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The kinematics of gas and stars in the solar neighbourhood Goulet, Thomas 1984

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THE KINEMATICS OF GAS AND STARS IN THE SOLAR NEIGHBOURHOOD by THOMAS GOULET B . S c , The U n i v e r s i t y of Sherbrooke, 1982 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Phys i c s ) We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA August 1984 © Thomas Goulet, 1984 In 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree t h a t the 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 study. I f u r t h e r agree 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 copying o f t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or 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 gain s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of P tf / S I C S The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date 2?/0% fSH ABSTRACT The kinematic p r o p e r t i e s of gas and s t a r s in the s o l a r neighbourhood are d e s c r i b e d in terms of the l i n e - o f - s i g h t component of a th r e e - d i m e n s i o n a l f i r s t order T a y l o r s e r i e s expansion of the l o c a l v e l o c i t y f i e l d . The types of o b j e c t analysed are: 1) 21-cm absorbing clouds, 2) HI i n t e r c l o u d medium, 3) main sequence B s t a r s , 4) B s t a r s of l u m i n o s i t y c l a s s ranging from I to IV, 5) main sequence A s t a r s , 6) K-giant s t a r s . The l e a s t squares f i t t i n g procedure used to d e r i v e the 10 c o e f f i c i e n t s d e s c r i b i n g the f i e l d s f o r v a r i o u s types of o b j e c t s was e s s e n t i a l l y the same so that a v a l i d comparison c o u l d be made. Marked departures from c i r c u l a r motion are found i n most cases but the only systematic trend i s a c o r r e l a t i o n between 9u/3x (u being the v e l o c i t y component along the x - a x i s d i r e c t e d towards the g a l a c t i c center) and s t e l l a r s p e c t r a l type, where the gas behaves l i k e a medium "younger" than the e a r l y type s t a r s . The a n a l y s i s of the gas i n d i c a t e s that the standard p l a n e - p a r a l l e l model p r o v i d e s a good d e s c r i p t i o n of the i n t e r c l o u d medium but i s inadequate fo r the absorbing c l o u d s . A v e l o c i t y e l l i p s o i d d e s c r i p t i o n of the r e s i d u a l s i s presented f o r each group of o b j e c t s . The i n f l u e n c e of the Gould b e l t and of nearby s p i r a l arms on l o c a l kinematics i s d i s c u s s e d . i i i TABLE OF CONTENTS Ab s t r a c t i i L i s t of Tables v i L i s t of F i g u r e s ix Acknowledgments x i v I n t r o d u c t i o n 1 1: The study of the kinematics of our Galaxy 4 a) m o t i v a t i o n and d i f f i c u l t i e s 4 b) the study of n o n - c i r c u l a r motion i n the l i t e r a t u r e 14 c) goals and c h a r a c t e r i s t i c s of t h i s study 27 2: Determination of the kinematic parameters 30 a) the v e l o c i t y f i e l d equation 30 b) the v e l o c i t y e l l i p s o i d model of v e l o c i t y d i s p e r s i o n 40 c) c o l l e c t i o n of the data 45 d) treatment of the data 47 i v 3: The kinematics of s t a r s 55 a) p r e s e n t a t i o n of the r e s u l t s 55 b) r e s u l t s f o r B s t a r s 58 c) r e s u l t s f o r AV s t a r s 66 d) r e s u l t s f o r K i l l s t a r s 72 e) r e s u l t s f o r FV s t a r s 76 4: The kinematics of the i n t e r s t e l l a r medium 113 a) treatment of the data 113 b) r e s u l t s f o r 21-cm emission data .....118 c) r e s u l t s f o r 21-cm a b s o r p t i o n data 121 d) r e s u l t s f o r s i n g l e a b s o r p t i o n f e a t u r e s 125 e) r e s u l t s f o r s t a r s and i n t e r s t e l l a r medium 130 5: D i s c u s s i o n 153 a) comparison of the r e s u l t s 153 b) the u n c e r t a i n t y i n d i s t a n c e determination for the gas 162 c) the constancy of the Oort c o e f f i c i e n t 164 d) the importance of the Gould b e l t 166 e) s p i r a l s t r u c t u r e and kinematics 168 f) the use of second order terms 170 g) the i n t e r p r e t a t i o n of the v e l o c i t y f i e l d s 171 Con c l u s i o n s 174 Appendix References v i LIST OF TABLES I A n a l y s i s of the d i s t r i b u t i o n s 32 II Values of the Vk term 36 III R e s u l t s f o r B s t a r s 60 IV R e s u l t s f o r B s t a r s using a 9 c o e f f i c i e n t f i t ....62 V R e s u l t s f o r B s t a r s using equation (1-15) 63 VI R e s u l t s for B I - I V s t a r s 65 V I I R e s u l t s f o r BV s t a r s with d > 200 pc 67 VIII R e s u l t s f o r BV s t a r s with d < 200 pc 68 IX R e s u l t s f o r AV s t a r s 70 X Comparison of our v e l o c i t y e l l i p s o i d with Gomez (1972) 71 XI R e s u l t s f o r A ( 0 - 4 ) V s t a r s 73 XII R e s u l t s f o r A(5-9)V s t a r s 74 XIII R e s u l t s f o r K i l l s t a r s 75 XIV Comparison of our v e l o c i t y e l l i p s o i d with Gomez (1972) 77 XV R e s u l t s f o r FV s t a r s 79 XVI R e s u l t s f o r 21-cm emission 120 XVII Comparison of our r e s u l t s with Takakubo(1967) ...122 XVIII R e s u l t s f o r 21-cm a b s o r p t i o n 124 XIX A l l a b s o r t i o n s p e c t r a comparison of two models of s p a t i a l d i s t r i b u t i o n .126 XX R e s u l t s f o r s i n g l e a b s o r p t i o n f e a t u r e s 128 XXI S i n g l e a b s o r p t i o n f e a t u r e s comparison of two models of s p a t i a l d i s t r i b u t i o n .129 XXII R e s u l t s f o r s t a r s and i n t e r s t e l l a r medium 132 v i i i XXIII Comparison of the r e s u l t s obtained for s t a r s and gas 154 XXIV The v a r i a t i o n of 9u/9x and 1 0 with age 158 XXV The i n f l u e n c e of the Gould b e l t 169 LIST OF FIGURES Fi g u r e 1 9 F i g u r e 2 9 F i g u r e 3 80 F i g u r e 4 81 F i g u r e 5 81 Figure 6 82 F i g u r e 7 82 F i g u r e 8 83 F i g u r e 9 83 F i g u r e 10 84 F i g u r e 11 84 F i g u r e 12 85 F i g u r e 13 86 F i g u r e 14 86 F i g u r e 15 87 F i g u r e 16 87 F i g u r e 17 88 F i g u r e 18 88 F i g u r e 19 • 89 F i g u r e 20 89 F i g u r e 21 90 Figure 22 90 Figure 23 91 X F i g u r e 24 91 F i g u r e 25 92 F i g u r e 26 92 F i g u r e 27 93 F i g u r e 28 93 F i g u r e 29 94 F i g u r e 30 94 F i g u r e 31 95 F i g u r e 32 95 F i g u r e 33 96 F i g u r e 34 96 F i g u r e 35 97 F i g u r e 36 97 F i g u r e 37 98 F i g u r e 38 98 F i g u r e 39 99 F i g u r e 40 99 F i g u r e 41 100 F i g u r e 42 100 F i g u r e 43 101 F i g u r e 44 101 F i g u r e 45 102 F i g u r e 46 102 F i g u r e 47 103 F i g u r e 48 103 F i g u r e 49 104 x i F i g u r e 50 1 0 4 F i g u r e 51 1 0 5 F i g u r e 52 1 0 5 F i g u r e 53 1 0 6 F i g u r e 54 1 0 6 F i g u r e 55 1 0 7 F i g u r e 56 1 0 7 F i g u r e 57 1 0 8 F i g u r e 58 1 0 8 F i g u r e 59 1 0 9 F i g u r e 60 1 0 9 F i g u r e 61 1 1 0 F i g u r e 62 1 1 0 F i g u r e 63 1 1 1 F i g u r e 64 1 1 1 F i g u r e 65 1 1 2 F i g u r e 66 1 1 2 F i g u r e 67 1 3 3 F i g u r e 68 1 3 3 F i g u r e 69 1 3 4 F i g u r e 70 1 3 5 F i g u r e 71 ...135 F i g u r e 72 1 3 6 F i g u r e 73 1 3 6 F i g u r e 74 1 3 7 F i g u r e 75 1 3 7 x i i F i g u r e 76 138 F i g u r e 77 . 1 38 F i g u r e 78 139 F i g u r e 79 139 F i g u r e 80 140 F i g u r e 81 141 F i g u r e 82 141 F i g u r e 83 142 F i g u r e 84 142 F i g u r e 85 143 F i g u r e 86 1 43 F i g u r e 87 144 F i g u r e 88 1 44 F i g u r e 89 145 F i g u r e 90 145 F i g u r e 91 1 46 F i g u r e 92 146 Figure 93 j 147 F i g u r e 94 147 Figure 95 148 Figure 96 148 Figure 97 149 Figure 98 ..150 Figure 99 1 50 Figure 100 151 Figure 101 151 x i i i F i g u r e 1 02 i 52 F i g u r e 103 152 F i g u r e 104 159 x i v ACKNOWLEDGMENTS I would l i k e to thank Dr. W.L.H. Shuter, my s u p e r v i s o r , f o r h i s constant guidance and a s s i s t a n c e thoughout t h i s p r o j e c t and f o r h i s f i n a n c i a l support. I would a l s o l i k e to thank my good f r i e n d s Robert Scharein, Andre Roberge, Debbie P h i l l i p s , Jim F i n l a y s o n and Steve Pearce who c o n t r i b u t e d to make the l a s t two years very pleasant and r i c h i n ex p e r i e n c e s . F i n a l l y , thanks i s due to Alec Sandy and C h r i s Chan whose help was g r e a t l y a p p r e c i a t e d . XV " Les gens ont des e t o i l e s q ui ne sont pas l e s memes. Pour l e s uns, qui voyagent, l e s e t o i l e s sont des guides. Pour d'autres e l l e s ne sont r i e n que des p e t i t e s l u m i e r e s . Pour d'autres qui sont savants e l l e s sont des problemes. Pour mon businessman e l l e s e t a i e n t de 1'or. Mais toutes ces e t o i l e s - l a se t a i s e n t . T o i , t u auras des e t o i l e s comme personne n'en a... " St-Exupery, Le P e t i t P r i n c e 1 INTRODUCTION The present p i c t u r e of our Galaxy i s one of a g i g a n t i c system composed of gas, s t a r s and dust i n which our s o l a r system i s embedded. I t can be s u b d i v i d e d i n t o 1) a c e n t r a l bulge where very e n e r g e t i c events take p l a c e ; 2) a s p h e r i c a l h a l o of g l o b u l a r c l u s t e r s and i n d i v i d u a l s t a r s that tend to be "metal" d e f i c i e n t and thus members of an o l d ge n e r a t i o n ; and 3) a very f l a t d i s k where most of the s t a r s , gas and dust l i e , where a continuous b i r t h of new s t a r s i s t a k i n g p l a c e and where, most probably, the b e a u t i f u l p a t t e r n of s p i r a l arms e x i s t s . The f l a t n e s s of the d i s k component i s c l o s e l y r e l a t e d to i t s s t a t e of r a p i d r o t a t i o n about the g a l a c t i c c e n t e r . There i s more and more evidence now that suggests a f o u r t h component of the Galaxy: the "dark h a l o " that would c o n s t i t u t e most of i t s mass but the nature of which i s s t i l l completely unknown. L o c a l l y , the o b s e r v a t i o n of the r e l a t i v e motion of s t a r s with r e s p e c t to the Sun has r e v e a l e d the f a c t that the di s k i s i n d i f f e r e n t i a l r o t a t i o n . On a l a r g e r s c a l e , i t i s p o s s i b l e to observe the 21-cm l i n e of n e u t r a l hydrogen (HI) throughout the Galaxy and to study i t s mor p h o l o g i c a l s t r u c t u r e . Since there i s no d i r e c t method of determining d i s t a n c e s f o r HI, a c l e a r g l o b a l kinematic p i c t u r e of the Galaxy i s necessary f o r c o r r e c t l o c a l i z a t i o n of the v a r i o u s 2 d e n s i t y f e a t u r e s . The model of kinematics which i s commonly used at t h i s stage i s the p e r f e c t c i r c u l a r r o t a t i o n of a l l the elements of the d i s k p o p u l a t i o n . However, t h i s model i s known to be i n c o r r e c t s i n c e n o n - c i r c u l a r motions are o f t e n observed in e x t e r n a l s p i r a l g a l a x i e s and are even found i n most s t u d i e s of the kinematics performed on v a r i o u s c l a s s e s of o b j e c t s of our Galaxy. In f a c t no one would expect pure c i r c u l a r motion i n a complex system where there are d e n s i t y waves, supernovae e x p l o s i o n s , a c t i v i t y i n the g a l a c t i c center and perhaps even t i d a l i n t e r a c t i o n with nearby companions. The problem with n o n - c i r c u l a r motions i s that they are very i r r e g u l a r and thus hard to d e s c r i b e using simple parameters on l a r g e s c a l e s . As a f i r s t step, v a r i o u s authors decided to study the kinematics of o b j e c t s i n the s o l a r neighbourhood and to determine the kind of n o n - c i r c u l a r motion that can be found. From the r e , attempts have been made to r e l a t e these "streaming motions" to the presence of s p i r a l arms or to other l o c a l subsystems. Since for d i f f e r e n t types of o b j e c t s the kinematic p r o p e r t i e s were i n v e s t i g a t e d and d e s c r i b e d in d i f f e r e n t ways, i t i s hard to have an overview of the v a r i o u s r e s u l t s . With t h i s need for a more u n i v e r s a l formalism f o r kinematic s t u d i e s i n mind, we decided to s y s t e m a t i c a l l y i n v e s t i g a t e d i f f e r e n t groups of o b j e c t s (HI c l o u d , i n t e r c l o u d gaseous medium, s t a r s of v a r i o u s s p e c t r a l types) to see i f any general trend can be detected i n the behaviour of n o n - c i r c u l a r motion parameters 3 as f u n c t i o n s of age, d i s t a n c e , l a t i t u d e , l o n g i t u d e , e t c . T h i s kind of study w i l l i n e v i t a b l y o v e r l a p some already p u b l i s h e d m a t e r i a l and w i l l t h e r e f o r e c o n t r i b u t e to v e r i f y and redetermine some known kinematic parameters such as the p e c u l i a r motion of the Sun and the Oort c o e f f i c i e n t A. I t w i l l a l s o present new r e s u l t s f o r many groups of o b j e c t s and, as we e x p l a i n e d , w i l l provide the p o s s i b i l i t y of comparing d i r e c t l y the v a r i o u s kinematic parameters of these groups. The o r g a n i z a t i o n of t h i s study i s as f o l l o w s . Chapter 1 d i s c u s s e s the importance and the f e a s a b i l i t y of k i n e m a t i c a l s t u d i e s of the Galaxy, presents the a c t u a l s t a t e of the re s e a r c h i n t h i s area and summarizes the d i s t i n c t i v e c h a r a c t e r i s t i c s of our treatment, i t s advantages and l i m i t a t i o n s . Chapter 2 e x p l a i n s the mathematical treatment of the data used to determine the kinematic p r o p e r t i e s of the v a r i o u s groups of o b j e c t s . T h i s d e r i v a t i o n and ex p l a n a t i o n of the mathematical formalism provides the s t r u t u r e f o r the p r e s e n t a t i o n of r e s u l t s f o r the s t a r s (Chapter 3) and f o r the n e u t r a l hydrogen (Chapter 4 ) . In Chapter 5 we d i s c u s s these r e s u l t s and t r y to i n t e r p r e t them. 4 CHAPTER 1 THE STUDY OF THE KINEMATICS OF OUR GALAXY a) M o t i v a t i o n s and d i f f i c u l t i e s The appearance of the M i l k y Way in the night sky suggests that we are embedded i n a f l a t s t e l l a r system that we c a l l our Galaxy. Since the work of the p i o n e e r s of g a l a c t i c astronomy in the l a t e n i n e t e e n t h century, we c o n t i n u o u s l y improved and r e f i n e d our knowledge of the s i z e , shape, composition and s t a t e of motion of our Galaxy. In the e a r l y t w e n t i e t h century the development of a s t r o n o m i c a l photography and the c o n s t r u c t i o n of l a r g e t e l e s c o p e s gave to people such as J.C. Kapteyn, P.J. van R h i j n , H. Shapley, E. Hubble and many oth e r s , a chance to complete the remarkable work of W. H e r s c h e l and to improve the comprehension of the s t r u c t u r e s of the Galaxy. They r e a l i s e d that the .Galaxy i s a huge e n t i t y , by f a r l a r g e r than the s o l a r system or even the s o l a r neighbourhood, where most of the a p p a r e n t l y b r i g h t s t a r s l i e . They a l s o r e a l i z e d t h a t i t i s only one of a l a r g e number of o b j e c t s that " f i l l " r a t h e r s p a r s e l y an enormous u n i v e r s e . The g a l a x i e s found in a v a r i e t y of s i z e s and types then appeared to be the fundamental b u i l d i n g blocks of the u n i v e r s e . The study of our Galaxy continued with B. L i n d b l a d and J.H. Oort (1926) who analysed the kinematics and dynamics of s t e l l a r systems 5 and developed a theory of d i f f e r e n t i a l r o t a t i o n f o r the d i s k component. Afterwards, the f i e l d of g a l a c t i c astronomy became extremely p r o l i f i c with the work of R.J. Trumpler (1930) on a b s o r p t i o n of l i g h t by dust p a r t i c l e s ; of W. Baade (1944) on the concept of s t e l l a r p o p u l a t i o n s ; of W.W. Morgan (1950) on s p i r a l arm t r a c e r s and of H.I. Ewen, E.M. P u r c e l l , W.N. C h r i s t i a n s e n , C A . M u l l e r and J.H. Oort (1951) on the d e t e c t i o n of 21-cm l i n e of n e u t r a l hydrogen, to name only few of them. In the l a t e f i f t i e s , 21-cm surveys were c a r r i e d out e x t e n s i v e l y i n order to produce a map of the Galaxy as seen in the r a d i o wave domain (see F . J . Kerr (1962) and r e f e r e n c e s t h e r e i n ) . In 1964, C.C. L i n and F.H. Shu p u b l i s h e d t h e i r important work on the d e n s i t y wave theory of s p i r a l arms. In 1965 M. Schmidt d e s c r i b e d the d i s t r i b u t i o n of mass in the Galaxy. Since then, more obs e r v a t i o n s have been c a r r i e d out, more data have become a v a i l a b l e and t h i s p r o l i f e r a t i o n of i n f o r m a t i o n about our Galaxy has helped us to r e a l i z e that we s t i l l have a l o t to l e a r n . For example, the r e a l i z a t i o n of the importance of n o n - c i r c u l a r motion c a s t doubt upon the r e l i a b i l i t y of the HI maps produced e a r l i e r . The study of the r o t a t i o n curve r e v e a l e d the inadequacy of the Schmidt's model of the Galaxy and suggested the e x i s t e n c e of a "dark h a l o " of unknown nature t h a t would d r a m a t i c a l l y i n c r e a s e the s i z e and the mass of the Galaxy. The recent a v a i l a b i l i t y of new "windows" on the u n i v e r s e l i k e m i l l i m e t e r waves, i n f r a - r e d , 6 u l t r a v i o l e t , x-rays and 7-rays provides an i n c r e d i b l e amount of new in f o r m a t i o n that has yet to be t r e a t e d and d i g e s t e d f o r a c l e a r e r understanding of t h i s system i n which we a r e . The study of the kinematics of the Galaxy i s bound to two major problems that even today prevent us from having a r e l i a b l e p i c t u r e of i t s o v e r a l l s t a t e of motion. One of these problems i s that there e x i s t s no method of determining the d i s t a n c e s to HI f e a t u r e s d e t e c t e d with the 21-cm l i n e . The same problem a p p l i e s to data obtained with molecular l i n e s and which d e s c r i b e s "dark" molecular c l o u d s . The other problem i s t h a t , i n c o n t r a s t to HI, the s t a r s cannot be observed at l a r g e d i s t a n c e s from the s o l a r system because t h e i r l i g h t i s s e v e r e l y s c a t t e r e d by the dust p a r t i c l e s which are h i g h l y c o n c e n t r a t e d i n the plane of the Galaxy. T h i s l i m i t s the range of d i s t a n c e s over which s t u d i e s of the kinematics of the s t e l l a r p o p u l a t i o n can be c a r r i e d . The advantage of a n a l y s i n g s t a r motions i s that there e x i s t some methods of determination of t h e i r d i s t a n c e s . For d<20 pc, t r i g o n o m e t r i c p a r a l l a x e s can be used while f o r l a r g e r d i s t a n c e s , we r e l y on s p e c t r o s c o p i c p a r a l l a x e s . T h i s l a t t e r method i s not as p r e c i s e because of the u n c e r t a i n t i e s on s p e c t r a l type determinations and on c o r r e c t i o n s f o r i n t e r s t e l l a r e x t i n c t i o n . The transparency of the Galaxy i n 21-cm i s due to the f a c t t h at the dust p a r t i c l e s are much smaller than 21 centimeters and thus i n e f f i c i e n t at s c a t t e r i n g r a d i a t i o n 7 with such a long wavelength. As mentioned e a r l i e r , t h i s transparency allows the o b s e r v a t i o n of the d i s t r i b u t i o n of HI on g l o b a l s c a l e but there i s a problem in the l o c a l i z a t i o n of the v a r i o u s s t r u c t u r e s . The method used in the l a t e f i f t i e s was to p o s t u l a t e the e x i s t e n c e of pure d i f f e r e n t i a l c i r c u l a r motion and to convert the v e l o c i t i e s a s s o c i a t e d with a measurement of the HI d e n s i t y i n t o a d i s t a n c e from the g a l a c t i c c e n t e r . F i g u r e 1 shows how the data appears on a VI vs 1 diagram and F i g u r e 2 shows the map that was c o n s t r u c t e d by Oort, Kerr and Westerhout i n 1958 from such data. There are two problems with t h i s method. F i r s t , d i s t a n c e determination i s ambiguous due to the f a c t t h a t , f o r most cases i n s i d e the s o l a r o r b i t , two l o c a t i o n s along the l i n e of s i g h t correspond to the same observed v e l o c i t y . Second, t h i s method i s very s e n s i t i v e to small d e v i a t i o n s (few %) from the assumed kinematic model. In f a c t , W.B. Burton (1973) showed that streaming motions of the order of 5 km/s c o u l d e x p l a i n the observed f e a t u r e s of F i g u r e 1 as w e l l as d e n s i t y c o n t r a s t s of 10 or 100 to 1. Since n o n - c i r c u l a r motions of that magnitude are observed i n e x t e r n a l g a l a x i e s and i n our own, the map of F i g u r e 2 became very q u e s t i o n a b l e . Thus, i t seems e s s e n t i a l to develop a more r e a l i s t i c model of the kinematics of the Galaxy. In t h i s regard, two types of i n v e s t i g a t i o n must be undertaken. The f i r s t i s a study of morphology and of kinematics of e x t e r n a l s p i r a l g a l a x i e s . The second i s a study of the 8 kinematics of the s o l a r neighbourhood. The study of e x t e r n a l s p i r a l g a l a x i e s i s d i f f i c u l t because only a few candidates are c l o s e enough to be r e s o l v e d i n d e t a i l . I n c l i n a t i o n must be l i m i t e d to intermediate values because low i n c l i n a t i o n s l i m i t c r i t i c a l l y the component of the v e l o c i t i e s along the l i n e of s i g h t while h i g h i n c l i n a t i o n s prevent a c l e a r i d e n t i f i c a t i o n of m o r p h o l o g i c a l s t r u c t u r e s l i k e the s p i r a l arms. However, s t u d i e s have been c a r r i e d out and, as reviewed by A . Bosma (1981), d i f f e r e n t types of n o n - c i r c u l a r motions have been observed. They can be c l a s s i f i e d as : 1) motions a s s o c i a t e d with the s p i r a l arms ; 2) l a r g e s c a l e symmetric d e v i a t i o n s (oval d i s t o r t i o n s or warp of the d i s k ) ; 3) l a r g e s c a l e asymmetries ( u s u a l l y a t t r i b u t e d to t i d a l i n t e r a c t i o n s with neighbouring g a l a x i e s ) ; and 4) small s c a l e asymmetries. A l l of these c o u l d be important i n the s o l a r neighbourhood but most l i k e l y 1) and 4) are dominant. I t i s i n t e r e s t i n g to note that n o n - c i r c u l a r motions a s s o c i a t e d with s p i r a l arms do not always behave as one would expect from the d e n s i t y wave theory. As mentioned by P.C. Van der K r u i t and R.J. A l l e n (1978) (see r e f e r e n c e s t h e r e i n ) the r e s u l t s from v a r i o u s s t u d i e s are somewhat c o n t r a d i c t o r y on t h i s i s s u e . S t u d i e s of the kinematics of o b j e c t s i n the s o l a r neighbourhood can be done on many d i f f e r e n t s c a l e s . Using HI data and b r i g h t young s t a r s that can be observed up to a few k i l o p a r s e c s , the presence of l o c a l s p i r a l f e a t u r e s c o u l d 9 t F i g u r e 1 : map of 21-cm b r i g h t n e s s temperature as a f u n c t i o n of 1 and VI. Darker areas i n d i c a t e higher temperatures. From Mihalas and Binney (1981). F i g u r e 2 : d i s t r i b u t i o n of n e u t r a l hydrogen d e n s i t i e s i n the g a l a c t i c plane as determined from the Dutch and A u s t r a l i a n surveys by Oort et a l . 1958. From Burton (1973). 