t h e r a d i a l v e l o c i t i e s o f a sample o f c a r b o n s t a r s i n t h e L a r g e and S m a l l M a g e l l a n i c C l o u d s and t h e n t o i n v e s t i g a t e t h e k i n e m a t i c s o f t h e s e two g a l a x i e s by f i t t i n g v a r i o u s m o d els t o t h e d e r i v e d s t e l l a r r a d i a l v e l o c i t i e s . The m a j o r i t y o f t h e pr o g r a m s t a r s were t a k e n from t h e c a t a l o g u e s o f W e s t e r l u n d , O l a n d e r , R i c h e r , and C r a b t r e e (1978, h e r e a f t e r WORC) and B l a n c o , M c C a r t h y and B l a n c o ( 1 9 8 0 , h e r e a f t e r BMB) w i t h a s m a l l sample o f c a r b o n s t a r s t a k e n from t h e S a n d u l e a k - P h i 1 i p c a t a l o g u e (1977 , h e r e a f t e r SP) . As a s t e l l a r g r o u p , c a r b o n s t a r s a r e p o t e n t i a l l y v e r y i m p o r t a n t i n t h e u n d e r s t a n d i n g o f g a l a c t i c d y n a m i c s . In r e c e n t c o m p a r i s o n s o f c a r b o n s t a r s i n t h e WORC and BMB c a t a l o g u e s w i t h g a l a c t i c c a r b o n s t a r s ( R i c h e r , O l a n d e r , W e s t e r l u n d 1979; A a r o n s o n and Mo u l d 1980; Cohen e t a l 1981) i t has been f o u n d t h a t t h e m a j o r i t y o f t h e s e s t a r s most c l o s e l y c o r r e s p o n d t o P o p u l a t i o n I c a r b o n s t a r s f o u n d i n t h e d i s k o f our own g a l a x y . In c o n t r a s t , many c a r b o n s t a r s t a k e n from t h e SP c a t a l o g u e more c l o s e l y c o r r e s p o n d t o P o p u l a t i o n I I c a r b o n s t a r s f o u n d i n t h e h a l o o f t h e M i l k y Way. Re c e n t s t u d i e s o f t h e LMC and SMC by BMB have shown t h a t t h e WORC and SP c a t a l o g u e s a r e o n l y a s m a l l sample o f t h e t o t a l number o f c a r b o n s t a r s t h a t a r e p r e s e n t i n t h e C l o u d s . F u r t h e r m o r e t h e s e s t a r s , f r o m a l l p r e s e n t i n d i c a t i o n s , a r e more l u m i n o u s t h a n M T=-3.5. S i n c e t h e d i s t a n c e modulus o f t h e LMC i s 2 18.6 ( A l l e n , 1981) a l l l a t e t y p e c a r b o n s t a r s p r e s e n t i n t h e LMC s h o u l d be d e t e c t a b l e u s i n g p r e s e n t t e l e s c o p e s and e q u i p m e n t . Carbon s t a r s a r e t h e r e f o r e b o t h l u m i n o u s and numerous enough t o o u t l i n e t h e k i n e m a t i c s o f b o t h t h e L a r g e and S m a l l M a g e l l a n i c C l o u d s i n f i n e d e t a i l . In t h e c a s e of P o p u l a t i o n I c a r b o n s t a r s , t h e s e o b j e c t s a r e p o s s i b l y t h e o l d e s t o b s e r v a b l e k i n e m a t i c i n d i c a t o r s f o r b o t h t h e d i s k of t h e M i l k y Way and t h e d i s k s of t h e LMC and SMC. P o p u l a t i o n I I c a r b o n s t a r s a r e a l s o i m p o r t a n t b e c a u s e t h e y a c c u r a t e l y d e s c r i b e t h e k i n e m a t i c s o f t h e h a l o s o f our g a l a x y and a l s o t h e h a l o s o f t h e LMC and SMC. T h e r e e x i s t s a t t h i s t i m e , i n a l l t h r e e o f t h e s e g a l a x i e s , no l a r g e p o p u l a t i o n o f o b s e r v a b l e v e l o c i t y i n d i c a t o r s w i t h w h i c h t o p r o b e t h e h a l o . A s m a l l sample o f v e r y o l d g l o b u l a r c l u s t e r s has been f o u n d i n t h e LMC by Cowley and H a r t w i c k ( 1 9 8 2 ) . They f i n d t h a t t h e s e c l u s t e r s a r e p a r t o f t h e h a l o p o p u l a t i o n o f t h e LMC. The number of t h e s e o b j e c t s however, i s much l e s s t h a n w o u l d be r e q u i r e d f o r a s e r i o u s s t u d y of t h e k i n e m a t i c s o f t h e h a l o o f t h e LMC. RR L y r a e s t a r s a r e a l s o a c a n d i d a t e f o r t h i s a r e a o f k i n e m a t i c s t u d y . RR L y r a e s , however, have an a b s o l u t e v i s u a l m a g n i t u d e o f ab o u t +1.0 and a r e t h e r e f o r e r a t h e r f a i n t i n t h e LMC and SMC. P o p u l a t i o n I I c a r b o n s t a r s , f o u n d p r e d o m i n a n t l y i n t h e SP c a t a l o g u e , c o u l d t h e r e f o r e be t h e much needed p r o b e f o r t h i s a r e a of k i n e m a t i c r e s e a r c h . T h i s t h e s i s i s b a s e d upon t h e d e r i v e d r a d i a l v e l o c i t i e s f o r a sample o f c a r b o n s t a r s t a k e n from t h e p r e v i o u s l y d e s c r i b e d c a t a l o g u e s . T h r o u g h p h o t o m e t r y and s p e c t r a l c l a s s i f i c a t i o n o f t h e s e s t a r s ( R i c h e r e t a l 1979 ; R i c h e r 1981) i t was f o u n d t h a t 3 a l l t h o s e s t a r s from t h e BMB and WORC c a t a l o g u e s were s i m i l a r t o g a l a c t i c P o p u l a t i o n I o b j e c t s . T h e s e s t a r s were t r e a t e d a s a s e p a r a t e g r o u p from t h e r e m a i n i n g SP s t a r s w h i c h were f o u n d t o be more l i k e P o p u l a t i o n I I s t a r s . T h r o u g h t h e e x a m i n a t i o n o f r o t a t i o n c u r v e s b a s e d upon two s i m p l e d e s c r i p t i o n s o f g a l a c t i c r o t a t i o n , i t was d e t e r m i n e d t h a t t h e d i s k t y p e c a r b o n s t a r s were s i m i l a r t o o t h e r s t e l l a r p o p u l a t i o n s i n t h e LMC i n t h a t t h e y u n d e r g o g a l a c t i c r o t a t i o n i n a s i m i l a r manner. R o t a t i o n c u r v e s b a s e d on o t h e r o b j e c t s i n t h e LMC were a l s o g e n e r a t e d f o r c o m p a r i s o n p u r p o s e s . T h e s e o b j e c t s , t a k e n f r o m t h e c a t a l o g u e o f F e i t z i n g e r and W e i s s ( 1 9 7 9 ) , i n o r d e r o f i n c r e a s i n g age were: HII r e g i o n s , l u m i n o u s e a r l y and l a t e - t y p e s t a r s , s t e l l a r c l u s t e r s and p l a n e t a r y n e b u l a e . The r o t a t i o n c u r v e s of t h e s e c l a s s e s o f o b j e c t s were compared w i t h t h e r o t a t i o n c u r v e s b a s e d on d i s k t y p e c a r b o n s t a r s . C a r b o n s t a r s a r e p r o b a b l y t h e o l d e s t g r o u p of o b j e c t s i n t h i s c o m p a r i s o n , p e r h a p s b e i n g e x c e e d e d i n age o n l y by t h e p l a n e t a r y n e b u l a e . A c o r r e l a t i o n between g r o u p age a n d t h e v e l o c i t y g r a d i e n t o f t h a t g r o u p ' s r o t a t i o n c u r v e was n o t e d . A s i m i l a r a n a l y s i s o f h a l o t y p e c a r b o n s t a r s i n d i c a t e s t h a t t h e s e o b j e c t s i n t h e LMC do n o t p a r t i c i p a t e i n t h e r o t a t i o n of t h e d i s k and do not r o t a t e a b o u t any o t h e r p r e f e r r e d a x i s i n t h e LMC. T h i s i s i n c o n t r a d i c t i o n t o e v i d e n c e f o u n d by Freeman (1982) t h a t t h e h a l o component o f t h e LMC as d e s c r i b e d by o l d g l o b u l a r c l u s t e r s showed e v i d e n c e o f d i s k - l i k e r o t a t i o n i n a p l a n e d i f f e r e n t from t h a t o f t h e y o u n g e r s t e l l a r p o p u l a t i o n . The f i n d i n g o f t h i s t h e s i s , t h a t t h e p o p u l a t i o n I I c a r b o n s t a r s do not p a r t i c i p a t e i n t h e r o t a t i o n o f t h e d i s k o f t h e LMC, was 4 t a k e n a s k i n e m a t i c e v i d e n c e t h a t s u b s t a n t i a t e d t h e s p e c t r o s c o p i c and p h o t o m e t r i c i d e n t i f i c a t i o n o f t h e s e c a r b o n s t a r s a s b e i n g P o p u l a t i o n I I h a l o o b j e c t s i n t h e LMC. A l l t h e S m a l l M a g e l l a n i c C l o u d c a r b o n s t a r s f o u n d i n t h i s s t u d y were t a k e n f r o m t h e BMB Bar f i e l d . The BMB Bar West f i e l d i n t h e LMC was a l s o s t u d i e d . In b o t h c a s e s t h e o b s e r v e d f i e l d s were 23' i n d i a m e t e r and t h e r e f o r e were s m a l l enough t h a t t h e y e x h i b i t e d no s y s t e m a t i c c h a n g e s i n r a d i a l v e l o c i t y a c r o s s t h e f i e l d due t o g a l a c t i c r o t a t i o n . T h i s p r e s e n t e d t h e o p p o r t u n i t y t o compare t h e v e l o c i t y d i s p e r s i o n s o f t h e c a r b o n s t a r s i n t h e s e two r e g i o n s . The BMB Bar West f i e l d i n t h e LMC c o n t a i n e d 24 c a r b o n s t a r s w i t h a v e l o c i t y d i s p e r s i o n o f 27.8 km/sec. The BMB Bar f i e l d s t u d i e d i n t h e SMC c o n t a i n e d 17 c a r b o n s t a r s w i t h a v e l o c i t y d i s p e r s i o n o f 43.6 km/sec. The d i f f e r e n c e i n v e l o c i t y d i s p e r s i o n c a n be u n d e r s t o o d i n terms of t h e d i f f e r e n t o r i e n t a t i o n of t h e p l a n e o f t h e s e two g a l a x i e s t o our l i n e o f s i g h t . 5 CHAPTER I_I DATA REDUCTION AND METHODOLOGY The p e r t i n e n t d a t a f o r a l l t h e s t a r s , b o t h s t a n d a r d s and pr o g r a m s t a r s , u s e d i n t h i s s t u d y a r e g i v e n i n T a b l e 1. T a b l e I D a t e T e l e s c o p e S t a n d a r d Program D i s p e r s i o n S p e c t r a l ) b t a i n e d S t a r s S t a r s Range Jan/78 ESO 3.6m 0 21 126 A/mm 5000-7200 A Dec/76 CTIO 4m 5 3 50 A/mm 5000-7000 A 9 8 90 A/mm 5000-7200 A Jan/78 CTIO 4m 6 6 90 A/mm 3000-5000 A 1 7 22 90 A/mm 5000-7200 A 10 5 25 A/mm 5600-6400 A Dec/78 CTIO 4m 9 20 90 A/mm 5000-7200 A Jan/80 CTIO 4m 10 20 90 A/mm 5000-7200 A O c t / 8 0 CTIO 4m 9 19 90 A/mm 5000-7200 A Of t h i s t o t a l sample o f program s t a r s , 46 o f t h e s p e c t r a a r e WORC s t a r s , 52 s p e c t r a a r e BMB s t a r s and 26 s p e c t r a a r e SP s t a r s . The a c t u a l number o f c a r b o n s t a r s f o r w h i c h r a d i a l v e l o c i t i e s were f i n a l l y d e r i v e d was s l i g h t l y s m a l l e r t h a n t h e above amount as some s p e c t r a d u p l i c a t e d t h e same s t a r s and a few s p e c t r a p r o v e d i r r e d u c i b l e f o r r a d i a l v e l o c i t y p u r p o s e s . To o b t a i n t h e r a d i a l v e l o c i t i e s o f t h e pr o g r a m s t a r s , t h e 6 p l a t e s were r e d u c e d i n t h e f o l l o w i n g manner. E a c h s e r i e s of p l a t e s was t a k e n on a s e p a r a t e o b s e r v i n g run and t h e r e f o r e was r e d u c e d i n d e p e n d a n t l y o f t h e o t h e r s , a s were s p e c t r a i n t h e same s e r i e s t a k e n a t d i f f e r e n t d i s p e r s i o n s . F i r s t l y , t h e s t a n d a r d c a r b o n s t a r s were measured on an Abbe C o m p a r a t o r . E m p l o y i n g t h e C o m p a r a t o r , t h e p o s i t i o n s o f s e l e c t e d HeNeAr a r c c o m p a r i s o n l i n e s on b o t h s i d e s o f t h e s p e c t r u m and a l l d i s c e r n i b l e l i n e s i n t h e s t e l l a r s p e c t r u m were measured. F i g u r e I shows t h e s p e c t r u m o f one p r o g r a m s t a r and on t h i s d e n s i t o m e t e r t r a c i n g some of t h e s t e l l a r l i n e s t h a t were i d e n t i f i e d a r e i n d i c a t e d . A computer p r o g r a m e n t i t l e d RADVEL was t h e n employed t o p e r f o r m t h e f o l l o w i n g t a s k s . A mean p o s i t i o n f o r t h e a r c l i n e s on e i t h e r s i d e o f t h e s t e l l a r s p e c t r u m was d e t e r m i n e d and a t h i r d o r d e r p o l y n o m i a l f i t t e d t o t h e s e p o s i t i o n s . T h i s y i e l d e d a r e l a t i o n s h i p between p o s i t i o n and w a v e l e n g t h . From t h i s r e l a t i o n s h i p , t h e w a v e l e n g t h s o f t h e s t e l l a r l i n e s were t h e n d e t e r m i n e d . As t h e r a d i a l v e l o c i t i e s were known f o r t h e s t a n d a r d s t a r s , t h e c a l c u l a t e d w a v e l e n g t h s o f t h e s t e l l a r l i n e s were t h e n c o r r e c t e d f o r o b s e r v e r m o t i o n by d o p p l e r s h i f t i n g t h e w a v e l e n g t h s t o h e l i o c e n t r i c r e s t w a v e l e n g t h s . The s t a n d a r d w a v e l e n g t h s from a l l t h e s t a n d a r d s t a r s were t h e n combined i n o r d e r t o d e t e r m i n e a mean r e s t w a v e l e n g t h f o r t h e s t e l l a r l i n e s . I f a l i n e was i d e n t i f i e d i n l e s s t h a n 50% o f t h e s t a n d a r d s t a r s o r i f t h e s t a n d a r d d e v i a t i o n o f t h e l i n e s i d e n t i f i e d r e s u l t e d i n a s t a n d a r d d e v i a t i o n i n v e l o c i t y o f g r e a t e r t h a n 30 km/sec, t h e l i n e was c o n s i d e r e d i n a c c u r a t e and was n o t u s e d . T h i s p r o c e s s r e s u l t e d i n a s e t o f a p p r o x i m a t e l y F i g . I. Spectrum of BMB Bar West 99 taVpn at an A A ™ TU-spectrum exhibits some of the features t$5L of carbo7ste S that were used for radial velocity determination! 