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A survey and comparison of various models of gravitational dynamical friction Elson, Rebecca 1982

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A SURVEY AND COMPARISON OF VARIOUS MODELS OF GRAVITATIONAL DYNAMICAL FRICTION by REBECCA A.W. ELSON A.B. Smith College, 1980 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES (Department of Physics) We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1982 (c) Rebecca A.W. Elson, 1982 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I f u r t h e r agree t h a t permission f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department or by h i s or her r e p r e s e n t a t i v e s . I t i s understood t h a t copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be allowed without my w r i t t e n p e r m i s s i o n . Department of P h y s i c s The U n i v e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Se^Jh. g \ 1<*ST-i i ABSTRACT Under c e r t a i n c i r c u m s t a n c e s , a massive o b j e c t t r a v e l l i n g t h r o u g h a medium may e x p e r i e n c e a net average d e c e l e r a t i o n due t o the g r a v i t a t i o n a l i n t e r a c t i o n between i t and the medium. T h i s s l o w i n g down e f f e c t i s c a l l e d d y n a m i c a l f r i c t i o n . There a r e a v a r i e t y of ways i n which d y n a m i c a l f r i c t i o n may be modeled, depending p r i m a r i l y on the n a t u r e of t h e medium, and on what a p p r o x i m a t i o n s a r e deemed t o be r e a s o n a b l e . T h i s t h e s i s i s dev o t e d t o r e v i e w i n g the v a r i o u s models, w i t h an emphasis on the assumptions u n d e r l y i n g each, t o p i n p o i n t i n g the source of any d i s c r e p a n c i e s between the models, and t o a s s e s s i n g the v a l i d i t y of each. For t h i s purpose the models a r e grouped i n t o two c a t e g o r i e s . The f i r s t c o m p r i s e s t h o s e i n which the medium c o n s i s t s of d u s t or gas. The models i n the second d e s c r i b e the i n t e r a c t i o n between a t e s t o b j e c t and a medium c o n s i s t i n g of o t h e r o b j e c t s of i t s own mass. These l a t t e r a r e of two g e n e r a l t y p e s : t h o s e w h i c h employ the two-body a p p r o x i m a t i o n , and tho s e which d e s c r i b e the i n t e r a c t i o n i n terms of a s t o c h a s t i c a l l y f l u c t u a t i n g f o r c e a r i s i n g from the v a r y i n g d i s t r i b u t i o n of f i e l d o b j e c t s s u r r o u n d i n g the t e s t o b j e c t . The models l e a d t o t h r e e d i f f e r e n t e x p r e s s i o n s f o r the d y n a m i c a l f r i c t i o n e x p e r i e n c e d by the o b j e c t : one i i i proportional to TlogX (X i s the distance from the object), one proportional to TlogvT (v i s the average v e l o c i t y of the members of the system), and one f i n i t e one proportional to TlogDe, (Do i s the mean inter-object spacing). It i s argued that the TlogD„ result i s based on an analysis which does not take into account the long range nature of g r a v i t a t i o n a l forces. The TlogX models rely on the behaviour of the medium being stationary, but i t i s demonstrated that i t takes an i n f i n i t e amount of time to estab l i s h such behaviour, so these models are not s t r i c t l y applicable to any real s i t u a t i o n . i v TABLE OF CONTENTS Abstract i i Table of Contents iv Acknowledgements v i Introduction 1 I . Dynamical F r i c t i o n Experienced by a Massive Object t r a v e l l i n g through an I n f i n i t e Cloud of Dust 5 Introduction 5 (a) Orbit Theory Based Models of Dynamical F r i c t i o n : C o l l i s i o n l e s s P a r t i c l e s with Negligible Thermal Motion 7 (b) Orbit Theory Based Models of Dynamical F r i c t i o n Including C o l l i s i o n s along the Downstream Axis 11 (c) Orbit Theory Based Models of Dynamical F r i c t i o n which Incorporate a Non-Zero Temperature 17 (d) Summary 25 I I . F l u i d Mechanical Models of Dynamical F r i c t i o n 27 Introduction 27 (a) Equations of F l u i d Mechanics 30 (b) Linearized F l u i d Mechanics 36 (c) Non-Linearized F l u i d Mechanics 44 (d) A F l u i d Mechanical Model of Dynamical F r i c t i o n Incorporating the Self-Gravity of the Medium 52 (e) Summary 60 I I I . Models of Dynamical F r i c t i o n Based on Binary Encounters 63 Introduction 63 (a) A Two-Body Calculation of Dynamical F r i c t i o n in which Complete C o l l i s i o n s are Assumed 65 (b) Time-Dependent C o l l i s i o n s 72 V ( c ) Summary 84 IV. S t o c h a s t i c M o d e l s o f D y n a m i c a l F r i c t i o n 86 I n t r o d u c t i o n 86 (a) F u n d a m e n t a l N o t i o n s o f t h e T h e o r y o f B r o w n i a n M o t i o n 89 (b) G r a v i t a t i o n a l B r o w n i a n M o t i o n and t h e C a l c u l a t i o n o f D y n a m i c a l F r i c t i o n 93 (c ) Summary 100 V. Summary and C o n c l u s i o n s 102 I n t r o d u c t i o n 102 (a) Summary o f R e s u l t s 103 (b) D i s c u s s i o n o f t h e D i f f e r e n c e s between t h e R e s u l t s 107 ( c ) S u g g e s t i o n s f o r F u r t h e r I n v e s t i g a t i o n s 111 A p p e n d i x A 113 A p p e n d i x B 120 A p p e n d i x C 122 B i b l i o g r a p h y 130 ACKNOWLEDGEMENTS I w o u l d l i k e t o thank my s u p e r v i s o r B i l l U n r u h f o r p a t i e n t l y p o i n t i n g me i n t h e r i g h t d i r e c t i o n s , and a l s o J o hn Cant f o r many f r u i t f u l d i s c u s s i o n s . c 1 INTRODUCTION Under c e r t a i n c i r c u m s t a n c e s , a m a s s i v e o b j e c t t r a v e l l i n g t h r o u g h a medium o f some d e s c r i p t i o n may e x p e r i e n c e a d e c e l e r a t i o n due t o t h e g r a v i t a t i o n a l i n t e r a c t i o n