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The time-dependence of the local stellar velocity distribution Byl, John 1972

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IS\00 THE TIME-DEPENDENCE OF THE LOCAL STELLAR VELOCITY DISTRIBUTION by JOHN BYL B . S c , U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1969 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department o f Astronomy and Geophysics We accept t h i s t h e s i s as conforming t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA November, 1972 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department The University of British Columbia Vancouver 8, Canada Date )f7I ( i ) ABSTRACT I t i s a well-known o b s e r v a t i o n a l f a c t t h a t the v e l o c i t y d i s t r i b u t i o n o f a group of s t a r s i s r e l a t e d to the s p e c t r a l c l a s s e s o f the s t a r s . In p a r t i c u l a r , a l a r g e v e r t e x d e v i a t i o n e x i s t s f o r s t a r s o f e a r l y s p e c t r a l t y p e s , which d i s a p p e a r s f o r s t a r s o f l a t e r s p e c t r a l t y p e s . A l s o , the v e l o c i t y d i s p e r s i o n s tend to i n c r e a s e w i t h l a t e r s p e c t r a l t y p e s . An e x a m i n a t i o n of the nearby s t a r s y i e l d e d r e l a -t i o n s between the v e l o c i t y d i s t r i b u t i o n and the s p e c t r a l t y p e . S i n c e i t was p o s s i b l e t o e s t i m a t e ages f o r a number of s t a r s , the dependence of the v e l o c i t y d i s p e r s i o n s on age c o u l d a l s o be d e t e r m i n e d . I t was proposed t h a t the observed e f f e c t s are due to s p i r a l d e n s i t y waves. I f account was taken a l s o o f the f a c t t h a t the o r b i t s o f the younger s t a r s are not y e t w e l l - m i x e d , then i t was found t h a t the p r e d i c t e d v a l u e s o f the v e r t e x d e v i a t i o n agree q u i t e w e l l w i t h the o b s e r -v a t i o n a l v a l u e s . The i n c r e a s e i n the v e l o c i t y d i s p e r s i o n can be e x p l a i n e d i f the s p i r a l p a t t e r n has d i s s o l v e d and reformed a number of t i m e s . From a comparison of the t h e o r e t i c a l age-dependence of the v e l o c i t y d i s p e r s i o n and the o b s e r v a t i o n a l curve i t was p o s s i b l e to e s t i m a t e the number of s p i r a l p a t t e r n s which have e x i s t e d , t h e i r a m p l i -tudes and t h e i r decay r a t e s . ( i i ) TABLE CF CONTENTS ABSTRACT ( i ) LIST OP TABLES ( i v ) LIST OF FIGURES (v) ACKNOWLEDGEMENTS ( v i ) CHAPTER 1 INTRODUCTION 1.1 The Problem 1 1.2 Review o f P r e v i o u s Work 1 1.3 Scope o f the T h e s i s 8 CHAPTER 2 AN ANALYSIS OF THE OBSERVATIONAL DATA 2.1 R e l a t i o n o f V e l o c i t y D i s t r i b u t i o n to S p e c t r a l Type 12 2.2 The D e t e r m i n a t i o n o f S t e l l a r Ages 17 2.3 R e l a t i o n o f V e l o c i t y D i s t r i b u t i o n t o Age 23 CHAPTER 3 THE RESPONSE OF STARS TO A SPIRAL GRAVITATIONAL FIELD 3.1 Changes i n the D i s t r i b u t i o n F u n c t i o n 27 3.2 The S o l u t i o n o f the Amplitude E q u a t i o n 40 3.3 The Elements o f the V e l o c i t y D i s t r i b u t i o n 43 3.4 The S o l u t i o n o f the G e n e r a l I n t e g r a l 48 3.5 R e s u l t s 53 CHAPTER 4 C0LLISI0NLES5 RELAXATION IM THE GALAXY 4.1 I n t r o d u c t i o n 57 4.2 E p i c y c l e Theory 58 4.3 M a t h e m a t i c a l F o r m u l a t i o n 62 4.4 R e s u l t s 69 ( i i i ) CHAPTER 5 THE VERTEX DEVIATION 72 CHAPTER 6 THE TIME-DEPENDENCE OF THE VELOCITY DISPERSION 6.1 I n t r o d u c t i o n 77 6.2 M a t h e m a t i c a l F o r m u l a t i o n 79 6.3 R e s u l t s 81 CHAPTER 7 CONCLUSIONS 85 BIBLIOGRAPHY 121 ( i v ) LIST OF TABLES Table I V a r i a t i o n of v e l o c i t y d i s t r i b u t i o n w i t h s p e c t r a l t y p e . 87 I I A d j u s t e d v e l o c i t y d i s t r i b u t i o n s . 88 I I I V e l o c i t y d i s t r i b u t i o n s from medians. 89 IV The zero-age main sequence. 90 V Age e s t i m a t e s f o r the e v o l v e d s t a r s . 91 VI Age e s t i m a t e s f o r the upper main sequence s t a r s . 93 V I I Q u a l i t y c l a s s e s . 95 V I I I C l u s t e r a g e s . 95 IX Age e s t i m a t e s f o r the g i a n t s . 96 X U n c e r t a i n t i e s i n the ages and v e l o c i t y d i s p e r s i o n s . 97 XI V e l o c i t y d i s t r i b u t i o n s of the age g r o u p s . 98 X I I I Parameters o f the s p i r a l p a t t e r n . 99 XIV Comparison o f R e s u l t s . 99 XV V a r i a t i o n o f the mean r a d i a l v e l o c i t y . 100 XVI V a r i a t i o n o f the mean t a n g e n t i a l v e l o c i t y . 100 XVII V a r i a t i o n o f the r a d i a l v e l o c i t y d i s p e r s i o n . 1 0 1 X V I I I V a r i a t i o n o f the a x i s r a t i o . 101 XIX V a r i a t i o n of the v e r t e x d e v i a t i o n . 102 XX V a r i a t i o n o f the d e n s i t y . 10 2 XXI Comparison o f a n g u l a r v e l o c i t i e s and e p i c y c l i c f r e q u e n c i e s . 10 3 (v) LIST OF FIGURES F i g u r e 1 Model t r a c k s from S c h l e s i n g e r ' s f o r m u l a e . 104 2 I s o c h r o n e s . 105 3 C l u s t e r Composite. 106 4 Age-dependence of the v e l o c i t y d i s p e r s i o n s . 107 5 Time-dependence o f r a d i a l v e l o c i t y . 108 6 Time-dependence o f t a n g e n t i a l v e l o c i t y . 109 7 Time-dependence of r a d i a l v e l o c i t y d i s p e r s i o n . 110 8 Time-dependence o f a x i s r a t i o . I l l 9 Time-dependence of v e r t e x d e v i a t i o n . 112 10 Time-dependence o f r a d i a l v e l o c i t y . 113 11 Time-dependence o f t a n g e n t i a l v e l o c i t y . 114 12 Time-dependence o f r a d i a l v e l o c i t y d i s p e r s i o n . 115 13 Time-dependence o f a x i s r a t i o . I l 6 14 Time-dependence o f v e r t e x d e v i a t i o n . 117 15 R e l a t i o n between v e l o c i t y d i s p e r s i o n and v e r t e x d e v i a t i o n . 118 16 Age dependence o f the r a d i a l v e l o c i t y d i s p e r s i o n . 119 17 R e v i s e d Age Dependence o f the R a d i a l V e l o c i t y D i s p e r s i o n 12 0 ( v i ) ACKNOWLEDGEMENTS I t i s w i t h p l e a s u r e t h a t I acknowledge the i n v a l u a b l e c o u n s e l and encouragement o f D r . M.W. Ovenden d u r i n g the development o f t h i s r e s e a r c h . I am a l s o g r a t e f u l t o D r s . J.R. Auman, G.A.H. Walker and G.G. Fahlman f o r h e l p f u l d i s c u s s i o n s and c r i t i c i s m s o f v a r i o u s phases of the work. S i n c e r e thanks t o C h r i s T u n s t a l l f o r a v e r y f i n e j o b o f t y p i n g the t h e s i s . I would a l s o l i k e t o exp r e s s my g r a t i t u d e t o the N a t i o n a l Research Counsel which has p r o v i d e d f i n a n c i a l s u p p o r t i n the form o f an N.R.C. S c h o l a r s h i p . CHAPTER 1 INTRODUCTION 1.1 The Problem I f the Galaxy were i n a steady s t a t e , the v e l o c i t y d i s t r i b u t i o n would be expected t o be o f the S c h w a r z s c h i l d t y p e — a n e l l i p s o i d w i t h the v e r t e x p o i n t i n g towards the g a l a c t i c c e n t r e . I t would a l s o be expected t h a t the v e l -o c i t y d i s t r i b u t i o n would be independent o f the ages o f the s t a r s . In f a c t , i t has been observed t h a t a l t h o u g h the v e l o c i t y d i s t r i b u t i o n does approximate an e l l i p s e , the v e r t e x d e v i a t e s from the g a l a c t i c c e n t r e . The v e r t e x d e v i a t i o n and the v e l o c i t y d i s p e r s i o n s are observed t o be c o r r e l a t e d t o the s p e c t r a l c l a s s e s o f the v a r i o u s sub-p o p u l a t i o n s . T h i s appears t o i m p l y t h a t the v e l o c i t y d i s t r i b u t i o n may be r e l a t e d t o the ages o f the s t a r s . I t i s the o b j e c t o f t h i s t h e s i s to examine the nearby s t a r s and hence t o determine a r e l a t i o n between the v e l -o c i t y d i s t r i b u t i o n and the age. An attempt w i l l then be made to d e r i v e an e x p l a n a t i o n o f the observed r e l a t i o n from the t h e o r e t i c a l p o i n t o f v i e w . 1.2 Review o f P r e v i o u s Work Although the l a r g e v e l o c i t y d i s p e r s i o n o f v e r y o l d s t a r s i s almost c e r t a i n l y a r e l i c o f the r a p i d i n i t i a l (2) c o l l a p s e o f the e a r l y G a l a x y , i t appears u n l i k e l y t h a t the g a l a c t i c gas d i s k can have been s u f f i c i e n t l y d i f f e r e n t 9 a few t i m e s 10 y e a r s ago from i t s p r e s e n t s t a t e . I t thus seems p r o b a b l e t h a t the v e l o c i t y d i s p e r s i o n s i n c r e a s e d a f t e r the s t a r s were formed. Some mechanism s h o u l d then be c o n t a i n e d i n the g a l a c t i c system which i s c a p a b l e of p r o d u c i n g the observed i n c r e a s e i n the v e l o c i t y d i s p e r s i o n . I t has been shown by Chandrasekhar (1942) t h a t s t a r - s t a r e n c o u n t e r s are not s a t i s f a c t o r y s i n c e the r e s u l t i n g r e l a x a t i o n time i s much l a r g e r than the ac-c e p t e d age of the G a l a x y . S p i t z e r and S c h w a r z s c h i l d (1953) suggested t h a t random momentum exchanges between l a r g e c l o u d aggregates and i n d i v i d u a l s t a r s c o u l d cause the observed i n c r e a s e i n the v e l o c i t y d i s p e r s i o n . However, B a r b a n i s and W o l t j e r ( 1 9 6 7 ) , f i n d i t d o u b t f u l t h a t enough massive c l o u d complexes do e x i s t i n i n t e r -s t e l l a r s p a c e . Toomre (1964) suggested t h a t g r a v i t a -t i o n a l i n s t a b i l i t i e s d e v e l o p i n g i n the g a l a c t i c d i s k might be r e s p o n s i b l e . A c c o r d i n g t o J u l i a n ( 1 9 6 7 ) , i n s t a b i l i t i e s t h a t might have some r e l a t i o n t o s p i r a l s t r u c t u r e would be e f f e c t i v e . T h i s t r e a t m e n t i s , however, o n l y l o c a l and has not been a p p l i e d to the g l o b a l s p i r a l p a t t e r n . I t was proposed by Marochnik and Suchkov (1969) t h a t the r e l a x a t i o n o f the younger s t a r s i s due t o the (3) fact that the system of Population I stars rotates faster than the system of Population II stars. Interactions be-tween the two systems cause Landau in s t a b i l i t i e s which produce density waves. The growing density waves then cause the velocity dispersions to grow. To obtain an increase of the observed magnitude i t was required to assume that 90 percent of the stars belonged to Population II and that the system of Population II stars was non-rotating. Observational evidence does not support these assumptions. Barbanis and Woltjer (1967) have shown that for stars with small peculiar velocities the gravitational action of spiral perturbations which decay exponentially with time can account for the increase in the velocity dispersion. This was done using first-order epicyclic theory. More recently, Pomagaev (1971) approached the problem in a similar manner but expanded the theory to include second-order terms. This approach differed in that he considered the spiral pattern to be growing exponentially. Although i t is possible to estimate in this manner the mean velocities due to the spiral waves, the rate at which the velocity dispersion changes with time is d i f f i c u l t to determine even i f i t is assumed that the stars are well-mixed and that averages are taken over a l l angles. (4) Although the v e r t e x d e v i a t i o n has been a t t r i -buted t o the presence of s p i r a l arms, two b a s i c a l l y d i f -f e r e n t approaches have been u s e d . In the f i r s t , i t i s assumed t h a t the d e v i a t i o n i s a r e f l e c t i o n o f i n i t i a l c o n d i t i o n s . S t a r s are formed i n the s p i r a l arms and m i g r a t e , the i n f l u e n c e o f s m a l l - s c a l e d e n s i t y f l u c t u a t i o n s not d e l e t i n g t h e i r memory o f o r i g i n . A c c o r d i n g t o Woolley (1970) the o r i g i n o f G l i e s e ' s A s t a r s can be t r a c e d back t o a t h i n s t r i p r e s e m b l i n g a s p i r a l arm. Yuan (1971) a l s o s t u d i e d the e f f e c t o f the o r i g i n o f the s t a r s . He assumed s t a r s were formed i n the s p i r a l arms. Then he c a l c u l a t e d o r b i t s f o r a number o f h y p o t h e t i c a l s t a r s t o a r r i v e at a s t a t i s t i c a l p r e d i c t i o n o f the v e l o c i t y d i s -t r i b u t i o n a t the Sun. For A s t a r s t h i s compared f a i r l y w e l l w i t h o b s e r v a t i o n . However, Yuan a l s o assumed t h a t at f o r m a t i o n the s t a r s are i n a w e l l - m i x e d s t a t e . T h i s appears t o be u n l i k e l y . In the second a p p r o a c h , the d e v i a t i o n i s pos-t u l a t e d t o be caused by a s t a t i o n a r y d i s t r i b u t i o n i n the presence o f the s p i r a l g r a v i t a t i o n a l f i e l d . K a l n a j s (1971) found t h a t i f the s p i r a l f i e l d i s 5% o f the mean g r a v i t a t i o n a l f i e l d i t i s i n s u f f i c i e n t t o produce the observed d e v i a t i o n . I n d e p e n d e n t l y , Mayor (1970) showed t h a t under c e r t a i n c o n d i t i o n s such a f i e l d c o u l d cause the d i s t r i b u t i o n of a group of s t a r s w i t h a s m a l l v e l o c i t y d i s p e r s i o n t o be d i s t o r t e d so t h a t i t becomes s i m i l a r t o (5) the observed d i s t r i b u t i o n . However, to do t h i s two assumptions were m a d e - - f i r s t l y , t h a t the l o w - d i s p e r s i o n had reached a w e l l - m i x e d s t a t e and s e c o n d l y , t h a t the Sun i s l o c a t e d on the i n n e r edge o f a major s p i r a l arm. Whether or not the l o c a l s p i r a l arm belongs t o the main two-arm p a t t e r n i s a c o n t r o v e r s i a l t o p i c . However, most o b s e r v a t i o n s f a v o r the l o c a l O r i o n arm as an i n t e r - a r m spur (Simonson ( 1 9 7 0 ) ) . Even i f the l o c a l arm were a major arm, the parameters o f t h s d e n s i t y wave would have to be chosen c o n s i s t e n t l y . A c c o r d i n g to Yuan, a s i m p l e r e -o r i e n t a t i o n o f L i n ' s p a t t e r n i s not p e r m i s s a b l e . R e c e n t l y , a number of a u t h o r s have c o n s i d e r e d the Galaxy as c o n s i s t i n g of a continuum of s t a r s . The e v o l u t i o n of the system i s then s t u d i e d by u s i n g hydro-dynamic e q u a t i o n s . These are d e r i v e d from the Boltzmann e q u a t i o n as e q u a t i o n s f o r the f i r s t few moments o f the d i s t r i b u t i o n f u n c t i o n . S i n c e the e q u a t i o n f o r the n^^ moment i n c l u d e s a term dependent on the s p a t i a l d e r i v a t i v e o f the (n+1)**1 moment, the moment e q u a t i o n s form an open c o u p l e d system. In a s m a l l number o f s p e c i a l cases the c h a i n o f e q u a t i o n s can be broken and a c l o s e d system o f e q u a t i o n s i s o b t a i n e d . These cases w i l l be b r i e f l y examined below. ( l ) I f c o l l i s i o n s dominate so t h a t one has l o c a l "thermodynamic" e q u i l i b r i u m , i t i s p o s s i b l e to a p p l y the Chapman-Enskog ex p a n s i o n t e c h n i q u e (Chapman and Cowling (6) (1953)). Since in the Galaxy collisions have a negli-gible effect (Chandrasekhar), this is not applicable. (2) Marochnik (1964) found that i t is possible to obtain a closed set of equations for motion perpendicular to the rotation axis i f one is interested in the evolution of the Galaxy over a time-scale very much longer than the period of rotation. He used the method developed by Chew, Goldberger, and Low (1956) who examined the analogous case of a collisionless plasma having a strong magnetic f i e l d . Since this method is not valid for timescales comparable to the rotation period, effects such as orbital mixing cannot be examined. Hence this method limits the description of the evolution of the system to a f i r s t -order approximation. (3) The last case is when the term containing the next higher moment can be neglected. This is true i f the spatial variation of the higher moment can be neglected (Hunter (1969)) or i f the characteristic velocity of the system is much greater than the velocity dispersion and the next higher moments are small compared to the lower-order moments. The latter is known as the low temperature approximation and is described in more detail by Bernstein and Trehan (1961). This method is valid only i f the stellar velocity distribution is at equilibrium. If this is not so, orbital mixing occurs and the spatial deri-vatives of the moments become very large. It should be (7) noted that "equilibrium" is used here to denote the state where orbital mixing has been completed and a l l memory of the origin of the stars has vanished. In spite of their shortcomings, the hydrodynamic equations have been used with some success. Kato (1968) was able to calculate the form of the velocity ellipsoid i f the stars had a systematic mean motion superposed on the circular velocity. This was limited to the steady-state case. The time variation of the velocity ellipsoid due to star-cloud encounters and due to galactic rotation was examined by Kitamura (1968). Near the 5un, the effect of star-cloud encounters was found to be negligible. Kitamura also found that i f i n i t i a l l y the vertex deviation or axis ratio were different from the equilibrium values, the velocity ellipsoid would osci l l a t e , without damping, about the equilibrium values. The period of oscillation was —10 years. However, Kitamura assumed that the stars had no systematic mean motions in addition to the circular velocity and that third and higher order moments could be neglected. These assumptions are valid only i f the velocity distribution is at a l l times at equilibrium, as is true locally when collisions are frequent. If the (8) i n i t i a l velocity distribution is not at equilibrium or i f systematic motions do exist, orbital mixing w i l l occur and equilibrium wi l l be approached. The oscillations about equilibrium wi l l hence be damped. The damping i s due to the fact that the spatial derivatives of the third and higher moments gradually become dominant because the stars have orbits with differing periods and thus get out of step with each other. Since Kitamura assumes that the i n i t i a l velocity distribution is not at equilibrium, his assumptions concerning mean motions and third and higher moments are not j u s t i f i e d . His hydrodynamic approach i s therefore invalid. In conclusion, i t was found that the hydrodynamic equations are applicable only i f the velocity distribution is time-independent or i f we are interested in the evolution of the system over very large time-scales. Due primarily to the phenomenon of orbital mixing, the method is inadequate for describing the time variation of the velocity ellipsoid caused by small perturbations from equilibrium. 