8 t h i r t y s t a n d a r d w a v e l e n g t h s f o r t h e l o w e r d i s p e r s i o n (90 and 126 A/mm) s p e c t r a and i n e x c e s s o f s i x t y s t a n d a r d w a v e l e n g t h s f o r t h e h i g h e r d i s p e r s i o n (50 and 25 A/mm) s p e c t r a . T h e s e s p e c t r a l l i n e s were c o n s i d e r e d t o be good v e l o c i t y i n d i c a t o r s and were employed t o d e t e r m i n e t h e v e l o c i t i e s o f t h e p r o g r a m s t a r s . The v e l o c i t i e s of t h e p r o g r a m s t a r s were d e t e r m i n e d by a g a i n e m p l o y i n g t h e p r o g r a m RADVEL. As was d e s c r i b e d p r e v i o u s l y , t h i s p r o g r a m f i t t e d a t h i r d o r d e r p o l y n o m i a l t o t h e a r c l i n e s and t h e n d e t e r m i n e d t h e w a v e l e n g t h s o f t h e l i n e s o b s e r v e d i n t h e s p e c t r u m o f t h e program s t a r . T h e s e w a v e l e n g t h s were t h e n compared w i t h t h e p r e v i o u s l y d e t e r m i n e d r e s t w a v e l e n g t h s o f l i n e s f o u n d i n t h e s t a n d a r d s t a r s . I f t h e l i n e i n t h e p r o g r a m s t a r was no f u r t h e r t h a n a p r e d e t e r m i n e d l i m i t from t h e p o s i t i o n of a s t a n d a r d l i n e ( t y p i c a l l y between 0 and +12 A n g s t r o m s ) , RADVEL c o n s i d e r e d t h e l i n e t o be due t o t h e same s p e c t r a l f e a t u r e . The d o p p l e r s h i f t between t h e two l i n e s was t h e n computed and t h e c o m p a r i s o n p r o c e s s c o n t i n u e d u n t i l a l l d o p p l e r s h i f t s f o r a l l p o s s i b l e l i n e p a i r s were d e t e r m i n e d . RADVEL t h e n c o n s i d e r e d a l l t h e r a d i a l v e l o c i t i e s and computed t h e mean v e l o c i t y and t h e s t a n d a r d d e v i a t i o n o f t h e mean. I n d i v i d u a l v e l o c i t i e s f u r t h e s t from t h e mean were t h e n d i s c a r d e d one a t a t i m e u n t i l a l l t h e r e m a i n i n g l i n e s were w i t h i n 1.5 s t a n d a r d d e v i a t i o n s f r o m t h e mean. T h i s v e l o c i t y was t h e n c o r r e c t e d t o a h e l i o c e n t r i c mean v e l o c i t y f o r t h a t s t a r . T h e s e measured v e l o c i t i e s , a l o n g w i t h t h e s t a n d a r d d e v i a t i o n i n v e l o c i t y and t h e number o f l i n e s u s e d i n t h e v e l o c i t y d e t e r m i n a t i o n a r e l i s t e d i n T a b l e I I . I t was n o t i c e d t h a t when t h e i n i t i a l r e d u c t i o n p r o c e s s was 9 a p p l i e d t o t h e s t a n d a r d s t a r s t h e r e were s y s t e m a t i c s h i f t s i n w a v e l e n g t h f o r t h e same s t e l l a r l i n e when t h e s t a n d a r d s t a r ' s s p e c t r u m was t a k e n t h r o u g h d i f f e r e n t n e u t r a l d e n s i t y f i l t e r s . T h e s e n e u t r a l d e n s i t y f i l t e r s were n e c e s s a r y t o r e d u c e t h e i n t e n s i t y o f t h e s t a n d a r d s t a r ' s s p e c t r u m t o a l e v e l a c c e p t a b l e f o r t h e i m a g e - i n t e n s i f i e r a t t a c h e d t o t h e s p e c t r o g r a p h . T h e s e f i l t e r s had t h e u n f o r t u n a t e e f f e c t of s h i f t i n g t h e s p e c t r u m o f t h e s t a n d a r d s t a r s r e l a t i v e t o t h e a r c c o m p a r i s o n l i n e s w h i c h had n o t p a s s e d t h r o u g h t h e n e u t r a l d e n s i t y f i l t e r . T h i s s h i f t i n t h e p o s i t i o n o f t h e s p e c t r u m , i f not c o r r e c t e d f o r , would r e s u l t i n s y s t e m a t i c v e l o c i t y s h i f t s from s e r i e s t o s e r i e s . To c o r r e c t f o r t h e v e l o c i t y s h i f t p r o d u c e d by t h e n e u t r a l d e n s i t y f i l t e r t h e f o l l o w i n g p r o c e d u r e was c a r r i e d o u t . The s e t of s t a n d a r d i d e n t i f i c a t i o n w a v e l e n g t h s d e t e r m i n e d f o r e a c h s e r i e s was compared w i t h t h e i d e n t i f i c a t i o n w a v e l e n g t h s from e v e r y o t h e r s e r i e s of p l a t e s . F o r t h o s e l i n e s p r e s e n t i n b o t h s e r i e s o f p l a t e s b e i n g compared, a r e l a t i v e f r e q u e n c y s h i f t and t h e r e f o r e a r e l a t i v e v e l o c i t y s h i f t f o r t h a t p a r t i c u l a r l i n e was d e t e r m i n e d . A l l t h e v e l o c i t y s h i f t s were t h e n summed t o y i e l d a mean v e l o c i t y s h i f t dV(m,n) between any two s e r i e s o f p l a t e s . The mean v e l o c i t y s h i f t between any two s e r i e s o f p l a t e s m k k and m' was g i v e n by (( I d v ( n , m ) ) / k - ( I d V ( n , m ' ) ) / k ) , where k was t h e t o t a l number of s e r i e s of p l a t e s i n v o l v e d . T h i s r e s u l t e d i n v a l u e s f o r t h e f i n a l r e l a t i v e v e l o c i t y s h i f t s o f e a c h s e r i e s of s t a r s . T h i s c o r r e c t i o n was t h e n a p p l i e d t o e a c h s e r i e s t o c o r r e c t f o r t h e e f f e c t of t h e n e u t r a l d e n s i t y f i l t e r s making t h e v e l o c i t i e s o f a l l t h e p r o g r a m s t a r s c o n s i s t e n t r e l a t i v e t o one 10 a n o t h e r . However, t h i s d i d not c o r r e c t f o r a z e r o - p o i n t e r r o r i n v e l o c i t i e s . T h i s f i n a l c o r r e c t i o n was a c c o m p l i s h e d by c o m p a r i n g t h e v e l o c i t y o f t h e s o l i d body r o t a t i o n c u r v e of t h e P o p u l a t i o n I c a r b o n s t a r s a t t h e R a d i o C e n t r e w i t h t h e w e i g h t e d mean v e l o c i t y a t t h e same p o i n t o f t h e r o t a t i o n c u r v e s o f o t h e r known r a d i a l v e l o c i t y i n d i c a t o r s i n t h e LMC ( F i e t z i n g e r and W e i s s 1979). The r a d i a l v e l o c i t i e s of t h e c a r b o n s t a r s were a d j u s t e d t o a g r e e w i t h t h e w e i g h t e d mean v e l o c i t y o f 274.5 km/sec a t t h e R a d i o C e n t r e . A l t h o u g h t h e s i z e o f t h e c o r r e c t i o n v a r i e d from s e r i e s t o s e r i e s , t h e a v e r a g e v e l o c i t y c o r r e c t i o n was o f t h e o r d e r o f 10 km/sec. T h i s f i n a l c o r r e c t i o n t o t h e v e l o c i t i e s l e d t o t h e f i n a l , a d o p t e d v e l o c i t i e s l i s t e d i n T a b l e I I . I t was t h e s e f i n a l c o r r e c t e d r a d i a l v e l o c i t i e s t h a t were employed i n t h e a n a l y s i s of t h e k i n e m a t i c s o f t h e L a r g e and S m a l l M a g e l l a n i c C l o u d s . 11 TABLE I_I RADIAL VELOCITIES OF LARGE MAGELLANIC CLOUD CARBON STARS WORC RA Dec. M e a s u r e d S t a n d a r d No. o f A d o p t e d No. (1975) (1975) V e l o c i t y D e v i a t i o n L i n e s V e l o c i t y 2 04 39.1 -70 10.0 275. 1 20.4 1 7 297.2 13 04 50.6 -70 25.6 254.3 22.6 8 240.8 40 04 59.0 -65 57.7 214.4 16.7 15 200.9 46 05 00.4 -70 15.8 256.5 28.5 7 243.0 51 05 01 .3 -66 52.9 315.3 27. 1 10 301 .8 54 05 01 .9 -70 33.8 210.1 29.7 13 232.3 56 05 02. 1 -70 53.3 207.0 20.2 18 229.2 64 05 02.7 -65 11.1 292.7 24.9 23 314.8 65 05 02.8 -66 26.5 279.9 24.6 1 5 302. 1 84 05 07. 1 -69 58.2 236.9 15.5 25 239.6 105 05 11.7 -65 16.1 322.7 19.7 1 7 309. 1 1 06 05 1 1 .7 -66 52.7 272.3 18.7 37 274.9 106 05 1 1 .7 -66 52.7 286.8 21.5 20 30'9.0 106 05 11.7 -66 52.7 268.3 13.7 54 276.7 107 05 11.8 -65 09.8 350.4 21.4 7 336.9 108 05 12.1 -65 47.6 287.5 29.6 1 2 273.9 1 1 2 05 12.9 -65 43.3 298.6 26.2 1 1 285.0 1 12 05 12.9 -65 43.3 313.5 19.6 15 300.0 115 05 13.8 -72 55. 1 261 .6 22.9 1 2 248.0 131 05 17.2 -64 48.5 314.6 11.2 13 301 .0 134 05 18.5 -70 57.7 264.5 25.9 1 1 251 .0 137 05 19.8 -65 04.2 353.7 30.4 1 4 340. 1 12 157 05 26. 1 -73 19.8 197.6 22.7 17 219.8 186 05 35.2 -71 01 .4 298.3 21.7 15 284.8 186 05 35.2 -71 01 .4 253.6 20. 1 20 256.2 190 05 38.0 -70 49. 1 187.5 30.5 22 209.7 192 05 38.2 -72 09.3 232. 1 34.9 10 218.6 210 05 49.6 -71 39.2 194.3 31.8 1 7 216.5 214 05 51 .8 -72 31.7 279.8 18.8 21 266.2 229 05 05.9 -69 27.2 238.3 23.8 21 260.5 237 06 08.8 -68 15.9 256. 1 38.8 14 242.5 239 06 09.2 -73 50. 1 272.0 27.0 1 2 258.6 239 06 09.2 -73 50. 1 278.3 20.5 1 2 264.7 248 06 12.0 -68 58.7 276.5 19.7 20 298.7 256 06 13.7 -67 28.4 323.3 30.8 1 4 309.8 271 06 16.9 -67 29.7 389.9 38.6 8 376.3 272 06 17.1 -68 13.2 376.4 37. 1 9 362.9 287 06 23.5 -68 35.9 296.7 29.5 1 3 283.2 294 06 24.6 -68 46.8 260.7 23.7 16 282.9 296 06 26.8 -71 41.8 311.6 19.7 17 298.0 300 06 29.6 -70 55.5 338.3 23.2 8 324.7 W 50 06 09.2 -73 50 234.8 32.8 10 221 .3 13 BMB Bar RA Dec. M e a s u r e d S t a n d a r d No. o f A d o p t e d St NO. (1950) (1950) V e l o c i t y D e v i a t i o n L i n e s V e l o c i ' 1 05 06.8 -69 10.4 288.1 44.1 6 254.7 9 05 07.5 -69 07.2 250.3 31 .9 9 216.9 1 1 05 07.5 -69 15.6 249.8 26.6 8 216.4 25 05 08. 1; -69 13.7 228.9 24.0 9 195.5 29 05 08.2 -69 03.7 209.5 46.9 7 206.3 31 05 08.3 -69 18.5 298. 1 26.8 7 264.7 32 05 08.3 -69 09.4 203.7 38.7 7 200.5 33 05 08.3 -69 19.2 270.2 39.4 6 236.8 35 05 08.4 -69 17.0 204.5 32.2 5 171.1 37 05 08.5 -69 02.0 210.1 19.5 7 1 76.7 41 05 08.6 -69 09.5 250.3 33. 1 9 216.9 47 05 08.8 -69 12.3 305.5 24.0 8 272. 1 54 05 09.0 -69 01 .3 264.8 19.4 6 261 .6 58 05 09. 1 -69 17.4 229.7 27.8 5 226.5 67 05 09.3 -69 18.6 234.2 35.5 5 230.9 70 05 09.5 -69 11.8 251.1 19.1 7 217.7 76 05 09.6 -69 15.2 244.5 23.9 6 241 .3 87 05 09.9 -69 11.8 227.7 27.6 6 194.4 89 05 09.9 -69 08.2 292.0 8.2 5 258.6 92 05 10.0 -69 07.7 290.7 36.2 7 257.3 99 05 10.6 -69 09.0 247.2 27.8 5 244.0 100 05 10.6 -69 03. 1 278. 1 47.7 8 244.7 105 05 10.8 -69 08.8 218.2 22.9 6 214.9 111 05 11.1 -69 05.7 213.0 15.0 6 209.8 14 SP No. RA Dec. M e a s u r e d S t a n d a r d No. o f A d o p t e d (1975) (1975) V e l o c i t y D e v i a t i o n L i n e s V e l o c i -12-1 04 32.3 -66 08 218.5 26.2 13 240.7 20-5 04 49.2 -65 48 202.9 23.8 1 5 225. 1 29-1 5 04 56.6 -67 26 223.6 26. 1 16 245.8 30-16 04 59.7 -68 27 288.0 19.6 27 296.3 30-1 6 04 59.7 -68 27 310.3 30.8 7 296.7 30-1 6 04 59.7 -68 27 343.8 21.2 4 330.3 33-7 05 03.0 -72 21 142.8 41 .9 7 129.2 39-18 05 13.9 -69 53 227.7 20.8 1 6 249.8 47-1 1a 05 26.2 -70 1 2 235.0 29.7 1 7 257. 1 51-21 05 36.7 -65 35 150.2 27.2 7 172.3 51-21 05 36.7 -65 35 1 52.5 14.8 17 160.8 53-31 05 41 .5 -68 47 248.7 22.3 23 270.9 54-51 05 42.8 -70 02 276.0 30.0 1 6 298. 1 54-59 05 44.9 -69 04 256.0 43.2 4 278.1 60-19 05 57.6 -67 35 353.0 13.4 8 319.6 61-9 05 49. 1 -68 42 222.6 24.5 15 244.8 64-5 05 57. 1 -71 55 335.9 40.6 4 302.5 64-5 05 57. 1 -71 55 358.6 33.2 6 325.2 69-1 7 06 08.9 -67 58 330.8 25.0 8 297.4 15 RADIAL VELOCITIES OF SMALL MAGELLANIC CLOUD CARBON STARS BMB Bar RA Dec. M e a s u r e d S t a n d a r d No. o f A d o p t e d No. (1950) (1950) V e l o c i t y D e v i a t i o n L i n e s V e l o c i 8 0 46.6 -73 29.5 158.8 29.1 2 1 55.6 8 0 46.6 -73 29.5 144.6 19.2 5 141 .4 8 0 46.6 -73 29.5 138.2 42.5 5 133.8 9 0 46.8 -73 36.0 53.3 35.1 5 48.9 1 1 0 47.0 -73 30.9 1 56. 1 21.4 6 152.9 1 1 0 47.0 -73 30.9 133.5 20.4 5 129. 1 20 0 47.4 -73 25. 1 233.5 50.2 4 229. 1 23 0 47.6 -73 24.7 164. 1 46.4 4 160.9 23 0 47.6 -73 24.7 181.0 28.6 8 176.6 30 0 47.8 -73 34.2 126.5 39.2 5 123.3 30 0 47.8 -73 34.2 116.0 28.8 13 M 1 .6 31 0 47.8 -73 25.0 86.5 24.7 6 83.3 31 0 47.8 -73 25.0 116.0 22.7 8 ' 111.6 31 0 47.8 -73 25.0 105.6 24.5 10 101.2 34 0 47.8 -73 33.2 168.6 38.3 10 164.2 35 0 47.9 -73 39. 1 90.4 36. 1 4 86.0 35 0 47.9 -73 39. 1 1 13.9 29.9 3 109.5 39 0 48. 1 -73 29. 1 1 43.9 35.1 6 139.5 40 0 48.3 -73 29.8 93. 1 30.6 5 89.9 40 0 48.3 -73 29.8 112.0 23.2 4 107.6 41 0 48.4 -73 22.9 133.5 10.5 5 1 29. 1 47 0 48.6 -73 39. 