between i t and t h e medium. T h i s e f f e c t i s c a l l e d d y n a m i c a l f r i c t i o n . In what f o l l o w s , v a r i o u s m o d e l s o f d y n a m i c a l f r i c t i o n w i l l be r e v i e w e d , compared, and a s s e s s e d . G r a v i t a t i o n a l d y n a m i c a l f r i c t i o n was f i r s t e x a m i n e d i n t h e 1940's and e a r l y 1950's i n two d i f f e r e n t c o n t e x t s . One t y p e o f model ( c f . B o n d i , H o y l e , L y t t l e t o n , Dodd, McCrea, Danby, and Camm) grew p r i m a r i l y o u t of an i n v e s t i g a t i o n o f t h e r a t e a t w h i c h s t a r s a c c r e t e m a t e r i a l f r o m t h e i n t e r s t e l l a r medium. The p h y s i c s u n d e r l y i n g t h e a c c r e t i o n p r o c e s s was a p p l i e d t o s u c h d i v e r s e p r o b l e m s as t e r r e s t r i a l i c e a g e s ( H o y l e and L y t t l e t o n , 1939), t h e f o r m a t i o n of c omets ( L y t t l e t o n , 1953), and o f i n t e r e s t h e r e , t h e d e c e l e r a t i o n o f a s t a r w h i c h i s i n m o t i o n w i t h r e s p e c t t o t h e i n t e r s t e l l a r medium ( c f . B o n d i and H o y l e ( 1 9 4 4 ) , Dodd and McCrea ( 1 9 5 2 ) ) . T h i s t y p e o f model o f d y n a m i c a l f r i c t i o n i s r e v i e w e d i n C h a p t e r I , and i s b a s e d h e a v i l y on o r b i t t h e o r y . The i n t e r s t e l l a r medium i s t a k e n t o be i n f i n i t e i n e x t e n t , and composed o f n o n - s e l f - g r a v i t a t i n g p a r t i c l e s w h i c h a r e c o l l i s i o n l e s s and have u n i f o r m d e n s i t y u p s t r e a m o f t h e s t a r . T h e s e p a r t i c l e s f o l l o w h y p e r b o l i c t r a j e c t o r i e s i n t h e 2 g r a v i t a t i o n a l f i e l d of t h e s t a r , and i n t h e frame o f r e f e r e n c e i n w h i c h t h e l a t t e r i s a t r e s t , t h e i r s t r e a m i n g m o t i o n i s assumed t o be s t a t i o n a r y . The s i m p l e s t model assumes t h a t t h e p a r t i c l e s r e m a i n c o l l i s i o n l e s s downstream of t h e s t a r , and have no t h e r m a l m o t i o n ( s e c t i o n ( a ) ) . In more c o m p l i c a t e d m o d e l s , a l l o w a n c e i s made f o r c o l l i s i o n s a l o n g t h e downstream a x i s ( s e c t i o n ( b ) ) , and f o r n o n - n e g l i g i b l e t h e r m a l m o t i o n among t h e p a r t i c l e s ( s e c t i o n ( c ) ) . The e x p r e s s i o n s f o r t h e d e c e l e r a t i o n of t h e s t a r y i e l d e d by t h e s e m o d e l s , d i v e r g e l o g a r i t h m i c a l l y w i t h d i s t a n c e . B e c a u s e o f t h e d i f f i c u l t i e s i n v o l v e d i n i n c o r p o r a t i n g t h e t h e r m a l m o t i o n o f t h e p a r t i c l e s i n t h e framework o f an o r b i t b a s e d m o del, f l u i d m e c h a n i c a l m o d els of t h e g r a v i t a t i o n a l i n t e r a c t i o n between a m a s s i v e o b j e c t and a g a s e o u s medium, have been d e v e l o p e d ( c f . S p i e g e l ( 1 9 6 9 ) , Ruderman and S p i e g e l ( 1 9 7 1 ) , Hunt ( 1 9 7 1 ) ) . T h e s e m odels a r e m o d i f i e d so t h a t t h e y may be a p p l i e d t o e x t e n d e d o b j e c t s ( i e . g a l a x i e s ) a s w e l l a s t o ' p o i n t ' o b j e c t s ( i e . s t a r s , b l a c k h o l e s ) . L i n e a r i z e d and n o n - l i n e a r i z e d f l u i d - m e c h a n i c a l a n a l y s e s of d y n a m i c a l f r i c t i o n a r e r e v i e w e d i n s e c t i o n s (b) and ( c ) of C h a p t e r I I . In s e c t i o n ( d ) , t h e l i n e a r i z e d model i s e x t e n d e d t o i n c l u d e t h e s e l f - g r a v i t y o f t h e medium. I t i s f o u n d t h a t i n t h e c a s e of s u b s o n i c m o t i o n , t h e r e i s no d y n a m i c a l f r i c t i o n , w h i l e i n t h e s u p e r s o n i c c a s e , t h e e x p r e s s i o n f o r t h e d y n a m i c a l f r i c t i o n d i v e r g e s l o g a r i t h m i c a l l y a s i n t h e o r b i t b a s e d m o d e l s . 3 At t h e same t i m e as t h e o r b i t b a s e d models o f d y n a m i c a l f r i c t i o n were b e i n g d e v e l o p e d , C h a n d r a s e k h a r ( s e e C h a n d r a s e k h a r ( 1 9 4 1 - 1 9 4 4 ) , C h a n d r a s e k h a r and von Neumann, (1 9 4 2 ) ) was l a y i n g t h e f o u n d a t i o n s o f a s t a t i s t i c a l t h e o r y o f s t e l l a r e n c o u n t e r s i n w h i c h a ' t e s t ' s t a r i s v i e w e d as a c t e d upon by a s t o c h a s t i c a l l y f l u c t u a t i n g f o r c e w h i c h a r i s e s f r o m t h e v a r y i n g d i s t r i b u t i o n o f ' f i e l d ' s t a r s s u r r o u n d i n g i t . The f u n d a m e n t a l i d e a s o f t h i s t h e o r y a r e d e s c r i b e d i n s e c t i o n (a) o f C h a p t e r I V . A l t h o u g h C h a n d r a s e k h a r s u g g e s t e d how a f u l l y s t o c h a s t i c model c o u l d be u s e d t o examine t h e d y n a m i c a l f r i c t i o n e x p e r i e n c e d by a t e s t s t a r , s u c h a c a l c u l a t i o n was n o t p e r f o r m e d u n t i l many y e a r s l a t e r , by Lee (1966) and K a n d r u p ( 1 9 8 0 ) . T h e i r a p p r o a c h e s a r e d e s c r i b e d i n s e c t i o n (b) of C h a p t e r IV. They a r e b a s e d on d i f f e r e n t a s p e c t s o f t h e s t a t i s t i c a l t h e o r y , and l e a d t o d i s p a r a t e r e s u l t s . Lee f i n d s t h a t t h e e x p e c t e d change i n t h e t e s t o b j e c t ' s v e l o c i t y , - - j. <AV>, i s z e r o , w h i l e t h e q u a n t i t y < ( i V) > d i v e r g e s l o g a r i t h m i c a l l y w i t h t i m e . K a n d r u p on t h e o t h e r hand, f i n d s t h a t b o t h < A V> and <( A V ) 1 > a r e p r o p o r t i o n a l t o T l o g