1.3 Scope of the Thesis Relationships between the velocity distributions and the ages of groups of stars wi l l be determined by examining the nearby stars in Gliese's (1969) catalogue. These stars were selected because their sample is (9) r e l a t i v e l y complete and because t h e i r d a ta are more a c c u r a t e than those f o r more d i s t a n t s t a r s . The s t a r s w i l l be d i v i d e d i n t o groups a c c o r d i n g t o s p e c t r a l t y p e . For each group the v e l o c i t y d i s t r i b u t i o n w i l l be then c a l c u l a t e d . T h i s w i l l f u r n i s h a r e l a t i o n between the v e l o c i t y d i s p e r s i o n and the v e r t e x d e v i a t i o n . S i n c e the s p e c t r a l type may be c o n s i d e r e d to be a rough i n d i c a t o r of age, t h i s w i l l p r o v i d e a l s o a z e r o t h - o r d e r e s t i m a t e of the age-dependence of the v e l o c i t y d i s t r i b u t i o n . Using Iben's (1966) s t e l l a r models i t i s p o s s i b l e to e s t i m a t e ( f o r a number of s t a r s ) t h e i r ages from t h e i r p o s i t i o n s on the c o l o r - m a g n i t u d e d i a g r a m . Hence, by d i v i d i n g the s t a r s i n t o v a r i o u s age g r o u p s , the age-dependence of the v e l o c i t y d i s t r i b u t i o n can be f o u n d . T h i s w i l l a l l be d e a l t w i t h i n Chapter 2 . I t i s emphasized t h a t the age e s t i m a t e s w i l l have u n a v o i d a b l e u n c e r t a i n t i e s . However, the importance to t h i s t h e s i s of the d e r i v e d f u n c t i o n a l v a r i a t i o n o f the v e l o c i t y d i s p e r s i o n w i t h age l i e s not i n the p r e c i s e d e t a i l s o f the f u n c t i o n but i n the d e m o n s t r a t i o n t h a t the t h e o r e t i c a l mechanism through which i t i s e x p l a i n e d i s i m p o r t a n t i n the dynamical e v o l u t i o n o f the G a l a x y . I t i s p o s t u l a t e d t h a t the observed age-dependence of the v e l o c i t y d i s t r i b u t i o n i s due p r i m a r i l y to the (10) presence of s p i r a l d e n s i t y waves. S i n c e the d e n s i t y waves proposed by L i n , Yuan and Shu (1969) appear to be su p p o r t e d o b s e r v a t i o n a l l y , the waves are assumed to be o f t h i s f o r m . However, Toomre (1969) has found t h a t u n l e s s the waves are r e -p l e n i s h e d they w i l l be damped due to i n t e r a c t i o n s . w i t h the s t a r s . H a r r i s o n (1970) has suggested t h a t s p i r a l waves are generated by the i n t e r a c t i o n o f r a d i a l f l o w from the g a l a c t i c c e n t r e w i t h the g a l a c t i c r o t a t i o n f l o w . The r e s u l t i n g s p i r a l waves t r a i l and are propagated o u t w a r d . These waves grow or decay e x p o n e n t i a l l y . In Chapter 3 the response of a s u b - p o p u l a t i o n of s t a r s to such d e n s i t y waves w i l l be d e t e r m i n e d . T h i s w i l l be done f o r the g e n e r a l case of a v e l o c i t y e l l i p s o i d w i t h a r b i t r a r y a x i s r a t i o and v e r t e x d e v i a t i o n . The d e n s i t y waves w i l l be assumed t o vary e x p o n e n t i a l l y w i t h t i m e . The r e s u l t s w i l l then be examined to t e s t the s e n s i t i v i t y o f the response to the v a l u e s o f the a x i s r a t i o and v e r t e x d e v i a t i o n . I f a s u b - p o p u l a t i o n o f s t a r s i n i t i a l l y has a v e l o c i t y d i s t r i b u t i o n which i s not at e q u i l i b r i u m , then the s t e l l a r o r b i t s g r a d u a l l y become "mixed" due to d i f f e r e n c e s i n o r b i t a l p e r i o d s . As m i x i n g reaches c o m p l e t i o n , the s t a r s w i l l be s a i d t o approach " e q u i l i b r i u m " . (11) In Chapter 4 such c o l l i s i o n l e s s relaxation i s studied in d e t a i l in order to determine how and how quickly equi-librium i s reached. This w i l l make i t possible to c a l -culate the axis r a t i o s and vertex deviations (and hence the response of the stars to density waves) for sub-populations of stars which have not yet reached the equilibrium state. Using the mixing theory and being able to deter-mine the response to density waves, i t i s now possible to calculate the perturbed v e l o c i t y d i s t r i b u t i o n of a sub-population whose unperturbed v e l o c i t y d i s t r i b u t i o n has not reached equilibrium. This w i l l be done in Chapter 5. A t h e o r e t i c a l r e l a t i o n between the vertex deviation and v e l o c i t y dispersion w i l l be obtained and w i l l be compared to the empirical r e s u l t s . In Chapter 6 the e f f e c t of density waves on the v e l o c i t y dispersion w i l l be examined. The waves w i l l be assumed to have decayed exponentially. Using the methods of Chapter 3, the mean velocity of a sub-population due to the wave can be calculated. As the amplitude of the wave diminishes the mean vel o c i t y also diminishes as mixing occurs. The resulting increase in the velocity dispersion can then be e a s i l y obtained. Relations between the v e l o c i t y dispersion and time w i l l then be determined for density waves of various i n i t i a l amplitudes and decay rates. These w i l l be compared with the empirical curve. CHAPTER 2 AM ANALYSIS OF THE OBSERVATIONAL DATA 2.1 R e l a t i o n o f V e l o c i t y D i s t r i b u t i o n to S p e c t r a l Type In o r d e r t h a t the samples o f s t a r s c o n s i d e r e d be r e l a t i v e l y c o m p l e t e , o n l y nearby s t a r s were s e l e c t e d . The s t a r s were a l l taken from G l i e s e ' s (1969) c a t a l o g u e of nearby s t a r s . Only those w i t h i n 22 p a r s e c s o f the Sun were used. T h i s l e f t 1105 s t a r s h a v i n g known v e l o c i t y components, a p p r o x i m a t e l y 900 o f these h a v i n g a known ( B - V ) . The mean v e l o c i t i e s and the v e l o c i t y d i s p e r s i o n s were determined from the f o l l o w i n g f o r m u l a e , r e s p e c t i v e l y V = i £ V. (1) n i = l 1 (fl - n=I X , ( V ^ «2> where n i s the number o f s t a r s i n the sample and V^ i s the v e l o c i t y of the i ^ s t a r . T h i s method of c a l c u l a t i n g v e l o c i t y d i s p e r s i o n s i s u n f o r t u n a t e l y h i g h l y s e n s i t i v e to s t a r s w i t h l a r g e r v e l o c i t i e s , g i v i n g them an unduly (13) h i g h w e i g h t . Woolley (1958) found t h a t f o r a Gaussian v e l o c i t y d i s t r i b u t i o n the v e l o c i t y d i s p e r s i o n i s g i v e n by (3 = 1.483 x median o f (V\-median v e l o c i t y ) (3) He p r e f e r r e d the median to o t h e r Gaussian c h a r a c t e r i s t i c s as i t a v o i d s g i v i n g h i g h weight to e x c e p t i o n a l l y h i g h v e l o c i t i e s . However, l a r g e numbers of s t a r s are needed i n t h i s a p p r o a c h . I f the number i s s m a l l and t h e r e are few s t a r s w i t h v e l o c i t i e s near the average v e l o c i t y , then the median may d i f f e r g r e a t l y from the a v e r a g e , c a u s i n g l a r g e e r r o r s inCf^. For groups o f s t a r s h a v i n g l a r g e v e l o c i t i e s the d i s t r i b u t i o n i n v i s not Gaussian but a s y m m e t r i c a l - - l a r g e n e g a t i v e v a l u e s o f v o c c u r i n g more f r e q u e n t l y than e q u a l l y l a r g e p o s i t i v e v a l u e s . T h i s i s due to the f a c t t h a t s t a r s w i t h l a r g e p o s i t i v e v ( g r e a t e r than 65 km/sec) escape from the G a l a x y . Hence the s i g -n i f i c a n c e o f O ^ i s weakened f o r the h i g h v e l o c i t y s t a r s . For groups c o n t a i n i n g o n l y a s m a l l f r a c t i o n o f h i g h v e l o c i t y s t a r s , the d i s t r i b u t i o n i s w e l l approximated by a Gaussian d i s t r i b u t i o n . The d i f f i c u l t y o f s t a r s w i t h l a r g e v e l o c i t i e s was a v o i d e d i n the f o l l o w i n g manner. Using e q u a t i o n (2) the v e l o c i t y d i s p e r s i o n s were c a l c u l a t e d f o r each c l a s s o f s t a r s . S t a r s h a v i n g v e l o c i t i e s d i f f e r i n g from the mean by more than t w i c e the v e l o c i t y d i s p e r s i o n were then e l i m i n a t e d . (14) Having e l i m i n a t e d the h i g h v e l o c i t y s t a r s , the v e r t e x d e v i a t i o n was found u s i n g the r e l a t i o n where u i s the v e l o c i t y component away from the g a l a c t i c c e n t r e and v i s i n the d i r e c t i o n o f r o t a t i o n . The s t a r s were f i r s t d i v i d e d i n t o s p e c t r a l c l a s s e s . In each c l a s s the g i a n t s , sub-dwarfs and w h i t e dwarfs were e l i m i n a t e d . The v e l o c i t y d i s p e r s i o n s were then c a l c u l a t e d f o r e a c h _ c l a s s and the h i g h v e l o c i t y s t a r s e l i m i n a t e d . The mean v e l o c i t i e s , the v e l o c i t y d i s p e r s i o n s and the v e r t e x d e v i a t i o n were then c a l c u l a t e d f o r each g r o u p . The r e s u l t s are shown i n T a b l e s I and I I . p r o p e r t i e s o f s t a r s are c o r r e l a t e d w i t h the s p e c t r a l t y p e . The phenomenon o f asymmetric d r i f t i s demonstrated by the change i n v. I t i s , however, p r i m a r i l y a f u n c t i o n o f the v e l o c i t y d i s p e r s i o n s . The v e l o c i t y d i s p e r s i o n s e x h i b i t a steady growth through the s p e c t r a l t y p e s , r e a c h i n g t h e i r maximums i n the w h i t e dwarf and sub-dwarf c l a s s e s . The v e r t e x d e v i a t i o n , on the o t h e r hand, has l a r g e v a l u e s f o r e a r l y s p e c t r a l t ypes and then g r a d u a l l y d i m i n i s h e s . A l t h o u g h Delhaye(1965) has observed much l a r g e r v e r t e x d e v i a t i o n s f o r young s t a r s these v a l u e s are not r e p r o -duced here because t h e r e are v e r y few e a r l y type (0-B) ta n 2$L = (4) I ( u . - u )2 - E ( v . - v )2 From Table I I i t i s apparent t h a t the k i n e m a t i c (15) s t a r s w i t h i n 22 p a r s e c s o f the Sun. The median v e l o c i t i e s and the v e l o c i t y d i s p e r s i o n s o b t a i n e d by then u s i n g e q u a t i o n (3) are l i s t e d i n Table I I I . The agreement between T a b l e s I I and I I I i s q u i t e good, s u g g e s t i n g t h a t a G a ussian d i s t r i b u t i o n w e l l r e p r e s e n t s the v e l o c i t y d i s t r i b u t i o n o f the s t a r s . The d i f f e r e n c e s i n the v a r i o u s v e l o c i t y d i s p e r s i o n s of the w h i t e dwarfs and sub-dwarfs are caused by the s m a l l numbers o f s t a r s i n these samples. I t i s emphasized t h a t i n the d e t e r m i n a t i o n o f the v e l o c i t y d i s t r i b u t i o n s s t a r s w i t h l a r g e v e l o c i t i e s were o m i t t e d o n l y because the v e l o c i t y d i s p e r s i o n s and the v e r t e x d e v i a t i o n were h i g h l y s e n s i t i v e to such s t a r s . S i n c e the v a s t m a j o r i t y of the s t a r s were s t i l l l e f t , i t was s t i l l p o s s i b l e t o o b t a i n good e s t i m a t e s o f the v e l o c i t y d i s p e r s i o n s and the v e r t e x d e v i a t i o n f o r each g r o u p . Table I I conveys the s u g g e s t i o n o f e q u i -p a r t i t i o n of e n e r g y . By t h i s i s meant t h a t the v e l o c i t y d i s p e r s i o n i n c r e a s e s w i t h s p e c t r a l type s i m p l y because the masses of the e a r l y - t y p e s t a r s are g r e a t e r than those o f o f the s t a r s o f l a t e r s p e c t r a l t y p e . However, Woolley (1958) found t h a t when the w - v e l o c i t y d i s p e r s i o n s are (16) c l a s s i f i e d a c c o r d i n g t o s t e l l a r mass t h e r e i s no s i g n i -f i c a n t v a r i a t i o n . Von Hoerner (I960) has found t h a t f o r s t a r s of s p e c t r a l type l a t e r than G5 no c o r r e l a t i o n e x i s t s between s t e l l a r mass and v e l o c i t y d i s p e r s i o n i n the p l a n e . I t thus appears u n l i k e l y t h a t e q u i - p a r t i t i o n of energy e x i s t s h e r e . I t has been shown t h a t the v e l o c i t y d i s t r i b u t i o n of the nearby s t a r s i s r e l a t e d t o the s p e c t r a l t y p e . S i n c e the s p e c t r a l type may be c o n s i d e r e d t o be a rough i n d i c a t o r of age, t h i s i s i n f a c t a z e r o t h - o r d e r e s t i m a t e of the age dependence of the v e l o c i t y p a r a m e t e r s . A more a c c u r a t e e s t i m a t e w i l l be attempted i n the next s e c t i o n . (17) 2.2 The Determination of Stellar Ages To estimate ages of stars from their positions on the colour-magnitude diagram, i t was f i r s t necessary to determine a number of isochronic lines for the diagram. The age of a star may then be determined by i t s position relative to these lines. The lines were found by the following procedure. Iben's (1966) evolutionary stellar model tracks for 1.0, 1.25, 1.5, 2.25 and 3 solar masses were used. These assumed an i n i t i a l composition of 70.8 percent hydrogen and 27.2 percent helium. Since only a small number of isochronic points could be determined from the five tracks, i t was necessary to interpolate between the models. Through such interpolation Schlesinger (1969) has derived some convenient formulae. He finds the zero age main sequence (ZAMS) to be given by log L = -0.1514 + 5.293M - 1.31 (log M ) 1 * 7 (1) log Te = 3.7486 + 0.678 (log M)0'8 (2) where L is the luminosity, T is the effective temperature and M is the mass in solar masses. The hydrogen burning times were found to be log tx = 9.5650 - 3.8075 log M + 1.31 (log M)1 , 7 (3) log t2 = 9.4817 - 4.293 log M + 1.6312 (log M)1*7 (4) where t^ is the time needed for half of the i n i t i a l central hydrogen to be depleted ( i . e . from Iben point 1 to 2) and (18) t2 is the time needed from there until overall contraction starts (from Iben point 2 to 3). If M is between 1 and 3 solar masses, these relations give log L, log Tg and log t to within 0.03, 0.01 and 0.02 respectively. The ZAMS thus obtained and the empirical main sequence determined by Johnson (1963) differ significantly. Adapting the method of Sandage and Eggen (1969), the two were reconciled by displacing the theoretical ZAMS horizon-tally ( i.e. in log T ) to f i t the empirical line while keeping the luminosity constant. Justification for such a shifting of the model tracks from the calculated effective temperature comes from Demarque's (1968) demonstration that ages are insensitive to differences in model r a d i