1 178.9 33.2 5 175.7 62 0 49.5 -73 28.5 118.2 19.5 6 113.8 65 0 49.7 -73 39.4 163.4 20.0 5 160 80 0 50.7 -73 38.2 177.0 28.5 5 172 81 0 50.7 -73 30.6 79.3 39.3 4 76 17 CHAPTER I I I ROTATION OF THE LMC AS DESCRIBED BY CARBON STARS The f i r s t a t t e m p t a t g e n e r a t i n g a r o t a t i o n c u r v e f o r t h e LMC, b a s e d upon t h e t o t a l sample o f c a r b o n s t a r s a v a i l a b l e , r e s u l t e d i n r o t a t i o n c u r v e s t h a t e x h i b i t e d r o t a t i o n a l e f f e c t s a t no s t a t i s t i c a l l y s i g n i f i c a n t l e v e l . I t was t h e n f o u n d t h a t by s i m p l y s e p a r a t i n g t h e SP c a r b o n s t a r s f r o m t h e r e s t of t h e sample o f c a r b o n s t a r s , r e s p e c t a b l e r o t a t i o n c u r v e s c o u l d be g e n e r a t e d f o r t h e r e m a i n i n g sample of c a r b o n s t a r s . The a c t u a l d e t a i l s of t h e s e r o t a t i o n c u r v e s w i l l be e x p l a i n e d l a t e r i n t h e t e x t . I t was t h e r e f o r e f o u n d t h a t t o i n v e s t i g a t e t h e s y s t e m a t i c r o t a t i o n o f c a r b o n s t a r s , t h e c a r b o n s t a r s as d e s c r i b e d i n T a b l e I I must be s e p a r a t e d i n t o two g r o u p s : Those s t a r s f o u n d t o p a r t i c i p a t e i n t h e r o t a t i o n o f t h e d i s k o f t h e g a l a x y and t h o s e s t a r s t h a t were f o u n d n o t t o show s y s t e m a t i c r o t a t i o n a l e f f e c t s . In o u r own g a l a x y , t h a t g r o u p o f s t a r s w h i c h p a r t i c i p a t e s i n t h e r o t a t i o n o f t h e d i s k o f t h e g a l a x y i s c o m p r i s e d a l m o s t t o t a l l y of P o p u l a t i o n I o b j e c t s . The g r o u p o f o b j e c t s i n our g a l a x y w h i c h shows no s y s t e m a t i c r o t a t i o n i n any p r e f e r r e d p l a n e i s c o m p r i s e d p r e d o m i n a n t l y of P o p u l a t i o n I I o b j e c t s . In t h e LMC, a s i n our own g a l a x y , t h e above d i v i s i o n was a l s o f o u n d t o e x i s t . A f t e r d i s c o v e r i n g t h a t t h e SP s t a r s a c t e d i n a d i f f e r e n t k i n e m a t i c a l manner t h a n t h e r e m a i n i n g s t a r s t a k e n from t h e WORC and BMB c a t a l o g u e s , and r e a l i z i n g t h a t t h e SP c a r b o n s t a r s were a n a l o g o u s t o P o p u l a t i o n I I o b j e c t s i n our own g a l a x y ( R i c h e r e t a l 1979), i t was d e c i d e d t o t r e a t t h e SP c a r b o n s t a r s a s a s e p a r a t e k i n e m a t i c a l g r o u p i n a l l of t h e f o l l o w i n g a n a l y s e s . T h e s e two g r o u p s , a l o n g w i t h o t h e r g r o u p s of r a d i a l 18 v e l o c i t y i n d i c a t o r s t a k e n from t h e c a t a l o g u e o f F e i t z i n g e r and W e i s s ( 1 9 7 7 ) , were t h e n a n a l y z e d f o r g a l a c t i c r o t a t i o n . The f i r s t s t e p i n t h i s a n a l y s i s was t o c o n v e r t t h e r i g h t a s c e n s i o n and d e c l i n a t i o n c o o r d i n a t e s of e a c h o b j e c t t o c o o r d i n a t e s i n an X-Y s y s t e m . In t h i s X-Y c o o r d i n a t e s y s t e m t h e a b s c i s s a was i n c r e m e n t e d i n d e g r e e s o f d e c l i n a t i o n and t h e o r d i n a t e i n c r e m e n t s were f o u n d by m u l t i p l y i n g t h e r i g h t a s c e n s i o n by t h e c o s i n e o f t h e d e c l i n a t i o n of t h e p o s i t i o n . T h i s r e s u l t e d i n X-Y c o o r d i n a t e s , measured i n i n c r e m e n t s o f d e g r e e s from some c h o s e n o r i g i n , f o r a l l o f t h e o b j e c t s s t u d i e d . The r e s u l t i n g maps, shown i n F i g u r e s I I t h r o u g h V I I , had two i m p o r t a n t f u n c t i o n s . P r i m a r i l y t h e X-Y c o o r d i n a t e s p o s i t i o n e d t h e o b j e c t s r e l a t i v e t o one a n o t h e r on t h e map i n t h e same manner t h a t t h e s e o b j e c t s a p p e a r e d r e l a t i v e t o one a n o t h e r i n t h e s k y . T h i s r e s u l t e d i n an a c c u r a t e p o s i t i o n f o r t h e o b j e c t s r e l a t i v e t o one a n o t h e r w h i c h i n t u r n a l l o w e d a c c u r a t e r o t a t i o n c u r v e s t o be c a l c u l a t e d . S e c o n d l y , t h e s e maps were employed t o i n d i c a t e t h e a c t u a l c o v e r a g e of t h e LMC a c h i e v e d by e a c h d a t a s e t . T h i s a l l o w e d c o m p a r i s o n between t h e r e s u l t s o f t h e r o t a t i o n c u r v e s and t h e v e l o c i t i e s g i v e n by i n d i v i d u a l d a t a p o i n t s i n s u c h d y n a m i c a l l y i n t e r e s t i n g a r e a s as t h e b a r of t h e LMC. The X-Y maps were c e n t r e d a t RA=5h30m, D e c = - 6 9 ° 0 0 ' , as t h i s was t h e a p p r o x i m a t e c e n t r e o f c o v e r a g e o f t h e s u r v e y . The r o t a t i o n c u r v e s however, were c e n t r e d a b o u t t h e R a d i o C e n t r e w h i c h was l o c a t e d a t RA=5h20m , D e c = - 6 8 ° 4 8 ' w h i c h i s c o n s i d e r e d t o be t h e d y n a m i c a l c e n t r e of t h e LMC ( F e a s t , 1 9 6 4 ) . The RC l a b e l on t h e X-Y maps i n d i c a t e s t h e p o s t i o n o f t h e R a d i o C e n t r e o f t h e F i g . II. x-Y map showing the s u r f a c e coverage o f WORC ' and BMB carbon s t a r s i n the LMC. THe approximate l o c a t i o n o f the Bar i s o u t l i n e d . e J f i S * 1 1 1 * X " Y m a p 6 h o w i n S the s u r f a c e coverage of S a n d u l e a k - P h i l i p carbon s t a r s i n the LMC. The approximate l o c a t i o n of the Bar i s o u t l i n e d . T.e„*-«FlgI X " Y m a p s h o w i n S the s u r f a c e coverage o f HII o u ? l i n e d ^ a ^ r o x i m a t e l o c a t i o n o f the Bar i s 5-i 4 -3-2 H d H Stars in the LMC -i 1-5 4 N -r 1 1 -r 2 1 0 -1 -2 "^3 RA*cos(dec) in Degrees ~ i 1 - 4 -5 -6 6 t a r s F i n ' t h e mc s h o w i n S t h ? s u r f a c e coverage o f luminous o u t l i n e d . approximate l o c a t i o n o f the Bar i s c l u s t e r s ' i n 1 rh^Lr^?°WlnS t h e s u r f a c e coverage o f s t e l l a r o u t l i n e d . approximate l o c a t i o n of the Bar i s F i g . V I I . X-Y map showing the s u r f a c e coverage o f o f a ? h ^ S L n f Ula!-, ln J h e L M C ' T h e a P P ^ x i m a t e l o c a t i o n 01 the Bar i s o u t l i n e d . 25 LMC. In o r d e r t o i n v e s t i g a t e t h e r o t a t i o n o f t h e LMC, two d i f f e r e n t models of g a l a c t i c r o t a t i o n were h y p o t h e s i s e d . A l l a v a i l a b l e v e l o c i t y d a t a was t h e n r e d u c e d and compared w i t h t h e d i f f e r e n t p r e d i c t i o n s e a c h model made ab o u t t h e manner i n w h i c h t h e LMC r o t a t e d . The f i r s t h y p o t h e s i s made was t h a t t h e o r b i t a l v e l o c i t y o f o b j e c t s i n t h e LMC i n c r e a s e d l i n e a r l y w i t h d i s t a n c e from t h e d y n a m i c a l c e n t r e o f t h e LMC. A model b a s e d upon t h i s s i m p l e i n i t i a l a s s u m p t i o n l a t e r p r o v e d t o be v e r y a c c u r a t e i n i t s c o r r e s p o n d a n c e t o t h e d a t a s e t s t h a t were e x a m i n e d . N e c e s s a r y t o t h i s model were v a l u e s f o r t h e i n c l i n a t i o n t o t h e l i n e o f s i g h t o f t h e d i s k o f t h e LMC, and t h e l o c a t i o n and o r i e n t a t i o n o f t h e major a x i s o f t h i s g a l a x y . B o t h o f t h e s e q u a n t i t i e s were d e r i v e d p r e v i o u s l y by F e a s t ( l 9 6 4 ) from HII o b s e r v a t i o n s . The o b s e r v e d a x i a l f l a t t e n i n g o f a r i n g of HII s u r r o u n d i n g t h e LMC was measured t o be 0.89 w i t h t h e major a x i s o f t h i s s t r u c t u r e o r i e n t e d a t 171° e a s t o f n o r t h . I f t h i s r i n g of HII i s assumed t o be c i r c u l a r i n t h e p l a n e o f t h e g a l a x y t h e o b s e r v e d a x i a l f l a t t e n i n g i m p l i e s t h e p l a n e of t h e g a l a x y i s i n c l i n e d a t an a n g l e o f 27° t o o u r l i n e o f s i g h t . T h e s e were t h e v a l u e s a d o p t e d i n t h i s t h e s i s . O t h e r a u t h o r s have d e r i v e d d i f f e r e n t v a l u e s f o r t h e s e two c o n s t a n t s and t h e s e v a l u e s a r e p r e s e n t e d by Freeman and de V a u c o l e u r s ( 1 9 7 2 ) . A l l of t h e s e v a l u e s however a r e n o t s u b s t a n t i a l l y d i f f e r e n t f r o m t h o s e v a l u e s f o r t h e i n c l i n a t i o n o f t h e d i s k and o r i e n t a t i o n o f t h e major a x i s o f t h e LMC as d e t e r m i n e d by F e a s t . A s s u m i n g t h a t o r b i t a l v e l o c i t y does i n c r e a s e l i n e a r l y w i t h d i s t a n c e f r o m t h e R a d i o C e n t r e , w h i c h was f o u n d t o be t h e 26 d y n a m i c a l c e n t r e o f t h e LMC by F e a s t ( 1 9 6 4 ) , t h e o b s e r v e d r a d i a l v e l o c i t y w i l l be t h e same f o r a l l o b j e c t s whose p e r p e n d i c u l a r t o t h e m a j o r a x i s o c c u r s a t t h e same p o i n t a l o n g t h e major a x i s . A l s o , t h e o b s e r v e d r a d i a l v e l o c i t y w i l l i n c r e a s e o r d e c r e a s e l i n e a r l y a l o n g t h e l e n g t h o f t h e major a x i s . The g e o m e t r i c a l p r o o f o f t h i s i s p r e s e n t e d i n A p p e n d i x I . To see i f t h i s was t r u e i n t h e LMC, t h e s i x d a t a s e t s of WORC c a r b o n s t a r s , SP c a r b o n s t a r s , p l a n e t a r y n e b u l a e , s t e l l a r c l u s t e r s , s t a r s and HII r e g i o n s were r e d u c e d i n t h e f o l l o w i n g manner. W i t h i n e a c h d a t a s e t , t h e d i s t a n c e i n d e g r e e s a l o n g t h e major a x i s f r o m t h e R a d i o C e n t r e t o t h e i n d i v i d u a l d a t a p o i n t s was c a l c u l a t e d . The major a x i s i n e a c h c a s e was c h o s e n t o be t h e e a r l i e r d e t e r m i n e d v a l u e o f 171° e a s t o f n o r t h . T h i s r e s u l t e d i n v e l o c i t y v s p o s i t i o n g r a p h s f o r t h e s e s i x d i f f e r e n t p o p u l a t i o n s of o b j e c t s i n t h e LMC. A t t h i s p o i n t a s p e c i a l n o t e s h o u l d be made a b o u t t h e WORC sample o f c a r b o n s t a r s . As was s t a t e d e a r l i e r , t h e BMB Bar West s t a r s were i n c l u d e d i n t h i s s a m p l e . However, s i n c e a l l t h e BMB s t a r s came from a v e r y s m a l l f i e l d , t o employ e a c h s t a r i n d i v i d u a l l y i n t h e c a l c u l a t i o n o f a r o t a t i o n c u r v e would g i v e undue w e i g h t t o t h i s one s m a l l r e g i o n i n t h e LMC. To c i r c u m v e n t t h i s p r o b l e m a mean v e l o c i t y and p o s i t i o n f o r t h i s g r o u p of s t a r s was c a l c u l a t e d and employed f o r a l l t h e c a r b o n s t a r r o t a t i o n c u r v e s g e n e r a t e d . However, a s t h i s one p o i n t was a s t a t i s t i c a l mean, i t s h o u l d be n o t e d t h a t i t s v e l o c i t y was much b e t t e r d e t e r m i n e d t h a n t h e o t h e r v e l o c i t i e s e mployed t o c a l c u l a t e t h e r o t a t i o n c u r v e . I f t h e o r b i t a l v e l o c i t y o f o b j e c t s i n t h e LMC does i n c r e a s e 27 l i n e a r l y w i t h o r b i t a l r a d i u s , t h e d a t a p o i n t s i n e a c h s e t s h o u l d show a l i n e a r c o r r e l a t i o n between o b s e r v e d r a d i a l v e l o c i t y and d i s t a n c e a l o n g t h e major a x i s . To t e s t f o r t h i s , a l i n e a r l e a s t s q u a r e s f i t was a p p l i e d t o t h e v e l o c i t y v s p o s i t i o n g r a p h of e a c h g r o u p o f o b j e c t s . In some d a t a s e t s t h e r e were d a t a p o i n t s l o c a t e d more t h a n t h r e e s t a n d a r d d e v i a t i o n s from t h e l i n e a r l e a s t s q u a r e s l i n e f i t t e d t o t h e d a t a s e t . I f t h i s o c c u r r e d t h e s e p o i n t s were c o n s i d e r e d t o be h i g h l y d i s c r e p a n t v e l o c i t y p o i n t s and were removed t o r e n d e r a b e t t e r f i t t o t h e r e m a i n i n g d a t a p o i n t s . In a l l c a s e s however t h e number o f d a t a p o i n t s removed was l e s s t h a n f i v e and t h e change made i n v e l o c i t y g r a d i e n t and R a d i o C e n t r e v e l o c i t y was l e s s t h a n t h e e r r o r g i v e n f o r t h e s e q u a n t i t i e s . The f i n a l r e s u l t o f t h i s a n a l y s i s i s g i v e n i n T a b l e I I I . The R a d i o C e n t r e v e l o c i t y c a n be i n t e r p r e t e d as t h e s y s t e m i c v e l o c i t y f o r t h e LMC. The c o r r e l a t i o n c o e f f i c i e n t g i v e n f o r e a c h d a t a s e t was t h e n o r m a l r, c o e f f i c i e n t f o r a l i n e a r l e a s t s q u a r e s f i t . A p e r f e c t c o r r e l a t i o n o f t h e d a t a p o i n t s i s a p p r o a c h e d as t h e a b s o l u t e v a l u e o f t h e c o r r e l a t i o n c o e f f i c i e n t a p p r o a c h e s 1.0. The t c o e f f i c i e n t g i v e n i n t h i s t a b l e was a n o t h e r measure o f how w e l l t h e d a t a s e t was c o r r e l a t e d . The t c o e f f i c i e n t i n d i c a t e s how many s t a n d a r d d e v i a t i o n s t h e d a t a s e t was removed from a t o t a l l y random, g a u s s i a n d i s t r i b u t i o n . An e x p l a n a t i o n o f how t h e s e c o e f f i c i e n t s were d e r i v e d c a n be f o u n d i n T o p p i n g ( 1 9 7 2 ) . 28 Table III Object S o l i d Body Rotation Curve Parameters Velocity Radio Centre Correlation SP Stars WORC Stars Planetary Nebulae S t e l l a r Clusters Stars in the LMC Gradient (km/sec/deg) 6.1±7.7 -10.4±2.2 - 1 1 - 6 ± 1 . 