D e , where D 0 i s t h e mean i n t e r - o b j e c t s p a c i n g . C h a n d r a s e k h a r (1943) d i d d e r i v e an e x p r e s s i o n f o r t h e d e c e l e r a t i o n o f a t e s t s t a r , b a s e d on t h e a p p r o x i m a t i o n t h a t i t u n d e r g o e s a s u c c e s s i o n o f c o m p l e t e b i n a r y e n c o u n t e r s w i t h t h e f i e l d s t a r s . T h i s c a l c u l a t i o n i s d e s c r i b e d i n C h a p t e r I I I , a l o n g w i t h two o t h e r b i n a r y e n c o u n t e r m o d e l s w h i c h , u n l i k e C h a n d r a s e k h a r ' s , do n o t assume c o m p l e t e c o l l i s i o n s , b u t r a t h e r , t a k e i n t o a c c o u n t t h e t i m e d e p e n d e n c e , 4 p a r t i c u l a r l y of d i s t a n t e n c o u n t e r s . These l a t t e r m o d e ls r e s u l t i n e x p r e s s i o n s f o r t h e d y n a m i c a l f r i c t i o n i n w h i c h t h e d i v e r g e n c e w i t h d i s t a n c e i s r e p l a c e d by a d i v e r g e n c e w i t h t i m e . In C h a p t e r V t h e v a r i o u s r e s u l t s a r e summarized and d i s c u s s e d . S u g g e s t i o n s a s t o t h e s o u r c e s o f t h e d i s c r e p a n c i e s between them a r e p u t f o r t h , and t h e v a l i d i t y o f e a c h model i s a s s e s s e d . 5 CHAPTER I DYNAMICAL FRICTION EXPERIENCED BY A MASSIVE OBJECT TRAVELLING THROUGH AN INFINITE CLOUD OF DUST Introduction In t h i s chapter, models of the dynamical f r i c t i o n experienced by a massive object moving through an i n f i n i t e cloud of dust, are presented. Their general structure i s as follows. The object i s considered to be at rest, while the cloud streams past i t . The p a r t i c l e s in the cloud follow hyperbolic t r a j e c t o r i e s in the gr a v i t a t i o n a l f i e l d of the object, and are 'focused' along the downstream axis (see figure a-1). The change in momentum of the p a r t i c l e s as they pass the object, i s calculated using o r b i t theory. The t o t a l change in the p a r t i c l e s ' component of momentum transverse to the dir e c t i o n of the object's motion, vanishes by symmetry. The- component p a r a l l e l to the dir e c t i o n of the object's motion does not vanish, and i s accompanied by an equal and opposite change in the object's momentum in t h i s d i r e c t i o n , with the result that i t slows down. 6 An a l t e r n a t e way o f v i e w i n g t h i s e f f e c t i s i n t e r m s of t h e d r a g f o r c e e x e r t e d on t h e o b j e c t by t h e r e g i o n of e n h a n c e d d e n s i t y w h i c h forms i n i t s wake. A v a r i e t y o f d i f f e r e n t a s s u m p t i o n s a r e made c o n c e r n i n g i n t e r a c t i o n s between t h e p a r t i c l e s . In a l l c a s e s t h e i r m u t u a l g r a v i t a t i o n i s i g n o r e d . In s e c t i o n (a) t h e s i t u a t i o n i n w h i c h t h e p a r t i c l e s a r e c o l l i s i o n l e s s and have no t h e r m a l m o t i o n , i s e x a m i n e d . S e c t i o n (b) s t u d i e s t h e c a s e where t h e r m a l m o t i o n i s n e g l i g i b l e , and t h e p a r t i c l e s a r e c o l l i s i o n l e s s u p s t r e a m o f t h e o b j e c t , b u t c o l l i d e a l o n g t h e downstream a x i s . In s e c t i o n ( c ) , t h e e f f e c t o f t h e r m a l m o t i o n among t h e p a r t i c l e s i s d i s c u s s e d . S e c t i o n (d) summarizes t h e r e s u l t s . 7 (a) Orbit Theory Based Models of Dynamical F r i c t i o n : C o l l i s i o n l e s s P a r t i c l e s with Negligible Thermal Motion The simplest orbi t theory based model of dynamical f r i c t i o n i s that in which the p a r t i c l e s do not c o l l i d e and have no thermal motion: the pressure and temperature of the cloud are n e g l i g i b l e . Such a model i s presented, for example, by Dodd and McCrea (1952). In the frame in which the object i s at rest, the p a r t i c l e s at upstream i n f i n i t y are assumed to have uniform density/> e , and uniform v e l o c i t y V, directed along the x-axis (see figure a-1). In an in t e r v a l of time At, a wedge of p a r t i c l e s passes through a plane P, at upstream i n f i n i t y . After a time T i t reaches plane Px at downstream i n f i n i t y , by which time i t s momentum has changed by Ap. The average rate of change in the momentum of the wedge i s therefore Ap/T. There are N=T/At such wedges between upstream i n f i n i t y and downstream i n f i n i t y . The t o t a l change in momentum of the cloud per unit time, i s thus the change per unit time of one wedge, times the number of wedges, or AT" T At This i s equal to the force exerted by the object on the cloud per unit time, which i s the negative of the force exerted on the object by the cloud, per unit time: 8 oLt The p a r t i c l e s describe hyperbolic o r b i t s around the object, and recede to downstream i n f i n i t y with v e l o c i t y r ^ (see figure a - 1 ) . At downstream i n f i n i t y , they have a l l picked up a component of v e l o c i t y transverse to the x-axis, and the i r component along t h i s axis has changed. These components may be calculated using orbi t theory, and the re s u l t s are (cf. Dodd and McCrea): \ -t s*Vu O,1 9 1/1 - c x - i When the changes i n momentum t h a t accompany t h e s e changes i n v e l o c i t y a r e summed over the i n d i v i d u a l p a r t i c l e s , the component t r a n s v e r s e t o the a x i s v a n i s h e s by symmetry, so o n l y the component p a r a l l e l t o t h e a x i s r e mains. The p a r t i c l e s p a s s i n g the o b j e c t t h u s p i c k up a component of momentum A (see f i g u r e a - 1 ) , so t o c o n s e r v e the momentum of the o b j e c t - c l o u d system, the o b j e c t must p i c k up an e q u a l and o