i , provided the luminosity is correctly given. The tables given by Schlesinger were used to convert luminosity and temperature to visual magnitude and (B-V) color. They consisted, in part, of Johnson's (1966) bolometric corrections and the (B-V, T ) relation due to Harris (1963). A value of 4.72 magnitudes was used for the solar bolometric magnitude. Using equations (1) to (4), model tracks were calculated for stars from 1.0 to 2.5 solar masses at intervals of 0.1 solar mass. These were then adjusted to f i t the empirical main sequence (see Table IV). The hydrogen burning times and the starting point on the ZAMS having been determined for each model, the tracks were then plotted on the color-magnitude diagram (Fig. 1) (19) by i n t e r p o l a t i n g from Iben's model t r a c k s . From the p l o t t e d t r a c k s i s o c h r o n e s were drawn by c o n n e c t i n g p o i n t s r e p r e s e n t i n g i d e n t i c a l a ges. The r e s u l t i n g i s o c h r o n e s are shown i n F i g 2. Ages f o r s t a r s which have e v o l v e d f a r from the main sequence c o u l d be e s t i m a t e d by n o t i n g t h e i r p o s i t i o n s on the c o l o r - m a g n i t u d e diagram w i t h r e s p e c t t o the i s o c h r o n e s . P r o b a b l e e r r o r s i n the ages were e s t i m a t e d by o b s e r v i n g the d e v i a t i o n s i n the p o s i t i o n s caused by the u n c e r t a i n t i e s i n the magnitudes and (B-V) c o l o r s . The u n c e r t a i n t y i n (B-V) was assumed t o be l e s s than 0.02 magnitudes. The u n c e r t a i n t y i n the a b s o l u t e magnitude was taken from G l i e s e and i n c l u d e d the p r o b a b l e e r r o r i n the p a r a l l a x . For each s t a r a maximum and a minimum age were e s t i m a t e d . The f o l l o w i n g assumptions were then made--(l) t h a t the r a t e of s t a r f o r m a t i o n had not changed a p p r e c i a b l y between the minimum and maximum age and (2) t h a t the s t a r s i n the s o l a r neighbourhood are w e l l - m i x e d , r e p r e s e n t i n g s t a r s of a l l a g e s . With these a s s u m p t i o n s , the s t a r c o u l d have any age between the minimum and maximum w i t h e q u a l p r o b a b i l i t y . S i n c e we are i n t e r e s t e d i n the mean age o f a group o f s t a r s , the age f o r each s t a r was e s t i m a t e d by t a k i n g the average o f the minimum and maximum age. E s s e n t i a l l y these assumptions i n t r o d u c e a c o r r e l a t i o n between age and s p e c t r a l t y p e . A c c o r d i n g t o von Hoerner (I960) the r a t e o f s t a r f o r m a t i o n has been a p p r o x i m a t e l y c o n s t a n t except f o r the (20) v e r y o l d s t a r s . The ages f o r the very o l d s t a r s must t h e r e f o r e be weighted a c c o r d i n g l y i f the minimum and maximum e s t i m a t e s d i f f e r g r e a t l y . The assumption t h a t the s t a r s are s p a t i a l l y mixed i s v a l i d o n l y f o r s t a r s o l d e r than 9 about 10 y e a r s . Hence, i n the f i n a l a n a l y s i s , o n l y the 9 s t a r s o l d e r than 10 y e a r s w i l l be u s e d . For s t a r s which have not y e t e v o l v e d f a r from the main sequence, ages were e s t i m a t e d by d e t e r m i n i n g the p o s i t i o n s of the s t a r s w i t h r e s p e c t t o the p o i n t s on the model t r a c k s c o r r e s p o n d i n g t o t ^ and S t a r s l y i n g more than one magnitude below the main sequence were d i s c a r d e d . The maximum ages o f s t a r s s t i l l b e i n g below o r on the ZAMS were taken t o be the c o r r e s p o n d i n g t ^ t i m e s . S i n c e the minimum ages, the c o n t r a c t i o n t i m e s , were s m a l l , the f i n a l age e s t i m a t e s were taken t o be h a l f o f the t ^ t i m e s . T h i s was j u s t i f i e d by our assumptions on the r a t e o f s t a r f o r m a t i o n . The age e s t i m a t e s f o r these s t a r s were c o n s i -d e r a b l y more u n c e r t a i n than those f o r the e v o l v e d s t a r s , p a r t i c u l a r l y f o r the main sequence s t a r s h a v i n g a (B-V) g r e a t e r than 0.50 magnitudes. For t h i s r e a s o n o n l y t h o s e main sequence s t a r s having a (B-V) o f l e s s than 0.5 magnitudes were c o n s i d e r e d . The e s t i m a t e d ages f o r the e v o l v e d s t a r s and f o r the upper main sequence s t a r s are l i s t e d i n T a b l e s V and VI r e s p e c t i v e l y . The f i r s t column c o n t a i n s the number o f the s t a r g i v e n i n G l i e s e ' s c a t a l o g u e w h i l e the f o u r t h (21) column l i s t s Q, the q u a l i t y o f the magnitude. The v a l u e s o f Q and the c o r r e s p o n d i n g u n c e r t a i n t y ranges are l i s t e d i n Table V I I . Age e s t i m a t e s f o r the g i a n t s were o b t a i n e d by u s i n g a composite of g a l a c t i c c l u s t e r sequences. A com-p a r i s o n o f the c l u s t e r sequences and the p r e v i o u s l y d e t e r -mined i s o c h r o n e s ( F i g . 2) p r o v i d e d e s t i m a t e s of the c l u s t e r a g es. The ages o f the g i a n t s c o u l d then be c a l c u l a t e d by n o t i n g t h e i r p o s i t i o n s w i t h r e s p e c t t o the c l u s t e r sequences. The c l u s t e r sequences which were used to form the composite ( F i g . 3) were those drawn by Sandage and Eggen. In Table V I I I , age e s t i m a t e s are g i v e n f o r the c l u s t e r s . Our e s t i m a t e s f o r NGC 188 and M 67 agree q u i t e w e l l w i t h those o f Sandage and Eggen and I b e n . However, our v a l u e s are c o n s i s t e n t l y much l a r g e r than L i n d o f f ' s (1968) e s t i m a t e s . T h i s i s p a r t l y due to the f a c t t h a t L i n d o f f used the t a b l e s i n A l l e n (1963) t o c o n v e r t ( L , T ) to (M , B-V). These t a b l e s , o u t d a t e d , are i n f e r i o r t o those of S c h l e s i n g e r . More i m p o r t a n t , L i n d o f f had not a d j u s t e d h i s t h e o r e t i c a l ZAMS to f i t the e m p i r i c a l ZAMS. S i n c e they d i f f e r s i g n i -f i c a n t l y , t h i s would cause l a r g e d i f f e r e n c e s i n the r e s u l t i n g age e s t i m a t e s . The i s o c h r o n e s o f Sandage and Eggen y i e l d e d age e s t i m a t e s o f NGC 3680 and NGC 7789 which were s i m i l a r t o o u r s . A comparison of the r e s u l t s o f H a r r i s and S c h l e s i n g e r suggested t h a t the u n c e r t a i n t i e s i n c o n v e r t i n g (L,Tg) t o (M , B-V) were no g r e a t e r than (0^05, oToi). (22) Prom the composite i n P i g . 3 i t can be seen t h a t at c e r t a i n p l a c e s the sequences come v e r y near t o each o t h e r , at some p o i n t s eves c r o s s i n g . At those p o s i t i o n s i t i s e x t r e m e l y d i f f i c u l t t o a r r i v e at a c c u r a t e age e s t i m a t e s . S m a l l u n c e r t a i n t i e s i n the magnitudes o r c o l o r s cause v e r y l a r g e p r o b a b l e e r r o r i n the ages t h e n . Hence i t was not p o s s i b l e t o o b t a i n ages f o r a l l o f the g i a n t s . E s t i m a t e s f o r the ages of the g i a n t s are g i v e n i n Table I X . Due to low met a l c o n t e n t , the subdwarfs l i e about one magnitude below the normal ZAMS. I f the met a l c o n t e n t of the o r i g i n a l i n t e r s t e l l a r m a t e r i a l were n e g l i g i b l e and i f the t i m e s c a l e f o r homogeneous m i x i n g o f the i n t e r s t e l l a r medium i s s h o r t compared t o the age o f the G a l a x y , then the subdwarfs must belong t o the v e r y o l d e s t s t a r s . The g i a n t s and the upper main sequence s t a r s are a l l r e l a t i v e l y b r i g h t and thus t h e i r samples can be assumed to be almost c o m p l e t e . The s u b d w a r f s , however, are q u i t e f a i n t and s e l e c t i o n e f f e c t s can e a s i l y g i v e r i s e t o a b i a s e d sample. The most i m p o r t a n t s e l e c t i o n e f f e c t , a c c o r d i n g to Von Hoerner (I960) i s the magnitude e f f e c t . P a i n t s t a r s are d i f f i c u l t t o observe and are put on o b s e r v i n g l i s t s o n l y i f they have n o t i c e a b l y h i g h proper m o t i o n s . Thus, on the a v e r a g e , the f a i n t s t a r s are expected t o show h i g h e r v e l o c i t i e s . Opposing t h i s i s the q u a l i t y e f f e c t . (23) Stars whose parallaxes have been overestimated wi l l appear to l i e below the main sequence. However, i f the parallax has been overestimated the space velocity w i l l be reduced. Hence this effect tends to give stars lower velocities. The subdwarfs were selected by considering a l l stars with (B-V) greater than 1.2 magnitudes and lying at least one magnitude below the ZAMS. For very faint stars the main sequence of Gliese (1968) was used. Since Gliese (1956) has shown that emission stars do not exhibit a dependence on spectral type and that they have smaller velocities that is normal, stars with emission spectra were omitted. To minimize the quality effect, only stars of quality classes (Q) 1, 2 and 3 were used. Finally, in order to reduce the magnitude effect, only stars whose velocities differed from the mean velocities by no more than twice the standard deviation for the class were considered. This l e f t only 18 subdwarfs. From their relative positions with respect to the isochrones, the oldest evolved stars appeared to be g about 12 x 10 years old. Hence this was taken to be the age of the subdwarfs. 2.3 Relation of Velocity Dispersion to Age The stars whose ages had been estimated were then divided into a number of groups corresponding to (24) v a r i o u s age r a n g e s . The subdwarfs and whi t e dwarfs were c o n s i d e r e d s e p a r a t e l y . For each group the v e l o c i t y d i s t r i b u t i o n was determined by u s i n g e q u a t i o n s (1) and (2) of s e c t i o n 2.1 a g a i n . The mean age was a l s o c a l c u l a t e d f o r each group but s i n c e each i n d i v i d u a l age had an u n c e r t a i n t y due to i n a c c u r a c i e s i n the s t a r ' s magnitude and c o l o r , the mean age l i k e w i s e had a p r o b a b l e e r r o r . T h i s u n c e r t a i n t y , ^ , was found u s i n g the r e l a t i o n n -where n i s the number o f s t a r s and (f^ i s the u n c e r t a i n t y i n the age of the i * *1 s t a r . The p r o b a b l e e r r o r s i n the v e l o c i t y d i s p e r s i o n s were e s t i m a t e d by assuming the samples to be from Gaussian p a p u l a t i o n s and then c a l c u l a t i n g 70% c o n f i d e n c e i n t e r v a l s . Our samples were too s m a l l t o o b t a i n r e a s o n a b l e e s t i m a t e s of the v e r t e x d e v i a t i o n s . The u n c e r t a i n t i e s are summarized i n Table X. The components o f the v e l o c i t y d i s t r i b u t i o n f o r each group are l i s t e d i n Table X I . Al t h o u g h u and w appear t o vary i r r e g u l a r l y w i t h age, v i n c r e a s e s s t e a d i l y i n magnitude (asymmetric d r i f t ) . The v e l o c i t y d i s p e r s i o n s , i n a l l t h r e e d i r e c t i o n s , a l s o i n c r e a s e w i t h age. T h i s may be seen more c l e a r l y i n F i g . 4. E r r o r b a rs have added (25) to o n l y the r a d i a l v e l o c i t y d i s p e r s i o n s i n c e they w i l l be s i m i l a r to the e r r o r bars f o r the o t h e r d i s p e r s i o n s . I t appears as i f the c u r v e s c o u l d be d i v i d e d i n t o two g i n t e r v a l s . U n t i l an age of about 2.5 x 10 y e a r s t h e r e i s a r a t h e r r a p i d i n c r e a s e , b u t from t h e r e on the i n c r e a s e c o n t i n u e s more s l o w l y . Once s t a r s have become w e l l - m i x e d they approach an " e q u i l i b r i u m " v e l o c i t y d i s t r i b u t i o n . Hence f o r o l d e r s t a r s i t can be assumed t h a t , a f t e r c o r r e c t i n g f o r the i n f l u e n c e of the s p i r a l arms, the v e l o c i t y d i s t r i b u t i o n i s near e q u i l i b r i u m . I f the v e l o c i t y d i s t r i b u t i o n i s at e q u i l i b r i u m , the r a t i o o f the v e l o c i t y d i s p e r s i o n s , 2 2 O" u/0^ » w i l l e q u a l 0.4 at the Sun. From Table I I i t can be seen t h a t f o r (f^20 km/sec e q u i l i b r i u m i s a l r e a d y approached. (26) The t h e o r y t h a t w i l l be developed can o f c o u r s e be a p p l i e d to any f u n c t i o n of age and v e l o c i t y d i s p e r s i o n d e r i v e d from o b s e r v a t i o n s . However, the a c t u a l v a r i a t i o n of v e l o c i t y d i s p e r s i o n w i t h age d e r i v e d here w i l l be s i m p l y i n t e r p r e t e d by the t h e o r y . The s i g n i f i c a n c e o f the t h e o r y i s not dependent on minor r e - i n t e r p r e t a t i o n s of the d a t a . The age e s t i m a t e s a r e , a d m i t t e d l y , s u b j e c t t o a number of u n c e r t a i n t i e s due to the assumptions r e g a r d i n g s t a r f o r m a t i o n r a t e s , s t e l l a r c o m p o s i t i o n and the e v o l u t i o n a r y t r a c k s , among o t h e r s . T h i s i s t r u e p a r t i c u l a r l y f o r s t a r s near the main sequence where the assumptions about s t a r f o r m a t i o n r a t e s become i m p o r t a n t . Here we have r e l i e d h e a v i l y on von Hoerner's r e s u l t s which i n d i c a t e d an almost 9 c o n s t a n t r a t e o f s t a r f o r m a t i o n over the l a s t ~6. x 10 y e a r s . T h i s c l e a r l y i n j e c t s an age-dependence on the s p e c t r a l type f o r main sequence s t a r s . H o p e f u l l y , the u n c e r t a i n t i e s may be reduced i n the f u t u r e . I t i s f e l t , however, t h a t the age e s t i m a t e s cannot be g r e a t l y improved upon w i t h the p r e s e n t l y a v a i l a b l e d a t a . CHAPTER 3 THE RESPONSE OF STARS TO A 5PIRAL GRAVITATIONAL FIELD 3.1 Changes i n the D i s t r i b u t i o n F u n c t i o n The response of a s t e l l a r d i s k s u b - p o p u l a t i o n t o a s p i r a l g r a v i t a t i o n a l f i e l d w i l l now be examined. In the i n i t i a l s t a g e s o f development the procedure w i l l resemble * t h a t o f L i n , Yuan and Shu (1969) and Mayor ( 1 9 7 0 ) . We have borrowed f r e e l y from these s o u r c e s . However, whereas L i n and Mayor c o n s i d e r e d o n l y the case o f a w e l l - m i x e d ( " e q u i l i b r i u m " ) s u b - p o p u l a t i o n i n a time-independent s p i r a l f i e l d , we s h a l l examine the more g e n e r a l case where the s t a r s need not be w e l l - m i x e d and the s p i r a l f i e l d may change w i t h t i m e . L e t r r e p r e s e n t 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 and l e t 9 be the a n g u l a r d i s p l a c e m e n t measured c l o c k w i s e from the g a l a c t i c r a d i u s p a s s i n g through the Sun. The g a l a c t i c c o - o r d i n a t e s o f a s t a r are then g i v e n by ( r , 6 ) • For an i n f i n i t e s i m a l l y t h i n d i s k the c o l l i s i o n l e s s Boltzmann e q u a t i o n i n such a c o - o r d i n a t e system i s g i v e n by 2 where f i s the d i s t r i b u t i o n f u n c t i o n , U i s the g r a v i t a t i o n a l ( 2 8 ) f i e l d and (1*^,(3^) are the components o f the t o t a l s t e l l a r v e l o c i t y . I f Q. i s the a n g u l a r v e l o c i t y o f the c i r c u l a r motion d e f i n e d by b a l a n c i n g the c e n t r i f u g a l a c c e l e r a t i o n T£L w i t h the s y m m e t r i c a l g r a v i t a t i o n a l f i e l d and i f the p e c u l i a r v e l o c i t y components o f the c e n t r o i d are g i v e n by ( r ^ j C S ^ ) , then the v e l o c i t y components r e l a t i v e t o the c e n t r o i d v e l o c i t y are n = n, - n (2) 1 o © = ©1 - ®o - r'ft (3) S i n c e the c e n t r o i d v e l o c i t y components a r e , i n g e n e r a l , s m a l l compared t o (IT,©) they w i l l be n e g l e c t e d . T h i s reduces equations ( 2 ) and (3) to n= n (4) © = @x - rft (5) In o r d e r to t r a n s f o r m the Boltzmann e q u a t i o n to (17»®)i the f o l l o w i n g r e l a t i o n s are needed: (29) The l a s t e q u a t i o n becomes E q u a t i o n ( l ) may now be r e w r i t t e n as where k*, the e p i c y c l i c f r e q u e n c y , i s d e f i n e d by K2 = 4&2(1 + (11) Suppose t h a t U c o n s i s t s o f a s y m m e t r i c a l g r a v i t a t i o n a l f i e l d Uq upon which i s superimposed a s p i r a l g r a v i t a t i o n a l f i e l d o f the form U1 = A(r,t)exp[i(mwt-m9+ < l(r)] (12) where A d e f i n e s the a m p l i t u d e o f the p a t t e r n , n i s the number o f arms, $ r e p r e s e n t s the phase and w i s the r e l a t i v e a n g u l a r v e l o c i t y o f the p a t t e r n . The s p i r a l i s then d e f i n e d by l i n e s o f c o n s t a n t phase through the r e l a t i o n n(0-eo) = $ ( r ) - $ ( r ) (13) where ( r ,Q ) i s the i n i t i a l r e f e r e n c e p o s i t i o n . L i n o o d e f i n e s the phase term as o (30) where i i s the i n c l i n a t i o n of the p a t t e r n . S i n c e the r a d i a l d e r i v a t i v e of i s g i v e n by 2 - r j f l , , e q u a t i o n (10) becomes, on s u b s t i t u t i n g U Q + f o r 3 S « 3 r+ ( f t +r) cTe+ (3r+ 2" +r)32+ (55e""znr > s ® " 0 { 1 5 ) For c o n v e n i e n c e , the l e f t - h a n d s i d e of t h i s e q u a t i o n w i l l be w r i t t e n s y m b o l i c a l l y as L ( f ) . I f o n l y a symmetric f i e l d e x i s t s ( i . e . U , s 0 ) i t w i l l be w r i t t e n as L ( f ) . 