8 -12.7±3.5 -15.2±1.2 HII Regions -16.4±0.8 Veloc i t y (km/sec) 260.6±17.5 274.5±11.5 276.313.8 266.4±5.7 275.811.9 270.8+1.2 Coefficient Coefficient 0.21 -0.61 -0.80 -0.65 -0.50 -0.90 0.90 4.64 7.04 4.73 12.5 19.8 The parameters found in the table were derived from the s o l i d body rotation curves shown in Figures VIII through XIII i n c l u s i v e . Of t h i s sample, only the SP stars did not show rotational motion at a s t a t i s t i c a l l y s i g n i f i c a n t l e v e l . The v e l o c i t y gradients given are the v e l o c i t y gradients that were measured on the sky and not in the plane of the galaxy. As the remaining data sets did have a linear correlation between v e l o c i t y and position, the following test was devised to uncover any small systematic variations in each data set from a simple linear c o r r e l a t i o n . Using the calculated v e l o c i t y gradient and intercept, a mean lin e a r velocity for every data point along the major axis was calculated. This calculated ve l o c i t y was then subtracted from the observed v e l o c i t y . These ve l o c i t y differences were then placed in bins 2 degrees wide F i g . V I I I . R a d i a l v e l o c i t y v s . p e r p e n d i c u l a r d i s t a n c e a l o n g the major a x i s , a t I ? l e a s t of n o r t h , f o r WORC carbon s t a r s . The p l o t t e d l i n e a r l e a s t squares f i t to the d a t a i s d e s c r i b e d i n T a b l e I I I . The o r i g i n i s l o c a t e d a t the Radio Centre of the LMC. The d a t a p o i n t f o r the mean p o s i t i o n o f the BMB Bar West f i e l d i s i n d i c a t e d by an i n s c r i b e d x. malor a x i s l t ! 91° vflo"ty vs . p e r p e n d i c u l a r d i s t a n c e a l o n g the ? h P i H 5 1 7 1 f a s t 0 f n o r t h » f o r S a n d u l e a k - P h i l i p carbon s t a r s T a b l e I I I d T i ' n e a r l 6 a S t f q U a r e s f i t t 0 t h e d ^ a i s d e s c r i b e d i n T a b l e i n . The o r i g i n l s l o c a t e d at the Radio C e n t r e of the LMC 4 2 0 4 0 0 3 8 0 3 6 0 3 4 0 •5 3 2 0 -2 3 0 0 > 2 8 0 "o 2 6 0 '~a o 2 4 0 2 2 0 •200 180 160 140 _HJI R e g i o n s r 1 1 , , , 1 • — cm -- P. I f V n n — — n 1 i i n 1 ' -5 -4 - 2 - 1 0 . 0 3 D e g r e e s A l o n g t h e M a j o r A x i s 4 at 1 7 1 ^ ! ; Radial velocity vs. p e r p e n d i c u l a r d i s t a n c e along the maior a x i s o CD > D 420 400 380 360 340 320 300 280 o 260 240 220 •200 180 160 140 - 5 - 4 ~3 -2 - 1 0.0 1 2 3 D e g r e e s A l o n g t h e M a j o r A x i s 4 5 at n i ^ i ! " Radial v e l o c i t y vs. perpendicular distance along the manor axis 420 400 380 360 >s 3 4 0 •5 3 2 0 -2 300 oj > 280 "5 260 '~o a 240 220 200 180 160 140 1 , _ S t e l l a r C l u s t e r s ! ! ~l 1 1 — • > 1 n a • -a n -nn % • n n -n n -D 1 i • a J !— 1 i , I 1 0 .0 D e g r e e s A l o n g t h e M a j o r A x i s 5 axis ^ t ' m ^ P ^ f S V e l ° , i i t l V S ' P e i P e n d i c ^ a r distance along the major £ £ a r e ? f i t to t h l Sft2°^ for s t e l l a r c l u s t e r s . The p l o t t e d l i n e a r l e a s t S ^ ^ r f X i Sc^ 5 ^^ i n T a b l e I H • T h e i s h e a t e d at CO 4 2 0 4 0 0 3 8 0 3 6 0 > N 3 4 0 -2 3 0 0 CD > 2 8 0 o 2 6 0 o 2 4 0 2 2 0 •200 180 160 140 "Planetary Nebulae - 5 - 4 - 3 - 2 - 1 0.0 1 2 3 4 5 D e g r e e s A l o n g i h e M a j o r A x i s F i g . XIII. Radial v e l o c i t y vs. perpendicular distance along the major axis, at 171 east of north, f o r planetary nebulae. The p l o t t e d l i n e a r l e a s t squares f i t to the data i s described i n Table I I I . The o r i g i n i s located at the Radio Centre of the LMC. 35 e v e r y 0.5 d e g r e e s a l o n g t h e major a x i s . The mean and s t a n d a r d d e v i a t i o n s o f t h e v e l o c i t y d i f f e r e n c e s i n e a c h b i n were t h e n c a l c u l a t e d . I n t h e c a s e o f t h e WORC c a r b o n s t a r s , p l a n e t a r y n e b u l a e , s t a r s and HII r e g i o n s , a s u b t l e S s h a p e d s t r u c t u r e was o b s e r v e d i n t h e mean d i f f e r e n c e p l o t s . The mean d i f f e r e n c e p l o t s of a l l t h e d a t a s e t s a r e shown i n F i g u r e s XIV t h r o u g h X V I I I i n c l u s i v e . T h i s S shape c o u l d be i n d i c a t i v e of some s y s t e m a t i c r o t a t i o n o f a n a t u r e o t h e r t h a n s o l i d body, s u c h as a f l a t r o t a t i o n c u r v e . In a l l c a s e s however, t h e mean d i f f e r e n c e between t h e o b s e r v e d v e l o c i t i e s and c a l c u l a t e d l i n e a r v e l o c i t i e s i n e a c h d a t a b i n was a l w a y s l e s s t h a n h a l f o f one s t a n d a r d d e v i a t i o n o f t h e o b s e r v e d v e l o c i t i e s o f t h e o b j e c t s i n t h a t b i n . T h i s i n d i c a t e d t h a t a l i n e a r f i t t o t h e o b s e r v e d v e l o c i t i e s , w h i c h i m p l i e s a l i n e a r r o t a t i o n c u r v e f o r t h e LMC was a v e r y good a p p r o x i m a t i o n f o r t h e r o t a t i o n of t h i s g a l a x y . The i n i t i a l s i m p l e a s s u m p t i o n o f a l i n e a r o r b i t a l v e l o c i t y v e r s u s r a d i a l d i s t a n c e r o t a t i o n c u r v e l e d t o t h e c a l c u l a t i o n o f t h e p a r a m e t e r s g i v e n i n T a b l e I I I . T h e s e p a r a m e t e r s l e a d t o t h e f o l l o w i n g c o n c l u s i o n s . As s t a t e d p r e v i o u s l y , t h e SP s t a r s show no s i g n o f s y s t e m a t i c r o t a t i o n a t t h i s major a x i s . F u r t h e r i n v e s t i g a t i o n a s s u m i n g a major a x i s a t an a n g l e o t h e r t h a n 171° e a s t of n o r t h a l s o f a i l e d t o p r o d u c e r o t a t i o n c u r v e s t h a t e x h i b i t e d r o t a t i o n a t a s t a t i s t i c a l l y s i g n i f i c a n t l e v e l . Two examples o f t h i s a r e F i g u r e s XIX and XX, w h i c h show r o t a t i o n c u r v e s f o r t h e SP s t a r s a t major a x e s o f 60° a n d 120° e a s t o f n o r t h . T h e s e r e s u l t s i m p l y t h a t t h e p l a n e o f t h e o r b i t s o f SP c a r b o n s t a r s a r e n o t i n t h e p l a n e o f t h e d i s k o f t h e LMC but r a t h e r a r e o r i e n t e d r a n d o m l y i n t h e h a l o o f t h e LMC. More 5 0 8 4 0 c CD 3 0 CD Q ±-! u CD > 15 TJ D rr c a CD 2 0 10 0.0 - 1 0 - 2 0 - 3 0 - 4 0 - 5 0 - 4 - 3 - 2 -1 Q.o 1 2 D e g r e e s A l o n g t h e M a j o r A x i s 3 s t a r s ^ ' ^ M n ? ! f ! ? r e n ? 6 ^ e t W e e n t h e m e a n r a d i a l v e l o c i t y of WORC carbon m L o ^ i n ^ g . m ^ T ^ i M ^ * * t h ? S e ^ e c t s \ y t h e R m e a a b r ° n v i i i as a function of distance along the major axis. cn rr c o OJ - 3 0 - 4 0 - 5 0 4 . i i - 4 - 3 - 2 - 1 0.0 1 2 3 D e g r e e s A l o n g t h e M a j o r A x i s F i g . XV. The d i f f e r e n c e between the mean r a d i a l v e l o c i t y o f HII regions i n 2 bins and the v e l o c i t y predicted f o r these objects by the l i n e a r l e a s t squares f i t shown i n F i g . X as a function o f distance along the major a x i s . - 4 - 3 - 2 - 1 0.0 1 2 3 4 D e g r e e s A l o n g t h e M a j o r A x i s F i g . XVI. The d i f f e r e n c e between the mean r a d i a l v e l o c i t y o f luminous stars i n 2 bins and the v e l o c i t y predicted f o r these objects by the l i n e a r l e a s t squares f i t shown i n F i g . XI as a function o f distance along the major axis. - 3 - 2 - 1 o . O 1 2 D e g r e e s A l o n g t h e M a j o r A x i s 3 cluster!'if ?:\?e d iJ f^ e n c e between the mean radial velocity of stellar Sear lit sho^ n^ S m ™ ! " * ? P " e d i C t | d £ ° r t h e s e objects by S shown m Fi g . XII as a function of distance along the major axis, to - 4 - 3 - 2 - 1 0.0 1 2 3 4 D e g r e e s A l o n g t h e M a j o r A x i s Fig. XVIII. The difference between the mean radial velocity of planetary nebulae in 2 bins and the velocity predicted for these objects by the linear f i t shown in Fig. XIII as a function of distance along the major axis. 420-400-380-360-340-| 320-'"2 300-o a^> 280 • 0 2 5 0 • 1 240-J 220-200-• , 180-160-140 120 4 Sanduleak-Phi l ip Carbon Stars A -6 -5 -4 " 3 - 2 - 1 0 1 2 3 Degrees Along the Major Axis m a 1 o / a X i s X I « ; fin°ia\Ve\OC±t* v s - P e r p e n d i c u l a r d i s t a n c e a l o n g the X ?Je Sc'. 6 0 e a 8 t ° f n ° r t h ' f ° r S a n d u l e a k - P h i ] i p S C a r b o n s t a r s 420-400-380-360-340-320->. 'o 300-o ty 280-> 260-~o o 240-or 220-200-180-160-140-120-— 6 Sanduleak-Philip Carbon Stars A A A A A A n 1 r--5 - 4 -3 - 2 - 1 0 1 2 3 Degrees Along the Major Axis 5 6 majo/axis^ t^ Llr'nf1^ It' 1*^™^™^ d i s t a n c e a l o n g the i n the LMC! f n ° r t h ' f ° r S a n d u l e a k - P h i l i p carbon s t a r s 43 s i m p l y , t h i s k i n e m a t i c s t u d y a g r e e s w i t h t h e p h o t o m e t r i c and s p e c t r o s c o p i c a n a l y s i s of SP s t a r s ( R i c h e r e t a l 1979) w h i c h c o n c l u d e s t h a t t h e y a r e P o p u l a t i o n I I i n n a t u r e and a r e e x p e c t e d t o be f o u n d i n t h e h a l o o f a g a l a x y . In an a t t e m p t t o e x p l a i n t h e l i n e a r r e l a t i o n s h i p between o r b i t a l v e l o c i t y and r a d i u s o b s e r v e d f o r t h e o t h e r o b j e c t s o b s e r v e d i n t h e LMC, t h e s i m p l e s t model t h a t c a n be employed i s one i n w h i c h t h e mass d i s t r i b u t i o n i s s p h e r i c a l l y s y m m e t r i c . T h i s r e s u l t s i n a c e n t r a l g r a v i t a t i o n a l f o r c e f i e l d i n w h i c h o n l y t h e mass, M , i n t e r i o r t o t h e o r b i t of a t e s t p a r t i c l e e x e r t s a n e t f o r c e on t h e t e s t p a r t i c l e . F o r c i r c u l a r o r b i t s , t h i s n e t f o r c e b a l a n c e s c e n t r i p e t a l f o r c e ; GMm/r 2=mv 2/r (1) T h i s t h e s i s f o u n d o b s e r v a t i o n a l l y t h a t t h e t a n g e n t i a l v e l o c i t y was p r o p o r t i o n a l t o r a d i u s , w h i c h i m p l i e s t h a t ; M o c r 3 (2) T h e r e f o r e , f o r a g a l a x y i n w h i c h t h e mass d i s t r i b u t i o n i s s p h e r i c a l l y s y mmetric and t h e v e l o c i t y o f c i r c u l a r o r b i t s i s p r o p o r t i o n a l t o r a d i u s , t h e g a l a x y h as a mass d e n s i t y w h i c h i s c o n s t a n t a t a l l r a d i i . E m p l o y i n g e q u a t i o n (1) and t h e a s s u m p t i o n of c i r c u l a r o r b i t s , d e r i v e d c o n s t a n t s o f t h e f i v e l i n e a r r o t a t i o n c u r v e s were u s e d t o c a l c u l a t e t h e t o t a l mass f o r t h e LMC c o n t a i n e d w i t h i n t h e r a d i u s g i v e n i n t h e f o u r t h column o f T a b l e V. The d e r i v e d v a l u e s f o r t h e mass of t h e LMC a r e l i s t e d i n t h e t h i r d column o f T a b l e V. The s i m p l i c i t y o f t h i s model i s i n d i c a t e d by a f a s c i n a t i n g c o r e l a t i o n i n T a b l e IV between v e l o c i t y g r a d i e n t a l o n g t h e major 44 a x i s f o r a g r o u p o f o b j e c t s and t h e age o f t h o s e o b j e c t s . I f a l l o b j e c t s f o l l o w e d c i r c u l a r o r b i t s , t h e v e l o c i t y g r a d i e n t would be t h e same f o r a l l o b j e c t s . The o b s e r v a t i o n t h a t v e l o c i t y g r a d i e n t i s a f u n c t i o n o f age i s an i n d i c a t i o n o f t h e d e g r e e t o w h i c h t h e o r b i t s o f o l d e r o b j e c t s d e v i a t e f r o m c i r c u l a r . S i n c e a s p h e r i c a l l y s y mmetric mass d i s t r i b u t i o n e x e r t s a c e n t r a l f o r c e , t h e a n g u l a r momentum o f any o b j e c t i s a c o n s e r v e d q u a n t i t y ( i . e . L = r X m v = c o n s t a n t ) . T h i s i n d i c a t e s t h a t f o r any o b j e c t f o l l o w i n g a n o n - c i r c u l a r o r b i t , t a n g e n t i a l v e l o c i t y i s i n v e r s e l y p r o p o r t i o n a l t o t h e r a d i a l d i s t a n c e f r o m t h e c e n t r e o f t h e g a l a x y . As t h e v e l o c i t i e s o f o b j e c t s a t p e r i g a l a c t u m and a p o g a l a c t u m a r e p u r e l y t a n g e n t i a l , t h e r a t i o o f t h e s e two v e l o c i t i e s i s t h e i n v e r s e r a t i o of t h e r a d i i of p e r i g a l a c t u m and a p o g a l a c t u m o f t h e o r b i t . The more e c c e n t r i c t h e o r b i t , t h e g r e a t e r w i l l be t h e c h a n c e of f i n d i n g an o b j e c t a t a p o g a l a c t u m , where i t has a s l o w e r v e l o c i t y and t h e r e f o r e spends a l a r g e r amount o f i t s t i m e . T h i s w i l l r e s u l t i n a g r e a t e r p r o b a b i l i t y o f f i n d i n g o b j e c t s w i t h v e l o c i t i e s l e s s t h a n t h e l o c a l c i r c u l a r o r b i t a l v e l o c i t y a t r a d i i f u r t h e r f r o m t h e d y n a m i c a l c e n t r e o f t h e g a l a x y a s t h e a v e r a g e o r b i t a l e c c e n t r i c i t y o f o r b i t s i n a g r o u p o f o b j e c t s i n c r e a s e s . T h i s i n t e r p r e t a t i o n o f t h e c o n s t a n t s f o u n d i n T a b l e IV a g r e e s w i t h t h e e a r l i e r f i n d i n g s o f Sandage, Eggen, and L y n d e n - B e l l ( 1 9 6 4 ) t h a t o l d e r o b j e c t s f o l l o w o r b i t s whose r a n g e o f e c c e n t r i c i t y i s g r e a t e r t h a n f o r y o u n g e r o b j e c t s . 45 T a b l e IV V e l o c i t y G r a d i e n t v s . Age o f t h e LMC O b j e c t O b j e c t V e l o c i t y G r a d i e n t Age (km/sec/deg) ( l o g ( y e a r s ) ) HII R e g i o n s 36.1±1.8 6.0 Luminous S t a r s 33.5±2.6 7.0 S t e l l a r C l u s t e r s 2 8.0±7.7 8.0 P l a n e t a r y N e b u l a e 2 5 . 