p p o s i t e component, B. The magnitude of t h i s component i s c a l c u l a t e d as f o l l o w s . The amount of ma t t e r c r o s s i n g an element of a r e a , sdsda?, of a p l a n e P, upstream of the o b j e c t , and p e r p e n d i c u l a r t o the x - a x i s , per u n i t t i m e , i s The change i n momentum i n t h i s d i r e c t i o n per u n i t mass, of p a r t i c l e s s t a r t i n g a t upstream i n f i n i t y , and r e c e d i n g t o downstream i n f i n i t y , i s so the t o t a l change i n the x-component of the momentum of the c l o u d , per u n i t t i m e , i s p.M seised) a-3 V -10 S u b s t i t u t i n g e x p r e s s i o n ( a - 1 ) f o r v v N , and i n t e g r a t i n g o v e r f r o m 0 t o 2rr, and s from 0 t o i n f i n i t y , g i v e s cit S-co C o n s e r v a t i o n o f momentum d i c t a t e s t h a t t h e momentum of t h e o b j e c t must change by an amount e q u a l and o p p o s i t e ( a - 5 ) . The d e c e l e r a t i o n o f t h e o b j e c t i s t h e r e f o r e S=c» S-o a-6 w h i c h d i v e r g e s l o g a r i t h m i c a l l y a s t h e im p a c t p a r a m e t e r s, goes t o i n f i n i t y . T h i s model i s b a s e d on t h e a s s u m p t i o n t h a t t h e p a r t i c l e s a r e c o l l i s i o n l e s s , however t h e f o c u s i n g e f f e c t o f t h e o b j e c t ' s g r a v i t a t i o n a l f i e l d may p r o d u c e a d e n s i t y enhancement a l o n g t h e downstream a x i s w h i c h i s s u f f i c i e n t l y l a r g e t h a t c o l l i s i o n s become i m p o r t a n t . In t h i s c a s e t h e a s s u m p t i o n b r e a k s down, and a model i n c l u d i n g c o l l i s i o n s a l o n g t h e downstream a x i s i s r e q u i r e d . Such a model i s p r e s e n t e d i n t h e n e x t s e c t i o n . 11 (b) Orbit Theory Based Models of Dynamical F r i c t i o n Including C o l l i s i o n s along the Downstream Axis Orbit theory based models of dynamical f r i c t i o n in which c o l l i s i o n s along the downstream axis are taken to be important, while the temperature of the cloud i s s t i l l assumed to be n e g l i g i b l e , are presented by Bondi and Hoyle (1944) and Dodd and McCrea (1952). In these models, the flow upstream of the object i s assumed to be c o l l i s i o n l e s s . However, streams of p a r t i c l e s are focused along the downstream axis, creating a region of enhanced density along th i s axis in which c o l l i s i o n s can no longer be ignored. The model which w i l l be described here i s s i m p l i f i e d in the following way. To begin with, i t i s assumed as in the previous section, that the density at upstream i n f i n i t y i s uniform. This implies that p a r t i c l e s a r r i v i n g at the downstream axis c o l l i d e in such a way that the component of their v e l o c i t y which is perpendicular to the downstream axis i s cancelled out, while the component along the axis i s conserved ( c . f . Dodd and McCrea). Cooperative behaviour i s established as p a r t i c l e s with v e l o c i t y greater than the escape v e l o c i t y from the object's g r a v i t a t i o n a l f i e l d move off to i n f i n i t y along the downstream axis, while those with v e l o c i t y less that the escape v e l o c i t y flow in towards the object, and are accreted (see figure b-1). Under these circumstances, incoming p a r t i c l e s w i l l c o l l i d e , not only 12 w i t h o t h e r i n c o m i n g p a r t i c l e s , b ut a l s o w i t h p a r t i c l e s a l r e a d y i n t h e ' a c c r e t i o n c o l u m n ' , so s t r i c t l y s p e a k i n g , more c o m p l i c a t e d e f f e c t s may be p r e s e n t . Dodd and McCrea c o n s i d e r t h e c a s e where t h e u p s t r e a m d e n s i t y v a r i e s r a d i a l l y , and B o n d i and H o y l e d i s c u s s some e f f e c t s t h a t a r i s e i n t h e c a s e where an a c c r e t i o n column f o r m s . N e i t h e r o f t h e s e c o n s i d e r a t i o n s i n f l u e n c e s t h e r e s u l t s s i g n i f i c a n t l y , so. f o r t h e sake o f s i m p l i c i t y , t h e y w i l l be l e f t o u t o f t h e d i s c u s s i o n . As i n s e c t i o n ( a ) , t h e d e c e l e r a t i o n of t h e o b j e c t i s f o u n d by c a l c u l a t i n g t h e change i n momentum of t h e p a r t i c l e s as t h e y p a s s t h e o b j e c t . The a s y m p t o t i c v e l o c i t y v N N a t downstream i n f i n i t y i s d e t e r m i n e d a s i n s e c t i o n (a) w i t h one d i f f e r e n c e : t h e i n i t i a l v e l o c i t y i s t a k e n t o be t h e v e l o c i t y t h e p a r t i c l e has i m m e d i a t e l y a f t e r a r r i v i n g a t t h e downstream a x i s , r a t h e r t h a n t h e v e l o c i t y a t u p s t r e a m i n f i n i t y . I m m e d i a t e l y a f t e r c o l l i d i n g on t h e x - a x i s , a p a r t i c l e has v e l o c i t y V i n t h e x - d i r e c t i o n , and i t s component o f v e l o c i t y t r a n s v e r s e t o t h e x - a x i s i s a n h i l a t e d . The r e s u l t i s y. In t h e l i m i t o f l a r g e i mpact p a r a m e t e r s , t h i s r e d u c e s t o ( a - 1 ) . M a t e r i a l a r r i v i n g w i t h k i n e t i c e n e r g y l e s s t h a n i t s p o t e n t i a l e n e r g y due t o t h e g r a v i t a t i o n a l f i e l d , 1 y 1 3 o r x<2x 0, where r e m a i n s g r a v i t a t i o n a l l y bound t o t h e o b j e c t . T h a t w i t h k i n e t i c e n e r g y g r e a t e r t h a n t h e p o t e n t i a l e n e r g y , x>2x„ , e s c a p e s t o i n f i n i t y . (See f i g u r e b-1.) The t o t a l amount of m a t e r i a l a r r i v i n g a t t h e downstream a x i s , p e r u n i t a r e a , p e r u n i t t i m e , i s as b e f o r e , ( s e e e x p r e s s i o n ( a - 3 ) ) V/o. sdsel<p