1 0 be d e t e c t e d by t h e i r i n f l u e n c e on kinematics as p r e d i c t e d by L i n , Yuan and Shu (1969). HI i n p a r t i c u l a r seems to be a good t r a c e r of s p i r a l arms because the streaming motions p r e d i c t e d f o r i t are l a r g e r than for s t a r s . On a smaller s c a l e (- 500 pc) i t c o u l d be p o s s i b l e to i n v e s t i g a t e the r e a l i t y of l o c a l subsystems l i k e the one which i s a s s o c i a t e d with the Gould b e l t and which authors such as M. Bonneau (1964) and J.R. Lesh (1968, 1972) d e s c r i b e d as being an expanding r i n g of young s t a r s . P.O. L i n d b l a d (1967) and H. Weaver (1974) claimed that the motion of the n e u t r a l hydrogen i s a l s o r e l a t e d to i t . A l o c a l subsystem l i k e t h i s one would correspond to what Bosma c l a s s i f i e d as a "small s c a l e asymmetry". Both the study of the p r e d i c t i o n s of the s p i r a l d e n s i t y wave theory and the study of the r e a l i t y of an expanding system are, as we w i l l see in the next s e c t i o n , bound to d i f f i c u l t problems. In both cases i t seems that these t h e o r i e s can e x p l a i n some of the observed kinematics but that they cannot e x p l a i n i t completely. There i s a l s o the problem which i s e s p e c i a l l y severe fo r the d e n s i t y wave theory, that there are too many a d j u s t a b l e parameters in the t h e o r e t i c a l models and that no unique s o l u t i o n can be der i v e d . On an even smaller s c a l e (= 200 p c ) , the f l a t n e s s of the s p a t i a l d i s t r i b u t i o n of gas and s t a r s along the g a l a c t i c mid-plane becomes l e s s important and i t i s p o s s i b l e to study 11 the motion of m a t e r i a l i n the d i r e c t i o n p e r p e n d i c u l a r to the plane ( z - d i r e c t i o n ) . Since the e a r l y work of J.H. Oort (1932) on the d e n s i t y of the d i s k as d e r i v e d from the motion of s t a r s towards the g a l a c t i c p o l e s , the a n a l y s i s and i n t e r p r e t a t i o n of such motions has remained p u z z l i n g . Oort showed t h a t the l o c a l d e n s i t y d e r i v e d from the study of the kinematics, using the Boltzmann equation, i s twice as l a r g e as the one i n f e r e d from an i n t e g r a t i o n of the s t e l l a r l u m i n o s i t y f u n c t i o n . T h i s "missing mass" problem i s a l s o present i n more recent s t u d i e s (Oort 1960, B a c h c a l l 1984). These r e s u l t s are based on the assumption that the s e l e c t e d group of s t a r s i s " r e l a x e d " , i . e . , that i t has reached a s t a t e of s t a t i s t i c a l e q u i l i b r i u m . I.R. King (1983) p o i n t e d out that some groups of s t a r s are known to be too young to be i n e q u i l i b r i u m (A s t a r s f o r example) and that some others are k i n e m a t i c a l l y inhomogeneous (such as K i l l s t a r s which may have v a r i o u s main sequence p r o g e n i t o r s ) . F and G s t a r s might be more s u i t a b l e i n t h i s sense but any type of object would be s u b j e c t to p e r t u r b a t i o n s of t h e i r k i n e m a t i c a l behaviour i f a systematic c i r c u l a t i o n of m a t e r i a l e x i s t e d in the s o l a r neighbourhood. H. Weaver (1974) i n d i c a t e d from h i s a n a l y s i s of the motion of HI that such a c i r c u l a t i o n c o u l d e x i s t . H i s idea was that the Gould b e l t i s a s s o c i a t e d with a plane of c o n t i n u o u s l y expanding gas, t h i s system being fed by a flow of incoming gas from the g a l a c t i c p o l e s . The r e a l i t y of such a system has not been f i r m l y e s t a b l i s h e d 12 but the p o s s i b i l i t y of i t s e x i s t e n c e s t r e s s e s the importance of a n a l y s i n g the three-dimensional v e l o c i t y f i e l d of the s o l a r neighbourhood. An i n t e r e s t i n g c h a r a c t e r i s t i c of the study of the kinematics of nearby gas i s that i t can pro v i d e some in f o r m a t i o n on the s p a t i a l d i s t r i b u t i o n of i n t e r s t e l l a r m a t e r i a l . The standard d e s c r i p t i o n of the HI s p a t i a l d i s t r i b u t i o n i s given by the " p l a n e - p a r a l l e l model" i n which the d e n s i t y of gas clouds and of the i n t e r c l o u d medium decreases with z (the e l e v a t i o n over the g a l a c t i c mid-plane) but does not depend on the g a l a c t i c l o n g i t u d e 1. The v a l i d i t y of t h i s model can be i n v e s t i g a t e d t o a c e r t a i n degree with a study of the kinematics. In f a c t we can t e s t the p r e d i c t i o n that the average d i s t a n c e of the i n t e r s t e l l a r m a t e r i a l along a given l i n e of s i g h t i s simply p r o p o r t i o n a l to cosecant b (b i s the g a l a c t i c l a t i t u d e ) . A f u r t h e r i n t e r e s t i n g p o i n t of a k i n e m a t i c a l study of a v a r i e t y of o b j e c t s i s that i t can he l p understand the e v o l u t i o n of the Galaxy. I t i s w e l l known f o r in s t a n c e that the v e l o c i t y d i s p e r s i o n of s t a r s i s an i n c r e a s i n g f u n c t i o n of t h e i r s p e c t r a l type ( r e l a t e d to t h e i r age). Such a f a c t suggests the e x i s t e n c e of some mechanism of d i f f u s i o n of s t e l l a r o r b i t s (see R. Wielen (1983) and re f e r e n c e s t h e r e i n ) . A comparative study of the d e v i a t i o n s from c i r c u l a r motion f o r d i f f e r e n t types of o b j e c t s having d i f f e r e n t ages c o u l d p r o v i d e some enlightment on the 13 e v o l u t i o n of the streaming motions. A w e l l e s t a b l i s h e d example of a c o r r e l a t i o n between a n o n - c i r c u l a r motion parameter and s p e c t r a l type i s the v a r i a t i o n of the v e r t e x angle. As w i l l be e x p l a i n e d i n the next chapter, the vertex angle i n d i c a t e s a type of departure from the expected p r o p e r t i e s of the r e s i d u a l v e l o c i t i e s of s t a r s . I t s value tends to decrease with the age of s t a r s . A systematic d e t e r m i n a t i o n of v a r i o u s n o n - c i r c u l a r motion parameters would allow one to i n v e s t i g a t e whether or not any other q u a n t i t y evolves i n a s i m i l a r way and c o u l d t h e r e f o r e h e l p to c l a r i f y the causes of the v a r i a t i o n of the vertex angle or r e v e a l new aspects of the e v o l u t i o n a r y c h a r a c t e r i s t i c s of g a l a c t i c kinematics. A l l these p i e c e s of i n f o r m a t i o n p o s s i b l y d e r i v a b l e from s t u d i e s of the kinematics of o b j e c t s i n the s o l a r neighbourhood have motivated astronomers to do a great deal of i n v e s t i g a t i o n s over the l a s t twenty years. Having s t a t e d the main goals and d i f f i c u l t i e s of such s t u d i e s , we w i l l now turn our a t t e n t i o n towards the l i t e r a t u r e to see what has been done r e c e n t l y in t h i s a r e a . T h i s w i l l h e l p us to c l a r i f y i n which sense the present study can improve the a c t u a l comprehension of the s t r u c t u r e of our Galaxy. b) The study of the departure from c i r c u l a r motion i n the 1i t e r a t u r e The f i r s t p art of t h i s s e c t i o n i s an i d e n t i f i c a t i o n of the main equations that v a r i o u s authors have used to d e s c r i b e n o n - c i r c u l a r motions. I t i s followed by a summary of the most important s t u d i e s c a r r i e d out over the l a s t twenty years on t h i s s u b j e c t . T h i s summary n e i t h e r pretends to be complete nor to d e s c r i b e i n d e t a i l the treatment and r e s u l t s of each study. Rather, i t s purpose i s to give a general idea of the s t a t e of the research on n o n - c i r c u l a r motions and to i n d i c a t e which d i r e c t i o n f u r t h e r s t u d i e s should take. A d i r e c t comparison of the outcomes of t h i s study with some p u b l i s h e d r e s u l t s i s d e a l t with i n Chapters 3 and 4. Kinematic s t u d i e s can be done e i t h e r with proper motions or with l i n e - o f - s i g h t v e l o c i t i e s . Proper motions are more i n f o r m a t i v e but t h e i r p r e c i s i o n decreases d r a m a t i c a l l y with i n c r e a s i n g d i s t a n c e s s i n c e they are measurements of angular displacements. Thus they are used mainly f o r s t u d i e s of nearby o b j e c t s . The O o r t - L i n d b l a d equations of proper motions are: (Ua cos5)" = f ( X s i n a - Y cosa) - C J , cosa sin5 (1-1) - u>2 s i n a sin6 + C J 3 cos6 + P(cos21 cosb costf> + V2 sin21 sin2b sin$) 1 5 (US)" = f(X cosa sin6 + Y N s i n a sinS - Z cos5) (1-2) + !>>! s i n a - io2 cosa + P ( c o s 2 l cosb s i n 0 - V2 sin21 sin2b cos^) with: a;, = Q si n f i s i n i w2 = "Q cosfl s i n i - An a>3 = Q cos i + AK (1-3) An = Ap, s i n e AK = An cotg e - (AX + Ae) where: x,y,z are the d i r e c t i o n s of the axes i n the e q u a t o r i a l system (x p o i n t s towards the v e r n a l equinox at a=0° and 6=0°, y towards a=90°, 6=0° and z towards 6=90°), X,Y,Z are the components of the s o l a r motion in the d i r e c t i o n s x,y,z r e s p e c t i v e l y i n second of arc per century, C L > 1 , W 2 , C O 3 are the components of the angular v e l o c i t y v e c t o r of r o t a t i o n about the axes x,y,z, f i s the p a r a l l a x f a c t o r , l , b are the angles of the g a l a c t i c c o o r d i n a t e system, 4> i s the p a r a l l a c t i c angle, P = A/47.4 i s r e l a t e d to the f i r s t Oort c o e f f i c i e n t A, Q = B/47.4 i s r e l a t e d to the second Oort c o e f f i c i e n t B, i i s the i n c l i n a t i o n of the g a l a c t i c plane to the equator, Q, i s the r i g h t ascension of the ascending node of the g a l a c t i c plane on the equator, Ap, i s the c o r r e c t i o n to the l u n i s o l a r p r e c e s s i o n , AX i s the c o r r e c t i o n to the p l a n e t a r y p r e c e s s i o n , An i s the c o r r e c t i o n to the p r e c e s s i o n i n d e c l i n a t i o n , Ak i s the c o r r e c t i o n to the p r e c e s s i o n i n r i g h t a s c e nsion, Ae i s the c o r r e c t i o n to a l l proper motion Ua due to a spurious motion of the equinox, e i s the o b l i q u i t y of the e c l i p t i c . N.B. For more e x p l a n a t i o n on any of these terms see W. F r i c k e (1971) and r e f e r e n c e s t h e r e i n . 16 A more complete d e s c r i p t i o n of n o n - c i r c u l a r motions with proper motion i s given by the Ogorodnikov-Milne equations which i n c l u d e compressions and shears i n three dimensions. They take the f o l l o w i n g form: (Ua c o s 6 ) " = f (X s i n a - Y cosa) (1-4) - a), cosa sin6 - C J 2 s i n a sin§ + u>2 cos5 + V2 M+ , , ( c o s 2 l sin2b sin</> - sin21 cosb cos<j>) + M +, 2 (cos21 cosb cos0 + V2 sin21 sin2b sintf>) - M +! 3 ( c o s l cos2b s i n ^ + s i n l s i n b cos<t>) + V2 M + 2 2 (sin21 cosb cos<p + s i n 2 l sin2b sine*) + M+ 2 3 ( sin2b sintf>) (US)" = f (X cosa sin6 + Y s i n a sin6 - Z cos6) (1-5) + a;, s i n a - u)2 cosa ~ V2 M+ , , (sin21 cosb sin</> + c o s 2 l sin2b cos^) + M*, 2 (cos21 cosb sin<£ - V2 sin21 sin2b cos<>) + M * u ( c o s l cos2b cos0 - s i n l s i n b sin<£) + M + 2 3 ( s i n l cos2b costf> + c o s l s i n b sin0) + V 2 M + 3 3 (sin2b cos</>) with: u, = M"21 s i n f i s i n i (x)2 = M" 2 1 cos®, s i n i - An (1-6) u>3 = M" 2 1 cos i + Ak where most of the terms have a l r e a d y been d e f i n e d except: M, 17 a 3 X 3 matrix d e f i n e d by: V = V 0 + M r = V 0 + M+ r + M" r (1-7) and c a l l e d the displacement tensor (r i s the p o s i t i o n v e c t o r ) . The terms M* n , M + 2 3 and M + 3 3 are the components of d i l a t a t i o n i n d i r e c t i o n of the p r i n c i p a l g a l a c t i c axes and the terms M*,2, M +, 3 and M + 2 3 are shears in the three normal planes of the g a l a c t i c system. The correspondence between two-dimensional and thre e - d i m e n s i o n a l f o r m u l a t i o n i s : M +, 2 = P (1-8) M" 21 = Q The + and - s u p e r s c r i p t s r e f e r to the f a c t that M i s decomposed i n t o a symmetric p a r t NT and in an antisymmetric p a r t M". The use of 1 i n e - o f - s i g h t v e l o c i t i e s i n s t e a d of proper motions allows one to study more d i s t a n t m a t e r i a l and the f a c t t h at they can be d e s c r i b e d with only one equation s i m p l i f i e s t h e i r treatment. Taking i n t o account only c i r c u l a r motion to the f i r s t order, the l i n e - o f - s i g h t v e l o c i t y VI i s expressed by: VI = - u 8 c o s l cosb - v e s i n l cosb - wQ s i n b (1-9) + A d sin21 c o s 2 b 18 where d i s the d i s t a n c e to the o b j e c t and A i s the f i r s t Oort c o e f f i c i e n t . u©, v 9 and w0 are the components of the s o l a r motion along the axes x, y and z of the g a l a c t i c c o o r d i n a t e system. In t h i s system, x p o i n t s towards the g a l a c t i c c e n t e r at 1=0°, b=0°, y i n the d i r e c t i o n of g a l a c t i c r o t a t i o n at 1=90°, b=0°, and z p o i n t s towards the g a l a c t i c n o r t h pole at b=90°. T h i s p e c u l i a r motion of the Sun i s measured i n a system where, on average, the m a t e r i a l of the s o l a r neighbourhood i s at r e s t . T h i s system i s c a l l e d the l o c a l standard of r e s t (LSR). N o n - c i r c u l a r motion i s o f t e n i n t r o d u c e d through the "nodal d e v i a t i o n " 1 0 as: V l = -u e c o s l cosb - v Q s i n l cosb - w0 s i n b (1-10) +A'd s i n 2 ( l - l 0 ) c o s 2 b The reason f o r t h i s approach seems to be p u r e l y h i s t o r i c a l . In f a c t , i n the o l d g a l a c t i c c o o r d i n a t e system, the g a l a c t i c center was not l o c a t e d at 1 = 0° and a term 1 0 had to be introduced. Equation (1-10) i s e q u i v a l e n t t o : VI = - u e c o s l cosb - v 0 s i n l cosb - w0 s i n b (1-11) +A''d sin21 c o s 2 b + C d cos21 c o s 2 b i f we d e f i n e : 1 0 = 1 / 2 arctan(-C/A'') A' 2 = A' ' 2 + C 2 (1-12) 19 The c o e f f i c i e n t s A' and A'' are not i n general equal to the f i r s t Oort c o e f f i c i e n t A because they can be a f f e c t e d by n o n - c i r c u l a r motions i n the plane of the Galaxy. Some authors add a Vk term to these equations s i n c e there might be a systematic e r r o r i n the c a l i b r a t i o n of the v e l o c i t i e s or a s h i f t of the p o s i t i o n of the s p e c t r a l l i n e s due to phenomena o c c u r i n g at the s u r f a c e of the s t a r s . T h i s p a r t i c u l a r q u e s t i o n w i l l be d i s c u s s e d i n more d e t a i l i n Chapter 2. Equation (1-11) can t h e r e f o r e be m o d i f i e d t o : VI = - u 9 c o s l cosb - v 9 s i n l cosb - we s i n b (1-13) + A''d sin21 c o s 2 b + C d cos21 c o s 2 b + Vk In order to d e s c r i b e a p o s s i b l e o v e r a l l d i f f e r e n t i a l expansion or c o n t r a c t i o n i n the g a l a c t i c plane, another term i s i n troduced and g i v e s : VI = Vk - u© c o s l cosb - v e s i n l cosb - w0 s i n b + A''d sin21 c o s 2 b (1-14) + C d cos21 c o s 2 b + K d c o s 2 b . Most of the s t u d i e s do not i n v o l v e c o e f f i c i e n t s other than those of equation (1-14) and o f t e n some of them are dropped. N e v e r t h e l e s s , some s t u d i e s use a more complete set of c o e f f i c i e n t s . From the d e s c r i p t i o n of (1-14), two p o s s i b l e developments are a v a i l a b l e . In studying r e l a t i v e l y d i s t a n t o b j e c t s , i t might be i n t e r e s t i n g to i n c l u d e some second 20 order terms and get: VI = Vk - u 9 c o s l cosb - v 0 s i n l cosb - w0 s i n b + d c o s 2 b [K + A ' s i n 2 ( l - l 0 ) ] (1-15) + d 2 c o s 3 b [K, s i n ( l - K 2 ) + R 3 s i n 3 ( l - K , ) ] These terms come from a T a y l o r expansion of the second order i n v o l v i n g only v e l o c i t y g r a d i e n t s that l i e along the plane of the Galaxy. For pure c i r c u l a r motion about the g a l a c t i c c e n t e r , the parameters would be: A'= 15 Km/s/Kpc K, = -A/4R0 (1-16) K 3 = 3A/4R 0 Vk = K = K 2 = K„ = 0 where R 0 i s the d i s t a n c e of the Sun from the g a l a c t i c center (see Ovenden, Pryce, Shuter (1983)). For the study of nearby o b j e c t s , i t i s more important to do a three-dimensional a n a l y s i s than to i n c l u d e second order terms. As w i l l be shown i n d e t a i l i n the next chapter, a three-dimensional f i r s t order T a y l o r expansion g i v e s : 21 VI = Vk - u 0 c o s l cosb - v 0 s i n l cosb - w0 s i n b + C, d c o s 2 l c o s 2 b + C 2 d s i n 2 b (1-17) + C 3 d s i n 2 l c o s 2 b + C„ d sin21 c o s 2 b + C 5 d c o s l sin2b + C 6 d s i n l sin2b For pure c i r c u l a r motion, a l l the C terms are zero except C« which equals 15 Km/s/Kpc, the standard value of the f i r s t Oort c o e f f i c i e n t . R e f e r i n g to these equations we w i l l now b r i e f l y summarize the work that has been done on n o n - c i r c u l a r motions dur i n g the l a s t 20 years or so. A l o t of s t u d i e s have been conducted on the kinematics of n e u t r a l hydrogen because HI i s the best a v a i l a b l e probe of l a r g e s c a l e n o n - c i r c u l a r motion. In 1961, H.L. H e i f e r t r e a t e d the kinematics of HI from 21-cm emission l i n e data in a very e l a b o r a t e way. He used equation (1-17) without the Vk term and determined the v a r i o u s c o e f f i c i e n t s from an a n a l y s i s of the l o c a t i o n of the zero l i n e - o f - s i g h t v e l o c i t y contour. In 1962, F . J . Kerr compared the r e s u l t s from two surveys, one performed i n Netherlands and the other in A u s t r a l i a , and he proposed an e x p l a n a t i o n f o r the systematic d i f f e r e n c e between the two surveys. H i s idea i n v o l v e d a Galaxy-wide outward flow of gas from the c e n t e r with the v e l o c i t i e s v a r y i n g with r a d i a l d i s t a n c e R as: 22 Vexp = 470 R-2 Km/s (R i n K i l o p a r s e c s ) . (1-18) Such an expansion p r e d i c t e d a p o s i t i v e v alue f o r K i n equation (1-14) which was observed in many subsequent s t u d i e s but f o r which many ex p l a n a t i o n s are p l a u s i b l e . P.O. L i n d b l a d (1967) observed 21-cm emission l i n e s i n the d i r e c t i o n of the g a l a c t i c a n t i c e n t e r and i d e n t i f i e d few " f e a t u r e s " i n VI vs 1 diagrams that c o u l d not be ex p l a i n e d with pure c i r c u l a r motion. T h i s important p i e c e of work l e d afterwards to the development of the idea of an expanding l o c a l gaseous subsystem r e l a t e d to the Gould b e l t . In the same issue of the j o u r n a l i n which L i n d b l a d p u b l i s h e d h i s paper, K. Takakubo (1967) presented an a n a l y s i s of the kinematics of HI us i n g a survey of 21-cm emission l i n e s (Van Woerden, Takakubo and Braes (1962)). Takakubo a l s o used equation (1-17) but h i s a n a l y s i s d i f f e r s from that of H e i f e r because he decomposed h i s s p e c t r a i n t o gaussian peaks and t r e a t e d each of them as an i n d i v i d u a l o b j e c t . Another d i f f e r e n c e between t h e i r work i s that Takakubo determined the values of h i s c o e f f i c i e n t s by using a l e a s t squares a l g o r i t h m . In 1970, J.W. Mast and S.J. G o l d s t e i n s t u d i e d 21-cm a b s o r p t i o n p r o f i l e s obtained towards e a r l y type s t a r s at high g a l a c t i c l a t i t u d e . T h e i r study, c a r r i e d out with the formalism of equation (1-13), r e v e a l e d that c o l d clouds (more l i k e l y t o be observed i n a b s o r p t i o n because of t h e i r h igh o p a c i t y ) are not i n pure c i r c u l a r motion. They a l s o 23 found that t h e i r d i s t r i b u t i o n i n space i s not w e l l d e s c r i b e d by the p l a n e - p a r a l l e l model i n which the d e n s i t y of clouds decreases e x p o n e n t i a l l y with z but does not vary with x or y. A.P. Henderson (1973) presented a paper i n which he analysed a set of 21-cm emission s p e c t r a obtained from a new survey c a r r i e d at Green Bank using a 140 foot t e l e s c o p e . His treatment took the form of equation (1-10). In c o n t r a s t to Takakubo, he d i d not e x t r a c t gaussian components from the s p e c t r a but used the average 1 i n e - o f - s i g h t v e l o c i t y as weighted by the observed b r i g h t n e s s temperature at d i f f e r e n t f r e q u e n c i e s . He gave a t h r e e - d i m e n s i o n a l f l a v o u r to h i s work by doing a comparison of the r e s u l t s obtained at d i f f e r e n t g a l a c t i c l a t i t u d e s . In 1978 J . C r o v i s i e r analysed ^ 300 HI clouds observed i n 21-cm a b s o r p t i o n and i d e n t i f i e d with gaussian components f i t t e d to h i s s p e c t r a . In h i s 1978 paper he used equation (1-10) but he r e c e n t l y ( B e l f o r t et C r o v i s i e r 1984) decided to add a Vk term and thus to use equation (1-13). His study was r e s t r i c t e d to b<l0° because of the complexity of the low l a t i t u d e s p e c t r a . Some authors t r i e d to use the observed n o n - c i r c u l a r motions of HI to c o n f i r m the v a l i d i t y of c e r t a i n t h e o r e t i c a l models. The p r e d i c t i o n s of the d e n s i t y wave theory f o r example were s t a t e d by C.C. L i n , L. Yuan and F.H. Shu (1969) and t e s t e d i n the i n v e s t i g a t i o n s of K. R o h l f s (1972), M. Creze (1973) and W.B. Burton and T.M. Bania (1974). These s t u d i e s showed that i t i s p o s s i b l e to c o n s t r u c t 24 p l a u s i b l e s p i r a l p a t t e r n s that would agree with the observed kinematics but the lack of independent determination of the a d j u s t a b l e parameters of the theory (number of arms, p o s i t i o n of the arms, p i t c h angle, amplitude, and so on) makes these models rather s p e c u l a t i v e . P.O. L i n d b l a d et a l . ( l 9 7 3 ) , H. Weaver (1974), F.M. Strauss et a l . d 9 7 9 ) and C A . Olano (1982) t r i e d to r e l a t e some streaming motions of HI to the expansion of a l o c a l subsystem a s s o c i a t e d with the Gould b e l t . These s t u d i e s were q u i t e s u c c e s s f u l i n d e r i v i n g an expansion age of the subsystem which i s i n agreement with the expansion age d e r i v e d from the kinematics of young s t a r s (^3 X 10 7 y e a r s ) . The problem with t h i s model i s that i t i s ad hoc s i n c e there i s no e x p l a n a t i o n f o r the a c t u a l e x i s t e n c e of such a system and no t h e o r e t i c a l model e x p l a i n s where the energy of expansion comes from. A l s o , as we w i l l see, the r e a l i t y of the s t e l l a r subsystem i t s e l f i s s t i l l q u i t e u n c e r t a i n . Concerning the kinematics of s t a r s , most work has been done on 0 and B s t a r s because of the l a r g e d i s t a n c e s at which these b r i g h t o b j e c t s can be observed. In 1964, V. Rubin and J . Burley s t u d i e d the l i n e - o f - s i g h t v e l o c i t i e s of 800 0 to B5 s t a r s with equation (1-14) where the Vk term was dropped. They found some s i g n i f i c a n t n o n - c i r c u l a r motion. The same year, M. Bonneau (1964) s t u d i e d 989 O-B s t a r s with equation (1-14) and r e l a t e d the observed expansion to the theory of expanding group developed by 25 A. Blaauw (1952). The e x i s t e n c e of departure from c i r c u l a r motion f o r these s t a r s was subsequently confirmed by the works of M.W. Feast and M. Shuttleworth (1965) and of R.M. P e t r i e and J.K. P e t r i e (1967) a l l of whom e s s e n t i a l l y used equation (1-14). In 1968, J.R. Lesh analysed the space v e l o c i t i e s of 464 0-B5 s t a r s b r i g h t e r than m=6.5(visual) and d e r i v e d some of the v e l o c i t y g r a d i e n t s that she compared to those that were p r e d i c t e d by d i f f e r e n t models of expanding groups. She showed that there e x i s t many non-zero v e l o c i t y g r a d i e n t s which d e s c r i b e departure from c i r c u l a r motion but that no model of a s i n g l e expanding group can e x p l a i n the value s of a l l of these g r a d i e n t s (see J.R. Lesh (1972)). In 1979, A.N. B a l a k i r e v used both 1 i n e - o f - s i g h t v e l o c i t i e s and proper motions of s t a r s from the FK4 catalogue (W. F r i c k e et a l (1963)) and of 193 others to do an a n a l y s i s of f i r s t order v e l o c i t y g r a d i e n t s i n three dimensions. He d e r i v e d from these g r a d i e n t s the d i r e c t i o n s and magnitudes of the p r i n c i p a l axes of expansion or c o n t r a c t i o n of the v e l o c i t y f i e l d . T h i s kind of in f o r m a t i o n can be d e r i v e d from equation (1-17) as w e l l . (A f u l l i n t r o d u c t i o n to the notion of deformation tensor w i l l be presented i n Chapter 2.) The same year J.A. F r o g e l and R. Stothe r s s t u d i e d the space motion of 0-B5 s t a r s with equation (1-13) where the C term was dropped and with equations (1-1) and (1-2). They d i s c u s s e d the r e a l i t y of the expansion of the Gould b e l t . R.J. Quiroga and R. T a r s i a (1983) s t u d i e d the z-motion of 26 some 400 O-B s t a r s and showed that the average r e s i d u a l v e l o c i t y i n z v a r i e s with d i s t a n c e . T h i s i n d i c a t e s the presence of what they c a l l e d "hydrodynamic motion", a systematic n o n - c i r c u l a r motion. M.W. Ovenden, M.H.L. Pryce and W.L.H. Shuter (1983) presented an a n a l y s i s of the l i n e - o f - s i g h t v e l o c i t i e s of 990 0 and B s t a r s . They used a second order T a y l o r s e r i e s i n two dimensions (equation (1-15) without the Vk term). As f o r HI data, some authors t r i e d to detec t the i n f l u e n c e of s p i r a l arms on the motion of e a r l y type s t a r s . In p a r t i c u l a r , K. R o h l f s (1972), M. Creze and M.V. Mennessier (1973) and J . B y l and M.W. Ovenden (1978) determined some c o n s t r a i n t s on the s p i r a l arm parameters but cou l d not r e a l l y prove that s p i r a l arms are necessary to e x p l a i n the streaming motion of young s t a r s . The i n t e r e s t in the kinematic p r o p e r t i e s of other types of s t a r s has been l i m i t e d mainly to the determination of the o r i e n t a t i o n of t h e i r v e l o c i t y e l l i p s o i d . Only a few authors have i n v e s t i g a t e d t h e i r departure from c i r c u l a r motion. R.M. Humphreys (1970) s t u d i e d a set of 400 supe r g i a n t s and found a s i g n i f i c a n t Vk term. In 1974 A.E. Gomez determined values of Vk f o r v a r i o u s types of s t a r s of the main sequence and of the gi a n t branch. In 1975, W. F r i c k e and A. Tsioumis used 1 i n e - o f - s i g h t v e l o c i t i e s (equation (1-13)) and proper motions (equations (1-1) and (1-2)) of the s t a r s of the FK4 ca t a l o g u e . They 27 showed that the s t a r s a s s o c i a t e d with the Gould b e l t do not y i e l d parameters d i f f e r e n t from the others but t h a t , i n g e n e r a l , young s t a r s depart s i g n i f i c a n t l y from c i r c u l a r motion. B. du Mont (1977) used a three-dimensional model (equations (1-4) and (1-5)) to i n v e s t i g a t e systematic motions i n the z - d i r e c t i o n . His a n a l y s i s was a l s o based on the FK4 catalogue so he c o u l d compare d i r e c t l y h i s r e s u l t s to those of F r i c k e and Tsioumis (1975). The FK4 catalogue i s so widely used s i n c e the proper motions of s t a r s of the catalogue have been c a r e f u l l y c o r r e c t e d f o r the e f f e c t s of p r e c e s s i o n . However, the catalogue c o n t a i n s only 512 s t a r s and thus i t s s u b d i v i s i o n i n t o groups of d i f f e r e n t s p e c t r a l types leads to small s t a t i s t i c a l samples. P. Brosche and H. Schwan (1981) s t u d i e d FK4 s t a r s but they used a s p h e r i c a l harmonic expansion of the v e l o c i t y f i e l d . T h e i r a n a l y s i s confirmed the e x i s t e n c e of some n o n - c i r c u l a r motion even f o r l a t e type s t a r s . W.L.H. Shuter and T. Goulet (1983) presented a f i r s t order T a y l o r expansion (equation (1-17)) a n a l y s i s of the l i n e - o f - s i g h t v e l o c i t i t i e s of 260 main sequence A s t a r s . T h i s p r e l i m i n a r y work i s completed and improved on i n the present study. c) Goals and c h a r a c t e r i s t i c s of t h i s study From what we have j u s t seen, i t seems that most of the work has been done on low g a l a c t i c l a t i t u d e HI and O-B s t a r s because of the l a r g e range of d i s t a n c e s over which 28 they can be observed. The kinematics of m a t e r i a l on a more l o c a l s c a l e , such as HI at high l a t i t u d e and s t a r s of s p e c t r a l type l a t e r than B, has r e c e i v e d l e s s a t t e n t i o n . We have a l s o seen that many d i f f e r e n t equations were used to d e s c r i b e n o n - c i r c u l a r motions. T h i s v a r i e t y of formalisms complicates any attempt to compare the r e s u l t s of d i f f e r e n t s t u d i e s . The present work attemps to check, complement and u n i f y the v a r i o u s i n v e s t i g a t i o n s l i s t e d i n the previous s e c t i o n . T h i s u n i f i c a t i o n can be acheived i f a standard method of t r e a t i n g the n o n - c i r c u l a r motion i s e s t a b l i s h e d and used s y s t e m a t i c a l l y with d i f f e r e n t groups of o b j e c t s . We chose s i x c l a s s e s of o b j e c t s to cover a range of d i s t a n c e s that go from few tens of parsecs to - '1 Kpc and to cover a l s o a c e r t a i n range of ages (up to = 5 X 10 9 y e a r s ) ; These c l a s s e s are: 1) i n t e r s t e l l a r HI c l o u d s , 2) gaseous (HI) i n t e r c l o u d medium, 3) B s t a r s of l u m i n o s i t y c l a s s smaller than V, 4) main sequence B s t a r s , 5) main sequence A s t a r s and 6) K-giant s t a r s . Lower l u m i n o s i t y main sequence s t a r s l i k e F and G c o u l d complement t h i s study on a very l o c a l l e v e l (d ^ 30 pc) and to a g r e a t e r spread in age; but our p r e l i m i n a r y i n v e s t i g a t i o n s show that the r e s u l t s obtained f o r n o n - c i r c u l a r motion parameters are not s i g n i f i c a n t f o r such small d i s t a n c e s . We have decided to use only 1 i n e - o f - s i g h t v e l o c i t i e s s i n c e p r e c i s e values of proper motions were not a v a i l a b l e f o r enough s t a r s and are impossible to o b t a i n f o r gaseous m a t e r i a l . The equation 29 that we have decided to use i s (1-17) i . e . a three-dimensional f i r s t order T a y l o r expansion with a Vk term. T h i s equation, we b e l i e v e , i s more s u i t a b l e than (1-15) i s s i n c e we want to study r e l a t i v e l y c l o s e m a t e r i a l . A second order T a y l o r expansion i n three dimensions c o u l d a l s o have been used but, as w i l l be d i c u s s e d i n Chapter 5, the i n c l u s i o n of second order terms improves very l i t t l e or not at a l l the goodness of the f i t . Since as a f i r s t step we want to study kinematics i n a simple and systematic way, we have decided not to i n c l u d e these higher order terms which would double the number of parameters, introduce severe problems of c o r r e l a t i o n and thus a f f e c t the c l a r i t y and the r e l i a b i l i t y of our r e s u l t s . We hope that the v a r i e t y of ages and of d i s t a n c e ranges that can be s t u d i e d with our s i x c l a s s e s of o b j e c t s w i l l h e l p to b u i l d a g l o b a l and e v o l u t i o n a r y p i c t u r e of the kinematics of the s o l a r neighbourhood. We a l s o hope that the study of the kinematics of gas w i l l p r ovide some inf o r m a t i o n on i t s l o c a l s p a t i a l d i s t r i b u t i o n and w i l l a llow an i n f o r m a t i v e comparison of the s t e l l a r and i n t e r s t e l l a r streaming motions. 