1 o Let f be the d i s t r i b u t i o n f u n c t i o n which the o s t a r s would have had i f no s p i r a l f i e l d e x i s t e d . The a d d i t i o n o f a s p i r a l f i e l d w i l l r e s u l t i n a p e r t u r b a t i o n yf i n the d i s t r i b u t i o n f u n c t i o n . I t i s our aim t o determine an e x p r e s s i o n f o r ^ . A knowledge of $ makes i t p o s s i b l e t o then c a l c u l a t e the p e r t u r b a t i o n s i n the components o f the v e l o c i t y d i s t r i b u t i o n . The p e r t u r b e d d i s t r i b u t i o n f u n c t i o n f i s g i v e n by f = fQ( l +^) (16) Assume t h a t f i s o f the form o fQ = c (0,r,t)exp[-q (0,r,t,©Jl)] (17) On s u b s t i t u t i n g e q u a t i o n s (16) and (17) i n t o e q u a t i o n (15), (31) we then o b t a i n Lo( ^ ) = - Lx( ^ ) + ( l + ^ L ^ Q ) (18) where L, ( ) - (19) 1 d r a n r d © d® Now, i f i s s m a l l , e q u a t i o n (18) may be l i n e a r i z e d to become Lo( ^ ) ^ ^(Q) (20) I f the s p i r a l f i e l d components are of the form d e s c r i b e d by e q u a t i o n ( 1 2 ) , then i t i s n a t u r a l l y expected t h a t 'p has a s i m i l a r form; $ = <f> (r,J7,©)exp[ i (mwt-me + $ ( r ) ) ] (21) where <f> d e f i n e s the a m p l i t u d e ofty . L i n has shown t h a t s o l u t i o n s o f t h i s form are p o s s i b l e . Mayor, however, has found t h a t the l i n e a r a p p r o x i m a t i o n i s no l o n g e r v a l i d i f the s p i r a l f i e l d i s l a r g e or i f the v e l o c i t y d i s p e r s i o n s o f the s t a r s are s m a l l . Hence n o n - l i n e a r terms must be i n c l u d e d . From e q u a t i o n (18) i t i s e v i d e n t t h a t the f i r s t n o n - l i n e a r e f f e c t s a s s o c i a t e d w i t h a p e r t u r b a t i o n of type (12) w i l l appear as terms p r o p o r t i o n a l t o exp[2i(mwt-m9+$)] (22) (32) The non-linear effects will therefore be estimated by assuming and ^ have the forms U * E b,(r,t) exp ij(mwt-m9+$) (23) 1 j>l J L oo ^ « Z ak(r,tfn,®)Bxp[ik(mwt-n9-4)] (24) If these are substituted into equation (18) i t is found that Lo( ak)+ i kak( -mf+, w+^ I ) = Ll k( Q )+E [anL (Q)-Ll n(a )]6k j + n (25) where ^ . indicates summation over a l l values of n and j , n»J <5^ j is the Kronecker delta, and L..( ) = - (ij bj S +-1l ) ^n i (26) Equation (25) may be simplified somewhat by introducing a change of variables, from (n»©) to ( T , l ) . These are related as follows: © = V2Tsini (27) J 7 = V1Tcosi (28) where V, = 24 ~ (29) 1 V2 = r f l (30) (33) With t h i s t r a n s f o r m a t i o n the d e r i v a t i v e s , i n terms o f the new v a r i a b l e s , become cM ) _ bJb( ) bib( ) dn blTST~ dn~dl~ (31) d( ) _ bTb( ) blb( ) b® " b&b~r~b®bi (32) bl ) b( )bTb( )b$b( ) ~bir = " 5 1 T + 5 r ~ 3 T - + 5 r - S T (33) U s i n g r e l a t i o n s (27) and ( 2 8 ) , e q u a t i o n s ( 3 l ) - ( 3 3 ) r e w r i t t e n cM ) l f _ . _ ^ ( ) s i J d( ) 1 _ = ^ [ C o s i l y — T J may be (34) at ) b® T d l n ( v 2 + V | ) d ( , i rQ.no d ( ) , c o s i a( ) i _ ( _ 3 I N ^ - ^ R - + _ 7 _ J (35) t a n i d l n ( V 2 + V | ) , dV2 a( r e x ) (36) Assuming t h a t the r e s i d u a l v e l o c i t i e s TT and 0 are s m a l l compared to the c i r c u l a r v e l o c i t y and i g n o r i n g s e c o n d - o r d e r t e r m s , the o p e r a t o r o f e q u a t i o n (26) becomes i j m b j c o s i ^ at ) + lCi js i n U V2 L i j ( ) . and - c . .COSX+ i & i s i n i j TT , / x _ a>( ) . a( ) . a( ) (37) (38) (34) The system (25) may now be r e w r i t t e n as da. da. kda, r ?r\2 >.X , ^ + - ^ - - ^ + i k a k [ r T . w - 2 r 2+( ^ ) (dX )+c o s i J = \{1 tK (39) where = L.. (Q)+.I! I a.L, (Q)-L...(a )1 F). (40) k Ik J»n<- j I n a j n Juk , j + n I f a^ changes o n l y s l o w l y w i t h r then e q u a t i o n (39) may be f u r t h e r s i m p l i f i e d t o K "1^ - ^ + i k [ / 7 + c x T c o s i ' ] ak = K"1S|< (41) where and V= (42) °<= n r~5? ( 4 3 ) We now depa r t from L i n and Mayor and c o n t i n u e as f o l -ows. To s o l v e e q u a t i o n (41) i t i s n e c e s s a r y f i r s t t o s p e c i f y the v a r i a t i o n o f IK w i t h t i m e . Knowing the dependence of the b.'s (the amp l i t u d e o f U.) on t i m e , one c o u l d then 3 a p p l y a L a p l a c e t r a n s f o r m and reduce e q u a t i o n (41) t o a d i f f e r e n t i a l e q u a t i o n i n one v a r i a b l e , ^ . In c e r t a i n c a s e s i t i s p o s s i b l e t o use a more d i r e c t a p p r o a c h . For example, i f the p a r t i a l d i f f e r e n t i a l e q u a t i o n i s o f the form Ida da a + IM - H » s U j h ( t ) (44) (35) and i f the r i g h t hand s i d e can be r e w r i t t e n i n the form g ( i ) h ( t ) = g1( i ) h1( t + J ) (45) where g,h,h^ and g^ are a r b i t r a r y f u n c t i o n s , then i f the s o l u t i o n o f a - ^ = 9 l( i ) (46) is a {ft) = g,<£) (47) the s o l u t i o n e q u a t i o n (44) i s a ( i , t ) = g2( i ) h1( t + | ) (48) T h i s r e s u l t h o l d s f o r a l l f u n c t i o n s h ( t ) which are o f the form h(t+^) = h1( t ) h2( i ) + b3(i) (49) I f i t i s assumed t h a t the a m p l i t u d e s b^ va r y e i t h e r e x p o n e n t i a l l y or s i n u s o i d a l l y , t h i s may be r e p r e s e n t e d by b. = b ^ e x p ^ t ) (50) where j3 i s i n v e r s e l y p r o p o r t i o n a l t o the t i m e s c a l e o f s p i r a l p e r s i s t e n c e and i s r e a l f o r e x p o n e n t i a l changes and i m a g i n a r y f o r s i n u s o i d a l v a r i a t i o n s . I t i s e v i d e n t t h a t f o r such f u n c t i o n s c o n d i t i o n (49) h o l d s . S i n c e e q u a t i o n s (37) and (36) (40) show 5^ to be p r o p o r t i o n a l to b^, 5^ may be w r i t t e n as s , - g ( i ) exp (fit) (51) To s a t i s f y e q u a t i o n ( 4 5 ) , i s r e a r r a n g e d as Sk = g ( i ) e x p ( 3 ^ ) e Xp[/3(t+|)] (52) Hence, i n s t e a d o f s o l v i n g e q u a t i o n ( 4 1 ) , o n l y the f o l l o w i n g s i m p l i f i e d e q u a t i o n need be s o l v e d : +/ak = A <53> where <f> = -iK(Z/+c<Tcosi ) h k - -Skaxp [ -#t + | f> ] E q u a t i o n (53) has the s o l u t i o n A au = e x p [ - J ^ d i ] ( ck + J) hke x p [ / V d l ] d i ) (54) where ck i s an i n t e g r a t i o n c o n s t a n t . S i n c e ak w i l l be p e r i o d i c i n ak( i ) = ak(^+2rf) (55) From e q u a t i o n s (54) and (55) i t i s p o s s i b l e to determine ck, which was found t o be g i v e n by (37) (56) From the d e f i n i t i o n s of </> ,h and c^, equation (54) may now be rewritten as (57) where = exp [ik (2/M-sini! )] Y2 = BXP[27T(iki/^) If k=l we have the l i n e a r approximation. For the non-linear case only the f i r s t two terms are important. Hence only h^ and h., are needed. These were found to be as follows: hl = -L1JL(a)exp[-/3[t+|)] (58) h 2 = - [ L i 2( Q ) + alLl l( Q ) " Ll l( al)] exp[-/5(t+|)] (59) To solve equation (57) i t i s necessary f i r s t to determine the d i s t r i b u t i o n function f . For a two-dimensional. o e l l i p s o i d a l d i s t r i b u t i o n i t can e a s i l y be shown that the d i s t r i b u t i o n function may be written as f = c expf-x/T 2 -x_@2 -2x_n0) o 1 C J (60) (38) where c , x^, x ^ t and x^ are d e f i n e d as f o l l o w s c = — (61) 7T xl ^ x = -±Ji (63) 61 -xl ^ n x3 = — (64) Here i s the s t a r d e n s i t y . I f the t r a n s f o r m a t i o n d e f i n e d by e q u a t i o n s (27) and (28) i s now made e q u a t i o n (60) becomes f = c exp(-T2(x1V1cos2i+x 2V2sin 2 J?+2x 3V 1V 2cos<?sinj3)) (65) The new parameters p, q and ^  are now i n t r o d u c e d by the r e l a t i o n s p c o s ^ = 0.5(xJ LV2 - x2V2) (66) p s i n j * = x3VlV2 ( 6 7 ) q - O . S ^ V2 + x2V2) (68) Using t r i g o n o m e t r i c r e l a t i o n s , e q u a t i o n (65) may be r e w r i t t e n fQ = c exp(-T2(q+p c o a ( 2 i - ^ ) ) ) (69) (39) The e x p r e s s i o n i n the e x p o n e n t i a l c o r r e s p o n d s to the Q i n t r o d u c e d i n e q u a t i o n ( 1 7 ) . To s o l v e e q u a t i o n (57) i t i s n e c e s s a r y f i r s t t o determine h^ and h^, and hence L ^ j ( Q ) . S i n c e by i t s d e f i n i t i o n ( e q u a t i o n (37)) L^^ i s dependent on the terms ^ and these w i l l be d e r i v e d f i r s t . From e q u a t i o n (69) Q = - T2( q + p c o s ( 2 j M ) ) (70) D i f f e r e n t i a t i n g i t i s found t h a t dQ 2Q (71) = T2[ ( xxV2 - x2V2) s i n li - 2 x3V1V2c o s 2i] (72) I f these e x p r e s s i o n s are now s u b s t i t u t e d i n e q u a t i o n ( 3 7 ) , i t f o l l o w s t h a t L± j(Q) = B ^ T c o s i + Bi j. T s i n i (73) where £ db. 4 i j b . x3V db 4 i j b . x V Bt j = - 2 x3V2( i j b . f + -g l ) + (75) With these r e l a t i o n s i t i s now p o s s i b l e t o determine h^ and h2 and hence a^ and a2. We s h a l l now proceed t o determine a s o l u t i o n f o r the a m p l i t u d e e q u a t i o n , e q u a t i o n ( 5 7 ) . (40) 3.2 The S o l u t i o n o f the Amplitude E q u a t i o n The l i n e a r case w i l l f i r s t be c o n s i d e r e d . From e q u a t i o n (57) of s e c t i o n 3.1 i t i s c l e a r t h a t the a m p l i t u d e a^ i s g i v e n by the r e l a t i o n i h, . Y„ h i fj. n i l i x = Y lexp(A + ^ d i + ?7 ' ^ d i 1 1 = ^ o / *T /K (1) where Yj^ = e x p [ i ( ^ J ? + 2 T s i n | )J Y2 = e x p [ 2 n ( i ^ + ^ ) ] h l = (B^TcosJ? + B2 1T s i n i ) e x p f - / ^ t + | ) ] I t i s e v i d e n t t h a t a s i n g u l a r i t y e x i s t s when Y2 e q u a l s u n i t y . For a time-independent wave (i.e./?=0) t h i s i m p l i e s t h a t ym ™^2£ m tn (2) Hence the s i n g u l a r i t y o c c u r s when - n K + 2a ,,x w = — (3) m I f the s p i r a l p a t t e r n has two arms ( i . e . T n = 2 ) , the s i n g u l a r i t i e s a t n=-l o c c u r when w =n i ^ ( 4 ) (41) These c o r r e s p o n d to the L i n d b l a d r e s o n a n c e s . Because o f the v e r y l a r g e r e s u l t i n g a m p l i t u d e s , the l i n e a r a p p r o x i m a t i o n i s no l o n g e r v a l i d at the resonance p o i n t s . We s h a l l , t h e r e f o r e , l i m i t the development to s i t u a t i o n s f a r away from the r e s o n a n c e s . S i n c e the Sun i s , f o r t u n a t e l y , not near a resonance p o s i t i o n , such a t r e a t m e n t w i l l be adequate f o r s t a r s i n the l o c a l v i c i n i t y . The form o f h^ may be s i m p l i f i e d somewhat by u s i n g the r e l a t i o n s C O S I = « » p ( i l ) + B X p ( - i l ) ( 5 ) s i n l = . j a x p d l ) - e x p ( - i l ) ( 6 ) The e q u a t i o n f o r h^ then becomes h1 = T exp(-/& + A j B x p f i i ) + A2e x p ( - i i ) (7) where Bl l + i B2 1 m = ^ (8) A, . B" ' (,) To s o l v e the i n t e g r a l s i n e q u a t i o n (1) i t i s c o n v e n i e n t to use B e s s e l f u n c t i o n s . The f o l l o w i n g r e l a t i o n i s g i v e n by Watson: oo exp (-icXTsin J (o(T)exp(-in h do) n=-o«n (42) where J (0(7) i s the B e s s e l f u n c t i o n d e f i n e d by n = E ( - l >nt o T / 2 )n*2'n J («T) = Z^n v-x ; (11) rn m=U , / \ , n!(n+m) ! S u b s t i t u t i n g e q u a t i o n s (7) and (10) i n t o e q u a t i o n ( l ) , the f i r s t i n t e g r a l becomes I J o T AjjaxpUJh + A 2exp(-iJ?)] J (o<T) exp (12) where = il> + (13) T h i s e x p r e s s i o n can be e a s i l y i n t e g r a t e d and becomes T ^ J (aT) A . e x p f - f e . + i n - i ] j^)+A.exp(-fe.+in+i] 0 - A - - A . I (14) where Al A- = —± r (15) 3 + i n - l A2 A. = % r (16) 4 + i n + I X ' S i m i l a r l y , the second i n t e g r a l i n e q u a t i o n (1) may be i n t e g r a t e d to y i e l d oo T n t ^ oJn( a T )[A3+ A4 ] [ « P C- 2 / I e ^ - l ] (17) s i n c e exp(in27T) = 1 (18) f o r a l l v a l u e s o f n. (43) I f e x p r e s s i o n s (14) and (17) are now s u b s t i t u t e d i n t o e q u a t i o n ( 1 ) , a number o f terms d i s a p p e a r . The f o l l o w i n g s i m p l i f i e d r e l a t i o n i s l e f t : V A-.exp(i£-in£)+A e x p ( - i i ( l + n ) ) ax = T e x p ^ t + i t f T s i n i h ^ ^ ^ t t T ) — : (19) In the l i n e a r a p p r o x i m a t i o n , the d i s t r i b u t i o n f u n c t i o n i s g i v e n by f as f (1 + a.exp i(mwt - m0 +$)]) (20) S i n c e an e x p r e s s i o n f o r a^ has now been o b t a i n e d , i t w i l l be p o s s i b l e t o s o l v e e q u a t i o n (20) f o r f . 3.3 The Elements o f the V e l o c i t y D i s t r i b u t i o n A knowledge o f the change i n the d i s t r i b u t i o n f u n c t i o n due to a s p i r a l d e n s i t y wave makes i t p o s s i b l e t o determine the c o r r e s p o n d i n g changes i n the v e l o c i t y d i s t r i b u t i o n . The p e r t u r b a t i o n s i n the s t a r d e n s i t y , the mean v e l o c i t i e s and the v e l o c i t y d i s p e r s i o n s are o f prime i n t e r e s t . L e t PQ denote the s t a r d e n s i t y i n the absence of an asymmetric g r a v i t a t i o n a l f i e l d . With the a d d i t i o n o f a s p i r a l f i e l d the p e r t u r b e d d e n s i t y , / } , becomes P= H ( l ) (1) (44) where H i s an o p e r a t o r d e f i n e d by oo .00 H(x) = J J x f (TI,@)drld0 -00 -00 S i m i l a r l y , the mean v e l o c i t i e s and the v e l o c i t y d i s p e r s i o n s may be w r i t t e n as fi = IS! (2) P § • H-f O ) 4 - ^ - ® 2 <=> As was noted i n the p r e v i o u s s e c t i o n , the d i s t r i b u t i o n f u n c t i o n f i s g i v e n by f = fQ( l + ai g i) (7) where g. = exp i(mwt-m8+ h) (8) I f the form of f i s taken to be t h a t o f e q u a t i o n (69) i n o s e c t i o n 3.1, e q u a t i o n (7) becomes. . f = c g2( l + a1g1) (9) (45) where g2 = exp [-T (q + p cos (2l-4))] (10) However, i f t h i s form of f i s used then the v a r i a b l e s IT and 0 must be t r a n s f o r m e d to e x p r e s s i o n s i n T andiL. R e c a l l t h a t J l = VjT cos! (11) S = V2T s i n i (12) I f e q u a t i o n s (1) to (6) are to be i n t e g r a t e d over T and Jl r a t h e r than Tl a n d ® , then the f o l l o w i n g r e l a t i o n must be used; oo J j ^ f U T . Q W J B ® = ^ ? f ( T j ) (f^*>) - ( ^ ^ |d^dT (13) Using e q u a t i o n s (11) and ( 1 2 ) , the r i g h t - h a n d s i d e reduces t o <^2/T/f(T,f)TV1V2dTdJ? (14) S u b s t i t u t i n g e q u a t i o n (9) and ( 1 4 ) , e q u a t i o n (1) may be r e w r i t t e n as P' V 2 C / o 7T al 9 l S 2d^d T + Po ( 1 5 ) E x p r e s s i o n s f o r the mean v e l o c i t i e s and the v e l o c i t y d i s p e r s i o n s can be o b t a i n e d i n a s i m i l a r f a s h i o n . A l l the r e s u l t i n g e q u a t i o n s w i l l have terms c o n t a i n i n g c o m b i n a t i o n s (46) of cos 1 and sin Ji after substituting for II and ® . The integration of the equations can be simplified slightly i f the following identities are used again: c o s i = e x p ( i l ) + e x p ( - i f ) s i n i = - ie xP( i^ - e x p ( - i f ) ( 1 7 ) On s u b s t i t u t i n g e q u a t i o n (19) of s e c t i o n 3.2 f o r a^ i n the e q u a t i o n s , i t was found t h a t , i n g e n e r a l , the f o l l o w i n g i n t e g r a l must be e v a l u a t e d : I (A,y) =~727fg,TAZ J («T)A exp [-iH(n-y) + io(T sin#]dl z oj oj c n=-oon z 1 ' , ( where z,A and y are integers. In terms o f i n t e g r a l s of t h i s f o r m , the components o f the v e l o c i t y d i s t r i b u t i o n may be w r i t t e n as I3( l , l ) + 1.(1,-1) yO= c V1V2g1e x p ( 0 t ) • g—- (19) I3( 2 , 2 ) + 1,(2,-2) + I3( 2 , 0 ) + I4( 2 , 0 ) n- c V1V2g1e x p ( ^ t ) - ^ , 1,(2,2) - 1,(2,-2) - I3( 2 , 0 ) + 1,(2,0) 0 = - x c V1V2g1e x p yCJt) (21) (47) -j n „ I-(3f3)+I.(3,l)+I-(3f-l)+I4(3,-3)+2I-(3,l)+ 4 = c v J v ^ . x p f Q o - L , 4 L — 2 21.(3,-1) 4K/0 - II ' (22) I3( 3 , 3 ) + I4( 3 , l ) + I3( 3 , - l ) + I4( 3 , - 3 ) - 2 l3( 3 , l ) ^ ® " - c V ^ g ^ x p ^ t ) — 21.