6 ± 4 . 0 9.5 WORC C a r b o n S t a r s 22.9±4.8 9.2 To o b t a i n a r e l a t i o n s h i p between t h e o b j e c t age and i t s v e l o c i t y g r a d i e n t i n t h e p l a n e of t h e LMC ( w h i c h i s t h e o b s e r v e d r a d i a l v e l o c i t y g r a d i e n t d i v i d e d by t h e i n c l i n a t i o n a n g l e o f t h e LMC) a l i n e a r l e a s t s q u a r e s f i t was a p p l i e d t o t h e v e l o c i t y g r a d i e n t v e r s u s t h e l o g a r i t h m o f t h e o b j e c t a g e . T h i s r e s u l t e d i n t h e r e l a t i o n s h i p ; S=57.7-3.58T (3) where S i s t h e v e l o c i t y g r a d i e n t a l o n g t h e majo r a x i s e x p r e s s e d i n km/sec/deg f and T i s t h e l o g a r i t h m o f t h e o b j e c t ' s a g e . As a f i r s t a p p r o x i m a t i o n , s p h e r i c a l symmetry p r o v i d e s a r o u g h e s t i m a t e f o r t o t a l g a l a c t i c mass and a l s o g i v e s some i n s i g h t i n t o t h e e f f e c t s o f o r b i t a l e c c e n t r i c i t y . The LMC however i s a d i s k t y p e g a l a x y and so t h e n e x t improvement i n t h e c a l c u l a t i o n s w o u l d be t o model t h e LMC a s a d i s k o f c o n s t a n t t h i c k n e s s w i t h a d e n s i t y t h a t c h a n g e s a s a f u n c t i o n o f r a d i u s . The r a d i a l d e n s i t y d i s t r i b u t i o n t h a t r e s u l t s i n a g r a v i t a t i o n a l f i e l d c a p a b l e o f g i v i n g o b j e c t s c i r c u l a r o r b i t a l v e l o c i t i e s t h a t i n c r e a s e l i n e a r l y w i t h r a d i a l d i s t a n c e i s g i v e n 46 by; c = ( 2 f l 2 , V 2 G ) V a z - r z (4) where cr i s t h e a r e a l mass d e n s i t y , a i s t h e r a d i u s a t w h i c h mass c u t o f f o c c u r s and Ti i s t h e a n g u l a r v e l o c i t y of o b j e c t s i n t h e p l a n e o f t h e g a l a x y ( S o b r i n o , 1982; Sweet, 1983). I n t e g r a t i o n of t h i s d e n s i t y d i s t r i b u t i o n o v e r t h e d i s k r e s u l t s i n an e x p r e s s i o n r e l a t i n g o r b i t a l a n g u l a r v e l o c i t y t o t h e t o t a l mass o f t h e g a l a x y . M = ( 4 a 3 A 2 ) / ( 3 7 t G ) (5) To e s t i m a t e t h e mass o f t h e LMC i t i s t h e r e f o r e n e c e s s a r y t o o b t a i n v a l u e s f o r a and A . A l l e n ( l 9 8 2 ) g i v e s a v a l u e o f 7 kpc as t h e l i n e a r d i a m e t e r o f t h e LMC ( t h i s c o r e s p o n d s t o an a n g u l a r d i a m e t e r o f 4 6 0 ' ) . The l i n e a r l e a s t s q u a r e s f i t o f t h e WORC v e l o c i t y d a t a i n d i c a t e s however t h a t t h e mass c u t o f f i s a t l e a s t as g r e a t a s 5° (4.6 kpc) f r o m t h e d y n a m i c a l c e n t r e o f t h e LMC. E m p l o y i n g a v a l u e o f a=4.6 kpc, t h e mass of t h e LMC d i s k was c a l c u l a t e d f r o m t h e v a r i o u s v e l o c i t y g r a d i e n t s and t h e d e r i v e d v a l u e s a r e l i s t e d i n t h e f i f t h c olumn o f T a b l e V. As i n t h e c a s e o f a s p h e r i c a l l y s y mmetric mass d i s t r i b u t i o n , t h e d e r i v e d mass i s b a s e d upon c i r c u l a r o r b i t s . I t i s most l i k e l y t h o u g h t h a t o n l y t h e y o u n g e s t o b j e c t s f o l l o w c i r c u l a r o r b i t s and t h e d e r i v e d mass i s a minimum v a l u e . T h i s i s most e a s i l y e x p l a i n e d by d e r i v i n g t h e g r a v i t a t i o n a l p o t e n t i a l of t h i s d i s k model from t h e g i v e n d e n s i t y d i s t r i b u t i o n . The d e r i v e d p o t e n t i a l i n c r e a s e s m o n o t d n i c a l l y a s r a d i u s i n c r e a s e s . F o r o b j e c t s i n an e c c e n t r i c o r b i t , t h e o r b i t a l r a d i u s w i l l change and t h e r e f o r e t h e p o t e n t i a l o f t h e o b j e c t w i l l c h a n g e . By c o n s e r v a t i o n o f e n e r g y , as any o b j e c t i n c r e a s e s i t s d i s t a n c e 47 from t h e c e n t r e of t h e g a l a x y i t s p o t e n t i a l e n e r g y w i l l i n c r e a s e and t h e r e f o r e i t s v e l o c i t y w i l l d e c r e a s e . Any d e c r e a s e i n o r b i t a l r a d i u s w i l l l i k e w i s e d e c r e a s e p o t e n t i a l e n e r g y o f t h e o b j e c t a n d i n c r e a s e i t s v e l o c i t y . In t h e same manner as i n t h e c a s e o f a s p h e r i c a l l y s y m m e t r i c g a l a x y , an o b j e c t f o l l o w i n g an e c c e n t r i c o r b i t i n t h i s d i s k - s h a p e d model g a l a x y w i l l s p end more t i m e a t s l o w e r v e l o c i t i e s a t a p o g a l a c t u m . T h i s w i l l w e i g h t t h e v e l o c i t y g r a d i e n t s t o w a r d i n c r e a s i n g l y l o w e r v a l u e s f o r o r b i t s o f i n c r e a s i n g e c c e n t r i c i t y . T a b l e IV i n d i c a t e s t h a t t h e o r b i t s o f c a r b o n s s t a r s a r e l i k e l y e c c e n t r i c and t h e r e f o r e t h e d e r i v e d v a l u e s f o r t h e mass o f t h e d i s k of t h e LMC s h o u l d be v i e w e d as l o w e r l i m i t s t o t h e t r u e v a l u e . The v e l o c i t y g r a d i e n t s shown i n t h i s t a b l e a r e t h e v a l u e s o f t h e v e l o c i t y g r a d i e n t i n t h e p l a n e o f t h e g a l a x y i f t h e LMC i s a t an i n c l i n a t i o n a n g l e o f 2 7 ° . 48 T a b l e V O b j e c t Mass of t h e L a r g e M a g e l l a n i c C l o u d V e l o c i t y S p h e r i c a l C o n t a i n e d Mass R a d i u s (Mo,e=0) (k p c ) G r a d i e n t (km/sec/deg) HI I R e g i o n s S t a r s S t e l l a r C l u s t e r s P l a n e t a r y N e b u l a e WORC S t a r s -36.1 -33.5 -28.0 -25.6 -22.9 7.5X10 9 6.4X10 9 4.5X10 9 3.7X10 9 14.0X10 9 2.7 2.7 2.7 3.6 4.6 D i s k Mass (Mo,e=0) 14.8X10 9 12.7X10 9 8.9X10 9 7.4X10 9 6.0X10 9 T h e s e c a l c u l a t i o n s p r o d u c e mass e s t i m a t e s f o r t h e LMC w h i c h a r e i n good agreement w i t h p r e v i o u s l y d e t e r m i n e d v a l u e s ( A l l e n , 1981). The s p h e r i c a l mass d e r i v e d from c a r b o n s t a r s i s l a r g e r b e c a u s e t h e y c o v e r a much g r e a t e r e x t e n t of t h e LMC. T h i s c a n e a s i l y be s e e n from t h e X-Y maps. T h e r e f o r e , a s s u m i n g t h e o r b i t s of t h e c a r b o n s t a r s were c i r c u l a r and t h e LMC mass d i s t r i b u t i o n was s p h e r i c a l l y symmetric a l o w e r l i m i t on t h e mass o f t h e LMC o f 14.0X10 9 Mo was d e r i v e d . A more a c c u r a t e e s t i m a t e however, i s b a s e d upon c i r c u l a r o r b i t s i n a d i s k - s h a p e d model w h i c h i m p l i e s t h a t t h e LMC has a mass o f 6.0X10 9 Mo. T h e s e r e s u l t s were a l l d e r i v e d from t h e i n i t i a l a s s u m p t i o n t h a t o r b i t a l v e l o c i t i e s i n c r e a s e l i n e a r l y w i t h d i s t a n c e from t h e d y n a m i c a l c e n t r e of t h e LMC. A l l t h e d a t a s e t s employed i n d i c a t e t h a t t h i s i s i n d e e d t h e manner i n w h i c h t h e LMC r o t a t e s . As was n o t e d e a r l i e r , however, t h e r e was some i n d i c a t i o n t h a t t h e LMC 49 m i g h t have a r o t a t i o n c u r v e t h a t was o t h e r t h a n s o l i d body. The n e x t most l i k e l y c a n d i d a t e i s a f l a t r o t a t i o n c u r v e . To i n v e s t i g a t e t h i s p o s s i b i l i t y , an a n g l e v s v e l o c i t y map was g e n e r a t e d f o r t h e s i x p r e v i o u s l y i n v e s t i g a t e d s t e l l a r g r o u p s . T h e s e maps were g e n e r a t e d by c a l c u l a t i n g e a s t o f n o r t h from t h e major a x i s , a b o u t t h e R a d i o C e n t r e , i n t h e p l a n e o f t h e LMC, t h e a n g l e 0 f o r e a c h o b j e c t i n t h e d a t a s e t . The g e o m e t r y o f t h i s model i s e x p l a i n e d i n A p p e n d i x I . T h i s a n g l e was t h e n p l o t t e d a g a i n s t t h e r a d i a l v e l o c i t y o f t h e c o r r e s p o n d i n g p o i n t . F i g u r e s XXI t h r o u g h XXVI e x h i b i t t h e r e s u l t s o f t h i s a n a l y s i s . I f t h e r o t a t i o n of t h e LMC were f l a t , t h e d a t a p o i n t s s h o u l d f o l l o w a r e l a t i o n s h i p o f t h e f orm; v=A-Bcos9 (6) where A i s t h e s y s t e m i c v e l o c i t y , B i s t h e v e l o c i t y o f r o t a t i o n and 9 i s t h e a n g l e of t h e o b j e c t i n t h e p l a n e of t h e LMC. To see i f t h e d a t a showed s i g n s o f a f l a t r o t a t i o n c u r v e , a t h i r d o r d e r F o u r i e r s e r i e s was f i t t e d t o t h e d a t a . I f a f l a t r o t a t i o n c u r v e was p r e s e n t , a f i r s t o r d e r f i t s h o u l d be a d e q u a t e w i t h a l l h i g h e r o r d e r t e r m s b e i n g n e g l i g i b l e i n m a g n i t u d e . The r e l a t i v e m a g n i t u d e o f t h e h i g h e r o r d e r terms was t h e r e f o r e employed as a t e s t o f t h e g o o d n e s s o f f i t o f t h e d a t a p o i n t s t o a f l a t r o t a t i o n c u r v e . The o b j e c t s t h a t were f i t t e d r e a s o n a b l y w e l l w i t h a l o w e s t o r d e r s i n e c u r v e were t h e WORC c a r b o n s t a r s , p l a n e t a r y n e b u l a e , s t a r s and HII r e g i o n s . In t h e c a s e o f t h e SP c a r b o n s t a r s and t h e s t e l l a r c l u s t e r s , terms h i g h e r t h a n f i r s t o r d e r d o m i n a t e d . T h i s i n d i c a t e d t h a t t h e s e o b j e c t s p o s s e s s e d l i t t l e o r none o f t h e m o t i o n i n d i c a t i v e of a f l a t r o t a t i o n c u r v e . The r e s u l t s o f t h e f i t s t o t h e o t h e r o b j e c t s a r e g i v e n i n 420-j 400-380-360-340-320-oci 300-> 280-'•6 260-o cr 240-220-200-180-160 -140-WORC Carbon Stars 60 90 120 150 180 210 Degrees About the Radio Centre T 1 1 1 1 24 0 2 7 0 3 0 0 3 3 0 3 6 0 f o r WORC Carbon S t a r s . 0 degrees c o r r e s p o n d s t o the major a x i s which ,, s i t u a t e d at 25" e a s t of n o r t h on the sky. The p l o t t e d s i n u s o i d the best f i r s t o r d e r f i t t o the d a t a d e s c r i b e d i n T a b l e V I . a n u i s C P O 420-j 4oo- Sanduleak-Philip Carbon Stars 380-360-340-320-300-280-260-A 240- A A 220-200-180 -160- A 140-120— , , A A A A A A A* A Degrees About the Radio Centre LMC i"'i.i3i!;.KIl:i1ip,'2isii;yst:?; o 9 d: Qr i n « h 7 ^ ^ r s r o g TJ o 420-i 400-380-360-340-320-300-280-260-240-220-200-180-160-1404 HII Regions ~ T — 30 I 60 90 120 150 18 0 210 24 0 2 7 0 3 0 0 3 3 0 3 6 0 Degrees About the Radio Centre LMC VoV H H R e q t o ^ s ^ a n g l e of the^ which i s sUuated 9It l< .1,?%"lAVS'ST*' 1° ^  m a ' o r »*» S i n u s o i d i s the best f i r s t order M % ^ T h e P l o t t « d d e s c r i b e d i n Ta b l e VI? e r  t l t t o t h e d a t a and i s cn o _o o 150 180 210 240 Degrees About the Radio Centre i 1 1 1 270 300 330 360 ™ ' - ' " a r s : S"di;j.::ln^^a:,,?iMiB t h e p i a n e °f th~ si tuated at 3° east of north on ^ ' m a ; i o r a x i s w h i c h is the best f i r s t order i t to the T ' P l o t ? e d s inusoid is c o t h e d a t a a"d is described in Table VI . 420-n 4 0 0 -380-360-3 4 0 -320-oci 3 00-> 280-260-O cn 240-220-2 0 0 -180-160-140 ~-Stellar Clusters A A A A A A i A A^ 30 60 90 120 150 18 0 210 24 0 2 7 0 3 0 0 3 3 0 3 6 0 Degrees About the Radio Centre w h i c h i s i t t r « . ^ ' i ; r = 2 9 . 