30 CHAPTER 2 DETERMINATION OF THE KINEMATIC PARAMETERS a) The v e l o c i t y f i e l d equation In order to d e s c r i b e the kinematic p r o p e r t i e s of nearby o b j e c t s we d e r i v e the v a r i o u s components of a v e l o c i t y f i e l d . The value V f l of the f i e l d v e l o c i t y at a given p o s i t i o n (l,b,d) or (x,y,z) i s the most probable value f o r the l i n e - o f - s i g h t v e l o c i t y of an o b j e c t at that p o s i t i o n . Of course, i n g e n e r a l , the a c t u a l l i n e - o f - s i g h t v e l o c i t y VI i s not equal to the f i e l d v e l o c i t y but the v a r i o u s components of the l a t t e r are determined i n such a way that the q u a n t i t y S 2 given by: N S 2 = I ( V l ( i ) - V f l ( i ) ) 2 (2-1 ) i = 1 i s minimized. Here " i " i s the l a b e l a s s o c i a t e d with the N o b j e c t s f o r which a v e l o c i t y f i e l d i s found. The use of a l e a s t squares method (equation (2-1)) i s j u s t i f i e d when the d i s t r i b u t i o n of r e s i d u a l s R = V l - V f l i s d e s c r i b e d by a Gaussian curve. The a n a l y s i s of the r e s i d u a l s f o r the v a r i o u s groups of s t u d i e d o b j e c t s i s shown i n Table I. One can see t h a t , f o r most groups, the f o u r t h moments of the d i s t r i b u t i o n s are too l a r g e due to the e x i s t e n c e of a "long t a i l " (only 0.3% of the p o i n t s are expected to exceed 3a). 31 For reasons that w i l l be d i s c u s s e d i n s e c t i o n d) of the present chapter, we decided to e l i m i n a t e , i n a second f i t t i n g procedure, the o b j e c t s that l e d to r e s i d u a l s l a r g e r than 3a i n the f i r s t f i t . As shown i n Table I, t h i s r e d u c t i o n of the samples rendered the shape of the d i s t r i b u t i o n s more Gaussian. A x 2 t e s t r e v e a l s t h a t , i n average, the d i s t r i b u t i o n s can be d e s c r i b e d by a Gaussian s i n c e i n 1% of the cases one would expect to f i n d a reduced X 2 l a r g e r than 3.3. For some groups the Gaussian d e s c r i p t i o n i s c e r t a i n l y adequate while f o r others i t i s not as good. C o n s i d e r i n g these r e s u l t s and the f a c t that the l e a s t squares method i s widely used i n s i m i l a r s t u d i e s , we f i n d i t s u i t a b l e f o r the present work. As e x p l a i n e d e a r l i e r , we assume that V f l i s smooth enough to be d e s c r i b e d by a f i r s t order T a y l o r s e r i e s . In order to d e r i v e the l i n e - o f - s i g h t v e l o c i t y equation, l e t us f i r s t c o n s i d e r the space v e l o c i t y of an o b j e c t near the Sun. The f i r s t order t h r e e - d i m e n s i o n a l expansion of the v e l o c i t y f i e l d i s given by: Vf * V f 0 + (x-x 0) 3_ 9x + ( y - y 0 ) 9_ » 9y + ( z - z 0 ) 9_ , 3z Vf (2 -2 ) where x 0, y 0 and z 0 are the c o o r d i n a t e s of the p o s i t i o n of the Sun. In order to s i m p l i f y the equation we choose a TABLE I A n a l y s i s of the d i s t r i b u t i o n s Wi th a l l o b j e c t s ( f i r s t f i t ) a f t e r e l i m i n a t i o n of R > 3a (second f i t ) Type of o b j e c t s 21 cm e m i s s i o n 21 cm a b s o r p t i o n BI-IV BV d > 2 0 0 pc BV d < 2 0 0 pc AV K i l l No . 4 0 6 284 46 1 5 3 2 334 1066 9 8 8 7 . 4 65 . 0 193 185 122 2 2 0 592 ma /mz -0.2 - 1 5 . 1 -1 . 0 - 0 . 8 - 6 0 . 0 - 1 5 . 5 3 . 2 ma / m 2 24 8 0 0 1979 1 183 536 3422 2 5 4 0 % >3o 0.0 1 . 1 2 . 2 2 . 3 1 .2 0 . 5 0 . 7 AK/<j . 4 6 4 . 4 2 0 . 4 5 4 . 4 1 7 . 4 5 0 . 4 2 7 7 . 4 4 4 . 4 123 132 102 181 5 3 7 m 3 /m t - 0 . 2 - 4 . 9 0 . 4 0 . 7 - 3 . 1 0 . 1 - 2 . 3 24 2 4 0 4 7 3 5 0 5 3 3 6 5 7 0 1767 X' ( i / = 4 ) 2 . 8 4 . 9 4 . 7 4 . 8 4 . 7 1 . 9 1 . 6 The q u a n t i t i e s m i, m3 and ma are the second, t h i r d and f o u r t h moments of the d i s t r i b u t i o n s . For a Gaussian, ra^s', m : = 0 , ma/mz =3 <s2 and 0 .3% of the p o i n t s are beyond 3 a • AK/cc i s the v a r i a t i o n of the v a l u e s of the c o e f f i c i e n t s from the f i r s t f i t to the second f i t d i v i d e d by t h e i r a s s o c i a t e d u n c e r t a i n t y . A xz t e s t i s ex p e c t e d to g i v e x ! > 3 . 3 ( w i t h y= 4 ) i n 1% of the cas e s . 32 3 3 c o o r d i n a t e s y s t e m i n w h i c h t h e S u n i s a t r e s t a n d l o c a t e d a t x o = Y o = z o = 0 . T h e t e r m v ' f o i s n o t n e c e s s a r i l y z e r o b e c a u s e t h e S u n m a y h a v e ( a n d i s k n o w n t o h a v e ) a p e c u l i a r v e l o c i t y w i t h r e s p e c t t o t h e " l o c a l s t a n d a r d o f r e s t " ( L S R ) . A s m e n t i o n e d e a r l i e r , t h e L S R i s a r e f e r e n c e f r a m e t h a t r o t a t e s a r o u n d t h e g a l a c t i c c e n t e r w i t h a v e l o c i t y e q u a l t o t h e a v e r a g e v e l o c i t y o f t h e o b j e c t s i n t h e s o l a r n e i g h b o u r h o o d ( s e e M i h a l a s a n d B i n n e y ( 1 9 8 2 ) , C h a p t e r 6 ) . T h e a x e s x , y a n d z a r e t h e p r i n c i p a l a x e s o f t h e G a l a x y a s d e f i n e d i n C h a p t e r 1 . I n t h i s f r a m e , e q u a t i o n ( 2 - 2 ) c a n b e w r i t t e n a s : V f = T V f x 0 + x 9 V f x 9 x + y 9 V f x + z 9 V f x , 9 z V f y 0 + x 9 V f y 9 x + y 9 V f y > 9y + z 9 V f y i 9 z ( 2 - 3 ) + K V f z 0 + x 9 V f z 9 x + y 9 V f z ay + z 9 V f z , 9 z o U s i n g s p h e r i c a l c o o r d i n a t e s : x = d c o s l c o s b y = d s i n l c o s b ( 2 - 4 ) z = d s i n b a n d t h e s p e c i a l c o n v e n t i o n 34 -u e. = V f x 0 - v 0 = V f y 0 -w0 = V f z 0 u = Vfx v = Vfy w = V f z and u, = 9u 9x v, = 9v 9x u 2 = 9u 3y v, = 9v u 3 = du 9z 9y , e t c . we can r e w r i t e equation (2-3) as: Vf = T [ - u 0 + d ( c o s l cosb u, + s i n l cosb u 2 +sinb u 3 ) ] + 3 I _ v 9 + d ( c o s l cosb v, + s i n l cosb v 2 (2-5) + s i n b v 3 ) ] + 1c [ -w0 + d ( c o s l cosb w, + s i n l cosb w2 +sinb w 3)] The l i n e - o f - s i g h t v e l o c i t y i s a p r o j e c t i o n of \7f given by: V f l = \ ? f ' ( c o s l cosb I + s i n l cosb J + sin b E) . (2-6) Using (2-5) f o r \?f i n (2-6) we get: V f l = - u e c o s l cosb - v e s i n l cosb - w0 s i n b (2-7) + d c o s l cosb ( c o s l cosb u, + s i n l cosb u 2 + s i n b u 3). + d s i n l cosb ( c o s l cosb v, + s i n l cosb v 2 +sinb v 3 ) + d s i n b ( c o s l cosb w, + s i n l cosb w2 + s i n b w 3) Introducing a Vk term f o r more g e n e r a l i t y and r e w r i t i n g 35 equation (2-7) we get: V f l = Vk - u© c o s l cosb - v 0 s i n l cosb - wG s i n b + d c o s 2 l c o s 2 b u, + d s i n l c o s l c o s 2 b u 2 + d c o s l s i n b cosb u 3 + d s i n l c o s l c o s 2 b v, (2-8) + d s i n 2 l c o s 2 b v 2 + d s i n l s i n b cosb v 3 + d c o s l s i n b cosb w, + d s i n l s i n b cosb w2 + d s i n 2 b w3 The i n t r o d u c t i o n of a Vk term i s j u s t i f i e d by the f a c t that there may be some systematic e r r o r s i n the dete r m i n a t i o n of VI. The Vk has been used by many authors i n t h e i r a n a l y s i s of the kinematics of v a r i o u s groups of o b j e c t s and has sometimes turned out to be s i g n i f i c a n t (see Table I I ) . The i n t e r p r e t a t i o n of the Vk term i s d i f f i c u l t s i n c e g r a v i t a t i o n a l r e d s h i f t , i n c o r r e c t measurement of the wavelengths, pressure s h i f t s i n the atmosphere of s t a r s as we l l as t u r b u l e n t i n f l o w s and outflows can a l l cause a systematic d e v i a t i o n of the measured l i n e - o f - s i g h t v e l o c i t i e s (see Smart (1968) p. 69). H i s t o r i c a l l y , the Vk term has a l s o been a s s o c i a t e d with c o n t r a c t i o n or expansion motion of the o b j e c t s with respect to the Sun s i n c e many authors f i t t e d the VI data with a reduced set of terms ( s o l a r motion- only or s o l a r motion and g a l a c t i c r o t a t i o n ) . In our case, the Vk term c o u l d be the r e s u l t of s i g n i f i c a n t terms of the second (or higher) order. The s i g n i f i c a n c e of the Vk term w i l l be d i s c u s s e d s e p a r a t l y f o r each group of 36 Table II Values of the Vk term i n Km/s S p e c t r a l a) b) c) Type No.Of Vk No.of Vk No.of Vk s t a r s s t a r s s t a r s B 284 4.9 645 4.7 5.3 A 500 1 .7 742 0.0 1 .4 F 1 99 0.3 523 -0.6 0.0 G 244 -0.2 433 -1.0 -0.5 K 687 0.3 1118 -0.2 0.0 M 234 0.7 222 0.0 0.4 No. Vk d) Dust clou d s 185 4.3±0.4 e) 21-cm emission 44 -0.8±0.7 a) Campbell and Moore (1928) b) Smart and Green (1936) c) A l l e n (1973) d) F r o g e l and S t o t h e r s (1978) e) Venugopal and Shuter (1967) 37 o b j e c t s . There are many ways of regrouping the terms of the r i g h t hand s i d e of equation (2-8). One of them i s to i d e n t i f y the v a r i o u s " F o u r i e r " terms l i k e s i n n l , cos n l , s i n nb, cos nb. Using the i d e n t i t i e s : c o s 2 9 = 1/2 (1 + cos20) s i n 2 0 = 1/2 (1 - cos20) (2-9) sinG cos© = 1/2 sin20 we can w r i t e : V f l = Vk - u 9 c o s l cosb - v 0 s i n l cosb - we s i n b + (u,/4 + v 2/4 + w3/2) d + (u,/4 + v 2/4 - w3/2) d cos2b + (u,/2 - v 2/2) d cos21 c o s 2 b (2-10) + (u 2/2 + v,/2) d sin21 c o s 2 b + (u 3/2 + w,/2) d cos21 sin2b + (v 3/2 + w2/2) d s i n l cos2b T h i s grouping i s i n t e r e s t i n g s i n c e a l l the terms are orthogonal to each other i f the o b j e c t s s t u d i e d are uni f o r m l y d i s t r i b u t e d with 1 ranging from 0° to 360° and b ranging from -90° to 90°. However, i n s p i t e of the a t t r a c t i v e n e s s of t h i s approach, we w i l l not use t h i s way of regrouping the terms due to of the strong c o r r e l a t i o n s that 38 e x i s t between the f i f t h , s i x t h and seventh terms. These terms are i n f a c t d i f f e r e n t combinations of the g r a d i e n t s u,, v 2 and w3 and i f one of these g r a d i e n t s i s much l a r g e r than the ot h e r s , the combinations would become h i g h l y interdependent. T h i s kind of c o r r e l a t i o n should be avoided as much as p o s s i b l e because i t c o n t r i b u t e s to the co v a r i a n c e of the v a r i o u s c o e f f i c i e n t s and thus i n c r e a s e s the g l o b a l u n c e r t a i n t y on the determination of V f l . Another way of regrouping the terms i s to i s o l a t e the d i f f e r e n t g r a d i e n t s as much as p o s s i b l e . T h i s g i v e s : V f l = Vk - u Q c o s l cosb - v 0 s i n l cosb - w9 s i n b + u, d c o s 2 l c o s 2 b + w3 d s i n 2 b + v 2 d s i n 2 l c o s 2 b + 1/2 ( u 2 + v,) d sin21 c o s 2 b (2-11) + 1/2 ( u 3 + w,) d c o s l sin2b + 1/2 ( v 3 + w 2) d s i n l sin2b which i s the equation that we use in t h i s study ( e q u i v a l e n t to equation (1—17)). As we s a i d e a r l i e r , our goal i s to use a l e a s t squares f i t to determine the v a r i o u s kinematic parameters that give the value of V f l f o r a given p o s i t i o n ( l , b , d ) . These kinematic parameters are the ten c o e f f i c i e n t s of equation (2-11), i . e . : 39 K(1 ) = Vk K(2) = - u e K(3) = - v 0 K(4) = -we K(5) = u, = = O u / a x ) 0 K(6) = w3 = : (9w/9z) 0 K(7) = v 2 = = ( 9 v / 9 y ) 0 K(8) = 1/2 (v, + u 2) = = 1/2 (9v/9x + 9u/9y) 0 K(9) = 1/2 ( u 3 + w,) = = 1/2 (9u/9z + 9w/9x) 0 K(10) = 1/2 ( v 3 + w 2) = - 1/2 (9v/9z + 9w/9y) 0 Among these terms one can f i n d the components of the p e c u l i a r v e l o c i t y of the Sun with r e s p e c t to the LSR (K(2) to K(4) with reversed s i g n s ) , the c o r r e c t i o n f o r systematic e r r o r on the measurement V l ( K ( 1 ) ) , the e f f e c t of d i f f e r e n t i a l r o t a t i o n about the g a l a c t i c center (K(8)) and the e f f e c t of v a r i o u s types of departures from c i r c u l a r motion (K (5) to K (7) ) and (K(9) and K(10)). One can a l s o n o t i c e that only 3 of the 9 g r a d i e n t s are i s o l a t e d and thus determined by the f i t . T h i s i s due to the f a c t that we use l i n e - o f - s i g h t v e l o c i t i e s e x c l u s i v e l y . Were we to work with proper motions and with l i n e - o f - s i g h t v e l o c i t i e s , we c o u l d determine the space v e l o c i t y and d e r i v e the values of the 9 f i r s t order g r a d i e n t s . As e x p l a i n e d i n the p r e v i o u s chapter, we have decided not to i n c l u d e the second order terms of the T a y l o r expansion. The f u l l development of the l i n e - o f - s i g h t 40 v e l o c i t y f i e l d equation of second order can be found in the Appendix. I t c o n t a i n s 20 terms i n s t e a d of the 10 of equation (2-11 ) . b) The v e l o c i t y e l l i p s o i d model of v e l o c i t y d i s p e r s i o n The measured l i n e - o f - s i g h t v e l o c i t y VI d i f f e r s i n general from V f l ( l , b , d ) , the most probable l i n e - o f - s i g h t v e l o c i t y at a p o s i t i o n 1, b and d, by an amount R, c a l l e d the r e s i d u a l . As d i s c u s s e d by Ogorodnikov (1965) and as v e r i f i e d i n our study, <R> i s approximately zero and the d i s t r i b u t i o n of R ( i ) i s c l o s e to a Gaussian of standard d e v i a t i o n o. When one uses the Gaussian d e s c r i p t i o n of the d i s t r i b u t i o n of r e s i d u a l s , one can w r ite the p r o b a b i l i t y * of measuring VI as: *(V1) = 1/(V/2¥ a ( l , b , d ) ) (2-13) • e x p - [ ( V l - V f l ( l , b , d ) ) 2 / 2 a 2 ( l , b , d ) ] From the d e f i n i t i o n of R = VI - V f l we can w r i t e : R = 7 , 3 ( x - xf) + 7 2 3 ( y " y f ) + 7 3 3 ( z " z f ) (2-14) where 7 , 3 , 7 2 3 and 7 3 3 are the usual d i r e c t i o n c o s i n e s in the d i r e c t i o n of the three axes x, y, z, i . e . , 41 7 1 3 = c o s l cosb 7 2 3 = s i n l cosb (2-15) 7 3 3 = s i n b and where x f , yf and zf are the most probable v a l u e s of the components of the v e l o c i t y along the three p r i n c i p a l g a l a c t i c axes. The standard d e v i a t i o n o = a(x,y,z) a s s o c i a t e d with R can then be expressed by: O2 = 7 2 1 3 U 2 0 0 + 7 2 2 3 U 0 2 0 + 7 2 3 3 " 0 0 2 (2"16) + 2 7 , 3 7 2 3 U 1 1 0 + 2 7 2 3 7 3 3 U 0 1 1 + 2 7 1 3 7 3 3 U , o i where the v a r i o u s u terms are the second order moments of the group of N o b j e c t s , i . e . : N u 2 0 o = 1/(N-1) L ( x ( i ) - x f ( i ) ) 2 i = 1 N u , 1 0 = 1/(N-1) L ( x ( i ) - x f ( i ) ) ( y ( i ) i=l e t c . These second order moments can be determined by l e a s t squares f i t t i n g of equation (2-16) where 7 1 3 , 7 2 3 ? and 7 3 3 (2-17) - y f ( i ) ) 42 are known f o r each o b j e c t and where the value of o2 can be r e p l a c e d by the v a r i o u s R 2 ( i ) s i n c e the average value of R 2 ( i ) i s a2 from: oo / (R 2/(v/2i a) exp-(R 2/2a 2) ) dR = a 2 . (2-18) — oo I f (x - x f ) , (y - y f ) and (z - z f ) were t r u l y independent, the terms u 1 1 0 , u , 0 i , and u011 would a l l be zero and the standard d e v i a t i o n s a(x), a ( y ) , a(z) would d e s c r i b e completely the departure of VI from V f l . As shown by J.H. Oort (1965) the a x i a l symmetry of the Galaxy demands t h a t , f o r a r e l a x e d set of o b j e c t s , a ( x ) , a(y) and a(z) be independent. For some unknown reason, t h i s i s not the case i n general and the t r u e axes of symmetry of the r e s i d u a l v e l o c i t y d i s t r i b u t i o n l i e i n other d i r e c t i o n s . In order to determine these d i r e c t i o n s , we w r i t e : a 2(a,b,c) = 7 2 ( a ) o 2 ( a ) + 7 2 ( b ) o 2 ( b ) + y2(c) a 2 ( c ) (2-19) and t r y to r e l a t e the parameters of (2-19) with those of (2-16). The d i a g o n a l i z a t i o n of the matrix of second order moments g i v e s the three e i g e n v a l u e s ( a 2 ( a ) , a 2 ( b ) , a 2 ( c ) ) and the o r i e n t a t i o n of the e i g e n v e c t o r s . T h i s procedure i s standard and w i l l not be d e s c r i b e d any f u r t h e r . For more infor m a t i o n see Trumpler and Weaver (1953) Chapter 1. T h i s process of e v a l u a t i n g o(a), a(b) and a(c) and 4 3 the d i r e c t i o n s of these p r i n c i p a l axes i s r e f e r e d to as the d e t e r m i n a t i o n of the v e l o c i t y e l l i p s o i d . T h i s name comes from the f a c t that one can express the p r o b a b i l i t y f o r an o b j e c t to have a r e s i d u a l v e l o c i t y V with components V ( a ) , V(b) and V(c) as: ¥(V(a),V(b),V(c)) = C exp [-1/2 ( V 2 ( a ) / o 2 ( a ) (2-20) + V 2(b)/<r 2(b) + V 2 ( c ) / a 2 ( c ) ) ] The resemblance between the argument of the e x p o n e n t i a l i n (2-20) and the equation of an e l l i p s o i d : x 2 / a 2 + y 2 / b 2 + z 2 / c 2 = 1 (2-21) i s r a t her s t r i k i n g ! In t h i s case, the e l l i p s o i d i n the " v e l o c i t y space". A standard s i m p l i f i c a t i o n of t h i s treatment i s to suppose that one of the p r i n c i p a l axes l i e s along the z - a x i s of the Galaxy. T h i s idea i s o b v i o u s l y i n s p i r e d by the a x i a l symmetry of the d i s k component of the Galaxy. Using t h i s s i m p l i f i c a t i o n , equation (2-19) becomes: a 2 = a 2 ( a ) c o s 2 b c o s 2 ( l - l v ) (2-22) + a 2 ( b ) c o s 2 b s i n 2 ( l - l v ) + a 2 ( c ) s i n 2 b where l v i s the angle between the x-axis and the f i r s t 44 p r i n c i p a l a x i s a. Equation (2-22) can be r e w r i t t e n as: o 2 = c o s 2 l c o s 2 b [ a 2 ( a ) c o s 2 l v + a 2 ( b ) s i n 2 l v ] + s i n 2 l c o s 2 b [ a 2 ( a ) s i n 2 l v + a 2 ( b ) c o s 2 l v ] (2-23) + sin21 c o s 2 b [ ( a 2 ( a ) - cr 2(b)) (1/2 • sin21v) ] + s i n 2 b [ a 2 ( c ) ] Returning to a l e a s t squares f i t method, one can determine the best values f o r the four q u a n t i t i e s i n b r a c k e t s and then e x t r a c t a 2 ( a ) , a 2 ( b ) , a 2 ( c ) , and l v . Most of the authors p r e f e r to give c 2 ( x ) , a 2 ( y ) , a 2 ( z ) and l v where the l a t t e r q u a n t i t y i s the angle between the l a r g e s t p r i n c i p a l a x i s ( a 2 ( a ) or a 2 ( b ) ) and the x - a x i s . A non-zero value of l v leads to what i s c a l l e d the d e v i a t i o n of the vertex because the l a r g e s t a x i s should, i n theory, l i e along the x-axis (see Chandrasekhar (1943)). The t r a n s f o r m a t i o n between ( a ( a ) , a(b), a(c)) and ( a ( x ) , a ( y ) , a(z)) i s given by: a 2 ( x ) a 2 ( y ) a 2 ( z ) = a 2 ( a ) c o s 2 l v + a 2 ( b ) s i n 2 l v = a 2 ( a ) s i n 2 l v + a 2 ( b ) c o s 2 l v = <x2(c) (2-24) 45 c) C o l l e c t i o n of the data A l l data on s t a r s and HI m a t e r i a l has been taken from e x i s t i n g l i t e r a t u r e . The s t e l l a r data has been e x t r a c t e d from SKY CATALOGUE 2000.0 Volume 1 e d i t e d by A. H i r s h f e l d and R.W. S i n n o t t i n 1982. T h i s r e f e r e n c e p r o v i d e s v a r i o u s c h a r a c t e r i s t i c s of s t a r s b r i g h t e r than m = 8 . 0 5 ( v i s u a l ) . For our study we used the p o s i t i o n (a and 6), the s p e c t r a l and l u m i n o s i t y types, the l i n e - o f - s i g h t v e l o c i t y VI and the d i s t a n c e d from the Sun. As e x p l a i n e d by the e d i t o r s of the c a t a l o g u e , the inf o r m a t i o n on s t a r s came from d i f f e r e n t sources (see r e f e r e n c e s t h e r e i n ) . For example, the d i s t a n c e d e t e r m i n a t i o n s have been done with d i f f e r e n t methods. For small d i s t a n c e s (d ^ 20 pc) t r i g o n o m e t r i c p a r a l l a x i s the most r e l i a b l e method and t h e r e f o r e data from the G l i e s e catalogue has been used while for l a r g e r d i s t a n c e s , s p e c t r o s c o p i c p a r a l l a x method has been used. T h i s l a t t e r c o n s i s t s of determining the d i s t a n c e of a s t a r from i t s s p e c t r a l type and i t s apparent magnitude. Of course, a b s o r p t i o n and reddening due to dust p a r t i c l e s must be taken i n t o account. In t h i s study, we r e f e r to the MK c l a s s i f i c a t i o n of s t a r s . Some s t a r s present i n the catalogue were known to have s p e c t r a l p e c u l i a r i t i e s such as "with emission l i n e s " , " v a r i a b l e s p e c t r a l type", "abnormally strong m e t a l l i c l i n e s " and many other v a r i a t i o n s . In order to keep our work 46 independent of the i n t e r p r e t a t i o n of these p e c u l i a r i t i e s , we s y s t e m a t i c a l l y r e j e c t e d any of these "abnormal" s t a r s as w e l l as those f o r which the s p e c t r a l or l u m i n o s i t y type was not determined. For the HI data, we used the r e s u l t s of a study of 21-cm emission by A.P. Henderson (1973) and the Nancay 21-cm Absorption Survey Catalogue ( J . C r o v i s i e r , I. Kazes, D. Aubry, 1978). The data from Henderson's work does not give the v e l o c i t i e s of the i n d i v i d u a l emission l i n e s but does provide a weighted average of the v a r i o u s measured l i n e - o f - s i g h t v e l o c i t i e s f o r each spectrum. T h i s weighted average i s given by: / V l ( u ) T(u) du <V1> = (2-25) / T(u) du where v i s the frequency and T(u) the measured b r i g h t n e s s temperature. His survey covers i n a systematic manner (one spectrum i s taken every 5°) that f r a c t i o n of the sky where -10° < 1 < 250° and -30° < b < 30°. The a b s o r p t i o n l i n e data i s d i f f e r e n t s i n c e a b s o r p t i o n i s observed a g a i n s t strong r a d i o sources which have t h e i r own d i s t r i b u t i o n in the sky. These sources are normally e x t r a g a l a c t i c and thus independent of the c o n c e n t r a t i o n of m a t e r i a l i n the d i s k of our Galaxy. The l a t i t u d e of the Nancay r a d i o t e l e s c o p e l i m i t e d the o b s e r v a t i o n s to d e c l i n a t i o n s 6 > -37° 30' (see 47 J . C r o v i s i e r et a l . ( l 9 7 8 ) ) . In C r o v i s i e r ' s work a b s o r p t i o n f e a t u r e s were i d e n t i f i e d by decomposing the s p e c t r a i n t o gaussian components. We t r e a t each of these components as an i n d i v i d u a l o b j e c t because they are a s s o c i a t e d with i s o l a t e d c o l d c l o u d s . Both s e t s of HI data d i d not i n c l u d e d i s t a n c e d e t e r m i n a t i o n s of the v a r i o u s f e a t u r e s s i n c e , as a l r e a d y mentioned, there e x i s t s no independent method of e v a l u t i o n of the d i s t a n c e of the n e u t r a l hydrogen observed with the 21-cm l i n e . d) Treatment of the data Equations (2-11), (2-16) and (2-23) are t r e a t e d with a l e a s t squares a l g o r i t h m . T h i s procedure p r o v i d e s a l l the values of kinematic parameters which minimize, the sum of the r e s i d u a l s . Since the formal u n c e r t a i n t i e s on the d i s t a n c e s and on the l i n e - o f - s i g h t v e l o c i t i e s were not a v a i l a b l e , we decided to perform an unweighted f i t . In order to reduce the i n f l u e n c e of "abnormal" o b j e c t s on the f i t , we used the standard procedure of e l i m i n a t i n g , i n a second f i t , the o b j e c t s that l e a d to r e s i d u a l v e l o c i t i e s l a r g e r than three times the standard d e v i a t i o n o of the f i r s t f i t . T h i s procedure i s o f t e n used i n the f i e l d of g a l a c t i c kinematics because one has to d e a l with "high v e l o c i t y " o b j e c t s (gas and s t a r s ) which seem to belong to another k i n e m a t i c a l group. For example, Feast and Shuttleworth (1964), i n t h e i r study of 0-B5 s t a r s , decided to e l i m i n