(3,-1) _ 2 — i - - Q 2 (23) 4K/3 I3( 3 , 3 ) ^ I4( 3 , l ) - I3( 3 , - l ) - I4( 3 , - 3 ) 0@TT- - " V ^ g ^ x p ^ t ) — -<§n ( 2 4 ) I t now remains to e v a l u a t e the g e n e r a l i n t e g r a l I-,(A,y). (48) 3•^  The S o l u t i o n of the G e n e r a l I n t e g r a l In o r d e r t o determine the elements of the v e l o c i d i s t r i b u t i o n i t i s ne c e s s a r y f i r s t to s o l v e the i n t e g r a l g i v e n by I2(A,y) =C^/27exp(-T2[q+ P c o s ( 2 ^ ) ] ) TA L J (O(T)A exp(ii(y-n)+i<XT s i n i ) d i d T n=-oon 2 (1) I t w i l l a g a i n be c o n v e n i e n t to use B e s s e l f u n c t i o n s . The f o l l o w i n g i d e n t i t i e s from Watson w i l l be used: oo exp[-T2p cos(2i-^)] = kZc >Ik( p T2) e x p [ i k ( 2 i - ^ + r r ) ] (2) oo exp iofl" s i n x J (cxT) e x p ( i m l ) (3) where Ik( z ) i s the B e s s e l f u n c t i o n o f an i m a g i n a r y argument, d e f i n e d by oo V2> = L nl (kin) l" <«> S u b s t i t u t i n g e q u a t i o n s (2) and (3) i n e q u a t i o n ( l ) and r e a r r a n g i n g t e r m s , i t i s found t h a t oo f oo co Iz(A,y) - J , T e x p ( - q T2)kEo <Ik( p T2)nIcJn( C < T ) Az oo •mS-ooi. m ]oTexp f1 ( k ( 2^+ T r ) + ( y ~n M +m 4)] dJ?dT (5) (49) C l e a r l y , the i n t e g r a l i n A w i l l v a n i s h u n l e s s y-n+2k+m = 0 (6) I f t h i s c o n d i t i o n i s s a t i s f i e d , then 2J e x p [ i k ( r r - ^ ) ] d l = 2n exp[ i k (n-<£)] (7) I n t r o d u c i n g t h i s i n t o e q u a t i o n ( 5 ) , the e q u a t i o n becomes j . oo oo I z ( A , y ) - ^ . x p ( - , T 2 > k E j k C p T * > n E J n ( « T ) A 2 OO mf- Jm( a T ) 2 n e x p [ i k ( 7 T - ^ ) ] d T (8) w i t h the c o n d i t i o n t h a t y-n+2k+m = 0 ( 9 ) To e v a l u a t e e q u a t i o n ( 8 ) , v a r i o u s p r o p e r t i e s and r e l a t i o n s of the B e s s e l f u n c t i o n s and h y p e r g e o m e t r i c f u n c t i o n s w i l l be u s e d . These can a l l be found i n Watson. The B e s s e l f u n c t i o n s Jn and may be combined by u s i n g the r e l a t i o n oo , t \ , r \ / / /-,xn+m+2s/ ,vS (n+m+2s) ! Jn{ x ) Jm( x ) - k 0( x / 2 ) s!(m+s)i(n + s)!(n-HTH-s)! ( 1 0 ) f o r n and m g r e a t e r than or e q u a l t o z e r o . When n or m i s n e g a t i v e , may be t r a n s f o r m e d t o the c o r r e s p o n d i n g (50) B e s s e l f u n c t i o n of p o s i t i v e n, J , by the i d e n t i t y -n J (x) = (-1)°J (x) (11) —n n E q u a t i o n (10) can now be g e n e r a l i z e d to h o l d f o r a l l v a l u e s of n and m i f i t i s r e w r i t t e n as V ^ V * ' - ko^  1121 where v = |n| + |m|+2s (13) E = ( - i )s2 "v( - j ~ )n( -1£r)m s! ( j n | + s ) ! (VM + s ) ! { v - a J ! U 4 ) I t w i l l be c o n v e n i e n t to r e p l a c e a l s o ^ ( x ) in e q u a t i o n ( 8 ) , by u s i n g the r e l a t i o n Ik( x ) = ( - i )kJk( i x ) (15) With these m o d i f i c a t i o n s , e q u a t i o n (8) now becomes CO Iz( A , y ) = l TAe x p ( - q T2)nEsE ( o ( T )vAz kLk< i p T2) ( - i )k 2nexp [ik(n-^)]dT (16) where oo co co (17) n,m,s n = - o o m=—co s=U The l a s t summation may be reduced w i t h the h e l p o f e q u a t i o n (11) by n o t i n g t h a t oo k Z - ^K( i p T2) ( - i )ke x p [ i k ( 7 T - ^ ) ] = k 4 1 ( i )k2 c o s kj> Jk( i p T2) + JQ( i p T2) (18) (51) If the following change of variables is introduced, x = T2 (19) then equation (16) is transformed to o o o o I2(A,y) = nn>5)SEAz0fV |xWexp(-qx) RS1ik2cos k ^ U p x ) + Jo(ipx)]dx (20) where w - (21) Watson has shown that o o // i i i /!• \ i m™l j i r- /m+n m+n+1 , b2»r~(m+n) -n uexp(-at) Jn(bt) t dt = 2F l ^ ~ t 2—; n + 1 ;— 2 ' rXn+lJ (22) where ^s the hypergeometric function defined by oo F h - r - . ^ _ T(c) ) [""U+njRb+nJ.n . 2F1(a,b,c,Z) = r{a) r ( b ) nJ p ( ;+ nj z (23) and P i s the well-known gamma function. If equation (22) is substituted for the integral in equation (21), i t is found that oo l z a ^ =nn , £ , sE^ \ 5 ]Ckc o^ where p2q-2)(-p/2)kqk-W-1 (24) U l = w + k + 1 (25) and (52) u2 = w + k + 2 (26) ck = 2 for k>0 (27) 1 for k=0 From the d e f i n i t i o n of the hypergecmetric function i t follows that c o F ( ) r w ) Y r(u1/2,Dnu2/2 , i ) A 2*V ' ~ r(u1/2)r\u2/2) 1^0 p(k + +Dn! ( p / q ) (28) For the gamma function there exists a duplication formula R2z) = ( 2 7 T ) - ^ 2( 2 z"^) Rz) Rz+D (29) Applying this twice to equation (29), 2F ^ i s si m p l i f i e d to oo 2Fi( > - fTuTT /=b p{k+ {P/ q ) ( 3 0 ) If use i s made of the fact that for positive integers n Rn+1) = n! (31) then equation (24) becomes oo > (w+k + 2Jg)t(p/qP+ZJL JT=0 Ik+iJI !i? 1 3 2 ' (53) w i t h the c o n s t r a i n t t h a t y-n+2k+m = 0 (33) I f e q u a t i o n (33) i s r e w r i t t e n as m = n-y-2k (34) t h e n , on s u b s t i t u t i n g t h i s i n e q u a t i o n (32) the summation m m a v b e r e p l a c e d by Hence the f i n a l form n j ni ^  s n ~ o o s—u o f e q u a t i o n (32) may be w r i t t e n as co -k -w-1 lz(A,y) -"XJ^o^hoV03^ <-2>'V CO . T (w+k+2l) ! , , ,k+2^ r-o(k^).»ii ( p / q ) ( 3 5 ) Although e q u a t i o n (35) s t i l l l o o k s somewhat f o r m i d a b l e , the s e r i e s do i n f a c t converge v e r y r a p i d l y . For t h i s r e a s o n i t i s much more e f f i c i e n t to determine the p e r t u r -b a t i o n s i n the v e l o c i t y d i s t r i b u t i o n by t h i s method than by s t r a i g h t n u m e r i c a l i n t e g r a t i o n . I f the s t a r s are w e l l -m ixed, then p v a n i s h e s . The e q u a t i o n then reduces t o o o o o Iz(A,y) - ^ X o o S i O E A ^ w l q - * "1 (36) 3.5 R e s u l t s Using the procedure d e s c r i b e d i n the p r e v i o u s s e c t i o n s , i t i s now p o s s i b l e to determine the d i s t o r t i o n due to a s p i r a l d e n s i t y wave on the v e l o c i t y d i s t r i b u t i o n ( 5 4 ) o f a s u b - p o p u l a t i o n of s t a r s . The r e s u l t i n g e q u a t i o n s are v a l i d f o r any a r b i t r a r y , u n p e r t u r b e d e l l i p s o i d a l v e l o c i t y d i s t r i b u t i o n and f o r any degree o f e x p o n e n t i a l o r s i n u s o i d a l time v a r i a t i o n o f the s p i r a l p a t t e r n . Thus f a r , o n l y the l i n e a r case has been c o n s i d e r e d . The n o n - l i n e a r terms can be found by u s i n g e q u a t i o n (57) i n s e c t i o n 3.1. S i n c e o n l y the f i r s t n o n - l i n e a r term i s i m p o r t a n t , the computation i s q u i t e s t r a i g h t f o r w a r d . A f t e r the l i n e a r term (a^) has been determined the n o n - l i n e a r term ( a2) can be r e a d i l y c a l c u l a t e d . The elements o f the v e l o c i t y d i s t r i b u t i o n are then found by the same method as used i n the l i n e a r c a s e . However, s i n c e the r e s u l t i n g e q u a t i o n s become q u i t e c o m p l i c a t e d , the develop ment i n t h i s t h e s i s w i l l be l i m i t e d t o a d i s c u s s i o n o f o n l y the l i n e a r c a s e . I t i s emphasized t h a t the l i n e a r a p p r o x i m a t i o n i s v a l i d o n l y i f the p e r t u r b a t i o n i n the d i s t r i b u t i o n f u n c t i o n i s s m a l l compared to the d i s t r i b u t i o n f u n c t i o n . T h i s w i l l not be the case when the s p i r a l f i e l d i s v e r y • s t r o n g o r i f the v e l o c i t y d i s p e r s i o n s of the s t a r s are v e r y s m a l l . However, B a r b a n i s and W o l t j e r , u s i n g f i r s t - o r d e r e p i c y c l i c t h e o r y r a t h e r than our method, were a b l e to show t h a t the mean v e l o c i t i e s of the s t a r s w i l l be d i r e c t l y p r o p o r t i o n a l t o the s t r e n g t h of the s p i r a l f i e l d . I f t h i s (55) i s s o , our e s t i m a t e s f o r the mean v e l o c i t i e s w i l l be v a l i d r e g a r d l e s s o f the s t r e n g t h o f the f i e l d o r the s i z e of the v e l o c i t y d i s p e r s i o n . In c o n s i d e r i n g p e r t u r b a t i o n s i n the v e l o c i t y d i s p e r s i o n s we w i l l be i n t e r e s t e d o n l y i n the s p i r a l p a t t e r n now e x i s t i n g . The l i n e a r a p p r o x i m a t i o n g i v e s r e l i a b l e r e s u l t s i f the v e l o c i t y d i s p e r s i o n i s g r e a t e r than about 20 km/sec. The use of the l i n e a r a p p r o x i m a t i o n i s , t h e r e -f o r e , j u s t i f i e d i n such c a s e s . The parameters o f the s p i r a l p a t t e r n were t a k e n t o be those used by L i n , Yuan and Shu. They are l i s t e d i n Table X I I I . Yuan (1969 I,II) has shown t h a t the parameters must be l i m i t e d t o q u i t e narrow r a n g e s . Hence the v a l u e s i n Table X I I I must be near t o the a c t u a l ones. I f the r e f e r e n c e p o s i t i o n i s (8.26 k i l o p a r s e c , 015 ) , then the c e n t e r of the n e a r e s t s p i r a l arm at the same g a l a c t o - c e n t r i c d i s t a n c e as the Sun w i l l be s i t u a t e d 70°.5 c l o c k w i s e from the Sun. Our method was checked by comparing our r e s u l t s w i t h those o b t a i n e d u s i n g L i n ' s a p p r o a c h . T h i s c o u l d , o f c o u r s e , be done o n l y f o r the case where the s t a r s were w e l l - m i x e d and the p a t t e r n was not time dependent. From Table XIV i t can be seen t h a t the agreement i s good, as s h o u l d be the c a s e . The s m a l l d i f f e r e n c e s are caused by (56) the fact that we have determinsd the mean v e l o c i t i e s to an accuracy of 3^, this being s u f f i c i e n t l y accurate for our purposes. To determine the s e n s i t i v i t y of the perturbed velo c i t y d i s t r i b u t i o n to the choice of i n i t i a l axis r a t i o and vertex deviation, the components of the perturbed ve l o c i t y d i s t r i b u t i o n were calculated for various values of the axis ratio and vertex deviation. The results are summarized in Tables XV to XX. The vel o c i t y dispersions and axis ratios in each table refer to the unperturbed values. It i s readily seen that the perturbed vel o c i t y d i s t r i b u t i o n i s highly sensitive to the values of the unperturbed axis r a t i o and vertex deviation. In a l l cases, the components are most sensitive to the unperturbed d i s t r i b u t i o n at low velocity dispersions. As the disper-sion increases, the s e n s i t i v i t y declines; CHAPTER 4 C0LLI5I0NLE55 RELAXATION J_N THE GALAXY 4 . 1 I n t r o d u c t i o n In the p r e v i o u s c h a p t e r i t has been shown t h a t the response of a s u b - p o p u l a t i o n of s t a r s to a p e r t u r b a t i o n i n the g a l a c t i c g r a v i t a t i o n a l f i e l d depends to a l a r g e e x t e n t on the parameters o f the u n p e r t u r b e d v e l o c i t y d i s t r i b u t i o n . The t i m e - v a r i a t i o n of the d i s t r i b u t i o n parameters w i l l now be examined. When s t a r s are formed, t h e i r v e l o c i t y d i s t r i b u t i o n i s p r i m a r i l y determined by the i n t e r n a l and s y s t e m a t i c motions o f the p a r e n t gas c l o u d s . However, due t o the f a c t t h a t s t a r s w i t h d i f f e r i n g v e l o c i t i e s w i l l have d i f -f e r i n g p e r i o d s , the s t e l l a r o r b i t s w i l l g r a d u a l l y get out o f s t e p w i t h each o t h e r . Such o r b i t a l m i x i n g causes the i n i t i a l c o n d i t i o n s t o be smoothed o u t . The approach t o a w e l l - m i x e d s t a t e w i l l be r e f e r r e d to as " r e l a x a t i o n " , w h i l e the w e l l - m i x e d s t a t e i t s e l f w i l l be d e s c r i b e d a l s o by the term " e q u i l i b r i u m " . By t h i s i s meant a dynamic, r a t h e r than a s t a t i s t i c a l , e q u i l i b r i u m . The r e l a x a t i o n of s t e l l a r s u b - p o p u l a t i o n s w i t h v a r i o u s i n i t i a l c o n d i t i o n s w i l l be examined. T h i s w i l l (58) be done by c a l c u l a t i n g the o r b i t s o f the i n d i v i d u a l s t a r s by means of f i r s t - o r d e r e p i c y c l e t h e o r y . The parameters of the v e l o c i t y d i s t r i b u t i o n w i l l then be determined f o r v a r i o u s v a l u e s o f time by i n t e g r a t i n g over a l l the o r b i t s a t a p a r t i c u l a r l o c a t i o n i n the G a l a x y . 4.2 The E p i c y c l e Theory Although the e p i c y c l i c t h e o r y has been d e s c r i b e d i n d e t a i l by M i h a l a s ( 1 9 6 8 ) , Chandrasekhar (1942) and o t h e r s , a b r i e f account w i l l be g i v e n h e r e . Motion o n l y i n the g a l a c t i c p l a n e w i l l be c o n s i d e r e d . C o n s i d e r a s t a r i n i t i a l l y w i t h c o - o r d i n a t e s ( RQ, 0 Q ) h a v i n g a t a n g e n t i a l v e l o c i t y © , e q u a l to the c i r c u l a r v e l o c i t y ®C( ^Q) but w i t h a s m a l l r a d i a l v e l o c i t y I7q . The o r b i t o f the s t a r w i l l now be c o n s i d e r e d u s i n g a r o t a t i n g c o - o r d i n a t e system c e n t e r e d at Rq and moving w i t h v e l o c i t y Qq about the g a l a c t i c c e n t e r . I f r i s the d i s t a n c e to the c e n t e r o f the Galaxy and i f U ( r ) i s the g r a v i t a t i o n a l p o t e n t i a l of the G a l a x y , then the r a d i a l a c c e l e r a t i o n of the s t a r i s g i v e n by r = -g- + r 0 (1) S i n c e at any a r b i t r a r y p o i n t 6 = r 0 (2) (59) and 2 dU = _©c dr r e q u a t i o n (1) may be r e w r i t t e n as (3) V - ®l-<i (4) r r I f we now w r i t e r = Rq + x (5) where x i s the change i n r due t o the r a d i a l v e l o c i t y Tl , then r = x (6) Due to c o n s e r v a t i o n o f a n g u l a r momentum 0 r = 0 R (7) o o Hence _ _ n2 R r I f we assume x i s s m a l l compared to r t h e n , on s u b s t i t u t i n g f o r r and c a r r y i n g o n l y f i r s t - o r d e r t e r m s , we f i n d 2 fo2 ®_ . ( i r ) ( i - j r > (9) o o I f x i s s m a l l the c i r c u l a r v e l o c i t y ©c may be expanded i n the s o l a r neighbourhood by a T a y l o r ' s s e r i e s . To f i r s t -o r d e r , we o b t a i n (60) 8c(p) . ®C( R0) • ' ( % ) „ x (10) O S q u a r i n g e q u a t i o n (10) and d i v i d i n g by r , we f i n d t o f i r s t -o r d e r t h a t 2 2 2 ® E ! i ! + 2(d® £ ) ( I D o o o R o On s u b s t i t u t i n g e q u a t i o n s ( 6 ) , (9) and (11) i n e q u a t i o n ( 4 ) , the e q u a t i o n may be reduced to "x" = -K2x (12) where k* i s c a l l e d the " e p i c y c l i c f r e q u e n c y " and i s d e f i n e d by n O O o A s o l u t i o n t o e q u a t i o n (12) i s x = ^ s i n Xt (13) x = TT = FT cos k*t (14) o Hence, knowing the i n i t i a l r a d i a l v e l o c i t y TiQ, i t i s p o s s i b l e t o determine the g a l a c t o c e n t r i c d i s t a n c e (RQ+ *) and the r a d i a l v e l o c i t y J"7 at any l a t e r t i m e . Using s i m i l a r a pproaches, the t a n g e n t i a l v e l o c i t y can a l s o be f o u n d . S i n c e a n g u l a r momentum i s c o n s e r v e d , i t f o l l o w s t h a t R @ /a, e . - 2 ^ , nr , ( Hr ) ( 1 5 ) r o o (61) A g a i n , we have n e g l e c t e d second-order t e r m s . The d i f f e r e n c e i n the a n g u l a r v e l o c i t y between the s t a r and the e p i c y c l i c frame i s 2 x 0o Ae = - ~ r ( 1 6 ) R* o S u b s t i t u t i n g e q u a t i o n (13) and c o n v e r t i n g t o a l i n e a r v e l o c i t y , t h i s becomes A @ = A r G = -2 s i n k t (17) o In determing the mean p e c u l i a r v e l o c i t y o f a group o f s t a r s which are at a c e r t a i n p o s i t i o n (r,0) and which have d i f f e r i n g i n i t i a l c o n d i t i o n s and o r b i t a l p a r a m e t e r s , we are not i n t e r e s t e d i n the mean A ® but i n the mean o f (®(r) - © ( r ) ) . To f i r s t - o r d e r , t h i s i s found t o be c g i v e n by R n K @(r ) - 0 ( r ) = - (-§—•) s i n k t (18) S i n c e e q u a t i o n s (14) and (18) p r o v i d e a v e r y s i m p l e f i r s t -o r d e r e s t i m a t e of the p e c u l i a r v e l o c i t y components o f a s t a r at any time they w i l l be very u s e f u l i n t h i s c h a p t e r . The main l i m i t a t i o n of the e p i c y c l e t h e o r y comes from the f a c t t h a t o n l y f i r s t - o r d e r terms have been c a r r i e d . A lthough t h i s causes some d i s c r e p a n c i e s between the a c t u a l and e s t i m a t e d o r b i t s when the e p i c y c l i c a m p l i t u d e i s l a r g e , the d i f f e r e n c e s are ve r y s m a l l f o r s t a r s w i t h s m a l l p e c u l i a r v e l o c i t i e s . (62) 4.3 M a t h e m a t i c a l F o r m u l a t i o n The g e n e r a l case w i l l be c o n s i d e r e d where a sub-p o p u l a t i o n of s t a r s i n i t i a l l y has mean v e l o c i t y components u and v i n the r a d i a l and t r a n s v e r s e d i r e c t i o n s , r e s p e c -t i v e l y . The i n i t i a l v e l o c i t y d i s t r i b u t i o n about the mean v e l o c i t i e s need not be at e q u i l i b r i u m but i s assumed t o be of the form f = c e x p ( - x J 72 - yo02 - 2z©TT) (1) where c i s a constant,IT a n d © are the r a d i a l and t r a n s v e r s e v e l o c i t y components w i t h r e s p e c t to the mean v e l o c i t y and x , y and z are d e f i n e d as f o l l o w s : o' Jo o X = . — j (2) y . - X4 (3) O The v e l o c i t y d i s p e r s i o n s , the