0 . f : ? r : r n S ? t r { : * r a s ^ h e m a j o r a x i s plotted s inusoid is the best f t r s t order f i t to t h ^ H „ T h e described in Table VI. 1 t 0 t h e d a t a a n d > s cn cn 56 T a b l e VI bel o w . T a b l e VI F l a t R o t a t i o n C u r v e P a r a m e t e r s f o r t h e LMC O b j e c t WORC S t a r s P l a n e t a r y N e b u l a e S t a r s HII R e g i o n s S y s t e m i c V e l o c i t y (km/sec) 277.9±15 269.6±11 278.2±8 274.6±5 R o t a t i o n a l V e l o c i t y (km/sec) 48.7±32 30.1±19 33.3134 30.2112 A n g l e o f t h e M a j o r A x i s ( E a s t o f N o r t h ) 25°130° 2 4 ° 1 3 0 ° 3°+20° 5°115° In a l l t h e above c a s e s t h e v e l o c i t y d i s p e r s i o n o f t h e d a t a p o i n t s a l o n g t h e f i t t e d s i n e c u r v e was t h e same o r worse t h a n f o r t h e p r e v i o u s l y d i s c u s s e d c a s e o f a l i n e a r l y i n c r e a s i n g r o t a t i o n c u r v e . An a n a l y s i s of t h e r o t a t i o n v e l o c i t y of t h e s e f o u r d a t a s e t s i s i n t r i g u i n g . The c a r b o n s t a r s show a v e l o c i t y o f r o t a t i o n w h i c h i s 60% h i g h e r t h a n f o r t h e p l a n e t a r y n e b u l a e , s t a r s o r HII r e g i o n s . The f a c t t h a t t h e p l a n e t a r y n e b u l a e r o t a t i o n a l v e l o c i t y a g r e e s w i t h t h e s t a r s and HII r e g i o n s so w e l l and t h e WORC c a r b o n s t a r ' s r o t a t i o n a l v e l o c i t y i s so d i s c r e p a n t i s n o t due t o any r e a l k i n e m a t i c a l r e a s o n . I t i s i n f a c t due t o a c o m b i n a t i o n o f t h e e x t e n t o f c o v e r a g e o f t h e LMC o f e a c h d a t a s e t and t h e r e a l k i n e m a t i c p r o p e r t y o f t h e g r o u p o f o b j e c t s ; i t s v e l o c i t y g r a d i e n t a l o n g t h e major a x i s . From t h e X-Y maps i t i s 57 se e n t h a t a l t h o u g h t h e p l a n e t a r y n e b u l a e have a s m a l l e r v e l o c i t y g r a d i e n t t h a n t h e s t a r s o r HII r e g i o n s , t h e y e x t e n d a l m o s t 3 d e g r e e s o u t w a r d from t h e R a d i o C e n t r e i n t h e i r c o v e r a g e o f t h e LMC. The s t a r s and HII r e g i o n s however o n l y e x t e n d o u t w a r d s a p p r o x i m a t e l y 2 d e g r e e s from t h e R a d i o C e n t r e . When a v e r a g i n g t h e v e l o c i t i e s a l o n g any g i v e n a n g l e , t h e l e s s s t e e p v e l o c i t y g r a d i e n t i s co m p e n s a t e d f o r by o b j e c t s t h a t a r e f u r t h e r away from t h e d y n a m i c a l c e n t r e . I t i s t h i s e f f e c t t h a t r e s u l t e d i n s i m i l a r r o t a t i o n a l v e l o c i t i e s f o r t h e s t a r s , HII r e g i o n s , a n d t h e p l a n e t a r y n e b u l a e . I t a l s o e x p l a i n s t h e s c a t t e r i n v e l o c i t i e s o b s e r v e d a t e a c h a n g l e a b o u t t h e d y n a m i c a l c e n t r e . The c a r b o n s t a r r o t a t i o n a l v e l o c i t y i s h i g h e r b e c a u s e a l t h o u g h t h e v e l o c i t y g r a d i e n t i s l e s s t h a n f o r any o t h e r o b j e c t , c a r b o n s t a r s a r e f o u n d f u r t h e r f r o m t h e R a d i o C e n t r e . The d i s t a n c e more t h a n c o m p e n s a t e s f o r t h e more s h a l l o w g r a d i e n t and t h e r e s u l t i s t h a t t h e WORC c a r b o n s t a r s a p p e a r t o have a h i g h e r r o t a t i o n a l v e l o c i t y t h a n t h e o t h e r o b j e c t s . T h i s s t u d y t h e r e f o r e c o n c l u d e s t h a t t h e b e s t d e s c r i p t i o n o f m o t i o n i n t h e LMC i s one i n w h i c h t h e o r b i t a l v e l o c i t i e s o f o b j e c t s i n c r e a s e i n d i r e c t p r o p o r t i o n t o t h e d i s t a n c e f r o m t h e d y n a m i c a l c e n t r e o f t h e LMC. 58 CHAPTER IV RADIAL VELOCITY DISPERSIONS IN THE LARGE AND SMALL MAGELLANIC CLOUDS The p r e v i o u s c h a p t e r e x amined s y s t e m a t i c m o t i o n s i n t h e LMC. S u p e r i m p o s e d upon t h e s e s y s t e m a t i c m o t i o n s however, were t h e i n d i v i d u a l random m o t i o n s o f t h e o b j e c t s c o m p r i s i n g t h e g r o u p u n d e r e x a m i n a t i o n . T h i s r a d i a l v e l o c i t y d i s p e r s i o n was of i n t e r e s t as a p r e v i o u s s t u d y i n our own g a l a x y has f o u n d t h a t p r o g r e s s i v e l y o l d e r o b j e c t s have an i n c r e a s i n g v e l o c i t y d i s p e r s i o n (Eggen , L y d e n - B e l l , Sandage 1962). T h i s s t u d y was b a s e d on s t a r s f o r w h i c h a l l t h r e e components of t h e i r s p a c e m o t i o n were known and i t was p o s s i b l e t o d e t e r m i n e t h a t i n c r e a s i n g e c c e n t r i c i t y of o r b i t was a l s o w e l l c o r r e l a t e d w i t h i n c r e a s i n g v e l o c i t y d i s p e r s i o n . I t was t h e r e f o r e o f i n t e r e s t t o see i f t h e o b j e c t s o b s e r v e d i n t h e LMC f o l l o w e d t h i s g e n e r a l r u l e a s w e l l . T h i s s c e n a r i o o f i n c r e a s i n g o r b i t a l e c c e n t r i c i t y w i t h o b j e c t age a g r e e s q u i t e w e l l w i t h o b s e r v e d d e c r e a s e i n v e l o c i t y g r a d i e n t of t h e r o t a t i o n c u r v e s i n t h i s s t u d y as p r o g r e s s i v e l y o l d e r o b j e c t s a r e u s e d as a sample g r o u p . The v e l o c i t y d i s p e r s i o n was a l s o e x p e c t e d t o i n c r e a s e w i t h i n c r e a s i n g o b j e c t age. T h i s would a p p e a r on t h e r o t a t i o n c u r v e s a s an i n c r e a s e i n s c a t t e r o f t h e d a t a p o i n t s i n t h e r a d i a l v e l o c i t y d i r e c t i o n f o r o l d e r o b j e c t s . T h i s g e n e r a l t r e n d was o b s e r v e d i n t h e l i n e a r l y i n c r e a s i n g r o t a t i o n c u r v e s by means o f m e a s u r i n g t h e s t a n d a r d d e v i a t i o n i n 59 r a d i a l v e l o c i t y a l o n g t h e r o t a t i o n c u r v e . T h i s was c a r r i e d o u t i n t h e same manner as was u s e d t o g e n e r a t e t h e mean r a d i a l v e l o c i t y d i f f e r e n c e p l o t s and r e s u l t e d i n F i g u r e s XXIV t h r o u g h XXIX. The r o t a t i o n c u r v e g e n e r a t e d f r o m s t a r s i n t h e LMC e x h i b i t e d a h i g h e r t h a n e x p e c t e d r a d i a l v e l o c i t y d i s p e r s i o n , b u t t h i s was l i k e l y c a u s e d by t h e i n c l u s i o n o f l a t e - t y p e s u p e r g i a n t s , and many p e c u l i a r s t a r s as w e l l as B s u p e r g i a n t s i n t h i s s a m ple. The l a t e - t y p e s u p e r g i a n t s and p e c u l i a r s t a r s would be e x p e c t e d t o i n c r e a s e t h e v e l o c i t y d i s p e r s i o n o f t h i s sample w h i l e t h e 0 and B s u p e r g i a n t s , s i n c e t h e y were v e r y young o b j e c t s , would t e n d t o have l e s s e c c e n t r i c o r b i t s i n c r e a s i n g t h e v e l o c i t y g r a d i e n t o f t h i s sample a l o n g t h e main a x i s . The r e s u l t o f t h e s e two o p p o s i n g a c t i o n s was a r o t a t i o n c u r v e w i t h t h e v e l o c i t y g r a d i e n t o f a r e l a t i v e l y young sample and t h e r a d i a l v e l o c i t y d i s p e r s i o n o f a much o l d e r sample o f s t a r s . The p l a n e t a r y n e b u l a e a l s o a p p e a r t o be d i s c r e p a n t w i t h t h e t h e o r y o f v e l o c i t y d i s p e r s i o n i n c r e a s i n g w i t h age, as t h e v e l o c i t y d i s p e r s i o n o f t h i s g r o u p a p p e a r e d t o be q u i t e s m a l l as compared t o t h a t o f t h e c a r b o n s t a r s o r s t e l l a r c l u s t e r s . T h i s m i g h t be a s t a t i s t i c a l e f f e c t due t o t h e s m a l l number o f p l a n e t a r i e s i n v o l v e d i n t h e sample and n o t a t r u e measure o f t h e v e l o c i t y d i s p e r s i o n o f t h e s e o b j e c t s . A p r e v i o u s s t u d y by F e a s t (1968) f o u n d a c o m p a r a b l e v a l u e o f 22 km/sec f o r t h e v e l o c i t y d i s p e r s i o n o f p l a n e t a r i e s i n t h e LMC. T h i s v a l u e however was b a s e d on 25 p l a n e t a r i e s , a s m a l l e r sample t h a n t h e 35 p l a n e t a r y n e b u l a e employed i n t h i s t h e s i s . I f more p l a n e t a r i e s c o u l d be f o u n d and measured, i t w o u l d be of i n t e r e s t t o see i f t h e v e l o c i t y d i s p e r s i o n of t h e s e o b j e c t s i s t r u l y a s low as b o t h o f > Q 10 CO o . o - 4 -3 - 2 -1 0 . 0 1 2 3 4 D e g r e e s A l o n g t h e M a j o r A x i s F i g . XXVII. Standard deviation i n r a d i a l v e l o c i t y about the f i t t e d l i n e a r v e l o c i t y of WORC carbon stars vs. distance along the major axis o f the LMC. O " 4 - 3 -2 -1 0 .0 1 2 3 D e g r e e s A l o n g t h e M a j o r A x i s velocity'of^ l'reeiSfvf *? V e l ° C i t^ a b o u t t h e t^ed linear u ^ x L y or mi rgions vs. distance along the major axis of the LMC. C T i ~ 4 -3 -2 -1 0 .0 1 2 3 4 D e g r e e s A l o n g t h e M a j o r A x i s v P l o H t 5 " . ? ( f X , - S t a n d a r d d e v i a t i o n ^ r a d i a l v e l o c i t y about the f i t t e d l i n e a r v e l o c i t y of luminous stars vs. distance along the major axis of the LMC ~ 4 -3 -2 -1 0.0 1 2 3 D e g r e e s A l o n g t h e M a j o r A x i s velocity 0f^ tenSdcIus^  a b o u t t h e £ i t t e d 1*"* * my or stellar clu ters vs. distance along the major axis of the LMC. -4 - 3 -2 - 1 0 . 0 1 2 3 D e g r e e s A l o n g t h e M a j o r A x i s Fig. XXXI. Standard deviation i n r a d i a l v e l o c i t y about the f i t t e d lin< v e l o c i t y o f planetary nebulae vs. distance along the major axis of the LMC 65 t h e s e s a m p l e s i m p l i e d . A n o t h e r c a u s e o f t h e o b s e r v e d d i s c r e p a n c i e s m i g h t s i m p l y be t h e d i f f e r i n g d e g r e e s o f a c c u r a c y t o w h i c h t h e v e l o c i t i e s of t h e s e o b j e c t s have been d e t e r m i n e d . The v e l o c i t i e s o f t h e HII r e g i o n s and t h e p l a n e t a r y n e b u l a e were a l l d e t e r m i n e d w i t h s t a n d a r d d e v i a t i o n s of a p p r o x i m a t e l y 5 km/sec. A l l of t h e o t h e r o b j e c t s , however, had s t a n d a r d d e v i a t i o n s a b o u t t h e mean v e l o c i t i e s o f between 15 and 25 km/sec. I t i s t h e r e f o r e p o s s i b l e t h a t t h e v e l o c i t y d i s p e r s i o n s of t h e HII r e g i o n s and p l a n e t a r y n e b u l a e i s i n d i c a t e d c o r r e c t l y by t h e s e p l o t s and t h e l a r g e v e l o c i t y d i s p e r s i o n e x h i b i t e d by t h e o t h e r d a t a s e t s i s due t o t h e l a r g e r s t a n d a r d d e v i a t i o n s i n t h e d e t e r m i n e d v e l o c i t i e s of t h e s e o b j e c t s . In e m p l o y i n g t h e r o t a t i o n c u r v e s t o d e r i v e v e l o c i t y d i s p e r s i o n s f o r a g r o u p of o b j e c t s i t must be remembered t h a t a l l c o n t r i b u t i o n s t o t h e v e l o c i t y d i s p e r s i o n due t o s u c h d y n a m i c a l l y d i s r u p t i v e b o d i e s a s t h e b a r o r arm o f t h e LMC have been i g n o r e d . F o r t h i s r e a s o n t h e p r e v i o u s a n a l y s i s was o n l y c a r r i e d o u t on a q u a l i t a t i v e l e v e l . To a t t e m p t t o d e r i v e q u a n t i t a t i v e v a l u e s f o r t h e v e l o c i t y d i s p e r s i o n f r o m t h e s e r o t a t i o n c u r v e s would o n l y y i e l d h i g h l y s u s p e c t o r e r r o n e o u s v a l u e s . To examine t h e v e l o c i t y d i s p e r s i o n i n a more q u a n t i t a t i v e manner, r o t a t i o n c u r v e s , o r any o t h e r method w h i c h samples v e l o c i t i e s o v e r a l a r g e e x t e n t o f t h e g a l a x y , must be abandoned. The p r o p e r a p p r o a c h i s t o c h o s e an a r e a i n t h e g a l a x y l a r g e enough f o r a d e q u a t e s t a t i s t i c s y e t s m a l l enough t h a t t h e a r e a i s u n a f f e c t e d by s y s t e m a t i c v e l o c i t y . c h a n g e s due t o g e n e r a l 66 r o t a t i o n of t h e g a l a x y o r s t r e a m i n g m o t i o n s a l o n g s u c h i n t e r n a l b o d i e s a s a b a r o r arm o f t h e g a l a x y . The LMC Bar West f i e l d and t h e SMC Bar f i e l d , t a k e n f r o m t h e BMB c a t a l o g u e , a r e two s u c h s t a r f i e l d s w h i c h f u l f i l l t h e above r e q u i r e m e n t s . In t h e LMC Bar West f i e l d , r a d i a l v e l o c i t i e s were d e t e r m i n e d f o r 24 o f t h e 70 c a r b o n s t a r s f o u n d i n t h i s a r e a . T h e s e v e l o c i t i e s were c o r r e c t e d i n t h e same manner a s t h o s e c a r b o n s t a r v e l o c i t i e s e mployed i n d e t e r m i n i n g t h e r o t a t i o n c u r v e . T h i s r e s u l t e d i n a mean v e l o c i t y o f 226.3 km/sec w i t h a v e l o c i t y d i s p e r s i o n o f 27.9 km/sec f o r t h e 24 c a r b o n s t a r s i n t h i s f i e l d . F o r t h e Bar f i e l d i n t h e SMC, 26 r a d i a l v e l o c i t i e s were d e t e r m i n e d f o r 17 d i f f e r e n t c a r b o n s t a r s . T h i s sample was o b t a i n e d f r o m a t o t a l sample o f 79 c a r b o n s t a r s f o u n d . F o r t h o s e s t a r s w i t h more t h a n one r a d i a l v e l o c i t y d e t e r m i n e d , a mean o f a l l d e t e r m i n e d r a d i a l v e l o c i t i e s was u s e d i n t h e c a l c u l a t i o n o f t h e v e l o c i t y d i s p e r s i o n f o r t h i s f i e l d . The r e s u l t was a mean v e l o c i t y of 133.8 km/sec and a v e l o c i t y d i s p e r s i o n o f 43.6 km/sec. The above q u o t e d v e l o c i t y d i s p e r s i o n s were s i m p l y t h e s t a n d a r d d e v i a t i o n s o f t h e r a d i a l v e l o c i t i e s . The d i s t r i b u t i o n of v e l o c i t i e s i n b o t h c a s e s was n o n - g a u s s i a n . T h i s c o u l d have been c a u s e d by i n a d e q u a t e s a m p l i n g and i f t h i s were t h e c a s e more c o m p l e t e s a m p l i n g of t h e f i e l d w o u l d have been o f i n t e r e s t t o c o n f i