a t e the abnormal long 48 t a i l of t h e i r d i s t r i b u t i o n of r e s i d u a l s which corresponded to R > 35 Km/s (=* 3a). Balona and Feast (1974) used the same procedure. C r o v i s i e r (1978) found necessary to set a " v e l o c i t y c u t o f f " i n order to e l i m i n a t e the high v e l o c i t y c l o u d s . H i s estimate of the best v e l o c i t y c u t o f f a l s o corresponds to 3a. One can see from Table I that the long t a i l of our d i s t r i b u t i o n s (more than 3a) represents =* 1.3% of the t o t a l number of o b j e c t s , that i s to say approximately four times more than what i s expected from a pure Gaussian curve! Table I a l s o shows that the e l i m i n a t i o n of these o b j e c t s l e d to a v a r i a t i o n of the c o e f f i c i e n t s which i s , i n average, equal to = 45% of t h e i r r e s p e c t i v e u n c e r t a i n t i e s (1a). U n c e r t a i n t i e s on the v a r i o u s c o e f f i c i e n t s have been determined in two ways. F i r s t , a standard procedure (see P.R. Bevington (1969)) g i v e s the a ( i ) ' s that correspond to the u n c e r t a i n t i e s on the v a r i o u s c o e f f i c i e n t s when they are c o n s i d e r e d to be l i n e a r l y independent. The sigmas obtained with t h i s method w i l l be quoted as standard d e v i a t i o n s of the c o e f f i c i e n t s throughout t h i s study. A second method c o n s i s t s of e v a l u a t i n g the c o v a r i a n c e matrix of the c o e f f i c i e n t s by means of a Monte C a r l o a n a l y s i s i n which the c o e f f i c i e n t s are evaluated many times with sets of data where the v e l o c i t i e s are the f i e l d v e l o c i t i e s to which random p e r t u r b a t i o n s P ( i ) ' s are added. The d i s t r i b u t i o n of P ( i ) ' s i s normal and centered on zero with the standard 49 d e v i a t i o n equal to the mean r e s i d u a l of the f i t . By l o o k i n g at the d i s t r i b u t i o n of c o e f f i c i e n t values found i n t h i s way, i t i s p o s s i b l e to evaluate the u n c e r t a i n t i e s on each c o e f f i c i e n t and to q u a n t i f y the v a r i o u s c o r r e l a t i o n s that e x i s t between them. The v a r i a n c e s found using t h i s method are very s i m i l a r to those found with the standard technique ( t h i s j u s t i f i e s i t s use to a c e r t a i n degree). The cov a r i a n c e s are small i n general but they are used i n the c a l c u l a t i o n of the u n c e r t a i n t i e s that determine the width of the i s o v e l o c i t y contour bands of the v e l o c i t y f i e l d p l o t s . As used by M. Creze' and M.O. Mennesier (1973) i n a s i m i l a r case, 50 seem to be an a p p r o p r i a t e number of i t e r a t i o n s f o r the Monte C a r l o technique s i n c e i t g i v e s s i g n i f i c a n t r e s u l t s but does not use up too much computer time. Along with the e v a l u a t i o n of the c o e f f i c i e n t s by l e a s t squares f i t t i n g , i t i s of course p o s s i b l e to e x t r a c t i n f o r m a t i o n on the d i s t r i b u t i o n of some parameters such as d, 1, b and VI f o r a given set of o b j e c t s . The most i n t e r e s t i n g of these d i s t r i b u t i o n s i s the r e s i d u a l d i s t r i b u t i o n because i t pr o v i d e s a check on the assumption of a gaussian d i s t r i b u t i o n of the departures of VI from V f l . It i s a l s o i n t e r e s t i n g to d e f i n e a " r e s i d u a l f u n c t i o n " R(l,b,d) to see i f any systematic dependence of R on 1, b or d has been l e f t behind i n which case the equation (2-11) would be incomplete. In the case of the v e l o c i t y e l l i p s o i d treatment, two 50 methods have been used i n t h i s study. In equations (2-16) and (2-23) we have s u b s t i t u t e d R 2 to a2 i n order to perform the f i t but R i s equal to VI - V f l and thus depends on the f i e l d equation used. In t h i s study, we c a l c u l a t e d the v e l o c i t y e l l i p s o i d parameters with both our d e t e r m i n a t i o n of V f l (equation (2-11)) and with a "reduced" V f l value t a k i n g i n t o account only the terms that d e s c r i b e the s o l a r motion as determined from our f i t . T h i s "reduced" v e r s i o n of V f l a l l o w s us to compare our r e s u l t s to the v e l o c i t y e l l i p s o i d parameters found i n the l i t e r a t u r e . I t i s a l s o i n t e r e s t i n g to see whether or not the i n t r o d u c t i o n of a r o t a t i o n term and of terms that d e s c r i b e departure from c i r c u l a r motion can i n f l u e n c e the p u z z l i n g d e v i a t i o n of the vertex found to be s i g n i f i c a n t f o r young s t a r s (see Mihalas and Binney ( 1982)). Another p i e c e of i n f o r m a t i o n that can be e x t r a c t e d from t h i s kind of study i s the d i r e c t i o n s and magnitudes of the c o n t r a c t i o n and expansion axes. As d i s c u s s e d by Ogorodnikov (1965) and a p p l i e d by Takakubo (1967), i t i s p o s s i b l e to decompose the g r a d i e n t matrix i n t o two m a t r i c e s , one being symmetric (the deformation matrix) and the other one being antisymmetric (the r o t a t i o n m a t r i x ) . T h i s can be seen i n the f o l l o w i n g way. I f one analyses the three components of equation (2-2) independently, one can w r i t e : 51 u = - u 9 + u,«x + u 2-y + u 3 ' Z v = - v 9 + v,-x + v 2 ' y + v 3 ' Z (2-26) w = -we + w , ' X + w2'Y + w 3 » z where the p r e v i o u s l y d e f i n e d convention f o r the p a r t i a l d e r i v a t i v e n o t a t i o n was used. From there, i t i s p o s s i b l e to draw an analogy with f l u i d mechanics f o r which i t i s u s e f u l to d e f i n e the "displacement t e n s o r " as: (D) u, u 2 u 3 V, v 2 v 3 w, w2 w3 (2-27) The f i e l d v e l o c i t y of a s t a r i s then given by: Vf = V f 0 + (D) r (2-28) where r i s the p o s i t i o n v e c t o r . The meaning of the term (D) r can be c l a r i f i e d by d e f i n i n g a conjugate matrix (Dc): (Dc) = u, v, w, u 2 v 2 w2 u 3 v 3 w3 (2-29) 52 and two combinations of (D) and (Dc): (S) = 1/2 ((D) + (Dc)) u, 1/2 (u 2+v,) 1/2 (v,+u 2) v 2 1/2 (w,+u3) 1/2 (v 3+w 2) (A) = 1/2 ((D) - (Dc)) 0 1/2 (u 2-v,) 1/2 (v,-u 2) 0 1/2 (w,-u 3) 1/2 (w 2-v 3) 1/2 (uj+w,) 1/2 (v 3+w 2) 1/2 (u 3-w,) 1/2 (v 3-w 2) 0 (2-30) Using these symmetric (S) and antisymmetic (A) m a t r i c e s , equation (2-28) becomes: = v f 0 + (S) r + (A) r (2-31 ) The matrix (A) i s easy to i n t e r p r e t s i n c e a l l i t s elements are the components of <3 =1/2 c u r l Vt. T h i s means t h a t : (A) r = 3 x r . (2-32) Equation (2-32) suggests s t r o n g l y that (A) should be c a l l e d the " r o t a t i o n matrix" s i n c e i t d e s c r i b e s the v o r t i c i t y of 5 3 the v e l o c i t y f i e l d . (S) can be i n t e r p r e t e d as a "deformation matrix" because (S) r i s the gr a d i e n t of F with F d e f i n e d as: F = 1/2 [u, x 2 + v 2 y 2 + w3 z 2 +(v 3 + w 2) yz + w, + u 3) xz + ( u 2 + v,) xy] (2-33) T h i s means t h a t , by analogy to f l u i d mechanics, the Helmholtz theorem can apply to the d i f f e r e n t i a l v e l o c i t y f i e l d of nearby o b j e c t s and take the form: Vf = vfo + ~Z x r + grad S . (2-34) Our study, even i f i t i s based s o l e l y on the a n a l y s i s of l i n e - o f - s i g h t v e l o c i t i e s , g i v e s us a l l the elements of the deformation matrix (S) (see (2-12)). T h i s means that we can eva l u a t e the d i r e c t i o n s and magnitudes of the p r i n c i p a l axes of deformation by d i a g o n a l i z i n g ( S). The magnitudes of expansion or c o n t r a c t i o n motions are given by the e i g e n v a l u e s of the matrix and t h e i r d i r e c t i o n s are r e l a t e d to the o r i e n t a t i o n of the e i g e n v e c t o r s . We c a l c u l a t e d these values f o r every set of o b j e c t s on which a f i t procedure was used. In order to remove the c o n t r i b u t i o n of the d i f f e r e n t i a l r o t a t i o n to the deformation matrix, we s u b s t r a c t e d , i n most cases, the standard value of 15 Rm/s/Kpc from the term 1/2 ( u 2 + v , ) . 54 T h i s chapter has d e a l t with the general treatment of the data and with the theory that j u s t i f i e s i t . Our next step i s to d e s c r i b e the r e s u l t s obtained f o r the v a r i o u s groups of o b j e c t s used i n t h i s study and to compare them to those found i n the l i t e r a t u r e . 55 CHAPTER 3 THE KINEMATICS OF STARS a) P r e s e n t a t i o n of the r e s u l t s R e s u l t s f o r a l l s e t s of s t a r s are presented together in t h i s chapter s i n c e they a l l have been obtained with the same method. T h i s method i n v o l v e s determining the kinematic parameters i d e n t i f i e d i n equation (2-12) that minimize the sum of the r e s i d u a l s when known values of VI, 1, b and d are a v a i l a b l e . For each group, two f i t t i n g procedures were performed. We give only the r e s u l t s of the second f i t i n which o b j e c t s that, l e d to r e s i d u a l s l a r g e r than 3a i n the f i r s t f i t were r e j e c t e d . For the purpose of comparison with the l i t e r a t u r e , we have a l s o used equation (1-13) from which we have d e r i v e d the value of the nodal d e v i a t i o n 1 0. A supplementary f i t was performed with only four terms: the Vk term and the three terms that d e s c r i b e the motion of the Sun with respect to the LSR. For each group the r e s u l t s are presented i n a t a b l e which g i v e s the number of o b j e c t s i n the group, the average d i s t a n c e of these o b j e c t s , the nodal d e v i a t i o n 1 0 (when c a l c u l a t e d ) and the d i r e c t i o n s and magnitudes of the three p r i n c i p a l axes of expansion or c o n t r a c t i o n obtained from the d i a g o n a l i z a t i o n of the deformation t e n s o r , (see equation (2-30) to (2-34)). A p o s i t i v e value of X corresponds to an expansion while X < 0 56 i n d i c a t e s a c o n t r a c t i o n . In g e n e r a l , a r o t a t i o n term equal to the standard Oort c o e f f i c i e n t A = 15 Rm/s/Kpc has been removed from the deformation t e n s o r . However, fo r some groups, the p e c u l i a r value of the r o t a t i o n term K(8) suggested that i t would have been meaningless to remove t h i s standard v a l u e . The t a b l e a l s o g i v e s the values d e r i v e d for the c o e f f i c i e n t s l i s t e d in equation (2-12) and t h e i r corresponding u n c e r t a i n t i e s ( t h e i r a s s o c i a t e d standard d e v i a t i o n ) , the average r e s i d u a l and the val u e s (and corresponding u n c e r t a i n t i e s ) of Vk and of the s o l a r motion terms obtained from a four c o e f f i c i e n t f i t . F i n a l l y the t a b l e d i s p l a y s the r e s u l t s of the v e l o c i t y e l l i p s o i d a n a l y s i s . These l a t t e r r e s u l t s are d i v i d e d i n two f a m i l i e s : one f o r which only the s o l a r motion terms has been removed from VI to c a l c u l a t e the r e s i d u a l s and one f o r which a l l the terms of the v e l o c i t y f i e l d have been used i n determining the values of the r e s i d u a l s . For each family the f o l l o w i n g q u a n t i t i e s are given: o ( a ) , a(b) and a(c) which are the r e s i d u a l s i n the d i r e c t i o n s of the three axes of the e l l i p s o i d when we f o r c e one of these axes to l i e along z; CT(X), a(y) and a(z) which are the p r o j e c t i o n s of these e l l i p s o i d axes along the three p r i n c i p a l axes of the Galaxy; and a ( i ) , M i ) and b ( i ) , the magnitudes and d i r e c t i o n s of the three axes of the e l l i p s o i d when they are f r e e to take any o r i e n t a t i o n . For each group of s t a r s , a number of p l o t s are 57 presented to d e s c r i b e the g l o b a l c h a r a c t e r i s t i c s of the v e l o c i t y f i e l d . Some of them, s p e c i f i c a l l y the three dimensional p l o t s , show the dependence of V f l on 1 and b at an a r b i t r a r i l y f i x e d d i s t a n c e of 1 Kpc. The other p l o t s show the shape and p o s i t i o n of the i s o v e l o c i t y contours f o r v a r i o u s s l i c e s of the v e l o c i t y f i e l d . The " h o r i z o n t a l " v e l o c i t y f i e l d s give the s t a t e of motion along the g a l a c t i c plane and the " v e r t i c a l " v e l o c i t y f i e l d s d e s c r i b e the kinematics i n planes that are p e r p e n d i c u l a r to the g a l a c t i c plane. The width of the bands g i v e s an idea of the u n c e r t a i n t i e s a s s o c i a t e d with the de t e r m i n a t i o n of the c o e f f i c i e n t s . Here not only the v a r i a n c e of each c o e f f i c i e n t has been used but the c o v a r i a n c e s a s s o c i a t e d with t h e i r c o r r e l a t i o n have a l s o been i n c l u d e d i n the deter m i n a t i o n of the u n c e r t a i n t y . For a l l p l o t s , p r i o r i t y has been given to the lowest i s o v e l o c i t y contour where one poi n t c o u l d belong to many of them. Where the probable e r r o r was too l a r g e , p l o t s have been produced with an a r b i t r a r y constant value of u n c e r t a i n t y . For most groups of s t a r s the f i e l d s are p l o t t e d with and without the removal of the standard value of K(8)= A =15 Km/s/Kpc which corresponds to the u n d e r l y i n g g a l a c t i c d i f f e r e n t i a l r o t a t i o n . We have presented a great many of these f i e l d s and, i n order to preserve the c o n t i n u i t y of the t e x t , we have decided to •present them at the end of the chapter. F i g u r e s 3 to 5 show the i s o v e l o c i t y contours f o r the 58 i d e a l case of pure d i f f e r e n t i a l c i r c u l a r motion with the standard value of 15 f o r the Oort c o e f f i c i e n t . The reader can compare these a r t i f i c i a l f i e l d s with the a c t u a l f i e l d s in order to ev a l u a t e , i n i n d i v i d u a l cases, the importance of departure from c i r c u l a r motion. F i g u r e s 6 to 11 show the dependence of V f l on 1 and b generated by every f i e l d term with a magnitude a r b i t r a r i l y f i x e d to equal 15 Km/s/Kpc. b) R e s u l t s f o r B s t a r s i ) A l l B s t a r s As we have seen in Chapter 1, B s t a r s are very i n t e r e s t i n g probes of the kinematics because of t h e i r high l u m i n o s i t y which allows one to observe them at l a r g e d i s t a n c e s and because of t h e i r small random motion r e l a t e d to t h e i r young age. U n f o r t u n a t l y , these s t a r s o f t e n appear in young a s s o c i a t i o n s where they were probably j u s t r e c e n t l y formed and thus have a lumpy s p a t i a l d i s t r i b u t i o n . T h i s lumpiness can be a problem because i t can render the d e s c r i p t i o n of the kinematics by a smooth v e l o c i t y f i e l d inadequate. We are n e v e r t h e l e s s c o n f i d e n t that t h i s e f f e c t i s not too important s i n c e Lesh (1972) d e r i v e d v e l o c i t y g r a d i e n t v a l u e s that were very s i m i l a r f o r f i e l d s t a r s and for a s s o c i a t i o n s t a r s . The r e s u l t s of our a n a l y s i s of B s t a r s are presented in Table III and i n F i g u r e s 12 to 18. T h e i r dominant motion i s a d i f f e r e n t i a l r o t a t i o n which leads to an Oort c o e f f i c e n t 59 of 10.4 ± 0.7 Km/s/Kpc, a somewhat low value i n comparison to the standard value of 15. B s t a r s show a remarkable c h a r a c t e r i s t i c i n that t h e i r v e l o c i t y e l l i p s o i d i s very s p h e r i c a l . T h i s w e l l known r e s u l t i s confirmed here and even strenghtened when a l l terms of the v e l o c i t y f i e l d are used to compute the r e s i d u a l s . Since the s p a t i a l d i s t r i b u t i o n of observed B s t a r s extends over r e l a t i v e l y l a r g e d i s t a n c e s , a second order f i t as d e s c r i b e d by equation (A-7) c o u l d be more a p p r o p r i a t e than our usual f i r s t order f i t . T h i s was t e s t e d and we found that the i n c l u s i o n of the a d d i t i o n a l 10 terms does not s i g n i f i c a n t l y improve the goodness of the the f i t . (An F - t e s t shows that the improvement i s not s i g n i f i c a n t to the 99 % l e v e l . ) We have decided not to present s u b d i v i s i o n s of B s t a r s i n t o e a r l y and l a t e s p e c t r a l types s i n c e we found that the r e s u l t s of the f i t are not c l o s e l y dependent upon s p e c t r a l type. Instead of t h a t , we have s u b d i v i d e d them i n t o three c l a s s e s : 1) BI-IV, 2) BV with d > 200 pc and 3) BV with d < 200 pc. These s u b d i v i s i o n s allow a comparison, over s i m i l a r d i s t a n c e ranges, of main sequence s t a r s with s t a r s of other l u m i n o s i t y c l a s s e s and a comparison of two se t s of the same type of o b j e c t s (BV) over d i f f e r e n t d i s t a n c e i n t e r v a l s . The l i m i t of 200 pc was f i x e d to c r e a t e a set of BV s t a r s that has an average d i s t a n c e s i m i l a r to the one of AV s t a r s , making then a d i r e c t 60 TABLE III Re s u l t s f o r B s t a r s No.of o b j e c t s <d> 1 301 438 pc -1° F i t with 10 c o e f f i c i e n t s F i t with 4 c o e f f i c i e n t s K(1) K(2) K(3) K(4) 3.4 -8.7 16.2 -6.2 + + + 0.4 Km/s 0.5 Km/s 0.5 Km/s 1.0 Km/s K ( 1 ) = 2.3 ± 0 . 4 Km/s K(2) = -8.0 ± 0.5 Km/s K(3) =-17.3 ± 0.5 Km/s K(4) = -7.6 ± 1.1 Km/s K(5) = -3. 2 + 1 .0 Km/s/Kpc K(6) = -5. 6 + 4 .5 Km/s/Kpc K(7) = -4. 8 + 1 .0 Km/s/Kpc K(8) = 10. 4 + 0 .7 Km/s/Kpc K(9) = 6. 3 + 2 .2 Km/s/Kpc K(10)= -10. 5 + 2 .5 Km/s/Kpc average r e s i d u a l a =11.5 Km/s P r i n c i p a l axes of the deformation tensor X, = -15.9 Km/s/Kpc 1 , = -81° b , = 31° X 2 = -7.5 Km/s/Kpc 1 2 = 215° b 2 = -37° X 3 = 9.9 Km/s/Kpc 1 3 = 162° b 3 = 38° V e l o c i t y e l l i p s o i d parameters a) With LSR only a(a) = 14. 4 a(b) = 11.6 a(c) = 9. 9 Km/s o(x) = 13. 6 a(y) = 12.6 a(z) = 9. 9 Km/s l v = -35° a, = 8.3 a 2 = 12.7 a 3 = 14.9 Km/s I i = 39° 1 2 = 76° 1 3 = -23° b, = 57° b 2 = -27° b 3 = -17° b) With a l l v e l o c i t y f i e l d terms o(a) = 11. 8 a(b) = 10.4 a(c) = 11. 0 Km/s a(x) = 11. 7 a(y) = 10.5 a(z) = 11. 0 Km/s l v = 14° a, = 9.2 a 2 = 10.6 a 3 = 13.4 Km/s l i = 241 ° 1 2 = 129° 1 3 = 205° b, = -47° b 2 = - 1 9 ° b 3 = 37° 61 comparison of these two types of o b j e c t s p o s s i b l e . Our r e s u l t s f o r B s t a r s can be compared to those of B a l a k i r e v (1977) i f we perform, as he d i d , a f i t without the Vk term. Table IV compares h i s r e s u l t s with ours f o r two l i m i t i n g d i s t a n c e s . From t h i s , one can see that the agreement i s , i n g e n e r a l , q u i t e good. Nevertheless we c o n s i d e r t h a t the Vk term should be introduced because i t s absence a f f e c t s the value of the K(6) term (the v e l o c i t y g r a d i e n t 9w/3z) which i s , because of the c o n c e n t r a t i o n of B s t a r s along z = 0, the most l i k e l y to absorb constant s h i f t s in VI caused by i n f l o w s or g r a v i t a t i o n a l r e d s h i f t s that can occur at the s u r f a c e of these young o b j e c t s . For i n s t a n c e , B s t a r s give a value of 3.4 Km/s f o r the Vk term and when t h i s term i s r e j e c t e d , K(6) changes from a negative to a p o s i t i v e v a l u e . For small d i s t a n c e s l i k e d < 250 pc, Vk i s small and t h i s e f f e c t i s not very important. One can a l s o note that our determination of 1 0 i s i n f a i r agreement with those of Rubin and Burley (1964) who obtained 1 0 = -5.5° f o r 0-B5 s t a r s and of P e t r i e and P e t r i e (1967) who got 1 0 = -4.9° f o r a set of B s t a r s . Table V shows the comparison between the two dimensional a n a l y s i s of Ovenden, Pryce and Shuter (1983) c a r r i e d out with 988 0 and B s t a r s and our r e s u l t s when we use t h e i r e q u a t i o n . A supplementary s o l u t i o n i s given i n which a Vk term has been introduced and one can see that t h i s term appears to be s i g n i f i c a n t . 62 TABLE IV Res u l t s f o r B s t a r s using a 9 c o e f f i c i e n t f i t a) f o r d < 250 pc B a l a k i r e v ( 117 s t a r s ) T h i s study ( 653 s t a r s ) K(2) =-11.5 ± 0.6 Km/s K(2) = -9.2 ± 0 . 8 Km/s K(3) = - 1 8 . 1 ± 1.0 Km/s K(3) =-17.7 ± 0 . 8 Km/s K(4) = -7.2 ± 0.7 Km/s K(4) = -6.5 ± 1.4 Km/s K(5) = 24 ± 7 Km/s/Kpc K ( 5 ) = 8.5 ± 6 . 1 Km/s/Kpc K(6) = 24 ± 1 2 Km/s/Kpc K(6) = 29 ± 1 6 Km/s/Kpc K(7) = 39 ± 1 1 Km/s/Kpc K(7) = 14.4 ± 5 . 8 Km/s/Kpc K ( 8 ) = 2 ± 1 3 Km/s/Kpc K ( 8 ) = 16.4 ± 4.9 Km/s/Kpc K(9) =-12 ± 14 Km/s/Kpc K(9) = 20.2 ± 7 . 8 Km/s/Kpc K(10)= -4 ± 1 9 Km/s/Kpc K(10)=-16.4 ± 8.5 Km/s/Kpc b) f o r d < 1500 pc B a l a k i r e v ( 183 s t a r s ) T h i s study (1260 s t a r s ) K(2) =-11.4 ± 0.8 Km/s K(2) = -8.6 ± 0.6 Km/s K(3) =-18.9 ± 0.8 Km/s K(3) =-16.2 ± 0.6 Km/s K(4) = -8.0 ± 0.6 Km/s K(4) = -7.0 ± 1.2 Km/s K(5) = -3 ± 3 Km/s/Kpc K(5) = 1.0 ± 1.8 Km/s/Kpc K(6) = 27 ± 7 Km/s/Kpc K(6) = 11.2 ± 8.0 Km/s/Kpc K(7) = 8 ± 3 Km/s/Kpc K ( 7 ) = 1.7 ± 1 . 6 Km/s/Kpc K(8) = 16 ± 4 Km/s/Kpc K ( 8 ) = 8.5 ± 1 . 5 Km/s/Kpc K(9) = -1 ± 9 Km/s/Kpc K ( 9 ) = 3.8 ± 3 . 8 Km/s/Kpc K(10)=-19 ± 9 Km/s/Kpc K(10)= -4.5 ± 3.9 Km/s/Kpc 63 TABLE V R e s u l t s for B s t a r s using equation (1-15)* Ovenden T h i s study T h i s study et a l . * * without Vk with Vk C O ) Km/s - - 3.1 ± 0.4 C(2) Km/s -7.4 -9.5 ± 0.5 -9.2 ± 0.5 C(3) Km/s -15.6 -15.7± 0.5 -16. 3± 0.5 C (4) Km/s/Kpc -1.3 0.1 ± 0.5 -3.1 ± 0.7 C(5) Km/s/Kpc 13.2 9.8 ± 0.8 9.1 ± 0.8 C(6) Km/s/Kpc -1 .7 0.6 ± 0.9 1 . 1 ± 0.9 C(7) Km/s/Kpc 2 -0.5 0.4 ± 0.5 0.9 ± 0.5 C(8) Km/s/Kpc 2 -0.3 1.1 ± 0.4 1.1 ± 0.4 C (9) Km/s/Kpc 2 0.9 0.8 ± 0.4 0.6 ± 0.4 COO) Km/s/Kpc 2 0.1 -1.4 ± 0.4 -1 .2 ± 0.4 * The equation was o r i g i n a l y : VI = Vk - u 0 c o s l cosb - v s s i n l cosb - wG s i n b + d c o s 2 b [ K + A ' s i n 2 ( l - 1 0 ) ] + d 2 c o s 3 b [ K, s i n ( l - K 2 ) +K3 s i n 3 ( l - K , ) ] but was transformed i n t o : VI = C O ) + C(2) c o s l cos b + C(3) s i n l cosb + C(4) d c o s 2 b + C(5) d sin21 c o s 2 b + C(6) d cos21 c o s 2 b + C(7) d 2 s i n l c o s 3 b + C (8 ) d 2 c o s l c o s 3 b + C(9) d 2 sin31 c o s 3 b + COO) d 2 cos31 c o s 3 b with w9 f i x e d to be 7.9 Km/s . ** Ovenden, Pryce and Shuter (1983) 64 i i ) BI-IV s t a r s Table VI shows the r e s u l t s of our a n a l y s i s of BI-IV s t a r s . Here again the use of a l l f i e l d terms i n the deter m i n a t i o n of the r e s i d u a l s reduces the e c c e n t r i c i t y of the v e l o c i t y e l l i p s o i d . One can see that the Vk term i s small f o r the four c o e f f i c i e n t f i t but stays l a r g e f o r the 10 c o e f f i c i e n t f i t . T h i s confirms the r e s u l t obtained long ago by P l a s k e t t and Pearce (1934) that the value of the Vk term decreases with i n c r e a s i n g l u m i n o s i t y when only four c o e f f i c i e n t s are used to f i t the data. However i t a l s o shows that t h i s v a r i a t i o n does not correspond to a true decrease of Vk but rather to a c a n c e l l a t i o n of Vk with the u n d e r l y i n g f i e l d terms at l a r g e d i s t a n c e s . T h i s tendency of having a d e c r e a s i n g value of Vk with i n c r e a s i n g d i s t a n c e i n a four c o e f f i c i e n t f i t was a l s o observed by F r o g e l and St o t h e r s (1977) who obtained Vk = 4.2 Km/s f o r d < 200 pc, Vk = 2.1 Km/s f o r d < 400 pc and Vk = 1.8 Km/s f o r d < 800 pc f o r a set of 0 and B s t a r s . F i g u r e s 19 to 26 show the shape of the v e l o c i t y f i e l d of BI-IV s t a r s . I t s s i m i l a r i t y with the f i e l d of a l l B s t a r s shows that the b r i g h t and d i s t a n t BI-IV s t a r s dominate the f i t . i i i ) BV s t a r s The r e s u l t s of the a n a l y s i s of BV s t a r s with 65 TABLE VI Re s u l t s f o r BI-IV s t a r s No.of o b j e c t s = 451 <d> = 692 pc lo = "2° F i t with 10 c o e f f i c i e n t s F i t with 4 c o e f f i c i e n t s Km/s Km/s Km/s Km/s K(1) = 3.4 + 0.8 Km/s K(1 ) = 0.8 + 0.7 K(2) = -8.2 + 0.9 Km/s K(2) = -6.1 + 1 . 1 K(3) = -17.4 + 0.8 Km/s K(3) =-19.3 + 1 .0 K(4) = -6.7 + 2.0 Km/s K(4) =-10.0 + 2.5 K(5) = -2.8 + 1 . 1 Km/s/Kpc K(6) = -6.9 + 5.6 Km/s/Kpc K(7) = -4.0 + 1 .2 Km/s/Kpc K(8) = 11.2 + 0.8 Km/s/Kpc K(9) = 5.4 + 2.9 Km/s/Kpc K(10)= -9.8 + 3.4 Km/s/Kpc average r e s i d u a l a =11.2 Km/s P r i n c i p a l axes of the deformation tensor X, = -15.6 Km/s/Kpc 1, = 260° b, = -30° X 2 = -6.3 Km/s/Kpc 1 2 = -38° b 2 = 39" X 3 = 8.1 Km/s/Kpc 1 3 = 16° b 3 = -36° o V e l o c i t y e l l i p s o i d parameters a) With LSR only a(a) = 17. 5 a(b) = 11.5 a(c) = 7. 6 Km/s a(x) = 15. 0 a(y) = 14.7 a(z) = 7. 6 Km/s l v = -43° o, = 3.3 a2 = 13.8 a 3 = 17.6 Km/s l i = 42° 1 2 = 233° 1 3 = 140° b, = 59° b 2 = 31° b 3 = 5° b) With a l l v e l o c i t y f i e l d terms o(a) = 10. 1 a(b) = 11.4 a(c) = 14. 1 Km/s a(x) = 10. 6 a(y) = 10.9 a(z) = 14. 1 Km/s l v = -40° a, = 9.3 a 2 = 10.4 a 3 = 15.1 Km/s l i = 105° 1 2 = 5° 1 3 = 243° b, = 27° b 2 = 20° b 3 = 56° 66 d > 200 pc are given in Table VII and d i s p l a y e d in F i g u r e s 27 to 36. These s t a r s l e a d to a very low value of the Oort c o e f f i c i e n t K(8) = A = 5.1 ± 2.2 Km/s/Kpc. Table VIII and F i g u r e s 37 to 42 d e s c r i b e the v e l o c i t y f i e l d obtained f o r BV s t a r s c l o s e r than 200 pc. The u n c e r t a i n t i e s are rather l a r g e with t h i s set of s t a r s because i t c o n t a i n s only 330 members and because the e f f e c t of the f i e l d terms i s l e s s important f o r such small d i s t a n c e s . c) R e s u l t s f o r AV s t a r s The r e s u l t s of our i n v e s t i g a t i o n of the kinematics of AV s t a r s are summarized in Table IX and i n F i g u r e s 43 to 48. The average d i s t a n c e of observed AV s t a r s i s small s i n c e these o b j e c t s are i n t r i n s i c a l l y f a i n t e r than B s t a r s . They a l s o show l a r g e r random motions (a = 13.5 Km/s) that can be r e l a t e d to t h e i r r e l a t i v e l y o l d e r age. The ten c o e f f i c i e n t f i t g i v e s a resonable value f o r the Oort c o e f f i c i e n t K(8) = A = 11.0 ± 7.4 Km/s/Kpc. T h i s r e s u l t i s , in a sense, s u r p r i s i n g because the set of AV s t a r s i s a c t u a l l y very inhomogeneous. I t can i n f a c t be d i v i d e d i n t o two subsets of e a r l y and l a t e s p e c t r a l type s t a r s that have extremely d i f f e r e n t kinematic p r o p e r t i e s . A l a r g e negative K(5) term (the v e l o c i t y g r a d i e n t 3u/9x ) in t r o d u c e s a f a i r l y important departure from c i r c u l a r motion that can be seen i n the p l o t s . As i n the case of nearby BV s t a r s , the Vk term i s almost z e r o . As shown in Table X, the v e l o c i t y e l l i p s o i d 67 TABLE VII R e s u l t s f o r B s t a r s with d > 200 pc No.of o b j e c t s =. 