mean v e l o c i t y components and the c o n s t a n t c may v a r y w i t h the s p a t i a l c o o r d i n a t e s . C o n s i d e r a s t a r i n i t i a l l y at (^ 0»6*Q) w i t h an i n i t i a l v e l o c i t y ( w i t h r e s p e c t t o the mean v e l o c i t y ^ of (IT , @o) . The t o t a l p e c u l i a r v e l o c i t y (I^j»©0) I s then g i v e n by (17+u, ©Q+ v ) and the p r o b a b i l i t y o f f i n d i n g such' (63) a s t a r i s f (9 , R , 17 , 0 ) . The d i s t a nee R i s r e l a t e d o o o o o to the g a l a c t o - c e n t r i c d i s t a n c e of the e p i c y c l i c frame, RX. by a(0Q+v) R o " Rl + '• R (5) where, i f X2 denotes the a n g u l a r v e l o c i t y o f the e p i c y c l i c f rame, a i s d e f i n e d as At a time t l a t e r , the p r o b a b i l i t y of such a s t a r e x i s t i n g w i l l , o f c o u r s e , s t i l l be the same. I t s p o s i t i o n and v e l o c i t y w i l l , however, have changed as f o l l o w s : 77 = (Tl + u ) c o s k t + a ( © + v ) s i n k t (6) o o R = (©_ + v ) c o s k t - i ( I 7 + u ) s i n k t (7) w o a o R' = RX + a ® ' ( 8 ) g ' = QQ + X U + |( 17'- TTQ _ U) ( 9 ) I f the p e r t u r b a t i o n s are such t h a t the v a r i a t i o n s o f the parameters w i t h Q are slow compared to the v a r i a t i o n s w i t h R, as i n the case o f s p i r a l waves w i t h s m a l l a n g l e s o f i n c l i n a t i o n , then Q may be approximated t o f i r s t o r d e r as 9 ' = Q +ftt (10) o ( 6 4 ) We now d e f i n e 1 7 = 1 7 cos k t + a s i n k t (11) o T T © = © cos k t 2 s i n k t (12) o a These c o r r e s p o n d t o the v e l o c i t y components the s t a r would have had i f i n i t i a l l y no mean v e l o c i t y ( u , v) e x i s t e d . In terms o f T7 and @ , e q u a t i o n s (6) to (7) become / T7 = 17 + u cos k t + av s i n k t (13) © = © + v cos k t - - s i n k t (14) a R' = R1 + | ( © + v cos k t - g s i n k t ) (15) The p r o b a b i l i t y o f f i n d i n g the s t a r may a l s o be w r i t t e n i n terms o f 17 and © , as f = f(Q-£bt, R -a (®+vcoskt--sinkt) ,I7coskt-a©sinkt,©coskt + 3 C s i n k t ) (16) 3 T h i s may be r e w r i t t e n more c o n v e n i e n t l y as f = c exp ( - x f l2 - y ©2 - 2z©n) (16a) where y y z x = i ( xQ 2 ) c o s 2k*t + i ( xQ + -|) 2 Si n 2k"t (17) a a a (65) y = i ( yo j J c o s 2kt + -J-(yo + -^) • zQa s i n 2Kt (18) z = -J-(_£ _ ax ) s i n 2k"t + z cos 2k"t (19) To determine the components o f the v e l o c i t y d i s t r i b u t i o n i t i s n e c e s s a r y t o f i n d the averages o f a l l the i n d i v i d u a l s t e l l a r v e l o c i t y components weighted by t h e i r r e s p e c t i v e p r o b a b i l i t i e s . From e q u a t i o n (16) i t i s e v i d e n t t h a t the p r o b a b i l i t y o f f i n d i n g a s t a r w i t h the v e l o c i t y (TT, © ) i s dependent on t i m e . In the case o f a w e l l - m i x e d i n i t i a l d i s t r i b u t i o n the time-dependent terms v a n i s h , as would be e x p e c t e d . The s t a r s w i l l be averaged at a p a r t i c u l a r p o s i t i o n (R , G ) . S i n c e the q u a n t i t i e s R and Q may t h e r e f o r e be c o n s i d e r e d to be known, i t remains t o f i n d Q , R , k and £1 f o r each v a l u e of (IT,® ) . o o I f Maarten Schmidt's (1965) mass model o f the Galaxy i s used , the a n g u l a r v e l o c i t y ^ and the e p i c y c l i c f r e q u e n c y k" can be determined f o r v a r i o u s v a l u e s o f R^. To s i m p l i f y the d e t e r m i n a t i o n o f the r e l a t i o n s h i p s , p o l y n o m i a l s were f i t t e d t hrough the v a l u e s o f D, and k by u s i n g a l e a s t squares f i t . The b e s t f i t s were found t o be g i v e n by the f o l l o w i n g p o l y n o m i a l s . ft (R) = 72.559 - 6.5874R + 0.18254R2 (20) k-(R) = 140.55 - 15.906R + 0.50856R2 (21) (66) A comparison between the v a l u e s from e q u a t i o n s (21) and (22) and those d e r i v e d d i r e c t l y from Schmidt's model i s shown i n Table X X I . The i n i t i a l d i s t a n c e R i s r e l a t e d t o the d i s t a n c e o R at time t by Ro = r / + £(® " v )<c o s k t - 1) + ^Tp3in k t <22J I f u and v depend on Rq w h i l e Q, and k" depend on R ^ , the s o l u t i o n f o r R and R , i s n o n - t r i v i a l . However, i f we o 1 c o n s i d e r the r a t i o o f the s p i r a l f o r c e t o the t o t a l g a l a c t i c g r a v i t a t i o n a l f o r c e t o be of the same o r d e r as the r a t i o o f the e p i c y c l i c a m p l i t u d e to R ^ , the v a r i a t i o n o f u and v over the e p i c y l i c o r b i t i s of second o r d e r and can thus be n e g l e c t e d . The v a l u e o f R ^ may then be found by s u b s t i t u t i n g e q u a t i o n s (20) and (21) i n t o e q u a t i o n (15) and i t e r a t i n g . Knowing R ^ , i t i s now p o s s i b l e t o c a l c u l a t e and 8 Q . The components o f the v e l o c i t y d i s t r i b u t i o n a t ( R , 0 ) may be found by i n t e g r a t i n g the i n d i v i d u a l v e l o c i t y components and c o m b i n a t i o n s t h e r e o f over a l l v a l u e s o f ( 7 7 , © ) . S i n c e R ^ ( and hence and 0 q ) are independent of EI, i n t e g r a t i o n o v e r T l i s t r i v i a l . U sing e q u a t i o n ( 1 6 a ) , the f o l l o w i n g i n t e g r a l s were e v a l u a t e d : (67) C O I rTfdn= - ^ ( J ) * exp((§- - y ) ©2) (23) - ' — C O X X X 2 J fdn* c ( S ) * exp ((§- - y ) ®2) (24) '"OO X X C O f 2 2 (25) S i n c e K, fl and QQ a l l depend o n ® , i n t e g r a t i o n o v e r © i s , u n f o r t u n a t e l y , not so t r i v i a l . N u m e r i c a l i n t e g r a t i o n t e c h n i q u e s must be a p p l i e d . The UBC L i b r a r y Subprogram COSIM was u s e d . T h i s program i n t e g r a t e s f u n c t i o n s by means o f a s t r a t i f i e d form o f Romberg i n t e g r a t i o n which uses Simpson's r u l e on s t r i p s of v a r y i n g w i d t h s . The components o f the v e l o c i t y d i s t r i b u t i o n were then found to be fV = H(- £ © + ucos kt + a v s i n k t ) (26) © ' = H(0 + vcos k t - - s i n k t ) (27) Cf^ = H(j}[l + 2 1 ® ] - 2^©[ucoskt + a v s i n k t ] + u2c o s2k t + a2v2s i n2k t + auvsin2kt)-17(28) (J* = H t ®2 + 2®fvcoskt - 2si n k t ] + v2c o s2k t + ^ s i n2k t -^ s i n 2 k t ) - ® '2 (29) a rf2 = H(- ? ©2 +@ u c o s k t + a v s i n k t - - v c o s k t + - - s i n k t + ©rr x L x xa J ^ f 2 u ^ l s i n 2 k t x ^'n' (30) uvcos2xt + [av - —J——) - c/ i I (68) wherB f oo j x exp - O O - yXS^Jd© J exp -OO (31) (69) 4.4 Results The relaxation of a number of sub-populations having various i n i t i a l velocity distributions was examined. In each case, equilibrium was approached after a time in 9 the order of 10 years. Two of the cases are discussed below in more deta i l . In the f i r s t case i t was assumed that the sub-population was at equilibrium and that a l l the stars then instantaneously received a velocity increment u of 10 km/sec. The i n i t i a l velocity dispersion was 20 km/sec. The variation of the velocity components with time is shown in Figures 5 to 9. The mean velocities oscillate with a 8 period of 2 x 10 years which corresponds to the epicyclic period. However, due to orbital mixing, the oscillations g are gradually damped. At 4 x 10 years they have decayed to half their original amplitudes. They have a l l but q vanished after 10 years. The velocity dispersion oscillates 8 with a period of 10 years but increases steadily. It is important to note that the vertex deviation and the 9 axis ratio also experience oscillations. After 10 years a l l oscillations have ceased, the stars have again reached equilibrium and the only change has been an increase in the velocity dispersion. For this case, the amplitudes of the oscillations are, in general, governed by the choice of u and v. The time for equilibrium to be reached ( i . e . (70) f o r m i x i n g t o be completed) depends s o l e l y on the i n i t i a l v e l o c i t y d i s p e r s i o n s . Here i t was assumed, f o r s i m p l i c i t y , t h a t u d i d not va r y w i t h 9 Q . I f u does depend on dQ, the same e q u a t i o n s can s t i l l be used but more parameters are needed to s p e c i f y the dependence o f u on 9 . For s p i r a l arms w i t h low i n c l i n a t i o n s , t h i s w i l l make l i t t l e d i f f e r e n c e i n the r e s u l t s . I t was a l s o assumed t h a t c was independent of 8 and R. In the second case under c o n s i d e r a t i o n , i t was assumed t h a t no mean v e l o c i t i e s e x i s t e d , but t h a t the r a d i a l v e l o c i t y d i s p e r s i o n e q u a l l e d the t r a n s v e r s e v e l o c i t y d i s p e r s i o n . I t was a g a i n assumed t h a t the e f f e c t was a x i - s y m m e t r i c . R e s u l t s are shown i n F i g u r e s 10 to 14 f o r an i n i t i a l d i s p e r s i o n o f 17 km/sec. The r e s u l t i n g c u r v e s show f e a t u r e s s i m i l a r t o those d i s c u s s e d i n the f i r s t c a s e . 9 E q u i l i b r i u m i s a g a i n reached a f t e r a time o f about 10 y e a r s and an i n c r e a s e i n the v e l o c i t y d i s p e r s i o n i s o b t a i n e d . From these s t u d i e s i t can be co n c l u d e d t h a t a s u b - p o p u l a t i o n o f s t a r s w i l l approach e q u i l i b r i u m even i n the absence of c o l l i s i o n s . The r e l a x a t i o n time f o r the system depends e n t i r e l y on the i n i t i a l v e l o c i t y d i s t r i b u t i o n . I t has been shown t h a t the c o l l i s i o n l e s s r e l a x a t i o n i s accompanied by an i n c r e a s e i n the v e l o c i t y d i s p e r s i o n and a l s o t h a t , f o r s u b - p o p u l a t i o n s not at e q u i l i b r i u m , the (71) mean v e l o c i t i e s and the v e r t e x d e v i a t i o n are i n g e n e r a l n o n - v a n i s h i n g . I t s h o u l d be noted t h a t the c o l l i s i o n l e s s r e l a x a t i o n does not depend on the masses o f the i n d i v i d u a l s t a r s . The phenomenon of o r b i t a l m i x i n g has a l s o been d i s c u s s e d by L y n d e n - B e l l ( 1 9 6 2 ) , among o t h e r s . His t r e a t m e n t was, however, l i m i t e d to a q u a l i t a t i v e d e s c r i p t i o n and d i d not i n c l u d e an e x a m i n a t i o n of the time v a r i a t i o n of the v e l o c i t y d i s t r i b u t i o n p a r a m e t e r s . CHAPTER 5 THE VERTEX DEVIATION A r e l a t i o n s h i p between the v e r t e x d e v i a t i o n and the v e l o c i t y d i s p e r s i o n has been found i n Chapter 1 (see Table I I ) . The r e s u l t s have been p l o t t e d i n F i g u r e 15. The c r o s s e s r e f e r to our data p o i n t s w h i l e the t r i a n g l e s r e p r e s e n t Delhaye's weighted mean v a l u e s f o r v a r i o u s s u b - p o p u l a t i o n s . As can be s e e n , t h e r e i s a good agreement. The dashed l i n e r e p r e s e n t s the expected r e l a t i o n s h i p f o r w e l l - m i x e d s t a r s p e r t u r b e d by the p r e s e n t s p i r a l d e n s i t y p a t t e r n . I t i s apparent t h a t the e s t i m a t e s from the wave t h e o r y are s i g n i f i c a n t l y l o w e r than the o b s e r v a t i o n a l v a l u e s . I f the o b s e r v a t i o n s are c o n s i d e r e d t o be r e l i a b l e , i t f o l l o w s t h a t some assumption i n the t h e o r e t i c a l approach must be i n v a l i d . The g r a v i t a t i o n a l f i e l d has been assumed t o be c o m p l e t e l y r e p r e s e n t e d by a two-armed s p i r a l d e n s i t y wave superimposed upon the a x i - s y m m e t r i c g r a v i t a t i o n a l f i e l d . I t i s , however, a well-known o b s e r v a t i o n a l f a c t t h a t l o c a l i r r e g u l a r i t i e s i n the s p i r a l arms do o c c u r . The Sun i s l o c a t e d i n such an i r r e g u l a r i t y , the i n t e r - a r m O r i o n s p u r . A c c o r d i n g to Yuan (19 7 1 ) , the presence o f the O r i o n spur (73) may cause a d e v i a t i o n of the v e r t e x of the r e l a t i v e l y o l d s t a r s but the v e l o c i t y d i s t r i b u t i o n of the younger s t a r s i s u n l i k e l y t o be i n f l u e n c e d by i t . S i n c e the nat u r e of the e f f e c t i s at p r e s e n t not w e l l understood from e i t h e r the o b s e r v a t i o n a l or t h e o r e t i c a l p o i n t o f v i e w , i t i s , u n f o r t u n a t e l y , i m p o s s i b l e to determine a r e a s o n a b l e e s t i m a t e o f any v e r t e x d e v i a t i o n i t may c a u s e . I t appears t o be p o s s i b l e , at any r a t e , t h a t the presence o f the O r i o n spur may e x p l a i n the observed v e r t e x d e v i a t i o n o f the o l d e r s t a r s . Such an e x p l a n a t i o n w i l l not s u f f i c e f o r the young s t a r s . However, i t has been assumed t h a t the s t a r s are w e l l - m i x e d . S i n c e i t was shown i n the p r e v i o u s c h a p t e r 9 t h a t about 10 y e a r s are r e q u i r e d b e f o r e m i x i n g can be co m p l e t e d , i t i s expected t h a t p o p u l a t i o n s younger than t h i s w i l l s t i l l have r e t a i n e d some memory of t h e i r o r i g i n . From Table X i t appears l i k e l y t h a t the f i r s t t h r e e age groups are not y e t w e l l - m i x e d . T h i s b r i n g s us to a maximum v e l o c i t y d i s p e r s i o n of 21 km/sec f o r group 3. I t must t h e r e f o r e be c o n s i d e r e d t h a t , i n d e t e r m i n i n g the response of s t a r s t o the s p i r a l f i e l d , the younger groups ( w i t h s m a l l v e l o c i t y d i s p e r s i o n s ) are not y e t w e l l - m i x e d . A c c o r d i n g t o Roberts ( 1 9 7 0 ) , s t a r s are formed i n the s p i r a l arms. The v e l o c i t y d i s p e r s i o n o f the newly born s t a r s w i l l be due p r i m a r i l y t o the t u r b u l e n t v e l o c i t y (74) o f the p a r e n t gas c l o u d . V e l o c i t i e s are a l s o i m p a r t e d t o the gas b a l l s , out of which the s t a r s are formed, by heat and r a d i a t i o n p r e s s u r e e m i t t e d from r e c e n t l y formed 0 - s t a r s . S i n c e these e f f e c t s are u n r e l a t e d t o the g a l a c t i c r o t a t i o n , the i n i t i a l v e l o c i t y d i s p e r s i o n w i l l be i s o t r o p i c ( i . e . the r a d i a l and t r a n s v e r s e v e l o c i t y d i s p e r s i o n s w i l l be e q u a l ) . The youngest s t a r s were found to have a v e l o c i t y d i s p e r s i o n o f about 15 km/sec. I f t h i s i s c o r r e c t e d f o r the e f f e c t due to the s p i r a l arm, i t c o r r e s p o n d s t o an unp e r t u r b e d v e l o c i t y d i s p e r s i o n o f about 17 km/sec (see Table X V I I I ) . I t i s assumed t h a t t h i s i s the v a l u e o f the i n i t i a l v e l o c i t y d i s p e r s i o n . At f o r m a t i o n , the mean v e l o c i t y o f the s t a r s w i l l e q u a l t h a t o f the parent gas complex. S i n c e , i n g e n e r a l , t h i s w i l l be s m a l l e r i n magnitude than the mean v e l o c i t y which the s p i r a l wave w i l l i m p a r t t o the s t a r s , no m i x i n g w i l l o c c u r as a r e s u l t o f the i n i t i a l mean v e l o c i t y . A f t e r f o r m a t i o n , the i s o t r o p i c v e l o c i t y d i s t r i -b u t i o n w i l l cause o r b i t a l m i x i n g . S i m u l t a n e o u s l y , the presence o f the s p i r a l waves w i l l i n f l u e n c e the s t e l l a r m o t i o n s . These e f f e c t s may be s e p a r a t e d i f the a m p l i t u d e o f the s p i r a l p a t t e r n does not change s i g n i f i c a n t l y d u r i n g the m i x i n g p r o c e s s . I f the am p l i t u d e d e c r e a s e s then a d d i t i o n a l m i x i n g w i l l o c c u r due to the r e s u l t i n g decrease i n the mean v e l o c i t y i m p a r t e d by the s p i r a l waves. T h i s (75) c o m p l i c a t e s the a n a l y s i s c o n s i d e r a b l y . For c o n v e n i e n c e , i t w i l l be assumed here t h a t the am p l i t u d e i s independent of t i m e . In the next c h a p t e r , where the time-dependence of the s p i r a l p a t t e r n w i l l be e s t i m a t e d , the v a l i d i t y of t h i s assumption w i l l be c h e c k e d . The time-dependence o f the v e l o c i t y d i s t r i b u t i o n was determined i n the f o l l o w i n g manner. The s t a r s were c o n s i d e r e d t o be formed near the c e n t r e s o f the s p i r a l arms. O r b i t a l m i x i n g was then assumed to o c c u r . T h i s was e n t i r e l y due t o the i n i t i a l i s o t r o p i c v e l o c i t y d i s t r i -b u t i o n which had a v a l u e o f 17 km/sec. N e g l e c t i n g the i n f l u e n c e o f the s p i r a l arms, the components of the v e l o c i t y d i s t r i b u t i o n were found f o r v a r i o u s times by the method d e s c r i b