The i m p a c t p a r a m e t e r s, i s r e l a t e d t o t h e d i s t a n c e downstream where t h e p a r t i c l e s t r i k e s t h e a x i s by ( c f . Dodd and McCrea) 14 The t o t a l rate of change of the momentum of the material i s , as in section (a) (equation (a-4)), d\ The asymptotic v e l o c i t y v x N i s given by (b-1). Assuming the dominant contribution to the drag force i s due to the cumulative e f f e c t of more distant p a r t i c l e s ( i e . x>>xe), (b-1) can be approximated as = v I . _ w . \ V f i ~ TC0 \ b - 7 Inserting t h i s in (b-6) and integrating over x from 2x 0 to i n f i n i t y gives The change in the momentum of the object must be equal and opposite t h i s , so = - ltrC*xhf>. ^ di V 1 G H b-S The assumption that i t i s the cumulative e f f e c t of distant material that i s primarily responsible for the deceleration of the object, may be j u s t i f i e d a p o s t e r i o r i by comparing (b-8) with expression (a-6). Substituting (b-4) into (a-6) y i e l d s 15 cU V 1 4 xx ^ The e f f e c t of p a r t i c l e s near the object was taken into account in deriving t h i s expression, and the only difference between i t and (b-8) i s that the argument of the logarithm i s m u l t i p l i e d by two and increased by one. In the case where ^ ^o, which i s i d e n t i c a l to expression (b-8). This suggests that c o l l i s i o n s along the downstream axis play an i n s i g n i f i c a n t role in the dynamical f r i c t i o n experienced by the object. The main conclusion to be drawn from sections (a) and (b), are: (1) the expression for the dynamical f r i c t i o n experienced by a massive object moving through a cloud of p a r t i c l e s which i s of i n f i n i t e extent, and has n e g l i g i b l e temperature, diverges logarithmically with distance; (2) t h i s r e s u l t i s i n s e n s i t i v e to whether or not c o l l i s i o n s occur along the downstream axis, and also to whether the object accretes any material. so (b-9) becomes oU 1/7 G, 1 V\p0 £A VC b- lo 16 I t now r e m a i n s t o examine t h e c a s e i n w h i c h t h e t e m p e r a t u r e o f t h e c l o u d i s n o t n e g l i g i b l e . T h i s i s d i s c u s s e d i n t h e n e x t s e c t i o n . 17 (c) Orbit Theory Based Models of Dynamical F r i c t i o n Which Incorporate a Non-Zero Temperature The discussion in t h i s section w i l l not provide an analytic expression for the dynamical f r i c t i o n suffered by the test object: the mathematical complexities encountered in incorporating the thermal motion of the p a r t i c l e s , do not allow such an expression to be derived. Rather, t h i s discussion i s included because i t provides a link between the models described so f a r , and those of the following chapters. Danby and Camm (1957) and Danby and Bray (1967) examine the case of a massive object t r a v e l l i n g through an i n f i n i t e , uniform cloud of p a r t i c l e s which are c o l l i s i o n l e s s , but which have non-negligible thermal motion. In the frame of reference in which the object i s at rest, the p a r t i c l e s at upstream i n f i n i t y move with mean v e l o c i t y V, and obey a Maxwellian d i s t r i b u t i o n . Using orbi t theory, Danby and Camm derive an expression for the perturbation in the density of the cloud, caused by the object. They assume, as in sections (a) and (b), that the flow i s steady in the rest frame of the object, and they calculate the number of p a r t i c l e s per unit time, impinging on a section of a cone as pictured in figure c-1. 18 V Figure c-1 . In the l i m i t i n g case of zero temperature, they find the expression c-\ This expression has the following features. ( 1 ) Since csc ( 0)=oo, the density i s i n f i n i t e along the downstream axi s . (2) At 6 = F , ( i e . along the upstream a x i s ) , i t becomes r = p. [ W * C - l In the l i m i t a GM t h i s reduces further to 19 P = £2 ( ^ MV 1 C^ Physically, in the immediate v i c i n i t y of the object, the density f a l l s off as 1/"iT . In the l i m i t ( i e . at positions a long way upstream of the object), P = /p" , as required. (3) The same l i m i t i n g expressions apply at positions away from the axis. Expression (c-1) was derived under the assumption that there are no c o l l i s i o n s along the downstream axis: p a r t i c l e s impinge on a from both sides, and pass unperturbed through the downstream axis. This i s the same si t u a t i o n as in section (a), and although no expression for the density of the cloud was derived there, i t was suggested that the focusing e f f e c t of the object's g r a v i t a t i o n a l f i e l d might produce a s u f f i c i e n t enhancement along the downstream axis, that c o l l i s i o n s would become important. Expression (c-1) indicates that t h i s i s indeed true: the density i s formally i n f i n i t e along the downstream axis. This suggests that the models of section (b), which incorporated c o l l i s i o n s along the downstream axis, i s more accurate. However, i t has already been demonstrated that the difference between the models of sections (a) and (b) i s n e g l i g i b l e . Likewise, i t i s straightforward to derive an expression equivalent to 20 (c-1), but in which p a r t i c l e s only impinge on the cone from one side ( i e . they do not cross the axis) ( c f . Danby and Camm). Under these circumstances, the expression for the density i s not altered s i g n i f i c a n t l y . Expression ( c - l ) for the density may be used to calculate the rate of deceleration of the object. The force on the object i s equal to i t s mass times i t s acceleration: F = tA e»V so _ F c - 4 The force i s given by Vo l orO^. Inserting (c-1) gives 41 = G,^ 3LGM + s » o * B There i s no <9 dependence, so c^c e (kM H si"o* 6 I - I ) ^ sioecJe