520 <d> = 414 pc 1 0 = "20° F i t with 10 c o e f f i c i e n t s F i t with 4 c o e f f i c i e n t s K(1 ) K(2) K(3) K(4) 3.0 ± -9.1 ± 14.1 ± -5.1 ± 1.1 Km/s 0.9 Km/s 1.0 Km/s 2.2 Km/s K ( 1 ) = 2 . 4 ± 0 . 6 Km/s K(2) = -9.0 ± 0.9 Km/s K(3) =-15.2 ± 0.9 Km/s K(4) = -6.8 ± 2.0 Km/s K(5) = -4. 2 + 3. 5 Km/s/Kpc K(6) = 7 + 1 1 Km/s/Kpc K(7) = -7. 0 + 3. 0 Km/s/Kpc K(8) = 5. 1 + 2. 2 Km/s/Kpc K(9) = 13. 5 + 4. 6 Km/s/Kpc K(10)= -11. 3 + 4. 7 Km/s/Kpc average r e s i d u a l a =11.6 Km/s P r i n c i p a l axes of the deformation tensor X, = -15.7 Km/s/Kpc 1, = 143° b , = 31° X 2 = -11.8 Km/s/Kpc 1 2 = 216° b 2 = -26° X 3 = 23.6 Km/s/Kpc 1 3 = -87° b 3 = 47° V e l o c i t y e l l i p s o i d parameters a) With LSR only a(a) = 13. 5 o(b) = 10.3 oic) = 12. 7 Km/s oix) = 13. 5 oiy) = 10.3 oiz) = 12. 7 Km/s l v = -4° o, = 9.7 o 2 = 11.7 03 = 15.3 Km/s 1, = 75° 1 2 = 138° 1 3 = 187° b, = 26° b 2 = -42° b 3 = 36° b) With a l l v e l o c i t y f i e l d terms a(a) = 13. 1 aib) = 10.6 oic) = 6. 8 Km/s oix) = 13. 0 oiy) = 10.7 oiz) = 6. 8 Km/s l v = -9° 0, = 4.4 o2 = 10.9 o3 = 14.4 Km/s 1, = 33° 1 2 = 96° la = 182° b, = 66° b 2 = -11° b 3 = 21 0 68 TABLE VIII R e s u l t s f o r B s t a r s with d < 200 pc No.of o b j e c t s = 330 <d> = 133 pc lo = 28° F i t with 10 c o e f f i c i e n t s F i t with 4 c o e f f i c i e n t s K(1) = 0. 7 + 1 .8 Km/s K(1) > = 2.8 + 0.6 K(2) = -9. 1 + 0.9 Km/s K(2) = -9.0 + 0.9 K(3) = -18. 2 + 1 .0 Km/s K(3) =-17.1 + 0.9 K(4) = -7. 0 + 1 .5 Km/s K(4) = -7.0 + 1 .4 K(5) = -2 + 1 4 Km/s/Kpc K(6) = 24 + 25 Km/s/Kpc K(7) = 30 + 1 4 Km/s/Kpc K(8) = 12. 7 + 6.8 Km/s/Kpc K(9) = 1 5 + 1 1 Km/s/Kpc R(10)= -29 + 1 2 Km/s/Kpc Km/s Km/s Km/s Km/s average r e s i d u a l a =10.3 Km/s P r i n c i p a l axes of the deformation tensor X, = -12.8 Km/s/Kpc 1, = 138° b , = 11° X 2 = 6.0 Km/s/Kpc 1 2 = 59° b 2 = -45° X 3 = 58.6 Km/s/Kpc 1 3 = 37° b 3 = 43° V e l o c i t y e l l i p s o i d parameters a) With LSR only oia) = 12. 4 oib) = 9 .7 oic) = 9 . 3 Km/s oix) = 1 1 . 8 oiy) = 10.5 oiz) = 9 . 3 Km/s l v = 30° " i = 7.0 o2 = 10.0 o3 = 14.0 Km/s l i = 67° 1 2 = 135° 1 3 = 37° b, = 57° b 2 = - 1 3 ° b 3 = - 2 9 ° >) With L a l l v e l o c i t y f i e l d terms oia) = 1 1 . 8 oib) = 9 .0 oic) = 7. 7 Km/s oix) = 11. 6 oiy) = 9 .4 oiz) = 7. 7 Km/s l v = 20° 0\ = 6.2 o2 = 9 .2 o2 = 13.0 Km/s l l = 32° 1 2 = 112° 1 3 = 20° b, = 64° b 2 = - 5 ° b 3 = - 2 6 ° 69 r e s u l t s are i n c l o s e agreement with those obtained by A.E. Gomez (1974). A remarkable c h a r a c t e r i s t i c of AV s t a r s i s that they l e a d to a very poor f i t ! In f a c t , when we use an F - t e s t to i n v e s t i g a t e how s i g n i f i c a n t the improvement of the goodness of the f i t i s when new terms are introduced, we f i n d t h a t , besides Vk and the s o l a r motion terms, no other term c o n t r i b u t e s to a s i g n i f i c a n t improvement. We b e l e i v e that t h i s i s due to the inhomogeneity mentioned b e f o r e . In f a c t , i f the set of AV s t a r s i s composed of two very d i s t i n c t sub-groups, a f i t with a l l AV s t a r s would only give the average p r o p e r t i e s of these sub-groups, a set of r e l a t i v e l y meaningless v a l u e s . The two sub-groups are: 1) the A(0~4)V s t a r s that c o n s t i t u t e the l a r g e s t p o r t i o n of the group of a l l AV s t a r s with 828 members; and 2) the 233 A(5~9)V s t a r s . The e a r l y AV s t a r s are, w i t h i n the u n c e r t a i n t i e s of our c a l c u l a t i o n s , b a s i c a l l y at r e s t when the e f f e c t of the p e c u l i a r motion of the Sun i s s u b s t r a c t e d . They do not even seem to be i n d i f f e r e n t i a l r o t a t i o n about the g a l a c t i c c enter s i n c e they l e a d to a very low value f o r the Oort c o e f f i c i e n t (K(8) = A =-2.0 ± 8.2 Km/s/Kpc). T h e i r "kinematic" p r o p e r t i e s are given i n Table XI and i n F i g u r e s 49 to 52. One can see that the v e l o c i t y f i e l d s are f i l l e d mostly with the l a r g e zero v e l o c i t y band. The l a t e AV s t a r s , on the other hand, give very l a r g e values of the v a r i o u s v e l o c i t y g r a d i e n t s with, i n p a r t i c u l a r , an Oort 70 TABLE IX R e s u l t s f o r AV s t a r s No.of o b j e c t s <d> lo = 1061 = 109 pc = 23° F i t with 10 c o e f f i c i e n t s F i t with 4 c o e f f i c i e n t s K(1) = -0.2 + 1 .0 Km/s K(1 ) = -0.3 + 0.6 Km/s K(2) =- 11.6 + 1 .0 Km/s K(2) =-10.4 + 0.8 Km/s K(3) =- 11.7 + 1 .3 Km/s K(3) =-11.5 + 0.9 Km/s K(4) = -6.8 + 1 . 1 Km/s K(4) = -6.3 + 0.9 Km/s K(5) = - 15 + 1 1 Km/s/Kpc K(6) = 8 + 15 Km/s/Kpc K(7) = 4 + 12 Km/s/Kpc K(8) = 11.0 + 7.4 Km/s/Kpc K(9) = 2.7 + 8.1 Km/s/Kpc K(10)= 1.8 + 8.5 Km/s/Kpc average r e s i d u a l a =13.5 Km/s P r i n c i p a l axes of the X, = -15.7 Km/s/Kpc X 2 = 4.8 Km/s/Kpc X 3 = 9.1 Km/s/Kpc deformation tensor 1, = -14° b, = 3° 1 2 = 78° b 2 = 19° 1 3 = 248° b 3 = 71° V e l o c i t y e l l i p s o i d parameters a) With LSR only a(a) = 18.0 a(b) = 10.5 a(c) = 10.0 Km/s a(x) = 17.5 a(y) = 11.2 a(z) = 10.0 Km/s l v = 16° a, = 9.6 a 2 = l 0 . 6 a 3 = 18.1 Km/s 1, = 127° 1 2 = 104° 1 3 = 196° b, = -7° b 2 = 18° b 3 = 7° b) With a l l v e l o c i t y f i e l d terms a(a) = 17. 9 a(b) = 10.4 a(c) = 9.9 Km/s a(x) = 17. 4 a(y) = 11.2 a(z) = 9.9 Km/s l v = 17° a, = 9.5 a 2 = 10.5 a 3 = 18.1 Km/s 1, = 128° 1 2 = 105° 1 3 = 1 97° b, = -71° b 2 = 18° b 3 = 7° 71 TABLE X Comparison of our v e l o c i t y e l l i p s o i d with Gomez (1972) V e l o c i t y e l l i p s o i d parameters of AV s t a r s a) our r e s u l t s = 9.5 0 " 2 = 10.5 o 3 = = 18.1 Km/s l i = 128° 1 2 = 105° 1 3 = = 197° b, = -71 ° b.2 = 18° b 3 = 7° b) r e s u l t s of Gomez = 8.3 o2 = 11.2 o3 = = 19.1 Km/s l i = 119° 1 2 = 106° 1 3 = = 199° b, = -64° b 2 = 26° b 3 = 5° 72 c o e f f i c i e n t K(8) of 51 ± 17 Km/s/Kpc! Table XII and F i g u r e s 53 to 56 show the c h a r a c t e r i s t i c s of the v e l o c i t y f i e l d f o r these s t a r s . In order to i n v e s t i g a t e the r e a l i t y of the i n f l u e n c e of s p e c t r a l s u b - c l a s s on the kinematics of AV s t a r s , we determined how a random s e l e c t i o n of about 800 s t a r s out of 1061 can a f f e c t the d e t e r m i n a t i o n of the v a r i o u s parameters. We found that a t y p i c a l v a r i a t i o n f o r a given parameter i s aproximately equal to one t h i r d of i t s a s s o c i a t e d sigma. From t h i s , i t seems that the v a r i a t i o n of K(8) from 11 ± 7 to -2 ± 8 Km/s/Kpc i s f i v e times more important than a t y p i c a l f l u c t u a t i o n and i s thus probably r e a l . With t h i s in mind, a l l r e s u l t s concerning the AV s t a r s should be i n t e r p r e t e d with c a r e . d) R e s u l t s f o r K i l l s t a r s The r e s u l t s of our a n a l y s i s of the kinematics of K i l l (or K-giants) s t a r s are given i n Table XIII and i n F i g u r e s 57 to 66. Owing to t h e i r o l d age, these s t a r s have la r g e random motions. T h i s can be seen by the l a r g e value of the average r e s i d u a l (o = 23.3 Km/s). For the same reason, the study of the kinematics of K i l l s t a r s can provide some i n f o r m a t i o n on the e v o l u t i o n of n o n - c i r c u l a r motion with time. L i k e f o r the case of AV s t a r s , the goodness of the f i t i s not s i g n i f i c a n t l y improved when new terms are added 73 TABLE XI Re s u l t s f o r A(0-4)V s t a r s No.of o b j e c t s = 828 <d> = 111 pc x o F i t with 10 c o e f f i c i e n t s F i t with 4 c o e f f i c i e n t s Km/s K(1) = -1.1 + 1 . 1 Km/s K(1 ) = -1.3 + 0.7 K(2) =-10.0 + 1.2 Km/s K(2) = -9.8 + 0.8 K(3) =-10.0 + 1.4 Km/s K(3) =-10.2 + 1 .0 K(4) = -7.0 + 1.2 Km/s K(4) = -6.3 + 1 .0 K(5) = -7 + 13 Km/s/Kpc K(6) = 1 1 + 16 Km/s/Kpc K(7) = -2 + 13 Km/s/Kpc K(8) = -2.0 + 8.2 Km/s/Kpc K(9) = 0.5 + 9.2 Km/s/Kpc K(10) = 5.9 + 9.5 Km/s/Kpc Km/s average r e s i d u a l a =13.3 Km/s P r i n c i p a l axes of the deformation tensor (no r o t a t i o n term removed here) X, = -8.0 Km/s/Kpc 1 , = 150° b, = -1° X 2 = -3.3 Km/s/Kpc 1 2 = 240° b 2 = 21° X 3 = 13.4 Km/s/Kpc 1 3 = 62° b 3 = 69° V e l o c i t y e l l i p s o i d parameters a) With LSR only a(a) = 17. 4 oib) = 9.9 oic) = 10. 6 Km/s oix) = 17. 1 oiy) = 10.6 oiz) = 10. 6 Km/s l v = 15° a, = 9.4 a 2 = 10.6 a 3 = 17.7 Km/s 1 , = -66° 1 2 = "87° 1 3 = 15° bi = 40° b 2 = -49° b 3 = -10° b) With a l l v e l o c i t y f i e l d terms o(a) = 17. 6 a(b) = 9.9 oic) = 10. 6 Km/s o(x) = 17. 2 oiy) = 10.6 oiz) = 10. 6 Km/s l v - 15° » 1 = 9.4 a 2 = 10.6 a 3 = 17.8 Km/s 1 , = 114° 1 2 = 95° 1 3 = 15° b, = -42° b 2 = 46° b 3 = -10° 74 TABLE XII Res u l t s f o r A(5-9)V s t a r s No.of o b j e c t s = 233 <d> = 104 pc 1 o F i t with 10 c o e f f i c i e n t s F i t with 4 c o e f f i c i e n t s Km/s K( 1) = 2. 4 ± 2.1 Km/s K(1) = 2. 4 + 1 .4 K(2) =-16. 1 ± 2.2 Km/s K(2) = -11. 8 + 1 .8 K(3) =-17. 4 ± 2.9 Km/s K(3) = -14. 9 + 2 . 1 K(4) = -7. 3 ± 2.7 Km/s K(4) = -6. 9 + 2 .0 K(5) =-49 ± 22 Km/s/Kpc K(6) = -3 ± 33 Km/s/Kpc K(7) = 43 ± 29 Km/s/Kpc K(8) = 51 ± 17 Km/s/Kpc K(9) = 15 ± 18 Km/s/Kpc K(10)= 1 ± 20 Km/s/Kpc average r e s i d u a l a =13.9 Km/s P r i n c i p a l axes of the deformation tensor (no r o t a t i o n term removed here) X, = -74 Km/s/Kpc 1, = 172° b, = 24° X 2 = -1 Km/s/Kpc 1 2 = -24° b 2 = 65° X 3 = 66 Km/s/Kpc 1 3 = 79° b 3 = 6° V e l o c i t y e l l i p s o i d parameters a) With LSR only o(a) = 19. 1 a(b) = 13.9 a(c) = 7. 0 Km/s tx(x) = 17. 5 a(y) = 15.9 a(z) = 7. 0 Km/s l v = .36° o, = 7.0 a2 = 13.9 a 3 = 19.1 Km/s 1 , = 159° 1 2 = 126° 1 3 = 216° b, = 87° b 2 = -2° b 3 = -1 0 b) With a l l v e l o c i t y f i e l d terms o(a) = 17. 7 a(b) = 12.8 a(c) = 7. 1 Km/s a(x) = 16. 8 a(y) = 14.0 a(z) = 7. 2 Km/s l v = 27° a, = 7.1 a 2 = 12.9 a 3 = 17.7 Km/s 1 , = 103° 1 2 = "63° 1 3 = 207° b, = 84° b 2 = 6° b 3 = 1 0 75 TABLE XIII R e s u l t s f o r K i l l s t a r s No.of o b j e c t s = 981 <d> = 167 pc lo = 36° F i t with 10 c o e f f i c i e n t s F i t with 4 c o e f f i c i e n t s K(1 ) = 2.5 + 1 .6 Km/s K(1) = 0.2 + 0.9 Km/s K(2) =-10.5 + 1 .5 Km/s K(2) = -9.4 + 1 .3 Km/s K(3) =-18.6 + 1 .7 Km/s K(3) =-18.3 + 1 .4 Km/s K(4) =-10.2 + 1 .7 Km/s K(4) = -8.5 + 1 .5 Km/s K(5) = -29 + 13 Km/s/Kpc K(6) = 6 + 1 5 Km/s/Kpc K(7) = -9 + 1 3 Km/s/Kpc K(8) = 4.9 + 7.8 Km/s/Kpc K(9) = 0.2 + 9.2 Km/s/Kpc K(10) = 16.8 + 9.3 Km/s/Kpc average r e s i d u a l a =23.3 Km/s P r i n c i p a l axes of the X, = -34.8 Km/s/Kpc X 2 = -15.1 Km/s/Kpc X 3 = 17.4 Km/s/Kpc deformation tensor 1 , = 150° b, = -7° 1 2 = 236° b 2 = 34° 1 3 = 70° b 3 = 55° V e l o c i t y e l l i p s o i d parameters a) With LSR only a(a) = 28. 6 oib) = 19.4 oic) = 1 9 . 7 Km/s oix) = 27. 4 oiy) = 21.1 oiz) = 1 9 . 7 Km/s l v = 23° a, = 17.4 a 2 = 21.4 a 3 = 28.6 Km/s 1 , = -63° 1 2 = -73° 1 3 = 202° bi = 43° b 2 = -41° b 3 = 5° >) With a l l v e l o c i t y f i e l d terms a(a) = 28. 4 a(b) = 19.1 oic) = 19. 9 Km/s a(x) = 27. 2 oiy) = 20.8 oiz) = 19. 9 Km/s l v = 23° a, = 17.4 a 2 = 21.2 a 3 = 28.5 Km/s l i = -63° 1 2 = -75° 1 3 = 202° b, = 40° b 2 = -50° b 3 = v6° 76 to the Vk term and to the three terms which d e s c r i b e the e f f e c t of the p e c u l i a r motion of the Sun with respect to the LSR. T h i s i s due to the f a c t that we are d e a l i n g with l a r g e random motions. As can be seen i n Table XIV, our determination of the v e l o c i t y e l l i p s o i d parameters i s not i n c l o s e agreement with the one of A.E. Gomez (1974). The d i s c r e p a n c i e s may be due to the l a r g e u n c e r t a i n t i e s present i n both a n a l y s e s . The main c h a r a c t e r i s t i c s of the v e l o c i t y f i e l d of K i l l s t a r s i s that the Vk term i s not zero (Vk = 2.5 ± 1.6 Km/s) and that the r o t a t i o n term i s not dominant. The term K(5) (the v e l o c i t y g r a d i e n t 3u/9x) i s l a r g e and negative and the nodal d e v i a t i o n i s l a r g e ( 1 0 = 36°). As we s h a l l see i n Chapter 5, these n o n - c i r c u l a r motion parameters tend to vary s y s t e m a t i c a l l y with the age of the o b j e c t s s t u d i e d and they take extreme values f o r K i l l s t a r s because of t h e i r o l d age. e) R e s u l t s f o r FV s t a r s We d i d not study FV s t a r s i n much d e t a i l because of t h e i r l a r g e random motion which, combined to t h e i r small d i s t a n c e range, makes the dete r m i n a t i o n of v e l o c i t y g r a d i e n t s very d i f f i c u l t . N e v e r t h e l e s s , we determined some of t h e i r kinematic c h a r a c t e r i s t i c s (see Table XV and found a l a r g e negative value f o r the K(5) term and a l a r g e p o s i t i v e value for the nodal d e v i a t i o n 1 0. 77 TABLE XIV Comparison of our v e l o c i t y e l l i p s o i d with Gomez (1972) V e l o c i t y e l l i p s o i d parameters of K i l l s t a r s a) our r e s u l t s = 17.4 °2 = 21.2 o3 --= 28.5 l i = -63° 1 2 = -75° 1 3 = = 22° b, = 40° b 2 = -50° b 3 = = -6° b) r e s u l t s of Gomez = 1 4 o2 = 1 6 a3 •• = 25 l i = -85° 1 2 = -43° 1 3 = 8° b, = 13° b 2 = -73° b 3 = = 1 1 ° Km/s Km/s 78 A more extensive a n a l y s i s , comparison and i n t e r p r e t a t i o n of the r e s u l t s presented i n t h i s chapter i s l e f t to Chapter 5 a f t e r the p r e s e n t a t i o n of the r e s u l t s obtained with 21-cm data. 79 TABLE XV R e s u l t s f o r FV s t a r s No.of o b j e c t s = 237 <d> = 47 pc lo = 42° F i t with 10 c o e f f i c i e n t s K(1 ) = -2. 7 + 3.1 Km/s K(2) = -14. 9 + 2.9 Km/s K(3) = -13. 2 + 3.6 Km/s R(4) = -6. 7 + 2.8 Km/s K(5) = -48 + 91 Km/s/Kpc K(6) = 57 + 84 Km/s/Kpc K(7) = 48 + 97 Km/s/Kpc K(8) = 32 + 56 Km/s/Kpc K(9) = 1 57 + 49 Km/s/Kpc K( 10) = 45 + 57 Km/s/Kpc average r e s i .dual a =18.6 K F i t with 4 c o e f f i c i e n t s K(1) = -1.5 ± 1.6 Km/s K(2) =-11.1 ± 2.2 Km/s K(3) =-12.2 ± 2.8 Km/s K(4) = -6.5 ± 2.2 Km/s 1=180° 80 = -12 Km/s = 12 Km/s = -9 Km/s 9 Km/s = -6 Km/s 6 Km/s = -3 Km/s 3 Km/s 0 Km/s 1= 0° F i g u r e 3: h o r i z o n t a l v e l o c i t y f i e l d f o r the i d e a l case of pure c i r c u l a r motion. The u n c e r t a i n t y on V f l i s f i x e d to be 0.5 Km/s. The box has a h a l f - w i d t h of 500 pc. b= 90° 81 1 = 225' i/iUiViUlUlU kJiuiUiUiUiui hiiuiL'iu'fUtli iiiiilUiUtiiilii BiiiiiuiuYd' ILTUYUIUIUI fuiLiUiUiUi MiliiiiiUiift uVuYuittflil liiuiiiiuiui auiiiiutui fej'tUiuiUi AJVUIUIW MiuiUiut Y -12 Km/s Z = 12 Km/s X = -9 Km/s + 9 Km/s = -6 Km/s X = 6 Km/s + = -3 Km/s A = 3 Km/s 0 = 0 Km/s b=-90° F i g u r e 4: v e r t i c a l v e l o c i t y f i e l d f o r the i d e a l case of pure c i r c u l a r motion. The u n c e r t a i n t y on V f l i s f i x e d to be 0.5 Km/s. The h a l f - w i d t h of the box i s 500 pc. b= 90' 1 = 315' Y - -12 Km/s Z = 1 2 Km/s X = -9 Km/s * = 9 Km/s <!> = -6 Km/s X = 6 Km/s + = -3 Km/s 3 Km/s CD = 0 Km/s lfc'tUiUiltiui| piVitiiiiital F i g u r e 5 : v e r t i c a l v e l o c i t y f i e l d f o r the i d e a l case of pure c i r c u l a r motion. The u n c e r t a i n t y on V f l i s f i x e d to be 0 .5 Km/s. The h a l f - w i d t h of the box i s 500 pc. 82 F i g u r e 6: dependence of V f l on 1 and b generated by a term 15 d c o s 2 l c o s 2 b (K(5) = 15 Km/s/Kpc) at a d i s t a n c e of 1 Kpc. Fi g u r e 7: dependence of V f l on 1 and b generated by a term 15 d s i n 2 b (K(6) = 15 Km/s/Kpc) at a d i s t a n c e of 1 Kpc. F i g u r e 8: dependence of V f l on 1 and b generated by a term 15 d s i n 2 l s i n 2 b (K(7) = 15 Km/s/Kpc) at a d i s t a n c e of 1 Kpc. Fi g u r e 9 : dependence o-f V f l on 1 and b generated by a term 15 d sin21 c o s 2 b (K(8) = 15 km/s/Kpc) at a d i s t a n c e of 1 Kpc. T h i s i s the case of pure c i r c u l a r motion with the standard value of 15 Km/s/Kpc f o r the Oort c o e f f i c i e n t A. 84 F i g u r e 10: dependence of V f l on 1 and b generated by a term 15 d c o s l sin2b (K(9) = 15 Km/s/Kpc) at a d i s t a n c e of 1 Kpc. Fi g u r e 11: dependence of V f l on 1 and b generated by a term 15 d s i n l sin2b (K(10) = 15 Km/s/Kpc) at a d i s t a n c e of 1 Kpc. 1=180° = - 1 2 Km/s = 12 Km/s = - 9 Km/s 9 Km/s = - 6 Km/s 6 Km/s = - 3 Km/s 3 Km/s 0 Km/s 1= 0° F i g u r e 12: h o r i z o n t a l v e l o c i t y f i e l d f o r B s t a r s , u n c e r t a i n t y on V f l i s the probable e r r o r . The box h a l f - w i d t h of 500 pc. The has a b= 90° = -12 Km/s = 12 Km/s = - 9 Km/s 9 Km/s = - 6 Km/s 6 Km/s = -3 Km/s 3 Km/s 0 Km/s F i g u r e 13: v e r t i c a l v e l o c i t y f i e l d f o r B s t a r s . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has h a l f - w i d t h of 500 pc. b= 90° = -12 Km/s 12 Km/s = - 9 Km/s 9 Km/s = - 6 Km/s 6 Km/s = -3 Km/s 3 Km/s 0 Km/s b = - 9 0 ° F i g u r e 14: v e r t i c a l v e l o c i t y f i e l d f o r B s t a r s . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has h a l f - w i d t h of 500 pc. 1=180° 87 1 = 270' Y = -6 .0 Km/s Z = 6 .0 Km/s X = -4 .5 Km/s * = 4 .5 Km/s <!> = -3 .0 Km/s X = 3 .0 Km/s + = -1 .5 Km/s = 1 .5 Km/s 0 = 0 Km/s 1= 0° F i g u r e 15: h o r i z o n t a l v e l o c i t y f i e l d f o r B s t a r s . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90° 1 = 240' Y = -12 Km/s Z = 1 2 Km/s X = -9 Km/s 4- = 9 Km/s <!> = -6 Km/s X = 6 Km/s + = -3 Km/s A = 3 Km/s o = 0 Km/s b=-90' F i g u r e 16: v e r t i c a l v e l o c i t y f i e l d f o r B s t a r s . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. F i g u r e 18: dependence of V f l on 1 and b f o r B s t a r s at a d i s t a n c e of 1 Kpc. A standard value (K(8) = 15 Km/s/Kpc) the r o t a t i o n term was removed. 1=180° 89 1=270' 1= 0° h a l f - w i d t h of 1000 pc. 1=180° 1=270' Y = -24 Km/s Z = 24 Km/s X = -18 Km/s ^ = 18 Km/s <!> = -12 Km/s X = 1 2 Km/s + = -6 Km/s A = 6 Km/s o = 0 Km/s I-IV The s t a r s . The box has a Y = -24 Km/s Z = 24 Km/s X = -18 Km/s 4> = 18 Km/s <S> = -12 Km/s X = 12 Km/s + = -6 Km/s A = 6 Km/s CD = 0 Km/s 1= 0' Fi g u r e 20: h o r i z o n t a l v e l o c i t y f i e l d f o r BI-IV s t a r s . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 1000 pc. b = 90' 90 1 = 2 2 5 ' Y = •• - 1 2 K m / s Z = 1 2 K m / s X = - 9 K m / s * = 9 K m / s <!> = - 6 K m / s X = 6 K m / s + = - 3 K m / s 3 K m / s o = 0 K m / s b = - 9 0 ° F i g u r e 2 1 : v e r t i c a l v e l o c i t y f i e l d f o r B I - I V s t a r s . T h e u n c e r t a i n t y o n V f l i s t h e p r o b a b l e e r r o r h a l f - w i d t h o f 1 0 0 0 p c . b = 9 0 ° 1 = 2 2 5 ' T h e b o x h a s Y = - 1 2 K m / s Z = 1 2 K m / s X = - 9 K m / s * = 9 K m / s o = - 6 K m / s X = 6 K m / s + = - 3 K m / s A = 3 K m / s CD = 0 K m / s b = - 9 0 ° F i g u r e 2 2 : v e r t i c a l v e l o c i t y f i e l d f o r B I - I V s t a r s . A s t a n d a r d v a l u e ( K ( 8 ) = 1 5 K m / s / K p c ) o f t h e r o t a t i o n t e r m w a s r e m o v e d . T h e u n c e r t a i n t y o n V f l i s t h e p r o b a b l e e r r o r . T h e b o x h a s a h a l f - w i d t h o f 1 0 0 0 p c . b= 90' 91 1=315° Y = -24 Km/s Z = 24 Km/s X = -18 Km/s ^ = 18 Km/s <!> = -12 Km/s X = 12 Km/s + _ - 6 Km/s A = 6 Km/s o = 0 Km/s b=-90° F i g u r e 23: v e r t i c a l v e l o c i t y f i e l d f o r BI-IV s t a r s . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 1000 pc. b= 90' 1=315° b = - 9 0 ° Y Z X X + A CD •2 4 Km/s 24 Km/s -18 Km/s 18 Km/s •12 Km/s 12 Km/s - 6 Km/s 6 Km/s 0 Km/s F i g u r e 24: v e r t i c a l v e l o c i t y f i e l d f o r BI-IV s t a r s . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 1000 pc. 92 F i g u r e 26: dependence of V f l on 1 and b f o r BI-IV s t a r s at a d i s t a n c e of 1 Kpc. A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. 1=180° 93 KnUtUtuiUtUtuiuiwt ftUiuiftuitilutuiUl 1=270' 51 VuXuXuVuWu.i 1= 0' Y = -24 Km/s Z = 24 Km/s X = -18 Km/s * = 18 Km/s O = -12 Km/s X = 12 Km/s + = -6 Km/s A = 6 Km/s O = 0 Km/s F i g u r e 27: h o r i z o n t a l v e l o c i t y f i e l d f o r BV s t a r s (d>200pc) The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has h a l f - w i d t h of 1000 pc. b= 90° 1 = 31 5' u lij'l ill utullJl 1/1U Miiuiiiiuiu/iiiujI/lU £|u|U|y(jlliilululu)UlUlU •iigill'lt. T T I T T (3D Y = = -36 Km/s Z = = 36 Km/s X = = -27 Km/s ^ = = 27 Km/s <!> = = -18 Km/s X = = 18 Km/s + ; = -9 Km/s A : 9 Km/s o : 0 Km/s b=-90° F i g u r e 28: v e r t i c a l v e l o c i t y f i e l d f o r BV s t a r s (d>200pc). The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has h a l f - w i d t h of 1000 pc. b= 90' 94 1=210' Y = = -24 Km/s Z = = 24 Km/s X = = -18 Km/s * = = 18 Km/s o = = -12 Km/s X = = 12 Km/s + = = -6 Km/s A = 6 Km/s o = 0 Km/s b = - 9 0 ° F i g u r e 29: v e r t i c a l v e l o c i t y f i e l d f o r BV s t a r s (d>200pc). The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 1000 pc. b= 90' 1=210' Y = = -12 Km/s z = = 12 Km/s X = : -9 Km/s * = 9 Km/s <+> = -6 Km/s X = 6 Km/s + -• -3 Km/s A = 3 Km/s o = 0 Km/s b=-90' F i g u r e 30: v e r t i c a l v e l o c i t y f i e l d f o r BV s t a r s (d>200pc). The u n c e r t a i n t y on V f l f i x e d to be 0.5 Km/s. The box has a h a l f - w i d t h of 1000 pc. 1=180° 95 1=270' XWrWrmrffiMWrm1 Y = : -24 Km/s 2 = 24 Km/s X = -• -18 Km/s * = • 18 Km/s <!> = = -12 Km/s X : = 12 Km/s -1- . = -6 Km/s A = 6 Km/s 0 s 0 Km/s 1= 0° F i g u r e 31: h o r i z o n t a l v e l o c i t y f i e l d f o r BV s t a r s (d>200pc) A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 1000 pc. b= 90° 1 = 315" Y = = -36 Km/s z = = 36 Km/s X = •- -27 Km/s + = = 27 Km/s 0 = = -18 Km/s X = = 18 Km/s 4 = = -9 Km/s & = 9 Km/s 0 = 0 Km/s b=-90' F i g u r e 32: v e r t i c a l v e l o c i t y f i e l d f o r BV s t a r s (d>200pc). A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 1000 pc. b= 90° 96 = -24 Km/s = 24 Km/s = -18 Km/s = 18 Km/s = -12 Km/s = 12 Km/s = -6 Km/s 6 Km/s 0 Km/s b = - 9 0 ° F i g u r e 33: v e r t i c a l v e l o c i t y f i e l d f o r BV s t a r s (d>200pc). A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 1000 pc. b= 90° 1 = 225' Y = -24 Km/s Z = 24 Km/s X = -18 Km/s f = 18 Km/s o = -12 Km/s X = 12 Km/s + = -6 Km/s A = 6 Km/s O = 0 Km/s b=-90' F i g u r e 34: v e r t i c a l v e l o c i t y f i e l d f o r BV s t a r s (d>200pc). A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s f i x e d to be 1.0 Km/s. The box has a h a l f - w i d t h of 1000 pc. 97 F i g u r e 36: dependence of V f l on 1 and b f o r BV s t a r s (d>200pc) at a d i s t a n c e of 1 Kpc. A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. 1=180' 98 1=270' Y = -24 Km/s z = 24 Km/s X = -18 Km/s = 18 Km/s o = -12 Km/s X = 12 Km/s + = -6 Km/s A = 6 Km/s O = 0 Km/s 1= 0' F i g u r e 37: h o r i z o n t a l v e l o c i t y f i e l d f o r BV s t a r s (d<200pc). The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 200 pc. b= 90' 1 = 240' Y- = -24 Km/s Z = 24 Km/s 1 X = -18 Km/s = 18 Km/s <? = -12 Km/s X = 1 2 Km/s + = -6 Km/s A = 6 Km/s 0 = 0 Km/s b=-90' F i g u r e 38: v e r t i c a l v e l o c i t y f i e l d f o r BV s t a r s (d<200pc). The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 200 pc. 1=180° 99 1 = 270' Y Z X + X + A CD •24 Km/s 24 Km/s •18 Km/s 18 Km/s -12 Km/s 12 Km/s -6 Km/s 6 Km/s 0 Km/s 1= 0 ( F i g u r e 39: h o r i z o n t a l v e l o c i t y f i e l d f o r BV s t a r s (d<200pc) A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 200 pc. b= 90' 1=280' Y- = -24 Km/s Z = 24 Km/s -18 Km/s 18 Km/s -12 Km/s 12 Km/s -6 Km/s 6 Km/s 0 Km/s b=-90' F i g u r e 4 0 : ' v e r t i c a l v e l o c i t y f i e l d f o r BV s t a r s (d<200pc). A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 200 pc. 100 F i g u r e 42: dependence of V f l on 1 and b f o r BV s t a r s (d<200pc) at a d i s t a n c e of 1 Kpc. A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. 1=180° b= 90' 1=330° b=-90' Y Z X * <!> X • + A CD -12 Km/s 12 Km/s -9 Km/s 9 Km/s -6 Km/s 6 Km/s -3 Km/s 3 Km/s 0 Km/s F i g u r e 44: v e r t i c a l v e l o c i t y f i e l d f o r AV s t a r s . The ^araoS 1^: t h e probable error- The b- h L * 1=180° 102 = -12 Km/s = 12 Km/s = -9 Km/s 9 Km/s = -6 Km/s 6 Km/s = -3 Km/s 3 Km/s 0 Km/s 1= 0° F i g u r e 45: h o r i z o n t a l v e l o c i t y f i e l d f o r AV s t a r s . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 200 pc. b= 90° = -12 Km/s = 12 Km/s = -9 Km/s 9 Km/s = -6 Km/s 6 Km/s = -3 Km/s 3 Km/s 0 Km/s b=-90° F i g u r e 46: v e r t i c a l v e l o c i t y f i e l d f o r AV s t a r s . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 200 pc. 103 F i g u r e 48: dependence of V f l on 1 and b f o r AV s t a r s at a d i s t a n c e of 1 Kpc. A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. 1=180° 104 1 = 270' QQQQGGGQGGGG GGGGGGGGGGGG QGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GQGGGGGGGGGG GGGGGGGGGGGG GQGGGGGGGGGG GQGGGGGGGGGG GGGGGGGGGGGG QGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG QGGQGGGGQGQG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGQGGGGQQGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GQGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG QQQQQQQQQQQQ GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG Y = -4 Km/s Z = 4 Km/s X = -3 Km/s * = 3 Km/s <J> = -2 Km/s X = 2 Km/s + = -1 Km/s A = 1 Km/s G = 0 Km/s 1= 0° Fig u r e 49: h o r i z o n t a l v e l o c i t y f i e l d for A(0-4)V s t a r s . The un c e r t a i n t y on V f l i s the probable e r r o r , h a l f - w i d t h of 200 pc. b= 90* 1=180' GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GQGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG QQ Ot>Ot> GGGGG >>> GGGGGGGG OOO GGGGGGGGGGGO GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GQGGGGGGGGGG GQGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG >GGGGGQGGGGG >00 GGGGGGGG >>> GGGGG >Ot>l> GG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG GGGGGGGGGGGG The box has Y = -12 Km/s Z = 1 2 Km/s X = -9 Km/s <f. = 9 Km/s o = -6 Km/s X = 6 Km/s + = -3 Km/s A = 3 Km/s CD = 0 Km/s b=-90° F i g u r e 50: v e r t i c a l v e l o c i t y f i e l d f o r A(0-4)V s t a r s . The un c e r t a i n t y on V f l i s the probable e r r o r . The box has a ha l f - w i d t h of 200 pc. b= 90° 105 1 = 270" QGGGGQGGQQQQ QGGGGQQGGGQG QQQGGGGQGGGQ QQGGGGGGGGQQ QGGGQGQGQGQG QGQGGQQGQQQQ GQGGQGGQGQGG QGQQQGGGGGQQ GGGGQGGQQGQG GQGGQQQGQGQQ QGQGGGGGQQQG QQGGQGGQQGQQ A AAAAAAAAAAA AAAAAAAAAAAO AAAAAAAAAAOO AAAAAAAAAOOO AAAAAAAOOOO AAAAOOOOO OOOOOO OOOOOOO oooooooo ooooooooo OOOOOOQOOO ooooooooooo OOOQOOOOOOO OOOOQOOOQO QOQOOOQQO OQOOOOOO (DOOOOOO OOQQOO OCDOOOAAAA Q CD CD CD A A A A A A A OOOAAAAAAAAA OOAAAAAAAAAA OAAAAAAAAAAA AAAAAAAAAAAA QQGGQGGQGQQG QGGGGGGGGGGG QQQQGQGGQQQQ QQQQQGGGQQQQ QGGGGGGGGGGG QGGGGQGGQQQQ QQGGQQGQQQQG QQGQQQGQQQGQ QQQQQQQQQQQQ GGQQGQGQGGQG GQQQGQGQGGQQ GQQQGQQQGGGG Y = -12 Km/s Z = 12 Km/s X = -9 Km/s = 9 Km/s <J> = -6 Km/s X = 6 Km/s + = -3 Km/s A = 3 Km/s O = 0 Km/s b=-90? Fi g u r e 51: v e r t i c a l v e l o c i t y f i e l d f o r A(0-4)V s t a r s . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 200 pc. Fig u r e 52: dependence of V f l on 1 and b f o r A(0-4)V s t a r s at a d i s t a n c e of 1 Kpc. 1=180° 106 1 = 270' Y = = -24 Km/s Z = = 24 Km/s X = = -18 Km/s * = = 18 Km/s 0 = = -12 Km/s X = = 12 Km/s + = = -6 Km/s A = 6 Km/s 0 s 0 Km/s 1= 0° F i g u r e 53: h o r i z o n t a l v e l o c i t y f i e l d f o r A(5-9)V s t a r s . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 200 pc. b= 90' 1 = 240' Y — -24 Km/s z = 24 Km/s = -18 Km/s = 18 Km/s = -12 Km/s X = 12 Km/s + = -6 Km/s = 6 Km/s CD = 0 Km/s b=-90' F i g u r e 54: v e r t i c a l v e l o c i t y f i e l d f o r A(5-9)V s t a r s . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 200 pc. ' b= 90° 107 1=330' b=-90° u n c e r t a i n t y on V f l i s the probable e r r o r , h a l f - w i d t h of 200 pc. Y = -24 Km/s Z = 24 Km/s X = -18 Km/s f = 18 Km/s o = -12 Km/s X = 12 Km/s + = -6 Km/s A = 6 Km/s CD = 0 Km/s 9)V s t a r s . The box has The F i g u r e 56: dependence of V f l on 1 and b f o r A(5-9)V s t a r s at a d i s t a n c e of 1 Kpc. 1=180° 108 1=270' Y = -24 Km/s Z = 24 Km/s X = -18 Km/s * = 18 Km/s <!> = -12 Km/s X = 12 Km/s + = -6 Km/s A = 6 Km/s O = 0 Km/s 1= 0° F i g u r e 57: h o r i z o n t a l v e l o c i t y f i e l d f o r K i l l s t a r s . Th° u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a" h a l f - w i d t h of 500 pc. b= 90' 1 = 350' Y = -24 Km/s z = 24 Km/s X = -18 Km/s = 18 Km/s o = " 12 Km/s X = 12 Km/s + = -6 Km/s A = 6 Km/s CD = 0 Km/s b=-90° F i g u r e 58: v e r t i c a l v e l o c i t y f i e l d f o r K i l l s t a r s . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90' 109 1=260' b=-90' Y Z X X + A CD - 2 4 Km/s 24 Km/s -18 Km/s 18 Km/s -12 Km/s 12 Km/s - 6 Km/s 6 Km/s 0 Km/s F i g u r e 59: v e r t i c a l v e l o c i t y f i e l d f o r K i l l s t a r s . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90' 1=260' Y = - 2 4 Km/s z = 24 Km/s X = -18 Km/s = 18 Km/s = - 1 2 Km/s X = 12 Km/s + = - 6 Km/s A = 6 Km/s CD = 0 Km/s b=-90° F i g u r e 60: v e r t i c a l v e l o c i t y f i e l d f o r K i l l s t a r s . The u n c e r t a i n t y on V f l i s f i x e d to be 1.0 Km/s. The box has a h a l f - w i d t h of 500 pc. 1=180 110 1 = 270' Y = -24 Km/s Z = 24 Km/s X = -18 Km/s = 18 Km/s = -12 Km/s X = 12 Km/s + = -6 Km/s = 6 Km/s CD = 0 Km/s 1= 0° F i g u r e 61: h o r i z o n t a l v e l o c i t y f i e l d f o r K i l l s t a r s . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90° 1=200' Y = -24 Km/s z = 24 Km/s X = -18 Km/s <r = 18 Km/s <J> = -12 Km/s X = 1 2 Km/s + = -6 Km/s A = 6 Km/s CD = 0 Km/s b=-90' F i g u r e 62: v e r t i c a l v e l o c i t y f i e l d f o r K i l l s t a r s . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90° 1=290' Y = -24 Km/s Z = 24 Km/s X = -18 Km/s = 18 Km/s o = -12 Km/s X = 12 Km/s + = -6 Km/s A = 6 Km/s CD = 0 Km/s b=-90° F i g u r e 63: v e r t i c a l v e l o c i t y f i e l d f o r K i l l s t a r s . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90° 1 = 290' UtUlUlUlUlUMlUlUlUlUlUlUlUllMV uiuiuiiiiUtuiuruiuiutuiuiuVuy|[igi^igigi|jti[iyrtj| UlJlUIUIUIUIUIuf|[|^tjj|jjr|ili| M V k V W I l W I l ' M swuwiww www 'itWH'M Y = -24 Km/s z = 24 Km/s X = -18 Km/s * = 18 Km/s o = -12 Km/s X = 12 Km/s + = -6 Km/s A = 6 Km/s o = 0 Km/s b=-90' F i g u r e 64: v e r t i c a l v e l o c i t y f i e l d f o r K i l l s t a r s . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s f i x e d to be 1.0 Km/s. The box has a h a l f - w i d t h of 500 pc. 112 1 13 CHAPTER 4 THE KINEMATICS OF THE INTERSTELLAR MEDIUM Th i s chapter presents the r e s u l t s of our a n a l y s i s of the kinematics of HI clouds and of the i n t e r c l o u d medium. L i k e Chapter 3, i t i s d i v i d e d i n t o s e c t i o n s that d e s c r i b e the r e s u l t s obtained f o r d i f f e r e n t s e t s of o b j e c t s and does not present an e x t e n s i v e a n a l y s i s and i n t e r p r e t a t i o n of these r e s u l t s . As i n Chapter 3, the f i g u r e s , because they are l a r g e i n number, are put together at the end of the chapter. a) Treatment of the data Since no method of determination of d i s t a n c e s i s a v a i l a b l e f o r the f e a t u r e s of 21-cm a b s o r p t i o n and emission s p e c t r a , the treatment of the data used f o r n e u t r a l hydrogen i s s l i g h t l y d i f f e r e n t from the one used for s t a r s . The same equation and the same l e a s t squares method are a p p l i e d but now a d i s t a n c e d must be estimated f o r each l i n e of s i g h t . In order to do t h i s , a model of the s p a t i a l d i s t r i b u t i o n of the gas i s assumed and used to p r e d i c t the v a r i a t i o n of the average d i s t a n c e <d> along a l i n e of s i g h t with 1 and b, the g a l a c t i c l o n g i t u d e and l a t i t u d e . The model normally used i s the " p l a n e - p a r a l l e l model" in which <d> does not vary with 1 (the axisymmetric case) but v a r i e s with b a c c o r d i n g to a cosecant law l i k e : 1 1 4 <d> = <|z|> cosec |b| . (4-1) T h i s v a r i a t i o n of <d> with b simply comes from the f a c t t h a t the gas i s d i s t r i b u t e d in a f l a t d i s k that extends on each s i d e of the g a l a c t i c mid-plane and t h a t , because of t h i s d i s t r i b u t i o n , more m a t e r i a l i s seen at low l a t i t u d e . T h i s model has been used by many authors because of i t s s i m p l i c i t y . The value of <|z|> i s the average e l e v a t i o n from the g a l a c t i c mid-plane and i s equal to $ f o r an exp o n e n t i a l v a r i a t i o n of the d e n s i t y with z l i k e : N(z) = N 0 exp(-z/$) (4-2) or to o v/2/7r f o r a gaussian d e n s i t y v a r i a t i o n l i k e : N(z) = n 0 e x p ( - z 2 / 2 o 2 ) . (4-3) Of course the value of <d> obtained with equation (4-1) i s an average d i s t a n c e and should be used only i n a s t a t i s t i c a l manner. For one given l i n e of s i g h t , i t prov i d e s one value of the d i s t a n c e around which the gaseous m a t e r i a l i s d i s t r i b u t e d . U n f o r t u n a t e l y , the i n t e r s t e l l a r medium cannot be de s c r i b e d as simply as t h i s s i n c e i t i s known now [see Shuter and Verschuur (1963), C l a r k (1965), Falgarone and 1 1 5 Lequeux (1973) and Baker and Burton (1975)] to be composed of at l e a s t two d i f f e r e n t types of m a t e r i a l : a c o l d and r e l a t i v e l y dense component c o n s i s t i n g of c l o u d s ; and a warmer and more r a r i f i e d component, the i n t e r c l o u d medium, that f i l l s the space between these c l o u d s . These two components can be d e s c r i b e d s e p a r a t l y by two p l a n e - p a r a l l e l models with d i f f e r e n t v a l u e s of <|z|>. These values are t y p i c a l l y 100 pc f o r the clouds and 190 pc f o r the i n t e r c l o u d medium (from Falgarone and Lequeux (1973)) but, as mentioned by C r o v i s i e r (1978), they are not p r e c i s e l y determined and d i f f e r from one study to another by as much as a f a c t o r of two. T h i s l e d us to determine our own values of <|z|> f o r both components by l o o k i n g at the v a r i a t i o n of A«<d> with the g a l a c t i c l a t i t u d e b (A being the usual Oort c o e f f i c i e n t a s s o c i a t e d with the term K(8) of equation (2-12)). Assuming the constancy of A over a wide range of b angles, the v a r i a t i o n of A«<d> with b i n d i c a t e s whether or not a cosecant law e x i s t s and, i f i t does, g i v e s a value of <|z|>. When t h i s method was used f o r 21-cm emission data, a remarkable agreement with the cosecant v a r i a t i o n was found and l e d to a value of 113 pc f o r <|z|>. Here the Oort c o e f f i c i e n t i s assumed to be equal to 15 Km/s/Kpc. T h i s assumption i s necessary s i n c e only the product A«<d> i s measured and the ex p r e s s i o n of <d> takes the form: <d> = <|z|> S cosec |b| (4 -4) 1 16 where S i s a s c a l i n g f a c t o r equal to 15/A. Our value of 113 pc f o r <|z|> i s i n f a i r agreement with <|z|> = 105 pc obtained by Baker and Burton (1975) f o r the i n t e r c l o u d medium but somewhat lower than values obtained from other s t u d i e s (see Mebold (1972), Falgarone and Lequeux (1973)). The reason why we a s s o c i a t e 21-cm emission with the i n t e r c l o u d medium i s t h a t , as mentioned before, the i n t e r c l o u d medium i s hot ( i t s temperature v a r i e s from 100 K to 10 5 K) and dominates the emission s p e c t r a i n which the c o l d c l o u d s , with t h e i r low temperature (a few tens of K), do not produce important peaks. Conversly, the a b s o r p t i o n s p e c t r a of 21-cm a g a i n s t background sources r e v e a l the e x i s t e n c e of c o l d clouds that have a l a r g e o p a c i t y because of t h e i r low temperature while the i n t e r c l o u d medium i s too hot to e f f i c i e n t l y absorb the 21-cm r a d i a t i o n . Our study of the v a r i a t i o n of A«<d> with b f o r the 21-cm a b s o r p t i o n data r e v e a l s that a cosecant law i s inadequate. T h i s was p r e v i o u s l y noted by Mast and G o l d s t e i n (1970) and by C r o v i s i e r (1978). Takakubo (1967) a l s o n o t i c e d that column d e n s i t y obeys a cosec |b| law f o r f e a t u r e s that have a l a r g e width but i s approximately constant over the whole sky f o r narrow f e a t u r e s that can be a s s o c i a t e d with low temperature c l o u d s . In c o n t r a s t to these authors who mentioned t h i s "anomaly" but n e v e r t h e l e s s used a p l a n e - p a r a l l e l model, we have decided to f i n d a b e t t e r d e s c r i p t i o n of the 117 d i s t r i b u t i o n of c l o u d s . A f t e r many t r i a l s , we have opted f o r a simple m o d i f i c a t i o n of the usual model which i s to introduce a constant term d 0 . T h i s g i v e s : <d> =( d 0 + z 0 cosec |b|)-S (4-5) were d 0 r e p r e s e n t s some s o r t of minimum d i s t a n c e between the Sun and the f i r s t absorbing c l o u d s . I t i s not evident at f i r s t glance whether or not d 0 corresponds to a r e a l l o c a l "hole" in the d i s t r i b u t i o n of c l o u d s s i n c e t h i s d i s t r i b u t i o n i s f a i r l y sparse with t y p i c a l i n t e r c l o u d d i s t a n c e s of a few hundred parsecs along a l i n e of s i g h t (see Radhakrishnan and Goss (1972)). However, Monte C a r l o s i m u l a t i o n s have i n d i c a t e d t h a t , even with a sparse and lumpy d i s t r i b u t i o n of clouds, a pure cosecant v a r i a t i o n of <d> with |b| i s to be expected and thus, the p o s i t i v e value of the d 0 term seems to suggest the e x i s t e n c e of a l o c a l lack of absorbing c l o u d s . In f a c t , t h i s lack of hydrogen i n the c l o s e v i c i n i t y of the Sun was mentioned by C r o v i s i e r (1978) i n h i s a n a l y s i s of h i s survey of a b s o r p t i o n f e a t u r e s and i n h i s i n t e r p r e t a t i o n of the work of Takakubo (1967), Mast and G o l d s t e i n (1970), Henderson (1973) and H e i l e s (1976). Our values are t y p i c a l l y 240 pc f o r d 0 and 75 pc f o r z 0 . Here the term z 0 i s not equal to the average e l e v a t i o n <|z|> unless d 0 = 0. In order to a v o i d the use of unclear s p e c t r a , we 118 have c r e a t e d a subset of a b s o r p t i o n f e a t u r e s that c o n t a i n s only the peaks of s i n g l e l i n e s p e c t r a . Even a f t e r J . C r o v i s i e r made a v a i l a b l e to us a l i s t of a b s o r p t i o n f e a t u r e s ( i n c l u d i n g f e a t u r e s taken from complex spectra) f o r which he c o u l d guarantee the r e l i a b i l i t y (see C r o v i s i e r (1981)), t h i s subgroup has remained i n t e r e s t i n g to work with s i n c e i t leads to r e s u l t s that are q u i t e d i f f e r e n t from those obtained with the more complete set of f e a t u r e s . b) R e s u l t s f o r 21-cm emission data The average l i n e - o f - s i g h t v e l o c i t i e s obtained f o r the i n t e r c l o u d medium have been taken from a survey c a r r i e d out by Henderson (1973). As mentioned before, the average d i s t a n c e along a l i n e of s i g h t i s w e l l d e s c r i b e d by: <d> = 0.113 S cosec |b| ( i n Kpc). (4-6) A f i t of the data with equation (4-5) g i v e s d 0 = 0.000 ± 0.050 Kpc and shows that the i n t r o d u c t i o n of t h i s term i s not necessary. Table XVI g i v e s the r e s u l t s of our a n a l y s i s of the 406 emission s p e c t r a . One can n o t i c e the very small value of the average r e s i d u a l (o = 2.7 Km/s) and the negative value of 1 0. In agreement with Takakubo (1967), we have decided not to i n c l u d e the Vk term i n our treatment because of i t s strong c o r r e l a t i o n with the terms K(5), K(6) and K(7), the v e l o c i t y g r a d i e n t s i n the 1 19 d i r e c t i o n s of the three p r i n c i p a l axes of the Galaxy. The f i t g i v e s p o s i t i v e values f o r K(5) (the term 9u/9x) and f o r K(6) (the term 9w/9z). In a four c o e f f i c i e n t f i t , these terms are absorbed by the Vk term which takes a l a r g e p o s i t i v e value as found e a r l i e r by F r o g e l and S t o t h e r s (1977). The r e s u l t s f o r the v e l o c i t y e l l i p s o i d are not given because they leads to imaginary values f o r o ( z ) . F i g u r e s 67 to 69 show how the average r e s i d u a l v a r i e s with 1, b and d when a four c o e f f i c i e n t f i t i s performed. These f i g u r e s show that 1) the r e s i d u a l s i n c r e a s e with d ( t h i s acknowledges the need for expansion terms); 2) the coverage of the g a l a c t i c poles i s very poor s i n c e only b angles between -30° and 30° are a v a i l a b l e ; and 3) there i s an obvious need f o r a r o t a t i o n term that produces a double sine curve when the average r e s i d u a l i s p l o t t e d a g a i n s t 1. V a r i o u s s l i c e s of the v e l o c i t y f i e l d are shown in F i g u r e 70 to 77. Our value of 1 0 i s i n f a i r agreement with the one obtained by Venugopal (1969) ( 1 0 = -12.9°). Takakubo (1967) and Henderson (1973) a l s o found small negative values f o r 1 0 and showed that i t becomes more and more negative as |b| i n c r e a s e s . Takakubo's a n a l y s i s (1967) was very s i m i l a r to ours but he decomposed h i s s p e c t r a i n t o i n d i v i d u a l components and then c l a s s i f i e d them i n t o three f a m i l i e s a c c o r d i n g to t h e i r width. He obtained r e s u l t s that are s i m i l a r to ours (see 1 20 TABLE XVI Re s u l t s f o r 21-cm emission No.of o b j e c t s = 406 <d> = 418 pc lo = "8° F i t with 10 c o e f f i c i e n t s K(1) = — K(2) =-10.1 ± 0.3 Km/s K(3) =-16.2 ± 0.3 Km/s K(4) =-11.7 ± 0.5 Km/s K ( 5 ) = 4.2 ± 0 . 6 Km/s/Kpc K(6) = 20.3 ± 5.4 Km/s/Kpc K(7) = -0.1 ± 0.9 Km/s/Kpc K(8) = 15.0 ± 0.6 Km/s/Kpc K ( 9 ) = 2.3 ± 0 . 9 Km/s/Kpc K(10)= 3.3 ± 1.1 Km/s/Kpc average r e s i d u a l a = 2.7 Km/s P r i n c i p a l axes of the deformation tensor X, = -0.7 Km/s/Kpc 1, = 266° b, = 8° X 2 = 3.9 Km/s/Kpc 1 2 = 175° b 2 = 9° X 3 = 21.1 Km/s/Kpc 1 3 = 36° b 3 = 78° F i t with 4 c o e f f i c i e n t s K(1) = 4 .5 + 0. 3 Km/s K(2) = -6 .8 + 0. 3 Km/s K(3) = -20 .9 + 0. 4 Km/s K(4) = -1 1 .6 + 0. 6 Km/s V e l o c i t y e l l i p s o i d parameters (not determined) 121 Table XVII) but h i s sample of s p e c t r a s u f f e r e d from the same poor coverage of high g a l a c t i c l a t i t u d e s which, as w i l l be explaned i n Chapter 5, a f f e c t s the de t e r m i n a t i o n of some v e l o c i t y g r a d i e n t s . c) R e s u l t s f o r 21-cm a b s o r p t i o n data In order to study the c o l d and lumpy component of the i n t e r s t e l l a r medium, people observe 21-cm ab s o r p t i o n l i n e s a g a i n s t e a r l y type s t a r s or e x t r a g a l a c t i c r a d i o - s o u r c e s . In 1978, C r o v i s i e r , Kazes and Aubry p u b l i s h e d the Nancay 21-cm Absorption survey. The same year, C r o v i s i e r analysed the kinematic p r o p e r t i e s of the clouds that he had observed. In the present study we use the same data but we introduce a d i f f e r e n t d e s c r i p t i o n of the d i s t r i b u t i o n of clouds with the i n c l u s i o n of a d 0 term that g i v e s : <d> = (0.234 + 0.074 cosec |b| )• S (4-7) when <d> i s expressed i n Kpc. Again, the s c a l i n g f a c t o r S i s equal to 15/A. Our a n a l y s i s a l s o d i f f e r s from C r o v i s i e r ' s because we use the f u l l development of a three-dimensional f i r s t order T a y l o r s e r i e s while he l i m i t s the d e s c r i p t i o n of n o n - c i r c u l a r motion to the determination of the nodal d e v i a t i o n 1 0. Table XVIII shows the r e s u l t s obtained from our a n a l y s i s of the o b j e c t s of the Nancay TABLE XVII Comparison of our r e s u l t s w i t h Takakubo (1967) r e s u l t s of Takakubo T h i s study al1 peaks <y<3 Km/s 3<d<7 Km/s <j>7 Km/s (21-cm em i ss i No. a n a l y s e d 544 182 191 171 406 K(5) Km/s/Kpc 0.9 ± 1.3 -2.2 ± 3.3 4 . 1 ± 2 . 4 -0.6 ± 1.8 4.2 ± 0.6 K(6) Km/s/Kpc 38 + 16 107 ± 36 -1 .6 ± 32 32 ± 18 20.3 + 5.4 K(7) Km/s/Kpc -3.5 + 1.8 -1.1 ± 4.4 -6.5 ± 3 . 3 -2.2 ± 2.2 -0.1 ± 0.9 K(8) Km/s/Kpc 15.0 ± 1.3 15.0 ± 2.8 15.0 ± 2.4 15.0 ± 1.4 15.0 ± 0.9 K(9) Km/s/Kpc 0.8 ± 2.0 2.2 ± 4.4 4.9 ± 4.0 0.0 ± 2.5 2.3 + 0.9 K( 10) Km/s/Kpc 0.0 ± 2.8 9 ± 1 1 " -2.8 ± 4 . 5 -2.2 ± 3.1 3.3 ± 1.1 122 123 Survey. As C r o v i s i e r d i d , we excluded low l a t i t u d e clouds (with |b| < 10°) to e l i m i n a t e many complex s p e c t r a . We f i n d a c l o u d - c l o u d v e l o c i t y d i s p e r s i o n of 6.8 Km/s, i n c l o s e agreement with the values which can be found i n the l i t e r a t u r e . Our value of 1 0 i s l a r g e r than those obtained by C r o v i s i e r (1978) ( 1 0 = -14°) and by Mast and G o l d s t e i n (1970) ( 1 0 = -13.5°) but s i m i l a r to the one of R o h l f s (1972) who obtained 1 0 = -22°. Our 9 c o e f f i c i e n t f i t g i v e s a p o s i t i v e K(5) term (the v e l o c i t y g r a d i e n t 9u/9x) and a negative K(6) or 9w/9z. F i g u r e s 78 to 80 show the dependence of the average r e s i d u a l on the v a r i a b l e s 1, b and d when only four c o e f f i c i e n t s are used. These f i g u r e s i n d i c a t e the need f o r an expansion term s i n c e the r e s i d u a l s i n c r e a s e with d, and the need f o r a r o t a t i o n term that would e x p l a i n the double s i n e v a r i a t i o n with 1. The average r e s i d u a l takes p o s i t i v e values f o r small b angles and becomes negative as |b| i n c r e a s e s . T h i s suggests that we should i n t e r p r e t the p o s i t i v e 9u/9x and the negative 9w/9z as i n d i c a t o r s of an i n f a l l i n g motion of the clouds from both g a l a c t i c p o l e s matched by a general expansion that takes p l a c e along the g a l a c t i c mid-plane. T h i s "polar compression" was mentioned by many authors i n the past and i s now a w e l l known c h a r a c t e r i s t i c of the l o c a l kinematics of atomic hydrogen [see McGee and Murray (1961), H e i f e r (1961), Weaver (1974) and Moles and Jaakkola (1977)]. In c o n t r a s t to the emission 1 24 TABLE XVIII R e s u l t s f o r 21-cm a b s o r p t i o n No.of o b j e c t s <d> lo 281 395 pc -26° F i t with 10 c o e f f i c i e n t s F i t with 4 c o e f f i c i e n t s K(1 ) K(2) K(3) K(4) 11.0 ± 0.8 Km/s 16.3 ± 1.1 Km/s -9.5 ± 1.2 Km/s K(1 ) K(2) K(3) K(4) 1 .6 -9.2 19.4 10.3 + + + + 0.5 Km/s 0.8 Km/s 1.0 Km/s 1.1 Km/s K(5) = 8. 1 + 2. 4 Km/s/Kpc K(6) = -11. 5 + 5. 9 Km/s/Kpc K(7) = - o . 2 + 3. 8 Km/s/Kpc K(8) = 15. 0 + 2. 6 Km/s/Kpc K(9) = 3. 1 + 2. 7 Km/s/Kpc K( 10) = 6. 9 + 4. 0 Km/s/Kpc average r e s i d u a l a = 6.8 Km/s P r i n c i p a l axes of the deformation tensor X, = -15.1 Km/s/Kpc 1, = 119° b, = -75° X 2 = 2.8 Km/s/Kpc 1 2 = 245° b 2 = -9° X 3 = 8.8 Km/s/Kpc 1 3 = -23° b 3 = -12° V e l o c i t y e l l i p s o i d parameters (not determined) 125 survey, the Nancay Survey covers the g a l a c t i c p o l e s q u i t e w e l l and thus g i v e s a more r e l i a b l e d e s c r i p t i o n of the motion along the z - a x i s . Once again, there seem to be problems with the v e l o c i t y e l l i p s o i d treatment which gi v e s imaginary values of a ( z ) . C r o v i s i e r avoided t h i s problem by r e j e c t i n g any fe a t u r e with VI > 25 or 15 Km/s but we c o n s i d e r such a s e l e c t i o n inadequate and do not wish to use i t . As we s h a l l see l a t e r , the d i f f i c u l t i e s with the v e l o c i t y e l l i p s o i d treatment are e x p l i q u a b l e . F i g u r e s 81 to 88 show d i f f e r e n t s l i c e s of the v e l o c i t y f i e l d with and without the r o t a t i o n term which i s a r b i t r a r i l y f i x e d by our d e f i n i t i o n of <d> to be 15 Km/s/Kpc. Table XIX giv e s a d i r e c t comparison of our best r e s u l t s with those obtained when a pure p l a n e - p a r a l l e l model was used to determine the average d i s t a n c e s (with <|z|> = 134 p c ) . The rather s t r i k i n g s i m i l a r i t y of these r e s u l t s shows that the choice of a model of s p a t i a l d i s t r i b u t i o n of the clouds i s not too c r i t i c a l i n determining the v e l o c i t y g r a d i e n t s . d) R e s u l t s f o r s i n g l e a b s o r p t i o n f e a t u r e s When we r e j e c t a l l a b s o r p t i o n s p e c t r a that are composed of many f e a t u r e s to keep only " s i n g l e a b s o r p t i o n f e a t u r e s " , we reduce the number of o b j e c t s from 281 to 161 and i n c r e a s e the average g a l a c t i c l a t i t u d e from 28° to 31° 1 26 TABLE XIX A l l a b s o r p t i o n s p e c t r a Comparison of two models of s p a t i a l d i s t r i b u t i o n f i t with d i n Kpc as: d = 0 .234 + 0. 074 cosec|b| K(2) = -1 1 .0 + 0.8 Km/s K(3) = -16 .3 + 1 . 1 Km/s K(4) = -9 .5 + 1 .2 Km/s K(5) = 8 . 1 + 2.4 Km/s/Kpc K(6) = -1 1 .5 + 5.9 Km/s/Kpc K(7) = -0 .2 + 3.8 Km/s/Kpc K(8) = 15 .0 + 2.6 Km/s/Kpc K(9) = 3 . 1 + 2.7 Km/s/Kpc K(10) = 6 .9 + 4.0 Km/s/Kpc o = 6. 76 Km/s f i t with d i n Kpc as: d = 0.134 cosec|b| K(2) = = -11. 0 + 0. 8 Km/s K(3) : = -16. 2 + 1 . 1 Km/s K(4) = = -9. 6 + 1 . 1 Km/s K(5) = 6. 7 + 2. 5 Km/s/Kpc K(6) : = -10. 8 + 9. 6 Km/s/Kpc K(7) = 0. 7 + 3. 8 Km/s/Kpc K(8) = 15. 0 + 2. 7 Km/s/Kpc K(9) = 4. 4 + 3. 5 Km/s/Kpc K(10) = 10. 4 + 4. 9 Km/s/Kpc a = 6.83 Km/s 127 because most of the m u l t i p l e s p e c t r a are at low l a t i t u d e |b|. The use of t h i s p a r t i c u l a r s e l e c t i o n c r i t e r i o n leads to many i n t e r e s t i n g r e s u l t s : 1) the r e s i d u a l s are g r e a t l y reduced (from 6.8 to 3.6 Km/s); 2) the v e l o c i t y e l l i p s o i d a n a l y s i s g i v e s a resonable value of a ( z ) ; and 3) the s p a t i a l d i s t r i b u t i o n of clouds i s changed. The new average d i s t a n c e i s best d e s c r i b e d by a constant value d 0 = 250 pc (the z 0 term determined by the v a r i a t i o n of A«<d>, i s equal to -20 ± 40 p c ! ) . Table XX summarizes the r e s u l t s obtained f o r the v a r i o u s kinematic parameters f o r t h i s group of absorbing clouds when a constant d i s t a n c e i s assumed. Table XXI gives a comparison of these r e s u l t s with those obtained when a p l a n e - p a r a l l e l model i s used with <|z|> = 67 pc. Again, although these two treatments i n v o l v e d very d i f f e r e n t d e s c r i p t i o n of the s p a t i a l d i s t r i b u t i o n of cl o u d s , the value s of the v a r i o u s v e l o c i t y g r a d i e n t s are q u i t e s i m i l a r in both cases. F i g u r e s 89 to 96 show the c h a r a c t e r i s t i c s of the v e l o c i t y f i e l d f o r our best f i t , i . e . when a constant d i s t a n c e of 0.250 Kpc i s assumed. Our success i n determining the v e l o c i t y e l l i p s o i d parameters f o r s i n g l e a b s o r p t i o n f e a t u r e s i n d i c a t e s the o r i g i n of our problems with the set of a l l f e a t u r e s . In f a c t , s i n c e m u l t i p l e f e a t u r e s p e c t r a i n t r o d u c e l a r g e r e s i d u a l s and s i n c e these s p e c t r a tend to be conc e n t r a t e d at low l a t i t u d e s , the value of a 2 decreases with i n c r e a s i n g |b| angles and thus with s i n 2 b . As can be seen i n equations 1 28 TABLE XX R e s u l t s f o r s i n g l e a b s o r p t i o n f e a t u r e s No.of o b j e c t s = 161 <d> = 251 pc 1 0 = "19° F i t with 10 c o e f f i c i e n t s K(1 ) = — K(2) = -11. 8 + 0 .6 Km/s K(3) = -15. 4 + 0 .8 Km/s K(4) = -9. 2 + 0 .8 Km/s K(5) = 18. 7 + 2 .8 Km/s/Kpc K(6) = -23. 8 + 4 .8 Km/s/Kpc K(7) = 2. 5 + 4 .6 Km/s/Kpc K(8) = 15. 0 + 2 .9 Km/s/Kpc K(9) = 3. 2 + 2 .3 Km/s/Kpc K( 10) = 4. 4 + 3 .5 Km/s/Kpc average r e s i d u a l a = 3.6 Km/s F i t with 4 c o e f f i c i e n t s K ( 1 ) = 0 . 8 ± 0 . 4 Km/s K(2) =-10.9 ± 0.6 Km/s K(3) =-16.3 ± 0.9 Km/s K(4) = -9.4 ± 0.8 Km/s P r i n c i p a l axes of the X, = -24.7 Km/s/Kpc X 2 = 3.2 Km/s/Kpc X 3 = 19.0 Km/s/Kpc deformation tensor 1, = 156° b, = -86° 1 2 = 261° b 2 = -1° 1 3 = "9° b 3 = -4° V e l o c i t y e l l i p s o i d parameters a) With LSR only o(a) = 5. 6 a(b) = 3.5 a(c) = 4. 9 Km/s a(x) = 4. 8 a(y) = 4.5 a(z) = 4. 9 Km/s l v = 41 ° a, = 1 .3 a 2 = 4.8 o3 - 5.8 Km/s l i = 1 33° 1 2 = -57° 1 3 = 220° b, = 31 0 b 2 = 59° b 3 = -4° b) With a l l v e l o c i t y f i e l d terms a(a) = 3. 7 a(b) = 2.9 a(c) = 4. 