e d i n the p r e v i o u s c h a p t e r . These were then c o r r e c t e d f o r the e f f e c t o f the s p i r a l arms. I f the Sun i s l o c a t e d 70°.5 c o u n t e r c l o c k w i s e from the c e n t e r o f the n e a r e s t s p i r a l arm at the same d i s t a n c e from the g a l a c t i c c e n t e r and i f the p a t t e r n has an a n g u l a r v e l o c i t y o f 13,5 km/sec/kpc, then the time r e q u i r e d f o r s t a r s t o m i g r a t e from the s p i r a l arm to the Sun i s 1.6 x 10^ y e a r s . S t a r s from the o t h e r arm a r r i v e a f t e r 4.2 x g 10 y e a r s . The v e l o c i t y d i s p e r s i o n s and v e r t e x d e v i a t i o n s c o r r e s p o n d i n g to these times are r e p r e s e n t e d by the dots i n F i g u r e 15. They appear to agree q u i t e w e l l w i t h the observed r e l a t i o n . (76) S i n c e the i n i t i a l d i s t r i b u t i o n f u n c t i o n v a r i e d w i t h the s p a t i a l c o - o r d i n a t e s , i t was i m p l i c i t l y assumed t h a t the v e l o c i t y d i s t r i b u t i o n had not changed a p p r e c i a b l y due to the m i x i n g of the o r i g i n a l group o f s t a r s w i t h n e i g h b o u r i n g s t a r s . On examining t h i s assumption i n g d e t a i l , i t was found t h a t a f t e r 4.2 x 10 y e a r s the d i s -t r i b u t i o n parameters had changed, through s p a t i a l m i x i n g , by no more than 10 per c e n t . The a p p r o x i m a t i o n i s t h u s g j u s t i f i e d f o r times l e s s than 4.2 x 10 y e a r s . I t i s c o n c l u d e d t h a t the l a r g e d e v i a t i o n o f the v e r t e x which has been observed f o r the young s t a r s may be e x p l a i n e d by the f a c t t h a t the o r b i t a l m i x i n g of t h e s e s t a r s has not y e t been c o m p l e t e d . When t h i s e f f e c t i s combined w i t h t h a t o f the s p i r a l arms the r e s u l t i n g r e l a -t i o n s h i p between the v e l o c i t y d i s p e r s i o n and the v e r t e x d e v i a t i o n agrees w e l l w i t h the o b s e r v a t i o n a l r e s u l t . CHAPTER 6 THE AGE DEPENDENCE OF THE VELOCITY DISTRIBUTION 6.1 I n t r o d u c t i o n The o r b i t a l m i x i n g o f i n i t i a l c o n d i t i o n s which was shown t o cause an i n c r e a s e i n the v e l o c i t y d i s p e r s i o n of young s t a r s cannot e x p l a i n the l a r g e d i s p e r s i o n s observed f o r o l d e r s t a r s . A l t h o u g h the presence of a s p i r a l d e n s i t y wave causes v a r i a t i o n s i n the v e l o c i t y d i s t r i b u t i o n o f a s t e l l a r s u b - p o p u l a t i o n , no net change i s r e c o r d e d when these are averaged over a l l a n g l e s . Thus f a r i t has been assumed t h a t the wave a m p l i t u d e s do not vary w i t h t i m e . However, Toomre (1969) has found t h a t u n l e s s the waves are somehow r e p l e n i s h e d , i n t e r a c t i o n s between the s t a r s and the waves w i l l cause the waves t o be damped. The e f f e c t o f such d e c a y i n g waves on the s t e l l a r v e l o c i t y d i s p e r s i o n w i l l now be examined. The f o l l o w i n g f i r s t - o r d e r model i s p r o p o s e d . S p i r a l waves are i n i t i a l l y caused by some unknown mechanism f a r from the l o c a l v i c i n i t y . As the waves propagate over the g a l a c t i c d i s k energy i s t r a n s f e r r e d from the s o u r c e . I t i s assumed t h a t w h i l e the wave i s growing the s t e l l a r v e l o c i t y d i s p e r s i o n i n c r e a s e s by a n e g l i g i b l e amount. T h i s q i m p l i e s t h a t the growth must be r a p i d (<C<10 y e a r s ) . In the Q model o f Marochnik and Suchkov o n l y a few times 10 y e a r s are 7 7 (77 A) needed f o r s p i r a l waves t o f o r m , A s i m i l a r t i m e s c a l e i s found by Toomre (1969) i f the s p i r a l waves are due t o a c l o s e e n c o u n t e r w i t h the Large M a g e l l a n i c C l o u d . The assumption o f r a p i d growth hence seems r e a s o n a b l e . S i n c e Pomagaev f i n d s t h a t the p e r t u r b a t i o n s of the s t e l l a r v e l o c i t y d i s t r i b u t i o n are determined p r i -m a r i l y by the g r a v i t a t i o n a l f i e l d when the waves have completed t h e i r growth, the assumption o f r a p i d growth w i l l not e f f e c t the f i n a l p e r t u r b a t i o n s . A c c o r d i n g t o Toomre ( 1 9 6 9 ) , n o n - l i n e a r e f f e c t s become i m p o r t a n t when the r a p i d growth i s c o m p l e t e d . I n t e r a c t i o n s between the s t a r s and the waves cause the waves to be damped. T h i s stems from a s l i g h t phase m i x i n g o f the p e r t u r b e d o s c i l l a t i o n s o f v a r i o u s s t a r s even i n the presence o f those c o l l e c t i v e f o r c e s t h a t m a i n t a i n the s p i r a l wave. As the waves decay the v e l o c i t y d i s p e r s i o n s o f the s t a r s w i l l i n c r e a s e by an amount e q u a l to the l o s s o f g r a v i t a -t i o n a l energy o f the wave. S i n c e the damping i s o f a Landau na t u r e i t i s assumed (as suggested by H a r r i s o n ) t h a t the time v a r i a t i o n be e x p o n e n t i a l . (78) If we were interested in the time-dependence of the velocity distribution of a particular sub-population, then orbital and spatial mixing would have to be taken into account. However, since a l l the observations were taken at the present epoch, no memory of mixing w i l l have remained except for the Q velocity increments obtained within the last 10 years. Mixing may thus be assumed to have been completed for stars older 9 than 10 years i f the most recent increments are neglected. The effect of orbital mixing may then be considered to occur instantaneously and the orbits may be averaged over a l l angles. It i s assumed that, as the spiral wave decays, the excess mean velocity of the gas in the galactic disk i s lost very rapidly due to collisions among the clouds. The mean velocity of the gas w i l l therefore be considered to be that due to the spiral f i e l d . It is assumed also that the turbulent velocity of the gas does not change appreciably, so that a l l stars w i l l have the same velocity dispersion at formation. (79) 6.2 M a t h e m a t i c a l F o r m u l a t i o n The a m p l i t u d e s o f the mean motions and the d e n s i t y f l u c t u a t i o n due to the s p i r a l f i e l d w i l l be denoted by u^, v ^  and J5^ > r e s p e c t i v e l y . At a time t the mean motions and the d e n s i t y may be w r i t t e n as Pit) = pQil + y°exp(i^ + A)) (1) u. e x p ( i ^ + 8t) u ( t ) = 1 ° P. (2) H u ( t ) v v ( t ) = — J J — i (3) 1 where i s the average density,J 3 r e p r e s e n t s the decay r a t e and $ i s the a n g u l a r p o s i t i o n w i t h r e s p e c t t o the s p i r a l arms. Averages w i l l now be taken over a l l a n g l e s ^1. S i n c e the v a r i a t i o n i n s t a r d e n s i t y w i t h ^ sugges t s t h a t s t a r s spend more time i n the denser r e g i o n s , the chances t h a t a s t a r w i l l o b t a i n an o r b i t a l v e l o c i t y increment t h e r e are c o r r e s p o n d i n g l y g r e a t e r . A w e i g h t i n g term ^- s h o u l d ro t h e r e f o r e be a p p l i e d . I t i s then found t h a t the average d e n s i t y e q u a l s the un p e r t u r b e d v a l u e and the average v e l o c i t i e s v a n i s h . I t i s assumed t h a t as the wave decays the l o s s o f g r a v i t a t i o n a l energy of the wave i s accompanied by an i n c r e a s e i n the energy of p e c u l i a r m o t i o n s . T h i s i s (80) e q u i v a l e n t to assuming t h a t the mean v e l o c i t i e s of the s t a r s due to the s p i r a l wave w i l l be c o n v e r t e d to a c o r r e s p o n d i n g i n c r e a s e i n the s t e l l a r v e l o c i t y d i s p e r s i o n s . A f t e r a v e r a g i n g over a l l a n g l e s 41 i t i s found t h a t u2( t ) = iu2exp(2/3t) (4) v2( t ) «= iv2exp(2/3t) (5') For s t a r s which had been formed b e f o r e the decay began, the r e s u l t i n g changes i n the o s c i l l a t i o n a m p l i t u d e s w i l l c o r r e s p o n d to an i n c r e a s e i n the v e l o c i t y d i s p e r s i o n s o f A CU = *<u2 + 2 . 5 v2) ( l - e x p ( 2 j 3 t ) ) (6) f o r s t a r s near the Sun. For s t a r s which were formed at a time t a f t e r the b e g i n n i n g o f the decay, t h i s becomes A C2, = T ( U2 + 2.5v2) (exp(2/3t0) - Bxp ( 2 j 3 t ) > ( 7> These e q u a t i o n s w i l l be used t o determine the t h e o r e t i c a l r e l a t i o n between age and v e l o c i t y d i s p e r s i o n . 6.3 R e s u l t s S i n c e f o r young s t a r s m i x i n g has not y e t been a c o m p l e t e d , o n l y s t a r s o l d e r than 10 y e a r s were used i n d e t e r m i n i n g the age-dependence o f the v e l o c i t y d i s p e r s i o n . (81) The q u a n t i t i e s i n T able XI" were c o r r e c t e d ( f o r the e f f e c t o f the p r e s e n t s p i r a l wave) to the u n p e r t u r b e d v a l u e s . The v e l o c i t y d i s p e r s i o n s were squared and p l o t t e d i n F i g u r e v e l o c i t y d i s p e r s i o n s of the o l d e r s t a r s c o u l d be e x p l a i n e d i f these s t a r s had been formed d u r i n g the i n i t i a l c o l l a p s e of the e a r l y Galaxy to i t s p r e s e n t f o r m . However, a c c o r d i n g to Eggen, L y n d e n - B e l l and Sandage (1962) the g c o l l a p s e was v e r y r a p i d , l a s t i n g o n l y about 10 y e a r s . S t a r s formed d u r i n g t h i s time would be e xpected to e x h i b i t v e r y l a r g e v e l o c i t i e s p e r p e n d i c u l a r t o the g a l a c t i c p l a n e . S i n c e no such v e l o c i t i e s are a p p a r e n t , i t f o l l o w s t h a t these s t a r s have most l i k e l y been formed a f t e r the i n i t i a l c o l -l a p s e . the d i s p e r s i o n i n c r e a s e s from 400 to 3000 km /sec i n 9 11 x 10 y e a r s . The p o s s i b i l i t y of such an i n c r e a s e b e i n g due t o j u s t one s p i r a l p a t t e r n w i l l now be i n v e s t i g a t e d . I f A denotes the a m p l i t u d e o f the p r e s e n t s p i r a l f i e l d , / ^ r e p r e s e n t s the decay r a t e and Aq i s the maximum a m p l i -9 tude o'f the s p i r a l f i e l d 12 x 10 y e a r s ago, then 16. At f i r s t s i g h t i t might appear t h a t the l a r g e From F i g u r e 16 i t i s e v i d e n t t h a t the square o f A = Aq exp(-12/3x 109) (1) (82) S t a r s i n i t i a l l y h a v i n g an i s o t r o p i c v e l o c i t y d i s p e r s i o n o f 17 km/sec w i l l soon become mixed, c a u s i n g the r a d i a l v e l o c i t y d i s p e r s i o n t o reach 22 km/sec. A s p i r a l f i e l d o f the p r e s e n t s t r e n g t h (5 per cent of the mean g r a v i t a t i o n a l f i e l d ) w i l l cause a mean v e l o c i t y ( u , v) o f an a m p l i t u d e of about (7, 1) km/sec. When the s p i r a l f i e l d has v a n i s h e d , t h i s w i l l c o r r e s p o n d to an i n c r e a s e i n the square o f the r a d i a l v e l o c i t y d i s p e r s i o n 2 2 of 13 km /sec . The d i f f e r e n c e i n the square o f the d i s p e r s i o n between the youngest and o l d e s t s t a r s i s then g i v e n by * ? 13(A*-1) ACT - 2600 = 2 (2) U A S o l v i n g e q u a t i o n s (1) and (2), i t i s e a s i l y found t h a t the i n i t i a l a m p l i t u d e o f the wave must have been 14 times the p r e s e n t a m p l i t u d e . The r e c i p r o c a l o f /3 i s then 4.4 x 10^ y e a r s . When the a g e - d i s p e r s i o n r e l a t i o n c o r r e s p o n d i n g to such a p a t t e r n ( curve a) i s compared to the o b s e r v a t i o n a l curve i n F i g u r e 16 i t i s e v i d e n t t h a t l a r g e d i s c r e p a n c i e s e x i s t between the two. I t was thus c o n c l u d e d t h a t more than one p a t t e r n must have e x i s t e d . I t was found t h a t i f two s p i r a l p a t t e r n s have e x i s t e d the r e s u l t i n g curve ( c u r v e b i n F i g u r e 16) c o i n -c i d e d q u i t e n i c e l y w i t h the o b s e r v a t i o n s . The maximum a m p l i t u d e s of the waves were 6 and 14 times the p r e s e n t (83) p a t t e r n a m p l i t u d e . The ,P r e c i p r o c a l s were found to be g g 1.3 x 10 and 3.0 x 10 y e a r s , r e s p e c t i v e l y . In d e t e r -m i n i n g the t h e o r e t i c a l c urve i t s h o u l d be taken i n t o c o n s i d e r a t i o n t h a t the mean v e l o c i t y caused by the s p i r a l wave i s dependent on the v e l o c i t y d i s p e r s i o n o f the s t a r s . T h i s i s i m p o r t a n t o n l y f o r the o l d e r s t a r s whose v e l o c i t y d i s p e r s i o n s have been s i g n i f i c a n t l y i n c r e a s e d by the f i r s t s p i r a l wave. S i n c e the u n c e r t a i n t i e s i n v o l v e d are l a r g e , i t i s , o f c o u r s e , p o s s i b l e t o f i t v a r i o u s c o m b i n a t i o n s o f p a t t e r n s t o the c u r v e . However, the break i n the c u r v e 9 10 -a t 2.4 x 10 y e a r s and the r a p i d i n c r e a s e at 10 y e a r s do suggest t h a t at l e a s t two major p a t t e r n s have e x i s t e d . o The break at 2.4 x 10 y e a r s was examined c l o s e l y t o ensure t h a t i t i s r e a l and not due t o u n c e r -t a i n t i e s i n the s t e l l a r a g e s . A f t e r e l i m i n a t i n g the main sequence s t a r s and o t h e r s h a v i n g l a r g e age u n c e r t a i n t i e s , i t was found t h a t the break s t i l l e x i s t e d , becoming perhaps even more d i s t i n c t . I t was t h e r e f o r e c o n c l u d e d t h a t the break was not a r e s u l t o f u n c e r t a i n t i e s i n the a g e s . F i g u r e 17-shows t h a t the same s p i r a l p a t t e r n s can be f i t t e d t o the r e v i s e d c u r v e . In the p r e c e d i n g c h a p t e r i t was assumed t h a t t h e a m p l i t u d e o f the s p i r a l wave had not changed ap p r e -c i a b l y d u r i n g the m i x i n g p r o c e s s . I f the p r e s e n t p a t t e r n i s the r e s i d u e o f a s p i r a l wave 6 times the a m p l i t u d e o f (84) the p r e s e n t wave and i f the decay s t a r t e d 2.4 x 10 y e a r s g ago, then the a m p l i t u d e of the wave 4.2 x 10 y e a r s ago would have been 1.28 times the p r e s e n t a m p l i t u d e . Such changes are s m a l l enough to be n e g l e c t e d . The decay was assumed to be e x p o n e n t i a l . I f , however, the decay were more r a p i d than assumed then i t f o l l o w s t h a t the p r e s e n t wave c o u l d not be a remnant of the wave whose 9 decay began 2.4 x 10 y e a r s ago. At l e a s t t h r e e s p i r a l waves are then needed to e x p l a i n the o b s e r v a t i o n s . I t was a l s o assumed t h a t a l l the wave energy i s e v e n t u a l l y t r a n s f o r m e d i n t o an i n c r e a s e i n the v e l o c i t y d i s p e r s i o n s of the s t a r s . I f , i n f a c t , o n l y a f r a c t i o n of the wave energy i s thus t r a n s f e r r e d (as suggested by Toomre) then the wave a m p l i t u d e s must be i n c r e a s e d c o r r e s p o n d i n g l y to account f o r the observed e f f e c t . The u n c e r t a i n t i e s i n the o b s e r v a t i o n a l curve and i n the assumptions r e g a r d i n g the s t e l l a r v e l o c i t y d i s p e r s i o n s at s t a r f o r m a t i o n and the growth and decay of s p i r a l waves suggest t h a t not too much emphasis s h o u l d be put on the a c t u a l n u m e r i c a l v a l u e s here o b t a i n e d . What i s i m p o r t a n t i s t h a t the proposed model can be used to e x p l a i n the observed age e f f e c t r a t h e r s i m p l y . I t i s hoped t h a t i n the f u t u r e new, more a c c u r a t e ' data and a deeper u n d e r s t a n d i n g o f the o r i g i n and e v o l u t i o n of s p i r a l waves w i l l p e r m i t a more p r e c i s e d e t e r m i n a t i o n of the h i s t o r y o f s p i r a l p a t t e r n s i n our G a l a x y . CHAPTER 7 CONCLUSION From the e v o l u t i o n a r y t r a c k s of s t e l l a r models i t was p o s s i b l e to e s t i m a t e ages f o r a number of s t a r s . The age dependence o f the v e l o c i t y d i s t r i b u t i o n o f the nearby s t a r s c o u l d then be f o u n d . A r e l a t i o n s h i p was a l s o found between the v e l o c i t y d i s p e r s i o n and the v e r t e x d e v i a t i o n . The phenomenon o f c o l l i s i o n l e s s r e l a x a t i o n due to a n o n - e q u i l i b r i u m i n i t i a l v e l o c i t y d i s t r i b u t i o n was i n v e s t i g a t e d . The s u b - p o p u l a t i o n o f s t a r s soon approached dynamic e q u i l i b r i u m as a r e s u l t o f o r b i t a l m i x i n g . A method f o r d e t e r m i n i n g the response of a non-e