To get an idea of the behaviour of the density at large distances, the integrand may be expanded in a Taylor series around l/a=0. The lowest order term vanishes: the f i r s t non-zero contribution i s CSC Q _ l _ x a 21 I t e x h i b i t s what w i l l be seen i n C h a p t e r I I , t o be a c h a r a c t e r i s t i c 1/a d e p e n d e n c e , w h i c h l e a d s t o t h e same l o g a r i t h m i c d i v e r g e n c e a l r e a d y e n c o u n t e r e d . F o r n o n - z e r o t e m p e r a t u r e , t h e e x p r e s s i o n f o r t h e d e n s i t y d o e s n o t become i n f i n i t e a l o n g t h e downstream a x i s . However, t h e m a t h e m a t i c s becomes s u f f i c i e n t l y c o m p l i c a t e d t h a t an a n a l y t i c a l s t u d y of t h e d e n s i t y enhancement i s n o t f e a s i b l e . A n u m e r i c a l a n a l y s i s ( c . f . Danby and B r a y ) y i e l d s t h e r e s u l t s p i c t u r e d i n f i g u r e c - 2 . The shape o f t h e i s o d e n s i t y c o n t o u r s depends m a i n l y on t h e q u a n t i t y H ere m i s t h e a v e r a g e mass of a p a r t i c l e , k i s B o l t z m a n ' s c o n s t a n t , and T i s t h e t e m p e r a t u r e o f t h e c l o u d a t i n f i n i t y . F o r <*=0 ( i e . f o r V=0) t h e c o n t o u r i s s p h e r i c a l l y s y m m e t r i c a b o u t t h e o b j e c t . F o r °C=0.5 i t i s s t i l l s p h e r i c a l l y s y m m e t r i c , but i t i s c e n t r e d on a p o i n t somewhat downstream o f t h e o b j e c t . F o r oC = 4 and g r e a t e r , t h e c o n t o u r s e x t e n d o n l y b e h i n d t h e o b j e c t , and become more and more e l o n g a t e d as oc i n c r e a s e s . 22 Q u a l i t a t i v e l y , figure c - 1 indicates that an object t r a v e l l i n g at any speed w i l l experience dynamical f r i c t i o n . However, in the absence of an analytic expression for either the density or v e l o c i t y f i e l d of the cloud, an analytic expression for the dynamical f r i c t i o n cannot be derived. Including a non-zero temperature i s not expected to influence the res u l t s obtained in the l a s t two sections, for the following reason. One of the e f f e c t s of a non-zero temperature i s to randomize the upstream flow of the p a r t i c l e s , and thus decrease the frequency of c o l l i s i o n s along the downstream axis. With no thermal motion, the p a r t i c l e s approaching from upstream i n f i n i t y with uniform v e l o c i t y V, w i l l be focused in such a way that they a l l pass exactly through the downstream axis. The density there w i l l therefore be high, and c o l l i s i o n s frequent. However, i f the 23 p a r t i c l e s have random t h e r m a l m o t i o n u p s t r e a m , some w i l l p a s s a s m a l l d i s t a n c e away f r o m t h e downstream a x i s , so t h e d e n s i t y w i l l n o t be a s h i g h . A n o n - z e r o t e m p e r a t u r e t h e r e f o r e has t h e e f f e c t o f r e d u c i n g t h e i m p o r t a n c e o f c o l l i s i o n s a l o n g t h e downstream a x i s . However, i t has a l r e a d y been shown i n s e c t i o n s (a) and (b) t h a t c o l l i s i o n s a l o n g t h e downstream a x i s p l a y an i n s i g n i f i c a n t r o l e i n t h e d y n a m i c a l f r i c t i o n e x p e r i e n c e d by t h e o b j e c t . I t w i l l be seen t h a t t h e c l a i m t h a t t h e r m a l m o t i o n among t h e p a r t i c l e s does n o t i n f l u e n c e t h e d y n a m i c a l f r i c t i o n e x p e r i e n c e d by t h e o b j e c t , i s s u b s t a n t i a t e d by t h e model p r e s e n t e d i n C h a p t e r I I I , s e c t i o n ( a ) . T h e r e a t e s t o b j e c t u n d e r g o i n g b i n a r y e n c o u n t e r s w i t h f i e l d o b j e c t s o f i t s own mass, w h i c h have p e c u l i a r v e l o c i t i e s s a t i s f y i n g a M a x w e l l i a n d i s t r i b u t i o n , i s e x a m i n e d , and t h e r e s u l t i s an e x p r e s s i o n f o r t h e t e s t o b j e c t ' s d e c e l e r a t i o n w h i c h d i v e r g e s l o g a r i t h m i c a l l y w i t h d i s t a n c e . The main v a l u e of t h e d i s c u s s i o n i n t h i s s e c t i o n i s t h e l i n k i t f u r n i s h e s between o r b i t t h e o r y b a s e d m o d e l s o f d y n a m i c a l f r i c t i o n , and f l u i d m e c h a n i c a l m o d e l s , w h i c h w i l l be p r e s e n t e d i n t h e n e x t c h a p t e r . The p r e s s u r e i n t h e c l o u d , due t o t h e t h e r m a l m o t i o n of t h e p a r t i c l e s , may be d e s c r i b e d ( a s i n t h e n e x t c h a p t e r ) by t h e p e r f e c t gas law: The q u a n t i t y oc i s t h e n ex. 24 But (P/^o) i s e q u a l t o t h e s p e e d o f sound, c, i n a gas u n d e r g o i n g an i s o t h e r m a l c h a n g e , so oc = _V c In f i g u r e c - 1 , t h e s p h e r i c a l c o n t o u r s t h e r e f o r e c o r r e s p o n d t o s u b s o n i c m o t i o n , and t h e f o o t b a l l - s h a p e d c o n t o u r s , t o s u p e r s o n i c m o t i o n . The d i f f i c u l t i e s e n c o u n t e r e d i n i n c o r p o r a t i n g a n o n - z e r o t e m p e r a t u r e s u g g e s t t h a t a more s a t i s f a c t o r y a p p r o a c h m i g h t be t o t r e a t t h e c l o u d as a f l u i d d e s c r i b e d by th e e q u a t i o n s o f f l u i d m e c h a n i c s , r a t h e r t h a n a c o l l e c t i o n of p a r t i c l e s whose m o t i o n s a r e d e s c r i b e d by o r b i t t h e o r y . A f l u i d m e c h a n i c a l a p p r o a c h t o d y n a m i c a l f r i c t i o n i s e x p l o r e d i n C h a p t e r I I . 