0 Km/s a(x) = 3. 0 a(y) = 3.6 a(z) = 4. 0 Km/s l v = 72° o, = 3.0 a2 = 3.6 a 3 = 4.3 Km/s l i = -14° 1 2 = 257° 1 3 = 186° b, = 17° b 2 = -6° b 3 = 72° 129 TABLE XXI S i n g l e a b s o r p t i o n spectra Comparison of two models of s p a t i a l d i s t r i b u t i o n f i t with d i n Kpc as: d = 0.251 K(2) =-11.8 + 0. 6 Km/s K(3) =-15.4 + 0. 8 Km/s K(4) = -9.2 + 0. 8 Km/s K(5) = 18.7 + 2. 8 Km/s/Kpc K(6) = -23.8 + 4. 8 Km/s/Kpc K(7) = 2.5 + 4. 6 Km/s/Kpc K(8) = 15.0 + 2. 9 Km/s/Kpc K(9) = 3.2 + 2. 3 Km/s/Kpc K(10) = 4.4 + 3. 5 Km/s/Kpc o = 3.60 Km/s f i t with d i n Kpc as: d = 0.067 cosec|b| K(2) =-11.7 ± 0.6 Km/s K(3) =-15.2 ± 0 .8 Km/s K(4) = -9.2 ± 0 .8 Km/s K(5) = 19.1 ± 3.4 Km/s/Kpc K(6) =-38 ± 12 Km/s/Kpc K(7) = -0.6 ± 5.5 Km/s/Kpc K(8) = 15.0 ± 3.7 Km/s/Kpc K ( 9 ) = 9.0 ± 4 . 2 Km/s/Kpc K(10)= 9 .8 ± 6.0 Km/s/Kpc o = 4.00 Km/s 130 (2-16) and (2-23), t h i s a r t i f i c i a l v a r i a t i o n of o2 with s i n 2 b i s i n t e r p r e t e d , because we are l i m i t e d to l i n e - o f - s i g h t v e l o c i t i e s , as a negative c o n t r i b u t i o n of the z-component of the r e s i d u a l v e l o c i t i e s , i . e . as a negative <r 2(z). When the m u l t i p l e f e a t u r e s p e c t r a are excluded, t h i s e f f e c t i s e l i m i n a t e d and the c h a r a c t e r i s t i c s of the v e l o c i t y e l l i p s o i d can be determined c o r r e c t l y . In t h i s case we obt a i n a r a t h e r s p h e r i c a l e l l i p s o i d . The i n t e r s t e l l a r gas being c o n s i d e r e d as the m a t e r i a l from which s t a r s are formed, t h i s r e s u l t i s c o n s i s t e n t with the small e c c e n t r i c i t y of the v e l o c i t y e l l i p s o i d obtained f o r young s t a r s . The p a r t i c u l a r s p a t i a l d i s t r i b u t i o n obtained f o r s i n g l e f e a t u r e s i s c o n s i s t e n t with our Monte C a r l o s i m u l a t i o n s that i n d i c a t e a small dependence of <d> on b f o r l i n e s of s i g h t that i n t e r s e c t only one c l o u d . e) R e s u l t s f o r s t a r s and i n t e r s t e l l a r medium When a l l s t a r s and a l l i n t e r s t e l l a r m a t e r i a l are put together, we have a group of 3964 o b j e c t s which can be used to i n v e s t i g a t e the g l o b a l p i c t u r e of the kinematics i n the v i c i n i t y of the Sun. Table XXII shows some of the r e s u l t s obtained f o r t h i s group. Of these, one should note the remarkable agreement of t h i s study with others f o r the deter m i n a t i o n of the p e c u l i a r motion of the Sun with respect to the LSR. The standard v a l u e s a re: u e = 10.4 Km/s, 131 v 0 = 14.8 Km/s and w9 = 7.3 Km/s. The v a r i a t i o n of the average r e s i d u a l with d i s t a n c e , when the r e s i d u a l s are obtained from a four c o e f f i c i e n t f i t , i n d i c a t e s the need f o r f i e l d terms because i t s a b s olute value i n c r e a s e s with i n c r e a s i n g d. We a l s o n o t i c e d that the r e s i d u a l s are always negative at the g a l a c t i c p o l e s ( t h i s i s true f o r a l l types of s t a r s and f o r gas), i n d i c a t i n g some tendency to have a p o l a r i n f l o w of m a t e r i a l i n the s o l a r neighbourhood. F i g u r e s 97 to 103 show d i f f e r e n t s l i c e s of the v e l o c i t y f i e l d f o r s t a r s and gas put together. One can see th a t , o v e r a l l , the departure from c i r c u l a r motion i s not very important. TABLE XXII R e s u l t s f o r s t a r s and i n t e r s t e l l a r m a t e r i a l No.of o b j e c t s = 3964 <d> = 280 pc 1 - __° F i t with 10 c o e f f i c i e n t s K(1) = 1 . 7 + 0 .3 Km/s K(2) = -9. 7 + 0 .4 Km/s K(3) = -15. 5 + 0 .4 Km/s K(4) = -7. 9 + 0 .5 Km/s K(5) = -1 . 4 + 1 .0 Km/s/Kpc K(6) = - o . 1 + 3 .9 Km/s/Kpc K(7) = -3. 4 + 1 .0 Km/s/Kpc K(8) = 10. 9 + 0 .7 Km/s/Kpc K(9) = 2. 7 + 1 .8 Km/s/Kpc K( 10) = -2. 2 + 2 .0 Km/s/Kpc average r e s i d u a l a =13.4 Km/s P r i n c i p a l axes of the deformation tensor X, = -6.6 Km/s/Kpc 1, = 142° b, = 39° X 2 = "2.7 Km/s/Kpc 1 2 = 204° b 2 = -30° X 3 = 4.5 Km/s/Kpc 1 3 = 269° b 3 = 37° V e l o c i t y e l l i p s o i d parameters (not determined) 133 » \ , , , , H I 1 1 I h 0.0 40.3 80.0 llrtl.O ICO.0 200 0 240.0 *BO .3 120.0 360.0 400.0 Angle 1 (degrees) F i g u r e 67: v a r i a t i o n of the average r e s i d u a l RES with 1 f o r the case of 21-cm emission. Here, only four terms ( s o l a r motion terms and Vk) were used i n the f i t . 6 = feci ? I 1 1 1 1 1 1 ' 1 I I •100.0 -60.0 60.r. « ) . 0 M.O 0.0 dO.O « .o GO.O flo.o JOO.O Angle b (degrees) F i g u r e 68: v a r i a t i o n of the average r e s i d u a l RES with b f o r the case of 21-cm emission. Here, only four terms ( s o l a r motion terms and Vk) were used i n the f i t . 134 e g g g E H H Z 0 . 9 1 0 D i s t a n c e (Kpc) F i g u r e 69: v a r i a t i o n of the average r e s i d u a l RES with d the d i s t a n c e from the Sun f o r the case of 21-cm emission. Here, only four terms ( s o l a r motion terms and Vk) were used i n the f i t . 1=180° 135 1=270' AVAVAVAWA H I I I11 + AAVAVAyAVA Y = -12 Km/s Z = 12 Km/s X = -9 Km/s * = 9 Km/s <!> = -6 Km/s X 6 Km/s + = -3 Km/s A = 3 Km/s O = 0 Km/s 1= 0° Fi g u r e 70: h o r i z o n t a l v e l o c i t y f i e l d f o r 21-cm em i s s i o n . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. 1=180' 1 = 270" U l U tuiufttUlU kj'iutiisuiL'iUU KJlifllHyiuiuiuiUtUlUi luiuuiuiwutviwuiulUlU' H^uiliiluvtuiinutiiiuiIiiulUlUlU' luiuiui|iiutut«nuti^y|ut}il{ii|itjitfiii4lUlUlU' WM'MWlt'lllBlitflili^ UIIJIUIUIUJU IfiUil^ UiUiUiUiUiUiUiUiUitJiUiUfll •Jiutuiifljjiu iiiutuitiiuiciiuiw'i|jii| luiuiuiuttitu uiutlltviuliifiitiy WK'K'K'K'K fi'H'H'H'tt'tf K'uiuiuiiiitJ'tiigrjjijj ~ £|iuitituiiiiy<|f ftjiuiui|j||i ~ Y = -4 Km/s z = 4 Km/s X = -3 Km/s = 3 Km/s = -2 Km/s X = 2 Km/s + = -1 Km/s = 1 Km/s O = 0 Km/s 1= 0' F i g u r e 71: h o r i z o n t a l v e l o c i t y f i e l d f o r 21-cm em i s s i o n . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90' 1 = 225' b=-90° F i g u r e 72: v e r t i c a l v e l o c i t y f i e l d f o r 21 u n c e r t a i n t y on V f l i s the probable e r r o r , h a l f - w i d t h of 500 pc. b= 90° 1 = 315' Y = -12 Km/s Z = 12 Km/s X = - 9 Km/s 4* = 9 Km/s o = -6 Km/s X = 6 Km/s + = -3 Km/s A = 3 Km/s CD = 0 Km/s zm emission T The box has Y = -24 Km/s z = 24 Km/s X = -18 Km/s ^ = 18 Km/s <j> = -12 Km/s X = 12 Km/s + = -6 Km/s A = 6 Km/s CD = 0 Km/s b=-90' F i g u r e 73: v e r t i c a l v e l o c i t y f i e l d f o r 21 u n c e r t a i n t y on V f l i s the probable e r r o r , h a l f - w i d t h of 500 pc. -cm emission The The box has a b= 90° 137 1=180' Y = -12 Km/s Z = 12 Km/s X = -9 Km/s * = 9 Km/s o = -6 Km/s X = 6 Km/s + = -3 Km/s A = 3 Km/s Q = 0 Km/s b=-90° F i g u r e 74: v e r t i c a l v e l o c i t y f i e l d . f o r 21-cm emission. A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90° 1=270' fiuiUlUtU AYu'iuYuYu'lUJll hYuYututuYututlMf wYuiuiuib'iUtu'iuYuiUlll feYuYuYulUibWiuiuibiiiiUltf luiuiu>^ ijj|L'iiiiui!iYuiuiuiUlUlU ijiuiuiMMA'iiIiUiUtuiHliiiuiiiiui lu'ruiL'itflVlUluiuiUiUiw^l l^uluiuiylUlU CO'O'Q'l O'O'O'OTO Y = -12 Km/s Z = 1 2 Km/s X = -9 Km/s = 9 Km/s o = -6 Km/s X = 6 Km/s + = -3 Km/s A = 3 Km/s CD = 0 Km/s b=-90' F i g u r e 75: v e r t i c a l v e l o c i t y f i e l d f o r 21-cm em i s s i o n . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. 138 Fi g u r e 77: dependence of V f l on 1 and b f o r 21-cm emission at a d i s t a n c e of 1 Kpc. A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. 139 i i i l 1 1 1 1 : 1 1 1 , °-° «°-0 80.0 120.0 160.3 200.0 240.0 280.0 320.0 360.3 400.3 Angle 1 (degrees) F i g u r e 78: v a r i a t i o n of the average r e s i d u a l RES with 1 f o r the case of 21-cm a b s o r p t i o n . Here, only four terms ( s o l a r motion terms and Vk) were used i n the f i t . Angle b (degrees) F i g u r e 79: v a r i a t i o n of the average r e s i d u a l RES with b f o r the case of 21-cm a b s o r p t i o n . Here, only four terms ( s o l a r motion terms and Vk) were used i n the f i t . 140 »-l 1 1 1 1 1 1 1 1 1 i 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.1 O.B O.S 1.3 D i s t a n c e (Kpc) F i g u r e 80: v a r i a t i o n of the average r e s i d u a l RES with d the d i s t a n c e from the Sun f o r the case of 21-cm a b s o r p t i o n . Here, only four terms ( s o l a r motion terms and Vk) were used i n the f i t . 1=180° 141 = -12 Km/s = 12 Km/s = - 9 Km/s 9 Km/s = -6 Km/s 6 Km/s = -3 Km/s 3 Km/s 0 Km/s 1= 0° F i g u r e 81: h o r i z o n t a l v e l o c i t y f i e l d f o r 21-cm a b s o r p t i o n . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. 1=180 .= -12 Km/s = 12 Km/s = - 9 Km/s 9 Km/s = -6 Km/s 6 Km/s = -3 Km/s 3 Km/s 0 Km/s 1= 0° F i g u r e 82: h o r i z o n t a l v e l o c i t y f i e l d f o r 21-cm a b s o r p t i o n . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90° 142 1 = 240' Y = -24 Km/s Z = 24 Km/s X = -18 Km/s = 18 Km/s <•> = -12 Km/s X = 12 Km/s + = -6 Km/s A = 6 Km/s CD = 0 Km/s b=-90° F i g u r e 83: v e r t i c a l v e l o c i t y f i e l d f o r 21-cm a b s o r p t i o n . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90' 1 = 300' Y = -12 Km/s Z = 12 Km/s X = -9 Km/s 4> = 9 Km/s o = -6 Km/s X = 6 Km/s + = -3 Km/s A = 3 Km/s CD = 0 Km/s b=-90' F i g u r e 84: v e r t i c a l v e l o c i t y f i e l d f o r 21-cm a b s o r p t i o n . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90° 143 = -12 Km/s = 12 Km/s = -9 Km/s 9 Km/s = - 6 Km/s 6 Km/s = -3 Km/s 3 Km/s 0 Km/s b=-90° F i g u r e 85 : v e r t i c a l v e l o c i t y f i e l d f o r 21-cm a b s o r p t i o n . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90° = -12 Km/s = 12 Km/s = -9 Km/s 9 Km/s = - 6 Km/s 6 Km/s = -3 Km/s 3 Km/s 0 Km/s b=-90° F i g u r e 86 : v e r t i c a l v e l o c i t y f i e l d f o r 21-cm a b s o r p t i o n . A standard value (K (8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. 144 F i g u r e 88: dependence of V f l on 1 and b f o r 21-cm a b s o r p t i o n , at a d i s t a n c e of 1 Kpc. A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. 1=180° 145 1 = 270' Y = -24 Km/s Z = 24 Km/s X = -18 Km/s * - 18 Km/s 0 = -12 Km/s X = 1 2 Km/s + = -6 Km/s A = 6 Km/s 0 = 0 Km/s 1= 0° F i g u r e 89: h o r i z o n t a l v e l o c i t y f i e l d f o r s i n g l e a b s o r t i o n f e a t u r e s . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. 1=180' 1=270' 7VA Y = = "12 Km/s Z = 1 2 Km/s X = -9 Km/s * = 9 Km/s o = -6 Km/s X = 6 Km/s + = -3 Km/s A = 3 Km/s o = 0 Km/s 1= 0' Fi g u r e 90: h o r i z o n t a l v e l o c i t y f i e l d f o r s i n g l e a b s o r p t i o n f e a t u r e s . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90° 1=225' Y = -24 Km/s Z = 24 Km/s X = -18 Km/s * = 18 Km/s <!> = -12 Km/s X = 12 Km/s + = -6 Km/s A = 6 Km/s CD = 0 Km/s b=-90° F i g u r e 91: v e r t i c a l v e l o c i t y f i e l d f o r s i n g l e a b s o r p t i o n f e a t u r e s . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90' 1 = 300' 441414 1414141414141. ,Hux"X,'X».u.i».<'X«Xi»Xi'X'*<*XuXi» .ilU.U.UU Y = -12 Km/s Z = 1 2 Km/s X = -9 Km/s * = 9 Km/s o = -6 Km/s X = 6 Km/s + = -3 Km/s A = 3 Km/s O = 0 Km/s b=-90' Figu r e 92: v e r t i c a l v e l o c i t y f i e l d f o r s i n g l e a b s o r p t i o n f e a t u r e s . The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90° 147 1=180' Hi] Ui'li U l U i t a " llluiuiyi liluiu'iyi tiluiuiig ' AJluruiu UiuiliiUiif itJiuiuitjtii Y = -24 Km/s Z = 24 Km/s X = -18 Km/s * = 18 Km/s = -12 Km/s X = 12 Km/s + = -6 Km/s A = 6 Km/s • = 0 Km/s b=-90° F i g u r e 93: v e r t i c a l v e l o c i t y f i e l d f o r s i n g l e a b s o r p t i o n f e a t u r e s . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90° 1=270' llWUtUiuiuiUiUiu ~ 4l,M'M,Hlii,H^llH,K,Kli'iH'H1*" ' ^iStgi|jiuiuiijttiii*iifi Y = -12 Km/s Z = 12 Km/s X = -9 Km/s * = 9 Km/s <!> = -6 Km/s X = 6 Km/s + = -3 Km/s A = 3 Km/s O = 0 Km/s b=-90' Fi g u r e 94: v e r t i c a l v e l o c i t y f i e l d f o r s i n g l e a b s o r p t i o n f e a t u r e s . A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. 148 Fi g u r e 96: dependence of V f l on 1 and b f o r s i n g l e a b s o r p t i o n f e a t u r e s at a d i s t a n c e of 1 Kpc. A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. 149 1=180° 1 = 270' HMD -H+t + - H - H -COD Y = -12 Km/s Z = 12 Km/s X = -9 Km/s <r = 9 Km/s o = -6 Km/s X = 6. Km/s + = - 3 Km/s A = 3 Km/s CD = 0 Km/s 1= 0' F i g u r e 97: h o r i z o n t a l v e l o c i t y f i e l d f o r s t a r s + gas The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90° 150 1 = 225' nwwm««< vnmmimm Y = -12 Km/s Z = 12 Km/s X = -9 Km/s 4* = 9 Km/s 0 = -6 Km/s X = 6 Km/s + = -3 Km/s A = 3 Km/s CD = 0 Km/s b=-90° F i g u r e 98: v e r t i c a l v e l o c i t y f i e l d f o r s t a r s + gas. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90° = -12 Km/s 12 Km/s = -9 Km/s 9 Km/s = -6 Km/s 6 Km/s = -3 Km/s 3 Km/s 0 Km/s b=-90° F i g u r e 99: v e r t i c a l v e l o c i t y f i e l d f o r .stars + gas. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. 1=180° 1 5 1 1=270' Y = - 6 .0 Km/s z = 6 .0 Km/s X = -4 .5 Km/s = 4 .5 Km/s = -3 .0 Km/s X = 3 .0 Km/s + = -1 .5 Km/s A = 1 .5 Km/s O = 0 Km/s 1= 0° Fi g u r e 100: h o r i z o n t a l v e l o c i t y f i e l d f o r s t a r s + gas. A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the probable e r r o r . The box has a h a l f - w i d t h of 500 pc. b= 90° 1 = 225' Y = -12 Km/s Z = 1 2 Km/s X = -9 Km/s = 9 Km/s = - 6 Km/s X = 6 Km/s + = -3 Km/s A = 3 Km/s CD = 0 Km/s b=-90' Fi g u r e 101: v e r t i c a l v e l o c i t y f i e l d f o r s t a r s + gas. A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. The u n c e r t a i n t y on V f l i s the. probable e r r o r . The box has a h a l f - w i d t h of 500 pc. 152 F i g u r e 103: dependence of V f l on 1 and b f o r s t a r s + gas at a d i s t a n c e of 1 Kpc. A standard value (K(8) = 15 Km/s/Kpc) of the r o t a t i o n term was removed. 153 CHAPTER 5 DISCUSSION In t h i s chapter, the v a r i o u s r e s u l t s obtained f o r s t a r s and n e u t r a l hydrogen are compared and the v a r i a t i o n s of the kinematic p r o p e r t i e s with age and d i s t a n c e range are i n v e s t i g a t e d . Some c r i t i c a l p o i n t s of our study are d i s c u s s e d such as the use of second order terms and the u n c e r t a i n t y on our d e t e r m i n a t i o n of the d i s t a n c e s f o r n e u t r a l hydrogen. We a l s o attempt to i n v e s t i g a t e the i n f l u e n c e of the Gould b e l t and of nearby s p i r a l arms on the l o c a l kinematics of gas and s t a r s . a) Comparison of the r e s u l t s R e f e r i n g to Table XXIII, we w i l l now compare the v a r i o u s r e s u l t s to s t r e s s the c h a r a c t e r i s t i c s of each group and to analyse the i n f l u e n c e of age and d i s t a n c e range on some kinematic parameters. The s i m i l a r i t y between the r e s u l t s obtained f o r BI-IV and f o r BV (d > 200 pc) s t a r s shows that the kinematics of these o b j e c t s i s not c l o s e l y r e l a t e d to the l u m i n o s i t y c l a s s . However, one can note that BV s t a r s with d > 200 pc give a r a t h e r small value f o r the Oort c o e f f i c i e n t (K(8) = 5.1 ± 2.2 Km/s/Kpc). T h i s i s q u i t e s u r p r i s i n g s i n c e the nearby BV s t a r s (with d < 200 pc) l e a d to a more resonable value of K(8) = 12.7 ± 6.8 Km/s/Kpc. TABLE XXIII Comparison of the r e s u l t s o b t a i n e d w i t h gas and s t a r s 21 -cm 2 1 -cm al 1 21-cm s i n g l e BI- IV BV d>200pc BV d<200pc AV K i l l emi ss i on abs . 1 i nes abs. 1i nes No . a n a l y s e d 406 281 161 451 520 330 1061 981 <d> (pc) 418 395 251 692 414 133 109 167 T a y l o r S e r i e s Coef f i c i e n t s K( 1 ) Km/s - - - 3.4 + 0 8 3 ,0± 1 . 1 0. 7± 1 .8 -0. 2± 1 .0 2 .5± 1 . 6 K<2) Km/s -10.1+ 0 3 - 11.0± 0. 8 -11.8± 0.6 -8 . 2± 0 9 -9. 1 + 0.9 -9 . 1± 0.9 -11.6+ 1 .0 -10.5± 1 . 5 K(3) Km/s - 16.2± 0 3 - 16.3± 1 . 1 -15.4+ 0.8 - 17 . 4± 0 8 - 14 . 1± 1 .0 -18.2± 1 .0 -11.7+ 1 . 3 -18.6+ 1 . 7 K(4) Km/s/Kpc -11.7+ 0 5 -9.5± 1 2 -9.2± 2.8 -6 . 7± 2 0 -5 . 1± 2.2 -7 .0± 1 .5 -6 . 8 + 1 . 1 -10.2+ 1 .7 K(5) Km/s/Kpc 4.2+0 6 8 . 1± 2 4 18.7+ 2.8 -2.8± 1 1 -4 . 2± 3.5 -2 ± 14 -15 ± 1 1 -29 + 13 K(6) Km/s/Kpc 20.3+ 5 4 -11 ,5± 5. 9 -23.8± 4.8 -6.9± 5 6 7 ± 1 1 24 ± 25 8 + 15 6 + 15 K(7) Km/s/Kpc -0.1± 0 9 -0. 2± 3 . 8 2.5+ 4.6 -4 .0± 1 2 -7 ,0± 3.0 30 ± 14 4 ± 12 -9 + 13 K(8) Km/s/Kpc 15.0+ 0 6 15 ,0± 2 . 6 15.0+ 2.9 11.2 + 0 8 5. 1± 2 . 2 12 . 7± 6.8 1 1 . 0± 7.4 4 . 9± 7.8 K(9) Km/s/Kpc 2.3± 0 9 3 . 1± 2 . 7 3.2± 2.3 5.4± 2 9 13.5 + 4.6 15 + 1 1 2 . 7± 8 . 1 0.2 + 9 . 2 K( 10) Km/s/Kpc 3.3+ 1 1 6.9 + 4 . 0 4.4± 3.5 -9.8 + 3 4 -11.3+ 4 . 7 -29 ± 12 1 .8± 8 . 5 16 . 8± 9.3 1 0 degrees -8 -26 -19 -2 -20 28 23 36 a Km/s 2 . 7 6.8 3.6 11.2 11.6 10.3 13.5 23 . 3 <r(x) Km/s - - 3.0 10.6 13.0 11.6 17.4 27 . 2 o(y ) Km/s - - 3 . 6 10.9 10.7 9 . 4 11.2 20. 8 C / U ) Km/s - - 4.0 14.1 6.8 7 . 7 9.9 19.9 1 V degrees - - - - -9 20 17 23 154 155 The comparison between BV s t a r s on d i f f e r e n t d i s t a n c e ranges shows that the only two important d i f f e r e n c e s between these groups l i e i n the determination of v e (the y-component of p e c u l i a r v e l o c i t y of the Sun) and of K(7) (the v e l o c i t y g r a d i e n t 3v/3y). These two d i f f e r e n c e s may a c t u a l l y be r e l a t e d to each other because v e can be c o n s i d e r e d as a constant term i n the v a r i a t i o n of v with y ( f o r which 3v/9y i s the s l o p e ) . AV s t a r s and BV (d < 200 pc) s t a r s are i n t e r e s t i n g groups to compare because they have s i m i l a r average d i s t a n c e s but d i f f e r e n t ages. Apart from the v 9 term which seems to be abnormal for BV (d <200 p c ) , the only d i f f e r e n c e between the kinematics of these two groups appears i n the K(10) terms. K(10) i s a v e l o c i t y shear term equal to V2 (3v/3z + 3w/3y). One can a l s o n o t i c e that the K(5) term (the v e l o c i t y g r a d i e n t 3u/3x) i s s m a l l e r (more negative) f o r AV than f o r BV even though t h e i r values are compatible w i t h i n the u n c e r t a i n t i e s . As t h e i r average r e s i d u a l i n d i c a t e s , the K i l l (or K-giants) s t a r s are o l d e r than AV and BV s t a r s . T h e i r kinematic parameters i s n e v e r t h e l e s s compatible with those obtained f o r AV s t a r s because of the l a r g e u n c e r t a i n t i e s attached to them. It i s i n t e r e s t i n g to n o t i c e f o r the case of K i l l s t a r s that the Oort c o e f f i c i e n t i s very low (K(8) = 4.9 ± 7.8) and that K(5) = 3v/9x = -29 ± 13, the l a r g e s t negative value obtained so f a r . 1 56 In regards to the i n t e r s t e l l a r medium, we f i n d a c l o s e agreement between the r e s u l t s obtained with a l l a b s o r p t i o n s p e c t r a and those obtained with s i n g l e a b s o r p t i o n f e a t u r e s o n l y . As explaned i n Chapter 4, these two s e t s of o b j e c t s have d i f f e r e n t s p a t i a l d i s t r i b u t i o n s and d i f f e r e n t average r e s i d u a l s . A comparison of the r e s u l t s obtained with 21-emision data and with 21-aborption data r e v e a l s that a l l kinematic parameters are compatible exept K(6), the v e l o c i t y g r a d i e n t 9w/9z, which takes values of o p p o s i t e s i g n s . We b e l i e v e that the negative value of K(6) obtained with a b s o r p t i o n data i s more r e l i a b l e than the that obtained with emission s p e c t r a because of the b e t t e r coverage of high g a l a c t i c l a t i t u d e s of the Nancay Survey. For Henderson's data, |b| i s never l a r g e r than 30° (and never s m a l l e r than 10°). T h i s small range in b angle g i v e s to the term K(6), the c o e f f i c i e n t of <|z|> s i n |b|, the r o l e of a constant term which takes a p o s i t i v e value and thus reduces the value of K(5) = 9u/9x that d e s c r i b e s the expansion along the g a l a c t i c mid-plane. As e x p l a i n e d i n Chapter 4, the i n t e r c l o u d medium, observed with the 21-cm emission l i n e , has a s t r a t i f i e d d i s t r i b u t i o n but the clouds, observed with 21-cm a b s o r p t i o n l i n e , do not seem to be w e l l d e s c r i b e d by the standard p l a n e - p a r a l l e l model. There seems i n f a c t to be a l a c k of clouds i n the immediate v i c i n i t y of the Sun (d < 200 p c ) . T h i s "hole" i n the d i s t r i b u t i o n of clouds must be more or 157 l e s s centered around the Sun but our data does not provide enough i n f o r m a t i o n to determine i t s a c t u a l shape and p o s i t i o n . The main d i f f e r e n c e between the kinematics of absorbing c l o u d s and of e a r l y type s t a r s seem to l i e i n the determination of K(5) = 9u/9x. In f a c t , i t i s i n t e r e s t i n g to n o t i c e that f o r a l l s t e l l a r groups the value of K(5) i s negative while i t i s p o s i t i v e f o r a l l i n t e r s t e l l a r groups. For n e u t r a l hydrogen, the p o s i t i v e value of K(5) i n d i c a t e s an expansion of the gas i n the g a l a c t i c plane. For s t a r s , the coresponding i n t e r p r e t a t i o n of the negative K(5) term would i n v o l v e a g e n e r a l c o n t r a c t i o n in the g a l a c t i c plane. In t h i s p i c t u r e , the importance of t h i s c o n t r a c t i o n i n c r e a s e s with time s i n c e we observe that K(5) decreases (becomes more and more negative) with i n c r e a s i n g age of the s t u d i e d o b j e c t s . T h i s c l e a r systematic trend can be seen i n Table XXIV where we c l a s s i f i e d o b j e c t s a c c o r d i n g to t h e i r age. F i g u r e 104 shows the v a r i a t i o n of K(5) with a2 which i s r e l a t e d to the age. One can see a general trend f o r K(5) to decrease with i n c r e a s i n g a2 but the e r r o r bars are q u i t e l a r g e ! The c o n s i s t e n c y of the values of K(5) f o r d i f f e r e n t groups of B s t a r s r e i n f o r c e s the v a l i d i t y of t h i s r e l a t i o n . In t h i s r e s p e c t , the p o s i t i v e v a l u e s of K(5) obtained f o r n e u t r a l hydrogen i n d i c a t e , as we a l r e a d y know, that the gas i s a m a t e r i a l "younger" that the s t a r s . The s t e l l a r ages were taken from Gomez (1972) and J a r e i s s (1983) 1 58 TABLE XXIV The v a r i a t i o n of 9u/9x and of 1 0 with age group of o b j e c t s age (yr.) 9u/9x (Km/s/Kpc) 1 0 (deg.) HI absorbing clouds 0 8.1 + 2.4 -26 i n t e r c l o u d medium 0 4.2 + 0.6 -8 a l l B s t a r s 10 7 -3.2 + 1 .0 -1 BI-IV - -2.8 + 1 . 1 -2 BV with d > 200 pc - -4.2 + 3.5 -20 BV with d < 200 pc - -2 + 1 4 28 a l l AV s t a r s 5- 1 0 8 -15 + 1 1 23 A(0-4)V - -7 + 1 3 --A(5-9)V - -49 + 22 K i l l s t a r s . 4- 10 9 -29 + 1 3 36 FV s t a r s 4« 10 9 -48 + 91 42 159 o a a: \ tn \ e *s 1 a in us I 1 6 0 0 120 0 400.0 « 0 . 