q u i l i b r i u m s u b - p o p u l a t i o n t o a s p i r a l d e n s i t y wave was d e v e l o p e d . The response was found t o be q u i t e s e n s i t i v e to the form o f the v e l o c i t y d i s t r i b u t i o n . I t was found t h a t the d e v i a t i o n o f the v e r t e x f o r young s t a r s c o u l d be caused by the e f f e c t of the p r e s e n t s p i r a l wave on a s u b - p o p u l a t i o n o f s t a r s whose o r b i t a l m i x i n g has not y e t been c o m p l e t e d . For the o l d e r s t a r s , (86) the v e r t e x d e v i a t i o n may be due to the presence o f the O r i o n s p u r . The i n c r e a s e o f the v e l o c i t y d i s p e r s i o n w i t h age was e x p l a i n e d by p o s t u l a t i n g t h a t the s p i r a l p a t t e r n has decayed and reformed a number of t i m e s . A good agreement w i t h o b s e r v a t i o n was o b t a i n e d i f two s p i r a l p a t t e r n s have e x i s t e d w i t h a m p l i t u d e s o f 6 and 14 t i m e s a the p r e s e n t a m p l i t u d e and w i t h decay r a t e s of 1.3 x 10 9 and 3 x 10 y e a r s , r e s p e c t i v e l y . TABLES TABLE I Variation of Velocity Distribution with Spectral Type Spectral Number Mean Velocities (km/sec) Velocity Dispersions (km/sec) Vertex Type u V w Cu Cfv Ow Deviation B5-A9 30 - 3.0 - 6.6 - 7.3 16.5 8.4 5.1 17°.3 F0-F4 41 -17.1 -10.4 - 6.9 20.8 15.3 12.8 22°.2 F5-F9 96 -10.7 -13.0 - 8.3 26.5 17.0 15.0 8°.4 G0-G4 118 -17.2 -21.4 - 4.3 36.4 24.6 20.8 12°.9 G5-G9 103 - 9.2 -25.2 -10.1 34.4 29.3 17.3 23°.9 K0-K4 167 -12.6 -19.7 - 7.5 35.6 23.1 18.6 14°.3 K5-K9 157 -11.1 -23.3 - 7.7 35.0 21.4 18.8 14°.0 M0-M9 321 - 6.8 -21.4 - 8.5 40.8 49.6 20.7 -12c.O sub dwarf 21 -10.4 -31.1 4.2 63.2 71.4 43.3 -38°.2 white 42* .4 dwarf 23 18.8 -82.1 - 3.4 112.5 119.9 39.1 TABLE II Adjusted Velocity Distributions Spectral Number Mean Velocities Velocity Dispersions Vertex Axis (km/sec) Deviation Ratio u V w 4, <*. . ( 0 B5-A9 30 - 3.0 - 6.6 - 7.3 16.5 8.4 5.1 17.3 0.26 F0-F4 39 -16.7 - 8.8 - 6.9 20.1 13.2 13.1 22.7 0.44 F5-F9 89 -11.0 -13.9 - 7.8 24.2 13.9 15.0 13.2 0.33 G0-G4 95 -12.9 -14.5 - 3.8 26.2 17.2 17.5 15.2 0.43 G5-G9 93 - 7.9 -18.8 - 9.4 26.3 19.6 15.6 15.6 0.55 K0-K4 154 -12.6 -17.7 - 7.1 31.4 19.1 18.1 8.2 0.37 K5-K9 140 - 8.4 -20.0 - 6.6 30.7 18.3 16.7 5.3 0.36 M0-M9 291 - 7.2 -15.5 - 9.0 31.4 19.3 18.8 5.7 0.38 white -8.8 0.27 dwarf8 22 - 2.0 -16.5 - 3.3 49.9 26.0 42.4 sub-0.36 dwarfs 18 -18.8 -34.8 -19.4 56.0 33.4 31.2 0.7 CD CD TABLE III Velocity Distributions from Medians Spectral Number Median Velocities (km/sec) Velocity Dispersions (km/sec) Axis Type u V w (from medians) Ratios du tfv B5-A9 30 - 4 - 8 - 9 17.8 8.9 4.5 0.25 F0-F4 39 -15 -11 -10 20.8 13.4 11.9 0.42 F5-F9 89 - 9 -16 - 9 28.2 13.4 10.4 0.23 GQ-G4 95 -18 -15 - 4 25.2 19.3 14.8 0.59 G5-G9 93 - 6 -18 - 9 26.7 19.3 10.4 0.52 K0-K4 154 -15 -16 - 8 31.1 14.8 14.8 0.23 K5-K9 140 -11 -19 - 8 26.7 14.8 14.8 0.31 M0-M9 291 - 9 -14 -10 35.6 16.3 14.8 0.21 white dwarf8 22 - 2 -15 - 4 59.3 14.8 32.6 0.06 sub-dwarf s 18 -18 -33 -21 56.4 38.6 13.4 0.47 TABLE IV The Zero -Age Main Sequence from Schlesinger1 s Formulas M log L Mb M V (B-V) tx (109 yrs)t2 (109 yrs) 1.0 0.0 4.72 3.8 3.0 1.1 0.062 4.565 4.64 0.57 2.6 2.1 1.2 0.250 4.095 4.16 0.49 1.9 1.45 1.3 0.419 3.67 3.73 0.43 1.45 1.08 1.4 0.572 3.29 3.35 0.37 1.13 0.83 1.5 0.712 2.94 3.01 0.31 0.92 0.65 1.6 0.841 2.62 2.695 0.25 0.75 0.52 1.7 0.960 2.32 2.41 0.18 0.62 0.42 1.8 1.071 2.04 2.16 0.13 0.53 0.35 1.9 1.175 1.78 1.95 0.08 0.45 0.30 2.0 1.272 1.54 1.75 0.04 0.39 0.25 2.1 1.363 1.31 1.59 0.01 0.34 0.215 2.2 1.449 1.08 1.42 -0.005 0.30 0.19 2.3 1.531 0.89 1.28 -0.02 0.25 0.16 2.4 1.608 0.70 1.19 -0.04 0.235 0.145 2.5 1.681 0.52 1.10 -0.06 0.21 0.13 For each mass the bolometric magnitude, M^ , was calculated. To f i t the main sequence the effective temperature was shifted. My and (B-V) are the starting positions (on the main sequence) and t^ and t2 are the hydrogen burning times. (91) TABLE V Age E s t i m a t e s f o r the E v o l v e d S t a r s Liese No. M B-V Q Age E s t i m a t e V (10* y e a r s ) 8.00 1.37 0.34 3 1.40 2 0.15 10.00 3.80 0.49 3 2.8 t 1.5 19.00 3.80 0.62 2 12.0 i 2.0 34.10 3.40 0.50 5 4.5 2 0.5 54.21 3.50 0.46 3 3.0 2 0.5 55.31 3.30 0.47 4 3.5 i 0.5 58.10 1.20 0.13 6 0.8 i 0.2 61.00 3.06 0.54 3 7.0 i 1.0 78.10 1.90 0.48 4 1.9 2 0.5 80.00 1.70 0.13 4 0.7 i o . i 83.00 2.10 0.28 6 1.0 t 0.4 105.60 3.30 0.59 5 8.0 I 2'5 124.00 3.72 0.60 3 10.0 127.00 3.30 0.51 3 5.7 J 0.5 143.21 3.40 0.39 3 0.7 - 0.5 147.00 3.21 0.57 3 7.0 i 1.5 155.00 3.10 0.42 3 2.5 i 0.5 177.10 4.20 0.65 4 12.0 - 2.0 197.0 3.84 0.62 3 10.0 2 2.0 240.10 3.80 0.50 3 3.5 i i . o 242.00 2.10 0.43 3 2.0 - 0.5 264.11 4.30 0.65 3 12.0 2 2.0 279.00 3.30 0.51 4 5.5 i i . o 280.01 2.64 0.42 1 2.8 I 0.1 2B4.00 3.80 0.77 4 10.0 i 2.0 297.10 3.50 0.43 3 1.5 299.21 3.80 0.49 4 3.5 305.00 2.50 0.40 5 2.3 i 0.4 335.01 3.40 0.49 4 4.0 i i . o 354.01 2.00 0.46 4 2.0 2 0.6 397.20 3.50 0.50 4 5.0 i I.O 449.00 3.60 0.55 2 7.5 2 0.5 527.01 3.50 0.48 3 3.5 t 0.7 549.01 3.22 0.50 3 5.0 2 0.6 550.21 3.10 0.74 4 6 t 2 580.20 3.60 0.53 3 6.0 t 0.7 609.10 2.30 0.52 4 3.0 t 0.1 657.00 2.10 0.40 4 1.9 J 0.4 681.00 0.96 0.15 2 0.9 I 0.05 721.00 0.50 0.0 2 0.4 2 0.05 725.20 2.80 0.46 3 3.5 - 0.5 743.11 3.70 0.52 3 5 759.00 4.00 0.78 4 12 i 2 767.11 3.35 0.46 3 3.4 I 0.7 794.00 3.70 0.53 4 6 i l (92) Table V (continued) iese No. M B-V Q Age Estimate V (10* years) 805.00 3.70 0.43 3 1.2 + 1.0 822.11 2.32 0.40 3 2.2 + 0.3 826.00 1.50 0.22 3 1.15 + 0.05 836.61 3.10 0.49 3 4.5 + 0.5 837.00 2.00 0.29 5 1.4 + 0.1 848.00 3.14 0.44 2 3.4 + 0.3 855.11 4.30 0.65 4 12 + 2 872.01 2.60 0.50 4 3.6 + 1.0 881.00 2.03 0.09 2 0.3 + 0.2 904.00 3.39 0.51 2 5.5 •f — 0.5 512.10 3.60 0.71 4 9 + 2 602.00 3.35 0.57 3 7.0 + 1.4 770.10 3.60 0.75 5 9 + 2 771.01 3.00 0.86 3 5 + 2 600.00 3.60 0.79 4 8 + 2 635.01 2.97 0.64 1 6.0 + 0.5 695.01 3.89 0.75 2 11 + 1 (93) TABLE Age E s t i m a t e s f o r the Upper Main Sequence S t a r s G l i e s e No. My B-V Q Age E s t i m a t e (10 y e a r s ) 6.00 4.20 0.47 4 1 .5 + 1 .0 20.00 2.80 0.17 4 0 .5 + 0 .3 31.11 2.60 -0.01 5 0 .3 + 0 .3 106.11 1.90 0.09 3 0 .3 + 0 .2 107.01 3.62 0.49 2 2 .0 + 0 .4 111.00 3.70 0.48 4 2 .5 + 1 .5 121.00 2.40 0.16 4 0 .5 + 0 .3 167.10 3.10 0.31 5 0 .5 + 0 .3 174.11. 2.90 0.34 4 0 ,8 + 0 .6 176.10 3.50 0.38 4 0 .8 + 0 .6 178.00 3.76 0.46 2 1 .4 + 0 .4 187.00 3.80 0.42 4 0 .7 + 0 .6 189.20 3.50 0.45 4 2. .0 + 0 .6 196.00 3.70 0.48 3 2, ,5 + 1 .5 209.10 3.60 0.46 4 1, .6 + 0 .6 217.10 1.80 0.10 3 0 .3 + 0 .2 219.00 2.50 0.17 3 0 .4 + 0 .3 225.00 2.80 0.33 4 0. .9 + 0 .5 244.01 1.42 0.00 1 0. a + 0 .05 248.00 2.10 0.21 4 0 .6 •f 0 .3 249.10 3.50 0.45 4 2 .0 + 0 .6 268.10 2.80 0.32 4 0, .8 + 0 .6 271.01 2.46 0.34 3 1, ,4 + 0, .3 274.01 2.84 0.32 3 0, .8 + 0 .6 278.01 1.14 0.04 2 0 .4 + 0 .1 303.00 4.10 0.47 3 1 .2 + 1 .0 306.00 4.30 0.46 3 1. 2 + 1 .0 321.31 0.50 0.04 4 0. .5 + 0. .05 331.01 2.24 0.19 3 0. 5 + 0, ,4 332.01 3.50 0.37 2 1. .1 + 0. .8 333.10 3.60 0.42 3 1. .2 + 0 .8 333.30 2.50 0.14 5 0. 3 + 0. .2 339.20 0.30 0.01 4 0. 5 + 0. .1 348.01 3.90 0.45 3 1. 4 + 1, .0 351.01 3.18 0.36 3 0. 8 + 0. .5 391.00 3.30 0.36 4 0, 8 + 0, .6 403.10 3.70 0.46 3 2. 0 + 1. .5 419.00 1.40 0.12 5 0. .7 + 0. .3 448.00 1.54 0.08 3 0. .5 + 0. .3 455.30 3.10 0.32 3 0. .5 + 0 .4 459.00 1.90 0.08 3 0 .3 + 0 ,2 471.20 2.90 0.37 5 1. 1 + 1 .0 482.01 3.46 0.36 3 0. .6 i 0, .5 501.01 3.69 0.45 3 1, 4 + 0 .8 (94) Table VI (continued) .iese No. M B-V Q Age | : s t i m a t e V (10- y e a r s ) 503.00 3.70 0.48 3 2.5 508.10 1.30 0.03 3 0.3 i o.i 525.10 2.70 0.38 4 1.3 i 0.3 557.00 3.20 0.37 4 0.6 t 0.5 560.01 2.00 0.24 3 0.8 - 0.3 563.40 3.90 0.41 4 0.8 i 0.6 564.10 1.50 0.15 4 0.7 I 0.2 578.00 3.70 0.43 3 1.0 3 0.8 594.00 3.40 0.41 3 1.5 J I.O 601.00 2.40 0.29 4 0.8 i 0.6 603.00 3.40 0.48 3 4.6 I 1.0 615.21 4.00 0.50 3 2.0 I 1-5 648.00 3.70 0.48 3 2.5 i 1.5 656.11 1.40 0.06 3 0.4 I 0.2 670.01 3.30 0.40 3 1.4 t 1-2 673.10 2.60 0.28 4 0.5 J 0.4 686.20 3.10 0.40 4 1.2 i I.O 692.00 3.60 0.47 4 2.5 i 2.0 694.11 3.00 0.43 3 2 11 700.11 3.50 0.39 5 1.0 i 0.8 708.10 3.40 0.40 3 1.2 11 713.00 4.13 0.49 2 2 11 760.00 2.60 0.32 3 0.8 i 0.2 764.20 3.70 0.50 5 2.5 i 2 765.01 3.20 0.39 4 1.0 X 0.8 768.00 2.24 0.22 1 0.6 i o.i 773.40 3.90 0.49 3 2.0 t 1.5 822.01 3.93 0.50 3 2.0 849.10 3.60 0.49 4 3.1 886.20 3.60 0.29 5 0.6 2 0.5 891.10 2.80 0.30 3 0.5 i 0.4 482.02 3.48 0.29 2 0.6 - 0.5 (95) TABLE V I I Q u a l i t y C l a s s e s Q u a l i t y (Q) U n c e r t a i n t y range 1 ^0.08 2 0.09—0.15 3 0.16--0.25 4 0.26—0.35 5 0.36—0.50 6 >0.50 TABLE V I I I C l u s t e r Ages C l u s t e r NGC 188 M 67 NGC 3680 NGC 7789 Hyades M 11 F i g . 3 10.0 6.0 3.2 2.0 1.4 0.3 Age (10 y e a r s ) L i n d o f f (1968) Sandage and Eggen Iben (1967) 6.4 4.0 0.9 0.7 9 5.5 i 1 11 $ 2 5.5 - 1 (96) TABLE IX Age E s t i m a t e s f o r the G i a n t s G l i e s e No. M B-V Q Age E s t i m a t e V (1Q9 y e a r s ) 31.00 0.70 1.01 3 2.0 + 0.7 81.01 2.70 0.85 4 4.5 + 1.0 84.30 0.60 1.15 3 1.4 + 0.5 150.00 3.77 0.92 1 10 + 1 154.20 2.30 1.13 5 10 + 2 167.30 3.50 1.06 4 10 + 1 171.11 -0.60 1.54 3 ? 194.01 0.09 0.80 2 0.7 + 0.2 224.01 3.00 1.05 6 7 + 2 286.00 0.98 1.00 2 1.5 + 0.4 321.10 2.50 0.87 5 4 + 1 412.10 3.30 1.03 5 8 + 2 534.00 2.72 0.58 2 5.0 + 0.5 539.00 1.20 1.00 4 2.0 + 0.5 541.00 -0.24 1.23 2 ? 624.11 1.00 0.91 5 1.6 + 0.4 626.10 2.70 0.91 4 4.5 + 1.0 636.00 1.80 0.92 4 3.0 + 0.5 639.10 1.10 1.16 5 ? 711.00 1.80 0.94 4 3.0 + 0.5 806.11 0.80 1.03 4 2.0 + 0.5 807.00 2.72 0.92 2 5.5 + 0.5 893.21 1.80 1.11 5 7 + 2 903.00 2.27 1.03 3 6 •f 1 835.10 2.30 0.99 5 4.5 + 2.0 239.10 2.60 1.05 3 7 + 1 596.20 1.10 1.17 3 4 + 2 355.10 2.90 0.77 4 4.5 1.5 894.21 3.60 0.79 4 8 + 2 355.20 3.20 1.02 4 9 + 2 532.10 4.60 1.02 6 10 + 2 TABLE X Uncertainties in the Ages and Velocity Dispersions Group Number 1 2 3 4 5 6 7 8 9 22 21 21 20 16 15 19 16 16 Mean Age (10 y years) 0.38 0.72 1.16 1.72 2.32 3.23 4.67 6.81 10.50 Age Uncertainty 0.07 0.12 0.18 0.18 0.38 0.21 0.26 0.37 0.45 Velocity Dispersion Lower Limit Upper (km/sec) (7055 confidence) Limit 15.5 16.8 18.3 21.6 28.4 30.8 30.3 35.3 46.1 13.-5 14 .6 16.0 18.8 24.2 26.2 26.0 30.1 39.2 18.7 20.5 22.2 26.3 35.7 39.1 37.3 44.4 58.0 —j white dwarfs sub-dwarf s 22 18 49.9 56.0 43.4 48.1 60.3 68.8 TABLE XI Velocity Distributions of the Mean Velocities (km/sec) Velocity u w - - 3.0 - 5.0 - 6.8 15.5 -12.8 - 6.6 - 8.2 16.8 -11.9 -11.9 - 8.4 18.3 -14.8 -10.5 -10.3 21.6 - 9.7 -12.1 - 5.6 28.4 -11.6 -11.6 -13.7 30.8 - 8.7 -20.1 - 4.8 30.3 - 8.4 -21.1 -14.6 35.3 -34.2 -35.1 -11.7 46.1 arious Age Groups Dispersions (km/sec) Vertex a rt Deviation v vw /o v 9.0 5.4 13.9 10.9 7.1 26.4 10.8 12.6 20.0 15.3 10.2 12.4 19.8 15.9 2.1 21.3 12.2 -14.6 22.9 16.3 22.5 20.5 21.2 3.2 30.4 19.9 20.1 ( 99) TABLE X I I I Parameters o f the S p i r a l P a t t e r n Parameter A ( S t r e n g t h of s p i r a l f i e l d ) i (Angle of i n c l i n a t i o n of arms) w ( P a t t e r n a n g u l a r v e l o c i t y ) rQ (Reference p o s i t i o n ) n (Number o f arms) N u m e r i c a l Value 5% o f the mean g a l a c t i c g r a v i t a t i o n a l f i e l d 6°.2 13.5 km/sec/kpc (8.26 k p c , 0° ) 2 TABLE XIV Comparison of R e s u l t s f o r the Time-independent Case V e l o c i t y Mean R a d i a l V e l o c i t y Mean T r a n s v e r s e V e l o c i t y D i s p e r s i o n (km/sec) (km/sec) (km/sec) Our r e s u l t s L i n Our R e s u l t s L i n 15 8.52 8.56 3.03 3.01 20 5.68 5.72 0.88 0.895 24 4.13 4.16 0.22 0.23 R a d i a l V e l o c i t y D i s p e r s i o n (km/sec) to CO ro 4= ro ro ro o CO ro CO ro ro 4= ro ro O CO 0.38 1 O • Ul - J I • ui o l U l • o Ul 1 m • CO 4= o B Ul VO o . _ft ON -0.32 I • o CO 1 ro . ro - J o . ro o o • U l 4? 1 o . U l U l 1 o • VO CO l to • o •cr 1 u> t CO - P o 0.05 i o U l 4r 1 o o 1 ro o o • U l o O U l t o ON 1 o ON cn 1 4= vO 1 ro vD ro o « u ) Ul o o U l -0. 30 1 o CO CO i *o o o • 4= o 0.27 1 o « o i o • vO t 1 • ft co i to • 4r o • ro cn 1 o o • . o ro 4=- cn 1 o • CO i • Ul o • Ul o 0.26 1 o • o CO 1 o » u> VO i o . vo CO 1 ro • o 0. 28 o . o 4= i o • ro 1 o • ON VO 1 ft • ON o o » ON o II II o R a d i a l V e l o c i t y D i s p e r s i o n (km/sec) ro to ro ro .—ft ro ro ro ro —ft CO ro o CO CO 4= ro o CO < 0) H —* H> U l Ul ON vO U l to U l 4T Ul - J o QJ . • • • . • • • • * • (-ft-4r ON —> Ul Ul Ul U l 4T o ro f J . CD «o o —» 4= -J 4T - » ro o O O —* -h U) 4= ON CO —ft ro 4r 4? ON CO o • • • * • • • • » • • > <+ o o o Ul - J o VO U l 4T U l X 3" ON o -J o 4=- O CO CO VO o H- CD CO 33 CD —* 0) 0) ro 4= Ul - J a ro 4r Ul - 0 vo o <+ D • • • • • • « . • • « V-vO U l ON ft vo ro 4=- o ON 4r O 33 U l Ul ON ro CO vo ro CO U l O 01 Q. H-0) —A 1-" ro 4? Ul -o ro 4= Ul -o o o • • • < • . • • » • • < 00 4= CO ro VO 4= ON Ul Ul CD ro U l ON u> ro _» CO 4= _* U l o o n H-— . * ^ c+ ro 4= Ul . ro 4= Ul o «< • • . • » » • . • • • CO U l vO 4? vO Ul CO vO ro CTv cn vo U l o ON CO Ul VO ON m o 3> tD I -m x < o II a (n i ) TABLE XVII V a r i a t i o n o f the R a d i a l V e l o c i t y D i s p e r s i o n A x i s R a t i o c o • H 10 m 18 17.30 15.45 13.57 11.61 9 . 3 6 i = 0° Q . 0) •<-t a — o >> to +> to • H x» u B o -* 0. 20 0. 30 0.40 0. 50 0.60    9. 36 20 20. 17 19. 05 17.98 17. 00 16.00 22 22. 54 21 . 78 21. 04 20. 39 19. 73 24 24, 71 24. 14 23.59 23 . 09 22.61 28 28. 75 28. 36 27.99 27. 65 27.42 18 16. 89 15. 66 14. 12 12. 43 10.35 20 21 . 25 19. 92 18. 82 17. 74 16. 66 22 23. 73 2 2 . 64 21 . 80 20. 96 20. 20 24 25. 75 24. 89 24. 18 23. 51 22. 90 28 29. 51 28. 86 28.34 2 7 . 92 27.37 i=io< > •-I ro • H -a ro cc TABLE X V I I I V a r i a t i o n o f the A x i s R a t i o A x i s R a t i o c o • H CO n CO Q . to •H a — o >> CO +> to • H \ O E o J * n o m~ 18 - 0 . 12 0. 18 0.52 1.03 2.03 ^ = 1 0° > 20 0.07 0.23 0.42 0.67 0.98 ^ 22 3.12 0.25 0.40 0.59 0.80 •H 24 0. 13 0.26 0.39 0.56 0. 73 28 0.16 0.27 0.40 0.53 0.67 cc 0.20 0.30 0.40 0.50 0. 60 18 -0.12 0. 11 0.48 1.11 2.39 20 0.01 0. 19 0.42 0.70 1.03 22 0.07 0.22 0.40 0.60 0.82 24 0.10 0.24 0.40 0.56 0.74 28 0.13 0.26 0.39 0.53 0.68 (102 ) c a •r4 CO U 0) a co •rl a — u >, to +> (0 • H \ u e o CO > ro • H T J ro cc TABLE XIX Variation of the Vertex Deviation Axis Ratio 0. 20 0. 30 0. 40 0. 50 0. 60 18 -4. 73 -10.47 -23. 32 -48. 64 -66. 30 20 -5. 10 -9.04 -15. 4 1 -27. 06 -46. 92 22 -4. 70 -7.45 -11. 22 -17. 42 -29. 39 24 -4. 12 -6.07 -8. 45 -12. 10 -18. 12 28 -3. 05 -4.08 -5. 13 -6. 49 -9. 05 18 21. 86 18. 54 19. 45 50. 86 -82. 82 20 11. 71 8.79 5. 94 1. 37 -40. 00 22 8. 32 6. 29 3. 84 2. 01 -5. 29 24 7. 35 5.91 4. 81 3. 74 2. 35 28 7. 33 6.74 8. 71 6. 53 7. 25 i = 10° c o • H CO u CO a CO • H a — o >> <u -p in • n \ o £ o rH > ro • H T J ro cc TABLE XX Variation of the Density Axis Ratio 0.20 0.30 0.40 0.50 0. 60 18 0.53 0.54 0. 55 0.55 0. 55 20 0.61 0.62 0. 62 0.62 0. 62 22 0.67 0.68 0.68 0.68 0. 69 24 0.72 0.73 0. 73 0.73 0. 74 28 0.80 0.80 0. 80 0.81 0. 81 18 0.46 0.52 0. 54 0.55 0. 55 20 0.57 0.61 0.62 0.63 0. 63 22 3.66 0.68 0. 69 0.69 0. 69 24 0.72 0.74 0.74 0.74 0. 74 28 3. 80 0.81 0.81 0.81 0. 81 i - o * i - 1 0 ' TABLE XXI Comparison of Angular V e l o c i t i e s and E p i c y c l i c Frequencies Galacto-centric Angular Velocity Si E p i c y c l i c Frequency K Distance R (km/sec/kpc) (km/sec/kpc) (kpc) Schmidt Polynomial (20) Schmidt Polynomial (21) 6 39.7 39.6 62.8 63.4 7 35.3 35.4 54.4 54.1 8 31.5 31.5 46.7 45.8 9 28.1 28.1 39.2 38.7 10 25.0 24.9 31.6 32.3 11 22.2 22.2 26.5 27.1 12 19.8 19.8 22.8 22.9 13 17.8 17.8 20.0 19.7 14 16.1 16.1 17.7 17.5 FIGURES (104 ) FIGURE 1 Model tracks from Schlesinger's Formulas r o 3aniJN9uw iBnsiA (105) FIGURE 2 I s o c h r o n e s ca T o * 0 0 * 7 0 * 2 O ' E 0 > Q ' S 3aniiN9cjw l a n s i A (106 ) F I G U R E 3 Cluster Composite (107) FIGURE 4 Age-dependence of the Velocity Dispersions o . 00 o . a o . to <->a\ LUin cn o < tn cn U J a. t—i . n O - 1 ° . U J a . a a ' ~ 1 1 1 2 . 0 4 . 0 o 6 . 0 AGE (10 YEARS) 1 0 0 . 0 8 . 0 A}TOOT9/\ TET PBa i° aouapusdap-siuxj_ ? 3 H H 91J (80T) ( 1 0 S * FIGURE 6 Time-dependence of Tangential Velocity RADIAL VELOCITY DISPERSION (KM/SEC) u o T S j a d s T Q ^^T D O X S/\ TBTpey 10 aquapuadap-auiTj^ L 3HH3IJ fDTT \ ( I l l ) FIGURE 8 Time-dependence of A x i s R a t i o (112) FIGURE £ Time-dependence of Vertex Deviation (113) FIGURE 10 Time-dependence of Radial Velocity Time (114) FIGURE 11 -dependence of Tangential Velocity (115) FIGURE 12 Time-dependence of Radial Velocity Dispersion (116 ) FIGURE 13 Time-dependence of A x i s R a t i o (117) FIGURE 14 Time-dependence of Vertex Deviation ( 1 1 8 ) FIGURE 15 R e l a t i o n between V e l o c i t y D i s p e r s i o n and Ve r t e x D e v i a t i o n \ * \ \ \ \ \ *\ \ . x* A ^ A* \ ^ \ V \ N A A A = D E L H A Y E ^ x x x =OUR DATA - =MIXED • = UNMIXED A i 1 1 1 1 1 ' 10.0 15.0 20.0 25.0 30.0 35.0 4a VELOCITY DISPERSION (KM/SEC) (121) BIBLIOGRAPHY A l l e n , C.W. 1963, A s t r o p h y s i c a l Q u a n t i t i e s , London, Athl o n e P r e s s . 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J . 158, 899 

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