r m a g a u s s i a n d i s t r i b u t i o n o f v e l o c i t i e s and improve t h e above c a l c u l a t e d v a l u e s f o r t h e r a d i a l v e l o c i t y d i s p e r s i o n s . T h e r e a l s o e x i s t e d t h e p o s s i b i l i t y t h a t t h e r a d i a l v e l o c i t y d i s t r i b u t i o n was n o n - g a u s s i a n even i f a t o t a l sample o f 67 v e l o c i t i e s was o b t a i n e d . In e i t h e r o f t h e above c a s e s t h e d e r i v e d v a l u e s f o r t h e d i s p e r s i o n were s t i l l v a l i d f o r c o m p a r i s o n p u r p o s e s but u n t i l t h e sample s p a c e c a n be i n c r e a s e d t h e above c a l c u l a t e d v a l u e s s h o u l d i n no way be c o n s i d e r e d as e x a c t v a l u e s f o r t h e r a d i a l d i s p e r s i o n s o f c a r b o n s t a r s i n t h e s e s t a r f i e l d s . The r a d i a l v e l o c i t y d i s p e r s i o n o f t h e LMC Bar West f i e l d was compared w i t h t h e r a d i a l v e l o c i t y d i s p e r s i o n o f t h e c a r b o n s t a r ' s s o l i d body r o t a t i o n c u r v e . I t was f o u n d t h a t b o t h methods y i e l d e d a v a l u e of a p p r o x i m a t e l y 30 km/sec f o r t h e v e l o c i t y d i s p e r s i o n . T h i s i n d i c a t e d t h a t t h e r o t a t i o n c u r v e s c o u l d be e mployed t o g i v e v e l o c i t y d i s p e r s i o n s t h a t w o u l d be e q u a l i n m a g n i t u d e t o t h e v e l o c t i y d i s p e r s i o n d e r i v e d f o r a s e t o f o b j e c t s i n a s m a l l f i e l d . The r a d i a l v e l o c i t y d i s p e r s i o n s of t h e two B l a n c o f i e l d s were a l s o e mployed t o compare t h e d y n a m i c s o f t h e LMC and SMC. A l t h o u g h t h e d i f f e r e n c e i n d i s p e r s i o n o f t h e s e two f i e l d s c o u l d be i n t e r p r e t e d i n many ways, t h e two s i m p l e s t and most p r o b a b l e e x p l a n a t i o n s w i l l be d i s c u s s e d . The f i r s t e x p l a n a t i o n i s b a s e d upon t h e a s s u m p t i o n t h a t t h e v e l o c i t y d i s p e r s i o n f o r any f i e l d i n a g a l a x y was of e q u a l m a g n i t u d e i n a l l d i r e c t i o n s . The m a g n i t u d e o f t h e v e l o c i t y d i s p e r s i o n would be e x p e c t e d t o change from g a l a x y t o g a l a x y . The l a r g e r r a d i a l v e l o c i t y d i s p e r s i o n o b s e r v e d i n t h e SMC would t h e n i n d i c a t e t h a t t h e m o t i o n s o f t h e c a r b o n s t a r s i n t h e SMC were much more r a n d o m i z e d t h a n t h e i r c o u n t e r p a r t s i n t h e LMC. S t a t e d more s i m p l y , t h e SMC i s k i n e m a t i c a l l y h o t t e r t h a n t h e LMC. On a r o t a t i o n c u r v e f o r c a r b o n s t a r s i n t h e SMC , t h i s 68 would a p p e a r a s a much l a r g e r d i s p e r s i o n i n t h e r a d i a l d i r e c t i o n t h a n was o b s e r v e d f o r a r o t a t i o n c u r v e of c a r b o n s t a r s f o u n d i n t h e LMC. A d i f f e r e n c e i n k i n e m a t i c t e m p e r a t u r e between t h e s e two g a l a x i e s i s l i k e l y o n l y a p a r t i a l e x p l a n a t i o n o f t h e d i f f e r e n c e i n v e l o c i t y d i s p e r s i o n between t h e s e two g a l a x i e s . The c a r b o n s t a r s s t u d i e s i n t h e LMC had o r b i t s t h a t were d e f i n i t e l y n o t random i n d i r e c t i o n as was c l e a r l y shown by t h e a b i l i t y of t h e i r r a d i a l v e l o c i t i e s t o d e f i n e a r o t a t i o n c u r v e . In t h e SMC t h e r e i s m o u n t i n g e v i d e n c e from o b j e c t s o t h e r t h a n c a r b o n s t a r s t h a t t h e SMC a l s o has a f a i r l y w e l l d e f i n e d r o t a t i o n c u r v e ( F e a s t 1970 , S m i t h Weedman 1973). I t i s most l i k e l y t h a t c a r b o n s t a r s i n t h e SMC a l s o p a r t i c i p a t e i n t h i s r o t a t i o n and t h e r e f o r e a l s o have o r b i t s c o n f i n e d t o t h e p l a n e o f t h e SMC. T h i s r e a s o n i n g l e d t o t h e s e c o n d p o s s i b l e e x p l a n a t i o n o f t h e d i f f e r e n c e i n d i s p e r s i o n o b s e r v e d between t h e s e two f i e l d s . S i n c e t h e s e c a r b o n s t a r s a r e d i s k o b j e c t s w h i c h p r o b a b l y have m o d e r a t e l y e c c e n t r i c o r b i t s , i t i s most p r o b a b l e t h a t t h e v e l o c i t y d i s p e r s i o n i n d i r e c t i o n s i n t h e p l a n e o f t h e g a l a x y i s much g r e a t e r t h a n i n a d i r e c t i o n p e r p e n d i c u l a r t o t h e p l a n e o f t h e g a l a x y . Under t h e s e c o n d i t i o n s , a g a l a x y o b s e r v e d a t i n c r e a s i n g i n c l i n a t i o n would g i v e an i n c r e a s i n g r a d i a l v e l o c i t y d i s p e r s i o n f o r s t a r s o b s e r v e d i n t h e same f i e l d . I f t h e LMC and SMC had c o m p a r a b l e d i s p e r s i o n s i n t h e t h r e e d i m e n s i o n a l s e n s e f o r t h e s e two f i e l d s , t h e d i f f e r e n c e i n t h e r a d i a l v e l o c i t y d i s p e r s i o n would t h e n be i n t e r p r e t e d as an i n c r e a s e d i n c l i n a t i o n o f t h e p l a n e o f t h e SMC as compared t o t h e LMC. The g r e a t e r i n c l i n a t i o n a n g l e o f t h e SMC r e s u l t s i n t h a t 69 g a l a x y b e i n g seen more edge on. T h i s edge on view r e v e a l s t h a t t h e v e l o c i t y d i s p e r s i o n i s g r e a t e r i n t h e p l a n e o f t h e g a l a x y t h a n p e r p e n d i c u l a r t o t h e p l a n e . The d i f f e r e n c e i n r a d i a l v e l o c i t y d i s p e r s i o n between t h e s e two f i e l d s w ould t h e n be c a u s e d s i m p l y by t h e d i f f e r e n c e i n i n c l i n a t i o n a n g l e o f t h e two f i e l d s . U n f o r t u n a t e l y , t h e o n l y way t o c o n f i r m i f t h e s e f i e l d s have c o m p a r a b l e d i s p e r s i o n s i s t o measure t h e t h r e e d i m e n s i o n a l s p a c e v e l o c i t i e s of a l l t h e s t a r s i n t h e s e two f i e l d s . I t i s most u n l i k e l y t h a t t h i s c o n d i t i o n w i l l e v e r be met a s t h e s e c a r b o n s t a r s a r e d e f i n i t e l y not s u i t a b l e c a n d i d a t e s f o r p r o p e r m o t i o n s t u d i e s . The t r u e i n t e r p r e t a t i o n o f t h e d i f f e r e n c e i n r a d i a l v e l o c i t y d i s p e r s i o n s between t h e s e two f i e l d s , i n l i g h t o f k i n e m a t i c i m f o r m a t i o n f r o m o t h e r o b j e c t s , i s l i k e l y a c o m b i n a t i o n o f t h e two above p o s s i b i l i t i e s as w e l l a s c o n t r i b u t i o n s f r o m l e s s e r s o u r c e s s u c h a s s t r e a m i n g m o t i o n s i n t h e b a r o r m o t i o n s due t o o t h e r l o c a l a n o m a l i e s i n t h e g r a v i t a t i o n a l f i e l d o f t h e g a l a x y . F e a s t (1970) i n t e r p r e t e d t h e o b s e r v e d b i m o d a l d i s t r i b u t i o n of r a d i a l v e l o c i t i e s a l o n g Hindmann's a x i s f o r t h e SMC as due t o o b s e r v i n g t h e Bar o f t h e SMC i n an end on c o n f i g u r a t i o n . In t h i s way s t r e a m i n g m o t i o n s o f o b j e c t s a t b o t h ends o f t h e Bar were o b s e r v e d and r e s u l t e d i n a b i m o d a l r a d i a l v e l o c i t y d i s t r i b u t i o n . T h i s b i m o d a l d i s t r i b u t i o n w o uld r e s u l t i n a g r e a t l y i n c r e a s e d r a d i a l v e l o c i t y d i s p e r s i o n f o r any f i e l d i n t h e Bar o f t h e SMC. T h i s e f f e c t m i g h t e x p l a i n t h e l a r g e r r a d i a l v e l o c i t y d i s p e r s i o n measured f o r t h e SMC b u t t h e sample o f c a r b o n s t a r s employed showed no e v i d e n c e o f a b i m o d a l d i s t r i b u t i o n i n r a d i a l v e l o c i t i e s . T h i s m i g h t have been 70 due t o i n a d e q u a t e s a m p l i n g of t h e f i e l d or i t may s i m p l y be t h a t t h e o r i e n t a t i o n o f t h e f i e l d c o n c e a l e d s u c h a b i m o d a l d i s t r i b u t i o n o f v e l o c i t i e s . In e i t h e r c a s e t h i s p r e s e n t sample of c a r b o n s t a r s d i d not w a r r a n t a model b a s e d upon t h o s e i d e a s f o r w a r d e d by F e a s t and f u r t h e r s a m p l i n g o f t h i s and a d j a c e n t f i e l d s i s n e c e s s a r y t o c o n f i r m o r deny t h e s e s p e c u l a t i o n s i n t h e c a s e o f c a r b o n s t a r s . 71 CHAPTER V CONCLUSIONS T h i s t h e s i s has u n c o v e r e d many k i n e m a t i c a l p r o p e r t i e s of c a r b o n s t a r s i n t h e L a r g e and S m a l l M a g e l l a n i c C l o u d s . T h e s e p r o p e r t i e s and some o f t h e i r i m p l i c a t i o n s a r e l i s t e d below. The f i r s t p r o p e r t y n o t e d was t h a t c a r b o n s t a r s l i s t e d i n t h e SP c a t a l o g u e a p p e a r e d t o be k i n e m a t i c a l l y q u i t e d i f f e r e n t f r o m t h o s e s t a r s l i s t e d i n t h e WORC and BMB c a t a l o g u e s . The SP c a r b o n s t a r s s t u d i e d showed no s y s t e m a t i c r o t a t i o n , w h i l e c a r b o n s t a r s f r o m t h e WORC c a t a l o g u e d i d . The i m p l i c a t i o n was t h a t t h e SP c a t a l o g u e was a good s o u r c e o f c a r b o n s t a r s t h a t were k i n e m a t i c a l l y a n a l a g o u s t o P o p u l a t i o n I I c a r b o n s t a r s i n our own g a l a x y . As t h o s e s t a r s t a k e n from t h e WORC c a t a l o g u e d i d e x h i b i t s y s t e m a t i c r o t a t i o n i n t h e LMC i t was f o u n d t h a t t h e s e s t a r s were k i n e m a t i c a l l y l i k e g a l a c t i c P o p u l a t i o n I c a r b o n s t a r s . T h i s d i v i s i o n was c o n f i r m e d t h r o u g h p h o t o m e t r y and s p e c t r o s c o p i c a n a l y s i s . BMB c a r b o n s t a r s a r e a l s o f o u n d t o be s i m i l a r t o g a l a c t i c P o p u l a t i o n I o b j e c t s (Cohen e t a l 1981) and were i n c l u d e d i n t h e sample w i t h s t a r s f r o m t h e WORC c a t a l o g u e . A l t h o u g h t h e WORC and SP c a t a l o g u e s e a c h l i s t a p p r o x i m a t e l y 300 c a r b o n s t a r s o v e r t h e f a c e o f t h e LMC, t h i s i s a s m a l l f r a c t i o n o f t h e t o t a l number of c a r b o n s t a r s p r e s e n t i n t h i s g a l a x y . T h i s p o i n t was made c l e a r by t h e BMB c a t a l o g u e w h i c h t o t a l l y s a m p l e d s e v e r a l 23 m i n u t e d i a m e t e r f i e l d s i n b o t h t h e LMC and t h e SMC and f o u n d a p p r o x i m a t e l y 70 c a r b o n s t a r s i n e a c h f i e l d . T h i s h o l d s g r e a t p r o m i s e f o r o u t l i n i n g t h e k i n e m a t i c s o f t h e LMC (and a l s o t h e SMC) i n g r e a t d e t a i l t h r o u g h f u t u r e s u r v e y s o f t h i s l a r g e sample o f c a r b o n s t a r s . 