0 S60.3 a 2 (Km 2/s 2) F i g u r e 104 : v a r i a t i o n of K(5) = 9u/9x with a2 which i s r e l a t e d to the age of the o b j e c t s . The l e n g t h of the e r r o r bars corresponds to one standard d e v i a t i o n a 160 and should be co n s i d e r e d as being only aproximate s i n c e the r a p i d i t y of s t e l l a r e v o l u t i o n depends on many f a c t o r s l i k e the i n i t i a l r e l a t i v e abundances of d i f f e r e n t elements and si n c e s t a r s can spend (depending on t h e i r mass) a c o n s i d e r a b l e ammount of time with a given s p e c t r a l type and l u m i n o s i t y c l a s s . As shown i n t a b l e XXIII, the v a r i a t i o n of 1 0 with age i s a l s o worth n o t i n g . The problem with 1 0 i s that i t i s r e l a t e d to both K(5) and K(7) terms. In f a c t , 1 0 i s given by: 1 0 = V2 arc tan (-C/A) (5-1) where C i s the c o e f f i c i e n t of d cos21 c o s 2 b or, e q u i v a l e n t l y , of 1 / 2 d c o s 2 l c o s 2 b - 1 / 2 d s i n 2 l c o s 2 b and A i s the Oort c o e f f i c i e n t . I t s v a r i a t i o n i s thus i n f l u e n c e d by the systematic v a r i a t i o n of K(5) with age but i s a l s o perturbed by the f l u c t u a t i o n of K(7) = 3v/9y which seems to vary rather randomly from one group to another. It i s a l s o i n t e r e s t i n g to n o t i c e the v a r i a t i o n of c e r t a i n kinematic parameters with d i s t a n c e range. I t seems q u i t e c l e a r f o r inst a n c e that the value of K(1) = Vk term i n c r e a s e s with d i s t a n c e . T h i s c o u l d be i n t e r p r e t e d as a r e l a t i o n between the absolute b r i g h t n e s s of s t a r s and the strenght of the mechanisms that cause, at t h e i r s u r f a c e , a s h i f t in the l i n e - o f - s i g h t v e l o c i t y . A c t u a l l y , t h i s i s reasonable s i n c e we expect an i n c r e a s e of the g r a v i t a t i o n a l 161 r e d s h i f t and of the athmospheric outflows with i n c r e a s i n g mass (and thus b r i g h t n e s s ) of the s t a r s . The Oort c o e f f i c i e n t a l s o seems to vary with d i s t a n c e . In f a c t we f i n d that K(8) - 12 Km/s/Kpc f o r nearby o b j e c t s and reduces to K(8) 5 f o r K i l l and d i s t a n t BV s t a r s before i t r e t u r n s to a value of about 11 f o r the the d i s t a n t BI-IV s t a r s . Here one must be c a r e f u l because of the l a r g e u n c e r t a i n t i e s . The value of K(8) = 4.9 ± 7.8 Km/s/Kpc obtained with K i l l s t a r s f o r example c o u l d be c o n s i d e r e d as low s i n c e i t i s not s i g n i f i c a n t l y d i f f e r e n t from zero but i t i s a l s o c o n s i s t e n t with 11 Km/s/Kpc o f t e n obtained i n t h i s study. Only the Oort c o e f f i c i e n t obtained with BV (d > 200 pc) i s s i g n i f i c a n t l y abnormal (K(8) = 5.1± 2.2 Km/s/Kpc). I t s i n t e r p r e t a t i o n i s d i f f i c u l t . F i n a l l y , we do not wish to dwell upon the r e s u l t s of the v e l o c i t y e l l i p s o i d a n a l y s i s but i t i s worth mentioning that they are in good general agreement with those obtained in other s t u d i e s (see Mihalas and Binney (1981)). I t seems that the i n c l u s i o n of the terms K(5) to K(10) i n the c a l c u l a t i o n of the r e s i d u a l s e x e r t s no great i n f l u e n c e on the value of v e l o c i t y e l l i p s o i d parameters. T h i s i s probably due to the f a c t t h a t , i n g e n e r a l , we d e a l with r e l a t i v e l y small d i s t a n c e s and thus the v e l o c i t y g r a d i e n t s do not c o n t r i b u t e very much to the f i e l d v e l o c i t y . The problem of the d e v i a t i o n of the vertex angle remains open, 1 62 even a f t e r t r y i n g to e l i m i n a t e the e f f e c t of n o n - c i r c u l a r motion as we d i d . b) The u n c e r t a i n t y on d i s t a n c e d e t e r m i n a t i o n f o r gas Since , f o r n e u t r a l hydrogen, an average d i s t a n c e had to be assumed f o r each l i n e of s i g h t , we expect a supplementary u n c e r t a i n t y to be introduced i n the treatment through a g e n e r a l l y i n c o r r e c t d i s t a n c e d e t e r m i n a t i o n . For a given o b j e c t , the assumed d i s t a n c e <d> and the r e a l d i s t a n c e d d i f f e r by an ammount Ad and t h i s q u a n t i t y should c o n t r i b u t e to increase the r e s i d u a l s . F o l l o w i n g Takakubo (1967), we can express o2 as: a2 = o 0 2 + e 2 (5-2) where o0 i s the " t r u e " v e l o c i t y r e s i d u a l r e l a t e d to the random motion of the gas and where e i s the c o n t r i b u t i o n of Ad. T h i s c o n t r i b u t i o n can be understood i n the f o l l o w i n g way. Were there no random motion of the clouds, the l i n e - o f - s i g h t v e l o c i t y c o u l d be p e r f e c t l y d e s c r i b e d by the v e l o c i t y f i e l d V f l . Then, an e r r o r Ad on the d i s t a n c e d e t e r m i n a t i o n would introduce a r e s i d u a l R given by: R = [ K(5) c o s 2 l c o s 2 b + K(6) s i n 2 b + K(7) s i n 2 l c o s 2 b + K(8) sin21 c o s 2 b + K(9) c o s l sin2b + K(10) s i n l sin2b ]-Ad (5-3) 163 In the case of a gaussian d i s t r i b u t i o n of Ad, the value of e 2 would be given by: e 2 = [ K(5) c o s 2 l c o s 2 b + K(6) s i n 2 b + K(7) s i n 2 l c o s 2 b + K(7) s i n 2 l c o s 2 b + K(8) sin21 c o s 2 b (5-4) + K(9) c o s l sin2b + K(10) s i n l sin2b ] 2 -<Ad2> . If the terms Ad are zero (no e r r o r i n the d i s t a n c e determination) the value of e 2 i s zero as w e l l . Thus, the determination of the r a t i o e2/o2 g i v e s an idea of the importance of the u n c e r t a i n t y on d. In order to o b t a i n t h i s r a t i o , we kept the square of the r e s i d u a l s and f i t t e d them to the square of the f i r s t order terms and to constant term o 0 2 . In doing t h i s we found a2 o 0 2 which suggests that the measured r e s i d u a l s are r e l a t e d to a true random motion and are not g r e a t l y a f f e c t e d by our assumptions concerning the s p a t i a l d i s t r i b u t i o n of the gas. T h i s r e s u l t was confirmed by a Monte C a r l o s i m u l a t i o n i n which the d i s t a n c e s were randomly perturbed to t e s t the s e n s i t i v i t y of the c o e f f i c i e n t d e t e r m i n a t i o n . We found that random p e r t u r b a t i o n s with standard d e v i a t i o n s as l a r g e as 100 pc and 200 pc do not i n c r e a s e the u n c e r t a i n t i e s on the i n d i v i d u a l c o e f f i c i e n t s by more than 20%. Thus i t seems that our method of assuming d i s t a n c e s a c c o r d i n g to a t h e o r e t i c a l model of s p a t i a l d i s t r i b u t i o n i s 1 64 s a t i s f a c t o r y and not c r i t i c a l i n the determination of the v a r i o u s kinematic parameters of the f i e l d . In f a c t , t h i s was s t r o n g l y suggested by the s i m i l a r i t y of the r e s u l t s obtained with d i f f e r e n t models of s p a t i a l d i s t r i b u t i o n . c) The constancy of the Oort c o e f f i c i e n t Regarding our a n a l y s i s of the i n t e r s t e l l a r medium, the Oort c o e f f i c i e n t has been assumed to be constant and equal to 15 Km/s/Kpc at any b angle. T h i s assumption was necessary because only the product A«<d> c o u l d be determined as a f u n c t i o n of b and used to study the v a r i a t i o n of <d> with b. Since any systematic v a r i a t i o n of A with b would a f f e c t d i r e c t l y the d e t e r m i n a t i o n of the s p a t i a l d i s t r i b u t i o n of n e u t r a l hydrogen, we have c o n s i d e r e d i t worthwhile to i n v e s t i g a t e the constancy of the Oort c o e f f i c i e n t over a wide range of g a l a c t i c l a t i t u d e s . We have found evidence that A does not vary with b. F i r s t , a c c o r d i n g to i t s d e f i n i t i o n , the Oort c o e f f i c i e n t should not vary very much over small d i s t a n c e s such as a few hundred pa r s e c s . The d e f i n i t i o n of A i s : A = 1 / 2 [ (6 0/R 0) " (9©/3R) 0 1 (5-5) where © i s the t a n g e n t i a l v e l o c i t y due to c i r c u l a r motion about the g a l a c t i c center and R i s the d i s t a n c e from the g a l a c t i c c e n t e r . (The s u b s c r i p t s 0 i n d i c a t e that the 1 65 q u a n t i t y i s eva l u a t e d at the p o s i t i o n of the Sun.) One can see that an e l e v a t i o n of few hundred parsecs over the g a l a c t i c mid-plane should not s i g n i f i c a n t l y a f f e c t the value of A s i n c e R 0 i s aproximatly equal to 10 Kpc. Secondly, the c l o s e agreement between the v a r i a t i o n of A«<d> and the expected cosecant law i n the case of the i n t e r c l o u d medium suggests that A i s constant over these l a t i t u d e s . In f a c t , i t would be a strange c o n s p i r a c y i f the v a r i a t i o n of A with b would compensate e x a c t l y f o r a p o s s i b l e departure from the p l a n e - p a r a l l e l d i s t r i b u t i o n of the i n t e r c l o u d medium. F i n a l l y and most•importantly, we have found that A does not seem to vary s y s t e m a t i c a l l y with b i n the case of s t a r s . T h i s r e s u l t has been obtained by a n a l y s i n g the dependence of A on b f o r d i f f e r e n t types of s t a r s and by combining the r e s u l t s together. We have found that p u t t i n g a l l s t a r s together and a n a l y s i n g them at once i s not a good method due to the inhomogeneity of the group which, i n t u r n , in t r o d u c e s a c o r r e l a t i o n between b and the dominant s t e l l a r type. The problem of assuming the standard value of 15 Km/s/Kpc f o r A i s that no way of checking t h i s value e x i s t s . The l o c a l s t a r s a c t u a l l y l e a d to a value that would be more l i k e 10 Km/s/Kpc and i f t h i s i s the case f o r gas as w e l l , then a l l the d i s t a n c e s . p r e v i o u s l y d e r i v e d should be m u l t i p l i e d by a f a c t o r 1.5. Thus, we are l e f t with l a r g e 1 66 u n c e r t a i n t i e s on the s c a l e h e i g h t s and d 0 of both components of the i n t e r s t e l l a r medium but one should keep i n mind that a s c a l i n g of the d i s t a n c e s would not a f f e c t the r e l a t i v e p r o p o r t i o n s of the v a r i o u s v e l o c i t y g r a d i e n t s . Nor would i t a f f e c t the r e a l i t y of the observed departure from the p l a n e - p a r a l l e l model f o r the absorbing c l o u d s . d) The importance of the Gould b e l t As mentioned i n Chapter 1, many authors have s t r e s s e d the importance of the Gould b e l t i n the d i s t r i b u t i o n and kinematics of gas and young s t a r s . In order to i n v e s t i g a t e the r e l i a b i l i t y of our study, we have p a i d s p e c i a l a t t e n t i o n to t h i s subsystem. Since we only use l i n e - o f - s i g h t v e l o c i t i e s f o r our a n a l y s i s , we have found i t d i f f i c u l t to study the Gould b e l t i t s e l f because the i s o l a t i o n of i t s "members" from the r e s t of the data reduces the range of 1 and b angles and intr o d u c e s systematic c o r r e l a t i o n between some of our terms. We have found i n f a c t that the study of a l i m i t e d part of the sky giv e s very l a r g e u n c e r t a i n t i e s . In order to a v o i d t h i s problem i n t e s t i n g the importance of the Gould b e l t on the l o c a l k i nematics, we have decided to do e x a c t l y the opposite which i s to r e j e c t the Gould b e l t ' s members to see how d i f f e r e n t the f i t would be with only the "normal" o b j e c t s . Our d e f i n i t i o n of the Gould b e l t was based on i t s recent d e s c r i p t i o n by Olano (1982) as an e l l i p s e with a 1 67 semi-major a x i s equal to 364 pc and a semi-minor a x i s equal to 211 pc. The center i s l o c a t e d at 1 = 131° , b = 0° and d = 166 pc. I t s o r i e n t a t i o n i s such that the major a x i s i s approximatly p a r a l l e l to the 1=52°-1=232° l i n e at b = 0°. T h i s d e s c r i p t i o n i s l i m i t e d to the g a l a c t i c mid-plane and i s a c t u a l l y a p r o j e c t i o n of the b e l t . In order to take i n t o account i t s w e l l known t i l t of 20° from the g a l a c t i c mid-plane, we decided to c o n s i d e r a l l m a t e r i a l w i t h i n the e l l i p s e and w i t h i n |b| < 20° as p a r t of the Gould b e l t . T h i s d e f i n i t i o n i s c e r t a i n l y not very good at d e s c r i b i n g the b e l t i t s e l f s i n c e i t i n c l u d e s too much m a t e r i a l but i s very good at r e j e c t i n g with confidence the " r e a l " members of the Gould b e l t . Table XXV shows the r e s u l t s f o r B s t a r s with and without the Gould b e l t members. One can see that the r e j e c t i o n of these s t a r s does not g r e a t l y a f f e c t the r e s u l t s of the f i t , i n d i c a t i n g that the v e l o c i t y f i e l d s r e a l l y d e s c r i b e the g l o b a l kinematics in the s o l a r neighbourhood and that they are not dominated by the p e c u l i a r motion of the expansion of a subsystem. B s t a r s were chosen here because they were the f i r s t type of o b j e c t s to be a s s o c i a t e d with the Gould b e l t . For each group of o b j e c t s that we have s t u d i e d , we have i n v e s t i g a t e d the importance of the Gould b e l t i n t h i s way and no s i g n i f i c a n t e f f e c t on the v e l o c i t y g r a d i e n t determination has been found. Even f o r 21-cm data for which, as was suggested by Weaver (1974), there i s an 168 expansion along the the g a l a c t i c mid-plane matched to an i n f a l l i n g motion from the g a l a c t i c p o l e s , the r e s u l t s were independent of the the i n c l u s i o n of the region of the Gould b e l t . Such c o n s i d e r a t i o n s do not exclude the p o s s i b i l i t y that the i n t e r n a l kinematics of the Gould b e l t i s p e c u l i a r and we do not c l a i m to have d i s p r o v e d the e x i s t e n c e of an expanding subsystem. We have j u s t shown that our r e s u l t s are not s i g n i f i c a n t l y i n f l u e n c e d by i t s p o s s i b l e e x i s t e n c e . e) S p i r a l s t r u c t u r e and kinematics As mentioned in Chapter 1, i t i s p o s s i b l e i n p r i n c i p l e to d e t e c t the presence of s p i r a l arms from t h e i r i n f l u e n c e on the l o c a l kinematics. K. R o h l f s (1972) showed that the value of the Oort c o e f f i c i e n t A and of the nodal d e v i a t i o n 1 0 vary across a s p i r a l arm. The value of 1 0 i n p a r t i c u l a r was used by t h i s author to suggest the e x i s t e n c e of nearby s p i r a l arms. He found the values 1 0 = -22° ± 5° f o r the gas and 1 0 = -4.2° ± 0.7° f o r young s t a r s which agree with the p r e d i c t i o n s of the d e n s i t y wave theory that 1 0 should be negative on the outer edge of an arm and that the e f f e c t of an arm would be more important on gas than on s t a r s . I t seems to us t h a t , although our study confirms these values of 1 0, the i n t e r p r e t a t i o n of t h i s term should be done c a r e f u l l y . We have found i n f a c t that 1 0 takes 169 TABLE XXV The i n f l u e n c e of the Gould b e l t A l l B s t a r s K(1) = 3. 4 + 0.4 Km/s K(2) = -8. 7 + 0.5 Km/s K(3) = -16. 2 + 0.5 Km/s K(4) = -6. 2 + 1 .0 Km/s K(5) = -3. 2 + 1 .0 Km/s/Kpc K(6) = -5. 6 + 4.5 Km/s/Kpc K(7) = -4. 8 + 1.0 Km/s/Kpc K(8) = 10. 4 + 0.7 Km/s/Kpc K(9) = 6. 3 + 2.2 Km/s/Kpc K(10)= -10. 5 + 2.5 Km/s/Kpc No. = 1301 Non-Gould B s t a r s K(1) = 3.0 + 0.6 Km/s K(2) = -8.1 + 0.6 Km/s K(3) = -15.8 + 0.6 Km/s K(4) = -6.7 + 1 . 1 Km/s K(5) = -3.1 + 1 .0 Km/s/Kpc K(6) = -4.3 + 4.7 Km/s/Kpc K(7) = -4.7 + 1 . 1 Km/s/Kpc K(8) = 10.3 + 0.7 Km/s/Kpc K(9) = 7.3 + 2.3 Km/s/Kpc K( 10) = -9.6 + 2.6 Km/s/Kpc No. 925 1 70 l a r g e p o s i t i v e values f o r some types of s t a r s (which would then put us on the inner edge of an arm) and we have dete c t e d t h a t , i n g e n e r a l , the values of 1 0 are r e l a t e d to those of K(5) = 3u/9x. Since K(5) seems to vary with the age of the s t u d i e d o b j e c t s , the value of 1 0 i s a l s o a f f e c t e d by the age. From t h i s , we would i n t e r p r e t the d i f f e r e n c e between the values of 1 0 f o r HI and B s t a r s as r e s u l t i n g from an e v o l u t i o n a r y process and not to the f a c t that the gas i s more i n f l u e n c e d than the s t a r s by the presence of d e n s i t y waves. Thus, i t seems to us that the argument used by R o h l f s to d e t e c t nearby s p i r a l arms i s r a t h e r dangerous. f) The use of second order terms As mentioned in Chapter 3 f o r the s p e c i a l case of B s t a r s , the i n t r o d u c t i o n of second order terms i n the f i t does not s i g n i f i c a n t l y improve the goodness of the f i t . This suggests that the v e l o c i t y f i e l d i s smooth enough over the range of d i s t a n c e s i n v o l v e d i n t h i s study to use only a f i r s t order approximation. I t was important to check t h i s p o i n t because nothing j u s t i f i e d a p r i o r i the r e s t r i c t i o n to a f i r s t order expansion i n the d e s c r i p t i o n of the l o c a l v e l o c i t y f i e l d . Equation (1-17) was chosen because i t seemed to be a good compromise s i n c e i t a l l o w s a f a i r treatment of the motion in three dimensions with a minimum number of terms. A c t u a l l y , second order terms c o u l d have been used 171 for the treatment of d i s t a n t B s t a r s s i n c e they were m a r g i n a l l y s i g n i f i c a n t but they were t o t a l l y i n s i g n i f i c a n t f o r AV and K III s t a r s . In order to have a uniform method throughout t h i s study, we have decided not to use them. Because of the problems a l r e a d y mentioned with the d i s t a n c e determination of i n t e r s t e l l a r m a t e r i a l , we d i d not t e s t the s i g n i f i c a n c e of second order terms f o r the 21-cm d a t a . g.) The i n t e r p r e t a t i o n of the v e l o c i t y f i e l d s A word should be s a i d here about the i n t e r p r e t a t i o n of the p l o t s that were presented i n Chapters 3 and 4. As shown i n F i g u r e 3, a pure c i r c u l a r d i f f e r e n t i a l r o t a t i o n can be i d e n t i f i e d i n terms of l i n e - o f - s i g h t v e l o c i t i e s by a very symmetrical f i e l d with four quadrants separated by a c r o s s of zero v e l o c i t y c o n t o u r s . T h i s p a t t e r n i s o f t e n seen i n the v a r i o u s v e l o c i t y f i e l d s that have been presented because c i r c u l a r motion i s o f t e n the dominant motion of the o b j e c t s of the s o l a r neighbourhood. The problem i s that a l i n e - o f - s i g h t v e l o c i t y f i e l d can be e x p l a i n e d by very d i f f e r e n t kinds of motion and thus the i n t e r p r t a t i o n of a f i e l d i s never unique. For example, the p a t t e r n of f i g u r e 3 c o u l d be obtained i f there were a shear along the x-axis or along the y - a x i s . T h i s comes from the f a c t that a pure c i r c u l a r motion i s d e s c r i b e d by equation (5-5) which c o u l d be r e w r i t t e n as: 1 72 A = 1 / 2 ( 9u/9y + 3v/3x ) 0 (5 -6 ) and c o u l d not be d i s t i n g u i s h e d from a pure 3u/3y or a pure 3v/3x term. In these cases, the motion (a v e l o c i t y shear) would be composed of two opposite flows p a r a l l e l to an a x i s having t h e i r v e l o c i t y v a r y i n g along the other a x i s . Such a motion i s very d i f f e r e n t from d i f f e r e n t i a l c i r c u l a r motion but g i v e s e x a c t l y the same l i n e - o f - s i g h t v e l o c i t y f i e l d ! T h i s ambiguity i s due to the f a c t that we are d e a l i n g with only one component of the t o t a l v e l o c i t y and ignore the behaviour of t a n g e n t i a l motion. Another example of such an ambiguity i s shown i n F i g u r e s 4 and 5 which c o u l d e a s i l y be i n t e r p r e t e d as expansion and c o n t r a c t i o n p a t t e r n s r e s p e c t i v e l y but which d e s c r i b e , s i n c e we have produced them that way, the e f f e c t of d i f f e r e n t i a l r o t a t i o n along planes that are p e r p e n d i c u l a r to the plane of r o t a t i o n . Thus, i t i s necessary to be very c a r e f u l i n the i n t e r p r e t a t i o n of these f i e l d s . The same remark a p p l i e s to the three-dimensional p l o t s that show the v a r i a t i o n of V f l with b and 1. In s p i t e of t h a t , we c o n s i d e r that these p l o t s are i n f o r m a t i v e because they show the l e v e l of co n f i d e n c e of the det e r m i n a t i o n of the kinematic parameters and because they can be used to compare g r a p h i c a l l y the d i f f e r e n t v e l o c i t y f i e l d s . The three-dimensional p l o t s are a l s o u s e f u l i n that 173 they i n d i c a t e the sign of the most probable value of the l i n e - o f - s i g h t v e l o c i t y i n any d i r e c t i o n and thus help to i d e n t i f y the zones of apparent expansion or c o n t r a c t i o n . 1 74 CONCLUSIONS In order to conclude, we w i l l now attemp to summarize the most important r e s u l t s obtained i n t h i s study. 1) The use of a three-dimensional f i r s t order expansion of the v e l o c i t y f i e l d seems to be q u i t e a p p r o p r i a t e f o r the study of the kinematics of o b j e c t s i n the s o l a r neighbourhood. On the one hand, i t g i v e s a b e t t e r d e s c r i p t i o n of n o n - c i r c u l a r motion than does the deter m i n a t i o n of only Vk and 1 0 (the nodal d e v i a t i o n ) because i t decouples the motions along the three p r i n c i p a l axes of the Galaxy. On the other hand, i t avoids the c o m p l i c a t i o n s that a r i s e when a three-dimensional second order expansion i s used. 2) The Vk term which d e s c r i b e s a systematic s h i f t i n the observed l i n e - o f - s i g h t v e l o c i t y i s p o s i t i v e f o r d i s t a n t s t a r s l i k e BI-IV but becomes i n s i g n i f i c a n t f o r nearby m a t e r i a l ( w i t h i n a few hundred p a r s e c s ) . 3) Our values of the components of the s o l a r motion along x,y and z are, i n g e n e r a l , i n c l o s e agreement with those found i n the l i t e r a t u r e . T h e i r d e t e r m i n a t i o n i s not very much a f f e c t e d by the i n c l u s i o n of the v a r i o u s v e l o c i t y g r a d i e n t s . 175 4) The v e l o c i t y g r a d i e n t 3u/3x shows a systematic t r e n d to vary with the "age" of the s t u d i e d o b j e c t s . I t goes smoothly from small p o s i t i v e values f o r i n t e r s t e l l a r m a t e r i a l to l a r g e negative v a l u e s f o r l a t e type s t a r s . The r e l a t e d parameter 1 0, the nodal d e v i a t i o n , a l s o v a r i e s with age but not as s y s t e m a t i c a l l y s i n c e i t i s i n f l u e n c e d by the v e l o c i t y g r a d i e n t 3v/3y which seems to vary randomly from one type of o b j e c t to another. 5) The motion p e r p e n d i c u l a r to the g a l a c t i c plane i s p a r t l y d e s c r i b e d by the v e l o c i t y g r a d i e n t 3w/3z which i s negative f o r absorbing c l o u d s (our most r e l i a b l e probe of t h i s kind of motion fo r the i n t e r s t e l l a r m a t e r i a l ) . T h i s negative value and the p o s i t i v e value of 3u/3x suggests the e x i s t e n c e of i n f l o w s of gas from the g a l a c t i c poles matched by a general expansion i n the g a l a c t i c plane. For s t a r s the term 3w/3z i s g e n e r a l l y not s i g n i f i c a n t l y d i f f e r e n t from zero and no s i m i l a r i n f a l l i n g motion has been detected. 6) Our best determination of the Oort c o e f f i c i e n t A leads to a value of 1 1 Km/s/Kpc, a somewhat low value as compared to the adopted standard of 15 Km/s/Kpc. For some reason, we f i n d that f o r BV d > 200 pc, A i s anomalously low (5.1 ± .2.2 Km/s/Kpc). 7) Even though marked departures from c i r c u l a r motion are found in most cases, no systematic "streaming motion" i s observed f o r a l l types of o b j e c t s . As a matter of f a c t , when a l l o b j e c t s are put together, they l e a d to a 176 s o l u t i o n i n which pure d i f f e r e n t i a l c i r c u l a r motion i s dominant. 8) Our r e s u l t s f o r the v e l o c i t y e l l i p s o i d parameters are i n good general agreement with those found in the l i t e r a t u r e . The i n t r o d u c t i o n of a l l v e l o c i t y f i e l d terms e x e r t s no s i g n i f i c a n t i n f l u e n c e on the determination of v e l o c i t y e l l i p s o i d parameters and, consequently, does not s o l v e the problem of the vertex d e v i a t i o n . 9) We found that the group of main sequence A s t a r s seems to be k i n e m a t i c a l l y inhomogeneous. In f a c t , i t can be d i v i d e d i n t o two d i s t i n c t subgroups a c c o r d i n g to the s p e c t r a l s u b c l a s s [ A(0-4)V and A(5-9)V ] which give very d i f f e r e n t v a l u e s of the v a r i o u s v e l o c i t y g r a d i e n t s . 10) We have not s t u d i e d the i n t e r n a l kinematics of the Gould b e l t but we can a s s e r t that p o s s i b l e anomalous kinematics w i t h i n the Gould b e l t does not s i g n i f i c a n t l y a f f e c t the v e l o c i t y f i e l d s that we determine. 11) Our r e s u l t s do not seem to show c l e a r evidence of the i n f l u e n c e of nearby s p i r a l arms on the l o c a l k inematics. 12) The usual p l a n e - p a r a l l e l model d e s c r i b e s very w e l l the s p a t i a l d i s t r i b u t i o n of the hot HI i n t e r c l o u d medium but seems inadequate fo r the 21-cm absorbing c l o u d s . In f a c t , we found an apparent lack of these o b j e c t s i n the v i c i n i t y of the Sun (d < 200 p c ) . 13) The v e l o c i t y e l l i p s o i d obtained f o r i n t e r s t e l l a r 177 m a t e r i a l i s q u i t e s p h e r i c a l and the average r e s i d u a l i s lower than that of e a r l y type s t a r s . T h i s i s c o n s i s t e n t with the w e l l known f a c t t h at the importance of random motions and the e c c e n t r i c i t y of the v e l o c i t y e l l i p s o i d both i n c r e a s e with the age of the s t u d i e d o b j e c t s . 1 78 APPENDIX D e r i v a t i o n of the v e l o c i t y f i e l d to the second order The i n c l u s i o n of second order terms i n the T a y l o r expansion of equation (2-2) g i v e s : v-f « v f 0 + (x-x 0) 3_ 9x + ( y - y 0 ) J L » 3y + ( z - z 0 ) 9_ 9z v - f + j_ 2 x 2 9 2 dx7 + y 2 a 2 9p" + z 2 9 2 IJZ"2" (A-1 ) + 2xy 9: 9x9y + 2xz 9 2 9x 9z + 2yz 9j 9y 9z v - f Since the f u l l development of the z e r o t h and f i r s t order terms was c a r r i e d i n d e t a i l s i n Chapter 2, we w i l l focus our a t t e n t i o n on the second order terms that we c a l l V f 1 2 ' . These terms have three components i n the d i r e c t i o n s of the g a l a c t i c axes x,y and z and these components take the form: V f n ( 2 ) = x 2 9 2Vfn + y 3 2Vfn 9y ; + Z ' 9 2Vfn + 2xy 9 2Vfn , 9x9y + 2xz 9 2Vfn 9x9z + 2yz 9 2Vfn , 9y 9z (A-2) where n may represent x,y or z. Using the g a l a c t i c s p h e r i c a l c o o r d i n a t e s as: 1 79 x = d c o s l cosb y = d s i n l cosb z = d s i n b (A -3 ) and the convention: u = Vfx v = Vfy w = V f z and u 1 2 = d2u 9x9y v 2 3 = 9 2u 9y 9z etc we can w r i t e : V f ( 2 ) = T (d 2/2) [ c o s 2 l c o s 2 b u n s i n 2 l c o s 2 b u 2 2 + s i n 2 b u 3 + sin21 c o s 2 b u 1 2 + c o s l sin2b u 1 3 + s i n l sin2b u 2 3 + 3 (d 2/2) [ c o s 2 l c o s 2 b v n s i n 2 l c o s 2 b v 2 2 + s i n 2 b v 3 + sin21 c o s 2 b v 1 2 + c o s l sin2b v 1 3 + s i n l sin2b v 2 3 + ¥ (d 2/2) [ c o s 2 l c o s 2 b w,, s i n 2 l c o s 2 b w 2 2 + s i n 2 b w3 + sin21 c o s 2 b w 1 2 + c o s l sin2b w 1 3 + s i n l sin2b w 2 3 (A-4) The l i n e - o f - s i g h t p r o j e c t i o n of V f { 2 ' i s given by: V f l ( 2 ) = V f ( 2 > ( c o s l cosb T + s i n l cosb J + s i n b Tc ) ( A - 5 ) 180 and takes the form: V f l ( 2 > = (d 2/2) [ c o s 3 l c o s 3 b u 1 2 + s i n 3 c o s 3 v 2 2 + s i n 3 b w 3 3 + c o s l s i n 2 l c o s 3 b ( u 2 2 + 2 v 1 2 ) + c o s l cosb s i n 2 b ( u 3 3 + 2w,3) + s i n l c o s 2 l c o s 3 b ( v n + 2 u 1 2 ) (A-6) + s i n l cosb s i n 2 b ( v 3 3 + 2w 2 3) + c o s 2 l s i n b c o s 2 b ( w n + 2 u 1 3 ) + s i n 2 l s i n b c o s 2 b (w 2 2 + 2 v 2 3 ) + s i n l c o s l s i n b c o s 2 b ( 2 u 2 3 + 2 v 1 3 + 2w 1 2) ] When these terms are added to the z e r o t h and to the f i r s t order terms of equation (2-11), they g i v e : V f l - K(1) + K(2) c o s l cosb + K(3) s i n l cosb + K(4) sin b + K(5) d c o s 2 l c o s 2 b + K(6) d s i n 2 b + K(7) d s i n 2 l c o s 2 b + K(8) d sin21 c o s 2 b + K(9) d c o s l sin2b + K(10) d s i n l sin2b (A-7) + K(11) d 2 c o s 3 l c o s 3 b + K(12) d 2 s i n 3 l c o s 3 b + K(13) d 2 s i n 3 b + K(14) d 2 c o s l s i n 2 l c o s 3 b + K(15) d 2 c o s l cosb s i n 2 b + K(16) d 2 s i n l c o s 2 l c o s 3 b + K(17) d 2 s i n l cosb s i n 2 b + K(18) d 2 c o s 2 l s i n b c o s 2 b + K(19) d 2 s i n 2 l s i n b c o s 2 b + K(20) d 2 s i n l c o s l s i n b c o s 2 b 181 with: K (1) = Vk K (11) = 1 / 2 u,, K (2 ) = -u, K(12) = 1 / 2 v 2 2 K(3) = -v, K(13) = 1 / 2 w 3 3 K(4) = -we K(14) = 1 / 2 (u,, + 2v, 2) K(5) = u, K(15) = V2 ( u 3 3 + 2w,3) (A-8) K(6) = w3 K (16) = V2 (v,, + 2u, 2) K(7) = v 2 K(17) = 1 / 2 ( v 3 3 + 2w 2 3) K(8) = V2 ( u 2 + v,) K(18) = 1 / 2 (w 1 t + 2 u 1 3 ) K(9) = 1 / 2 ( u 3 + w,) K(19) = 1 / 2 (w 2 2 + 2 v 2 3 ) K(10) = V2 ( v 3 + w 2) K(20) = u 2 3 + v 1 3 + w 1 2 182 REFERENCES A l l e n C.W., A s t r o p h y s i c a l Q u a n t i t i e s , 3rd ed., Athlone Press, London, 1973. 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