25 (d) Summary In s e c t i o n (a) i t was f o u n d t h a t an o b j e c t t r a v e l l i n g t h r o u g h a c l o u d of c o l l i s i o n l e s s , n o n - s e l f - g r a v i t a t i n g p a r t i c l e s w h i c h have no t h e r m a l m o t i o n , i s d e c e l e r a t e d a t a r a t e S e c t i o n ( b ) , i n w h i c h c o l l i s i o n s a l o n g t h e downstream a x i s were i n c l u d e d , gave t h e r e s u l t T h i s was c a l c u l a t e d a s s u m i n g x t o be l a r g e . In t h e l i m i t of l a r g e x, ( d - 1 ) i s i d e n t i c a l t o ( d - 2 ) . The c o n c l u s i o n s t o be drawn from t h i s a r e : ( i ) t h e e x p r e s s i o n f o r t h e d y n a m i c a l f r i c t i o n s u f f e r e d by t h e o b j e c t d i v e r g e s l o g a r i t h m i c a l l y w i t h d i s t a n c e ; ( i i ) c o l l i s i o n s a l o n g t h e downstream a x i s have a n e g l i g i b l e e f f e c t on t h e r a t e a t w h i c h t h e o b j e c t i s d e c e l e r a t e d ; ( i i i ) t h e d o m i n a n t c o n t r i b u t i o n t o t h e d y n a m i c a l f r i c t i o n comes from t h e c u m u l a t i v e e f f e c t o f d i s t a n t CIV r - XT7 G , X M A I •x = c » 26 p a r t i c l e s . Ignoring the e f f e c t of p a r t i c l e s in the v i c i n i t y of the object does not change the re s u l t s s i g n i f i c a n t l y . (iv) The accretion process, which i s associated with, p a r t i c l e s near the object, does not influence the rate at which the object i s decelerated by dynamical f r i c t i o n . (v) Although (d-1) and (d-2) were derived in the special case where the density at upstream i n f i n i t y i s uniform, i t has been shown by Dodd and McCrea that allowing the density to vary r a d i a l l y does not a l t e r these r e s u l t s . Section (c), where a non-zero temperature was included, did not y i e l d an expression for the dynamical f r i c t i o n , but for reasons already discussed, thermal motion among the p a r t i c l e s i s not expected to influence the deceleration s i g n i f i c a n t l y . The main value of section (c) i s that i t represents a link between orbit based models, and the f l u i d mechanical models of dynamical f r i c t i o n described in the next chapter. 27 CHAPTER I I FLUID MECHANICAL MODELS OF DYNAMICAL FRICTION I n t r o d u c t i o n In s e c t i o n ( c ) of^ C h a p t e r I , a model o f d y n a m i c a l f r i c t i o n was p r e s e n t e d i n w h i c h t h e t e m p e r a t u r e of t h e c l o u d s t r e a m i n g p a s t an o b j e c t was i n c l u d e d . However, b e c a u s e o f t h e m a t h e m a t i c a l c o m p l e x i t i e s i n v o l v e d i n i n c o r p o r a t i n g n o n - n e g l i g i b l e t h e r m a l m o t i o n among t h e p a r t i c l e s i n an o r b i t b a s e d t h e o r y , a f l u i d m e c h a n i c a l a p p r o a c h t o d y n a m i c a l f r i c t i o n i s e x p l o r e d i n t h i s c h a p t e r . The b a s i c i d e a o f t h i s a p p r o a c h i s t h a t t h e g r a v i t a t i o n a l f i e l d of t h e o b j e c t c a u s e s an enhancement i n t h e d e n s i t y o f t h e medium s u r r o u n d i n g i t w h i c h , under some c i r c u m s t a n c e s , may e x e r t a d r a g f o r c e on t h e o b j e c t , c a u s i n g i t t o slow down. The e q u a t i o n s o f f l u i d m e c h a n i c s u s e d t o d e s c r i b e t h e b e h a v i o u r o f t h e f l u i d i n t h e p r e s e n c e of t h e o b j e c t a r e p r e s e n t e d i n s e c t i o n ( a ) , where t h e f l u i d i s assumed t o be n o n - v i s c o u s and n o n - s e l f - g r a v i t a t i n g . In s e c t i o n (b) b o t h s u b s o n i c and s u p e r s o n i c m o t i o n a r e exa m i n e d by a n a l y s i n g t h e l i n e a r i z e d e q u a t i o n s o f f l u i d 28 m e c h a n i c s . I t t u r n s o u t t h a t i n t h e s u b s o n i c c a s e , t h e d e n s i t y enhancement c r e a t e d by t h e o b j e c t i s s y m m e t r i c f o r e and a f t , so t h e r e i s no n e t d r a g f o r c e . T h i s c o n t r a d i c t s t h e r e s u l t s o f Danby and B r a y ( s e e C h a p t e r I , s e c t i o n ( c ) ) , and a r e a s o n f o r t h i s d i s p a r i t y i s s u g g e s t e d . In t h e s u p e r s o n i c c a s e an e n h a n c e d d e n s i t y wake forms b e h i n d t h e o b j e c t , and t h e e x p r e s s i o n f o r t h e d r a g f o r c e e x e r t e d on t h e o b j e c t by t h i s wake d i v e r g e s l o g a r i t h m i c a l l y w i t h d i s t a n c e . In C h a p t e r I i t was a r g u e d t h a t i t i s t h e c u m u l a t i v e e f f e c t of d i s t a n t m a t e r i a l t h a t p l a y s t h e d ominant r o l e i n s l o w i n g down t h e o b j e c t , and t h a t e f f e c t s i n t h e v i c i n i t y of t h e o b j e c t , s u c h as a c c r e t i o n , a r e i n s i g n i f i c a n t as f a r a s t h e d e c e l e r a t i o n p r o c e s s i s c o n c e r n e d . The l i n e a r i z e d m o d e ls s h o u l d p r o v i d e an a c c u r a t e d e s c r i p t i o n of t h e b e h a v i o u r of t h e m a t e r i a l f a r f r o m t h e o b j e c t : t h e r e i s no r e a s o n t o e x p e c t n o n - l i n e a r e f f e c t s t o be i m p o r t a n t anywhere but i n t h e immediate n e i g h b o u r h o o d o f t h e o b j e c t , and i n t h e v i c i n i t y of t h e s h o c k s w h i c h o c c u r i n t h e c a s e of s u p e r s o n i c m o t i o n . N e i t h e r o f t h e s e r e g i o n s i s e x p e c t e d t o have a s i g n i f i c a n t i n f l u e n c e on t h e d r a g f o r c e e x e r t e d by t h e medium on t h e o b j e c t . However, f o r t h e sake of c o m p l e t e n e s s , a n o n - l i n e a r i z e d t r e a t m e n t i s d i s c u s s e d b r i e f l y i n s e c t i o n ( c ) . I n s e c t i o n (d) t h e l i n e a r i z e d model o f s e c t i o n (b) i s e x t e n d e d t o i n c l u d e s e l f - g r a v i t y . I t i s d e m o n s t r a t e d t h a t , a l t h o u g h s e l f - g r a v i t y f u r n i s h e s an o u t e r c u t o f f f o r t h e d i v e r g e n t i n t e g r a l s a t t h e J e a n s l e n g t h , t h i s c u t o f f i s more m a t h e m a t i c a l t h a n p h y s i c a l , and i n c l u d i n g s e l f - g r a v i t y d o e s 29 n o t g e t r i d o f t h e l o g a r i t h m i c d i v e r g e n c e s . F i n a l l y , t h e v a r i o u s r e s u l t s a r e summarized and d i s c u s s e d i n s e c t i o n ( e ) . 