72 The c a r b o n s t a r s t h a t were employed i n t h i s t h e s i s , a l o n g w i t h o t h e r s t e l l a r g r o u p s , r e v e a l e d two i n t e r e s t i n g f a c t s a b o u t t h e manner i n w h i c h t h e LMC r o t a t e d . F i r s t l y , t h e r a d i a l v e l o c i t i e s of t h e s e o b j e c t s i m p l i e d t h e b e s t i n t e r p r e t a t i o n was t h a t o r b i t a l v e l o c i t i e s o f o b j e c t s i n t h e LMC i n c r e a s e l i n e a r l y w i t h d i s t a n c e f r o m t h e d y n a m i c a l c e n t r e . I t was a l s o f o u n d t h a t t h e v e l o c i t y g r a d i e n t o f t h e s o l i d body r o t a t i o n c u r v e s f o r t h i s g a l a x y became l e s s s t e e p as o l d e r o b j e c t s were employed f o r t h e d a t a s a mple. T h i s was i n t e r p r e t e d as an i n c r e a s e i n o r b i t a l e c c e n t r i c i t y o f t h o s e o b j e c t s o f i n c r e a s i n g age. T h i s h y p o t h e s i s a g r e e d f a v o u r a b l y w i t h e a r l i e r f i n d i n g s i n o u r own g a l a x y by Eggen , L y n d e n - B e l l and Sandage ( 1 9 6 2 ) . In an a t t e m p t t o q u a n t i f y t h i s r e l a t i o n s h i p between o b j e c t age and v e l o c i t y g r a d i e n t i n t h e LMC, a p p r o x i m a t e l i f e t i m e s were o b t a i n e d f o r t h e o b j e c t s employed t o g e n e r a t e t h e r o t a t i o n c u r v e s ( T a y l e r 1975). The numbers f o u n d were e x h i b i t e d i n T a b l e IV and a i t was a l s o f o u n d t h a t t h e v e l o c i t y g r a d i e n t was p r o p o r t i o n a l t o t h e l o g a r i t h m o f t h e o b j e c t s age. The g r a d i e n t o f t h e c a r b o n s t a r s o l i d body r o t a t i o n c u r v e a l o n g w i t h t h e p o s i t i o n s o f t h e s t a r s i n t h i s s u r v e y was a l s o e m ployed t o d e t e r m i n e a minimum mass f o r t h e LMC. A p p l y i n g t h e a s s u m p t i o n o f c i r c u l a r v e l o c i t i e s f o r t h e s e o b j e c t s l e d t o a minimum mass f o r t h e d i s k o f 6.0X10 9 Mo. E m p l o y i n g t h e l i n e a r o r b i t a l v e l o c i t y v e r s u s r a d i a l d i s t a n c e r o t a t i o n c u r v e s and t h e BMB f i e l d s i n t h e LMC and SMC, i n s i g h t i n t o t h e v e l o c i t y d i s p e r s i o n o f t h e s e two g a l a x i e s was a l s o g a i n e d . As was p r e v i o u s l y s t a t e d , t h e o r y p r e d i c t e d t h a t t h e v e l o c i t y d i s p e r s o n o f a g r o u p o f o b j e c t s s h o u l d i n c r e a s e a s t h e 73 age o f t h o s e o b j e c t s i n c r e a s e d . The d a t a base e m p l o y e d i n t h i s t h e s i s c o u l d n e i t h e r c o n f i r m nor deny t h i s h y p o t h e s i s . I t was however f o u n d t h a t t h e v e l o c i t y d i s p e r s i o n o f t h e BMB Bar West f i e l d was a p p r o x i m a t e l y t h e same as t h e v e l o c i t y d i s p e r s i o n o f c a r b o n s t a r s a l o n g t h e l e n g t h of t h e r o t a t i o n c u r v e t h a t t h e s e o b j e c t s d e f i n e d f o r t h e LMC. T h i s i m p l i e d t h e r a d i a l v e l o c i t y d i s p e r s i o n o f c a r b o n s t a r s was a g l o b a l q u a n t i t y s u b j e c t t o o n l y m i n o r v a r i a t i o n s w i t h p o s i t i o n a c r o s s t h e f a c e of t h e LMC. C o m p a r i s o n o f t h e BMB f i e l d s r e v e a l e d t h a t t h e r a d i a l v e l o c i t y d i s p e r s i o n of t h e Bar West f i e l d i n t h e LMC was l e s s t h a n t h e d i s p e r s i o n of t h e Bar f i e l d i n t h e SMC. T h i s d i f f e r e n c e i n d i s p e r s i o n was l i k e l y c a u s e d by s e v e r a l f a c t o r s . T h i s t h e s i s p r o p o s e s t h a t t h e v e l o c i t y d i s p e r s i o n was g e n e r a t e d by t h e s t e e p e r i n c l i n a t i o n of t h e SMC o v e r t h e LMC t o o u r l i n e o f s i g h t . The h i g h e r i n c l i n a t o n of t h e SMC would i n c r e a s e t h e r a d i a l v e l o c i t y d i s p e r s i o n by p r e s e n t i n g t o our l i n e o f s i g h t t h a t o r i e n t a t i o n o f t h e c a r b o n s t a r o r b i t s w i t h t h e g r e a t e s t v e l o c i t y d i s p e r s i o n . T h i s h i g h e r v e l o c i t y d i s p e r s i o n would be c a u s e d by t h e h i g h e r v e l o c i t y d i s p e r s i o n i n a d i r e c t i o n p a r a l l e l t o t h e p l a n e o f t h e g a l a x y . The d a t a sample of c a r b o n s t a r s t a k e n from t h e f i e l d i n t h e SMC was examined f o r a b i m o d a l r a d i a l v e l o c i t y d i s t r i b u t i o n a s t h i s would i n d i c a t e s t r e a m i n g m o t i o n s i n o p p o s i n g d i r e c t i o n s f r o m b o t h ends o f t h e b a r o f t h e SMC. The b i m o d a l d i s t r i b u t i o n was n o t o b s e r v e d but t h e r a d i a l v e l o c i t y d i s t r i b u t i o n was v e r y f l a t . F u r t h e r s a m p l i n g o f t h i s f i e l d w o u l d be r e q u i r e d t o c o n f i r m or deny i f t h e r a d i a l v e l o c i t y d i s t r i b u t i o n i n t h i s f i e l d was o f a b i m o d a l n a t u r e . I f s t r e a m i n g m o t i o n s i n t h e bar 74 of t h e SMC were f o u n d t h r o u g h s u c h a r a d i a l v e l o c i t y d i s t r i b u t i o n , t h i s would h e l p t o c o n f i r m and improve upon models by Kormendy ( 1 9 8 1 ) , and o t h e r s , t h a t p r e d i c t s u c h s t r e a m i n g m o t i o n s a r e n e c e s s a r y f o r t h e s t a b i l i t y o f a bar s y s t e m . In c o n c l u s i o n , t h e f u t u r e f o r c a r b o n s t a r s as r a d i a l v e l o c i t y i n d i c a t o r s l o o k s v e r y p r o m i s i n g as t h e s e o l d P o p u l a t i o n I o b j e c t s have t h e a b i l i t y t o o u t l i n e t h e k i n e m a t i c s o f t h e L a r g e and S m a l l M a g e l l a n i c C l o u d s i n f i n e d e t a i l . I t i s hoped t h a t t h i s t h e s i s and f u t u r e s t u d i e s b a s e d upon t h e k i n e m a t i c s o f c a r b o n s t a r s w i l l y i e l d new i n s i g h t s i n t o t h e k i n e m a t i c s and e v o l u t i o n of k i n e m a t i c s i n t h e L a r g e and S m a l l M a g e l l a n i c C l o u d s . 75 A p p e n d i x I_ G e o m e t r i c a l P r e d i c t i o n s o f R a d i a l V e l o c i t y The two r o t a t i o n a l m o d e ls p r o p o s e d i n t h i s t h e s i s a r e b a s e d upon t h e f o l l o w i n g a s s u m p t i o n s . G e o m e t r i c a l l y i t was assumed t h a t t h e d i s k o f t h e LMC.was a t h i n , f l a t d i s k and t h e o r b i t s o f a l l o b j e c t s were p u r e l y c i r c u l a r and i n t h e p l a n e of t h e d i s k . L i n e a r d i s t a n c e s i n t h e p l a n e o f t h e d i s k were a p p r o x i m a t e d v e r y c l o s e l y by a n g u l a r d i s t a n c e s i n c e t h e l a r g e s t a n g u l a r d i s t a n c e s m easured were l e s s t h a n 6 ° . F o r t h e o r i e n t a t i o n a nd p o s i t i o n of t h e d i s k o f t h e LMC t h e v a l u e s d e r i v e d by F e a s t (1964) were a d o p t e d . F e a s t f o u n d t h e d y n a m i c a l c e n t r e of t h e LMC t o be l o c a t e d a t RA=5h20m, Dec=-6 8 . 8 ° , t h e n o r m a l t o t h e p l a n e o f t h e LMC i s i n c l i n e d a t 27° t o t h e l i n e o f s i g h t and t h e major a x i s o f t h e LMC i s a t 171° e a s t o f n o r t h . F o r t h e p o s i t i o n s of t h e o b j e c t s u s e d i n t h i s t h e s i s , t h e WORC and SP s t a r s were e p o c h 1975 and a l l o t h e r p o s i t i o n s were e p o c h 1950. T h i s d i f f e r e n c e i n ep o c h had n e g l i g a b l e e f f e c t on t h e WORC p o s i t i o n s a s t h e maximum e f f e c t o f p r e c e s s i o n between t h e two epoc h s i s l e s s t h a n 5' f o r any WORC o b j e c t (Smart 1977). Under t h e above a s s u m p t i o n s t h e g e o m e t r i c a l m o d e ls were for m e d a s f o l l o w s . O b j e c t s i n t h e p l a n e o f t h e g a l a x y were a s s i g n e d p o s i t i o n s i n t h e f o l l o w i n g manner; x'=-rsin0 y = r c o s 9 where x',y',r and 6 a r e a l l q u a n t i t i e s measured i n t h e p l a n e o f t h e g a l a x y and 0 i s mea s u r e d c o u n t e r - c l o c k w i s e from t h e +ve y 76 a x i s a s i s i l l u s t r a t e d i n F i g . X X X I I . I f the p l a n e o f t h e g a l a x y i s r o t a t e d by t h e i n c l i n a t i o n a n g l e i a b o u t t h e y - a x i s , t h e p o s i t i o n o f t h e o b j e c t s i n t h e p l a n e o f t h e sky i s g i v e n by; x = - r s i n 9 c o s i y = r c o s 9 z = - r s i n 8 s i n i What i s now o b s e r v e d a s t h e m a j o r a x i s o f t h e g a l a x y i s t h e y -a x i s . I f t h e major a x i s i s a t an a n g l e o f 0„ e a s t o f n o r t h on t h e p l a n e o f t h e s k y , t h e new c o o r d i n a t e s a s seen on t h e sky a r e g i v e n by; X=-rsin9cosicos0-rcos9sin0 o Y=-rsin9cosisin0> r c o s 9 c o s 0 „ Z=-rsin9sini The X and Y c o o r d i n a t e s c o i n c i d e w i t h s t e l l a r p o s i t i o n as g i v e n by ( R A - r a ) * c o s ( d e c - D E C ) and (dec-DEC) r e s p e c t i v e l y , where (RA,DEC) i s t h e p o s i t i o n o f t h e c e n t r e o f r o t a t i o n o f t h e d i s k and ( r a , d e c ) i s t h e r i g h t a s c e n s i o n and d e c l i n a t i o n o f t h e i n d i v i d u a l o b j e c t . A s s u m i n g m o t i o n i n t h e p l a n e o f t h e g a l a x y i s p u r e l y a n g u l a r i n n a t u r e , t h e Z component c a n be d i f f e r e n t i a t e d t o p r e d i c t t h e o b s e r v e d r a d i a l v e l o c i t y . rift Z=-rsimcos9gY L i n e a r O r b i t a l V e l o c i t y v s . R a d i u s R e l a t i o n s h i p . F o r a l i n e a r v e l o c i t y v s . r a d i u s r e l a t i o n s h i p f o r o b j e c t s i n t h e p l a n e o f t h e LMC, t h e f o l l o w i n g q u a n t i t i e s a r e c o n s t a n t ; s i n i = c o n s t . ^ = c o n s t . The o b s e r v e d r a d i a l v e l o c i t i e s s h o u l d be o f the form; R a d i a l V e l o c i t y = y * c o n s t . 77 F i g . XXXII Geometry of D i s k M o d e l s . V t h e 78 T h i s i m p l i e s t h a t t h e o b s e r v e d r a d i a l v e l o c i t y s h o u l d be l i n e a r l y p r o p o r t i o n a l t o t h e d i s t a n c e a l o n g t h e major a x i s of t h e g a l a x y . F l a t R o t a t i o n C u r v e F o r a g a l a x y w i t h a f l a t r o t a t i o n c u r v e ; s i n i = c o n s t . r d r § = c o n s t . T h e r e f o r e t h e o b s e r v e d r a d i a l v e l o c i t y s h o u l d be o f t h e form; R a d i a l V e l o c i t y = c o s 9 * c o n s t . I n o r d e r t o i n v e s t i g a t e t h i s r e l a t i o n s h i p i t c a n be shown; tan9=tan(0-0,)/cosi where tan0=-X/Y T h i s e q u a t i o n was employed t o r e l a t e an o b j e c t ' s X-Y p o s i t i o n t o i t s 9 p o s i t i o n . The a d o p t e d f i t t o t h e d a t a was o b t a i n e d by a d j u s t i n g 0 O u n t i l a b e s t c o s 0 f i t o f v e l o c i t y v s . 9 was f o u n d . 79 B i b l i o g r a p h y A a r o n s o n , M. , Mould, J . , 1980, Ap. J . , 240, 804. A b e l l , C O . , 1975, E x p l o r a t i o n o f t h e U n i v e r s e 3 r d e d . , H o l t , R i n e h a r t and W i n s t o n . A l l e n , C.W., 1981, A s t r o p h y s i c a l Q u a n t i t i e s 3 r d e d . , U n i v e r s i t y of London, The A t h l o n e P r e s s . A r d e b e r g , A., M a u r i c e , E . , 1979, A s t r o n . A s t r o p h y s . , 77, 277. B l a n c o , B.M., B l a n c o , V.M., M c C a r t h y , M.F., 1978, N a t u r e 27J_, 638. B l a n c o , V.M., M c C a r t h y , M.F., B l a n c o , B.M., 1980, Ap. J . , 24_2, 938. Cohen, J.G., F r o g e l , J.A., P e r s s o n , S . E . , E l i a s , J.H., 1981, Ap. J . , 249, 481 . Cowley, A.P., H a r t w i c k , F.D.A., 1981, Do m i n i o n A s t r o p h y s i c a l O b s e r v a t o r y P r e p r i n t 81-0687D. Dean, C.A., 1976, A. J . , 8J_, 364. Eggen, O.J., L y n d e n - B e l l , D., Sandage, A.R., 1962, Ap. J . , J_36, 748. E v a n s , L.T., 1980,'M.N.R.A.S., J_93, 97. F e a s t , M.W., 1964, M.N.R.A.S., J_27» 195. F e a s t , M.W., 1968, M.N.R.A.S., JUO, 345. F e a s t , M.W., 1970, M.N.R.A.S., 149, 291. F e i t z i n g e r , J.V., 1979, P h o t o m e t r y , K i n e m a t i c s and Dynamics o f G a l a x i e s , e d. D.S. E v a n s , D e p t . o f Astronomy, U n i v e r s i t y o f T e x a s a t A u s t i n . F e i t z i n g e r , J.V., S c h m i d t - K a l e r , T., 1982, Ap. J . , 257, 587. F e i t z i n g e r , J.V., W e i s s , G., 1979, A s t r o n . A s t r o p h y s . S u p p l . , 37, 575. Freeman, K.C., 1982, P r i v a t e C o m m u n i c a t i o n . F r o g e l , J.A., B l a n c o , V.M., M c C a r t h y , M.F., Cohen, J.G., 1982, Ap. J . , 252, 133. Kormendy, J . , 1981, The S t r u c t u r e and E v o l u t i o n o f Normal G a l a x i e s , e d. S.M. F a l l , D. L y n d e n - B e l l , Cambridge U n i v e r s i t y P r e s s . 80 van d e r K r u i t , P.C., A l l e n , R . J . , 1978, Ann. Rev. A s t r o n . A s t r o p h y s . , J_6, 103. M i h a l a s , D., B i n n e y , J . , 1981, G a l a c t i c Astronomy 2nd e d . , W.H. Freeman and Company. Moore, C , 1981, UBC CURVE-Curve F i t t i n g R o u t i n e s , C omputing C e n t r 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 . Ovenden, M., 1983, P r i v a t e C o m m u n i c a t i o n . P r e v o t , L., 1973, A s t r o n . A s t r o p h y s . , 28, 165. R i c h e r , H.B., 1981, Ap. J . , 24_3, 744. R i c h e r , H.B., O l a n d e r , N., W e s t e r l u n d , B.E., 1979, Ap. J . , 230, 724. S a n d u l e a k , N., P h i l i p , A.G.D., 1977, Pub. Warner and Swasey O b s e r v a t o r y , V o l . 2, No. 5. S e a r l e , L., W i l k i n s o n , A., B a g n o u l o , W.G., 1980, Ap. J . , 239, 803. Smart, W.M., 1977, T e x t b o o k on S p h e r i c a l Astronomy, C a m b r i d g e U n i v e r s i t y P r e s s . S m i t h , M.G., Weedman, D.W., 1973, Ap. J . , J_79, 461. S o b r i n o , L . , 1982, P r i v a t e C o m m u n i c a t i o n . Sweet, P.A., 1983, P r i v a t e C ommunication t h r o u g h M. Ovenden. T a y l e r , R . J . , 1975, The O r i g i n o f t h e C h e m i c a l E l e m e n t s , Wykeham P u b l i c a t i o n s (London) L t d . T o p p i n g , J . , 1972, E r r o r s o f O b s e r v a t i o n and T h e i r T r e a t m e n t , Chapman and H a l l . de V a u c o u l e u r s , G., Freeman, K.C., 1973, V i s t a s i n A s t r o n . , J_4, 163. W a l l e r s t e i n , G., 1973, Ann. Rev. A s t r o n . A s t r o p h y s . , _n_, 115. W e s t e r l u n d , B.E., O l a n d e r , N., R i c h e r , H.B., C r a b t r e e , D.R., 1978, A s t r o n . A s t r o p h y s . S u p p l . , 3J_, 61. 

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