30 (a) E q u a t i o n s of F l u i d M e c h a n i c s ( i ) E q u a t i o n s o f M o t i o n and C o n s e r v a t i o n : The t h r e e b a s i c e q u a t i o n s d e s c r i b i n g t h e b e h a v i o u r of a n o n - v i s c o u s , n o n - s e l f - g r a v i t a t i n g f l u i d i n t h e p r e s e n c e of a m a s s i v e o b j e c t a r e E u l e r ' s e q u a t i o n ( e q u a t i o n o f m o t i o n ) t h e e q u a t i o n o f c o n s e r v a t i o n o f mass and an e q u a t i o n o f s t a t e , w h i c h w i l l be d i s c u s s e d l a t e r . "H* i s t h e g r a v i t a t i o n a l f i e l d of t h e o b j e c t of mass M: y - Gvh a- 3 a and sk i s t h e E u l e r d e r i v a t i v e : p, v, and jo a r e t h e p r e s s u r e , v e l o c i t y , and d e n s i t y o f t h e f l u i d . An a l t e r n a t e s e t o f e q u a t i o n s ' c o n s i s t s o f t h e 31 c o n s e r v a t i o n e q u a t i o n s f o r mass, momentum, and e n e r g y , t o g e t h e r w i t h an e q u a t i o n o f s t a t e . C o n s e r v a t i o n of momentum i s f o u n d by c o m b i n i n g c o n s e r v a t i o n o f mass and E u l e r ' s e q u a t i o n : To be c o m p l e t e , a C o r i o l i s f o r c e O L and a c e n t r i f u g a l f o r c e o. s h o u l d be added t o t h e r i g h t - h a n d s i d e o f t h e momentum c o n s e r v a t i o n ( o r E u l e r ' s ) e q u a t i o n . C o n s e r v a t i o n of e n e r g y i s imposed by f o r m i n g an e q u a t i o n o f e n e r g y b a l a n c e c o m p r i s i n g t h e i n t e r n a l p l u s k i n e t i c e n e r g y o f a f l u i d e l e m e n t , t h e work done on t h e e l e m e n t by p r e s s u r e and by g r a v i t a t i o n a l f o r c e s , and t h e e n e r g y l e a v i n g t h e f l u i d t h r o u g h c o n d u c t i o n o r r a d i a t i o n . H e r e , 32 U i s t h e i n t e r n a l e n e r g y o f t h e f l u i d , and X i s i t s t h e r m a l c o n d u c t i v i t y . T h i s l a t t e r s e t i s p a r t i c u l a r l y c o n v e n i e n t i n n u m e r i c a l t r e a t m e n t s of s u p e r s o n i c f l o w , b e c a u s e f l u x e s o f mass, momentum, and e n e r g y a r e c o n s e r v e d a c r o s s s h o c k s ( L a n d a u and L i f s h i t z , 1959) so s h o c k s a p p e a r a s s t e e p r i s e s o v e r s e v e r a l i n t e r v a l s , and n o t as d i s c o n t i n u i t i e s . A n o t h e r u s e f u l e x p r e s s i o n i s B e r n o u l l i ' s e q u a t i o n , w h i c h f o l l o w s f r o m t h e s t e a d y s t a t e e q u a t i o n s o f c o n s e r v a t i o n of mass and e n e r g y . In t h e a b s e n c e of r a d i a t i o n o r c o n d u c t i o n of h e a t , B e r n o u l l i ' s e q u a t i o n i s /> The c o n s t a n t on t h e r i g h t i s d e t e r m i n e d from t h e b o u n d a r y c o n d i t i o n s a t i n f i n i t y . ( i i ) E q u a t i o n o f S t a t e : The e q u a t i o n o f s t a t e , d e s c r i b i n g t h e r e l a t i o n between p r e s s u r e and d e n s i t y i n t h e f l u i d , i s i m p o r t a n t i n d e t e r m i n i n g t h e n a t u r e of t h e d e n s i t y p e r t u r b a t i o n c a u s e d by t h e o b j e c t i n t h e s p h e r i c a l l y s y m m e t r i c c a s e , and i n t h e r e g i o n n e a r t h e o b j e c t i n t h e n o n - l i n e a r i z e d c a s e . The f l u i d i s assumed t o be a p e r f e c t monatomic gas i n t h e r modynamic e q u i l i b r i u m . The v a r i a b l e s p , y ° , and T a r e 33 therefore related by the perfect gas law, p> = £ 1° ~^ on where k i s Boltzman's constant, and m i s the average mass of a p a r t i c l e . A second r e l a t i o n between these three variables may be imposed by assuming that the gas undergoes a polytropic change in the presence of the object's g r a v i t a t i o n a l f i e l d . A polytropic change i s a change that i s ca r r i e d out in such a way that (a) the change i s qua s i - s t a t i c ( r e v e r s i b l e ) : the system remains i n f i n i t e s i m a l l y close to thermodynamic equilibrium at a l l times, and (b) the derivative dQ/dT varies in a s p e c i f i e d way throughout the change. In a perfect monatomic gas undergoing a polytropic change, the density and pressure are related according to r where T i s the polytropic index, and p „ and /°° are the unperturbed pressure and density. The two p a r t i c u l a r cases of interest here are that in which the gas undergoes an adiabatic change, and that in which i t undergoes an isothermal change. In the isothermal case, dT=0, so dQ/dT=oo f and T =1 (c.f . Cox and G i u l i ( 1 968)). Physically, the molecules in the gas radiate energy s u f f i c i e n t l y rapidly 34 t h a t the t e m p e r a t u r e of the gas remains c o n s t a n t t h r o u g h o u t . In the a d i a b a t i c case dQ=0, so dQ/dT=0, and p / cv=5/3. c p and c v a r e the s p e c i f i c h e a t s of the f l u i d a t c o n s t a n t p r e s s u r e and volume r e s p e c t i v e l y ( c . f . Cox and G i u l i ) . The assumption u n d e r l y i n g t h i s i s t h a t c o o l i n g i s u n i m p o r t a n t : no energy i s co n d u c t e d or r a d i a t e d by the gas as i t streams p a s t the o b j e c t . Thus, the c h o i c e of p o l y t r o p i c index embodies as s u m p t i o n s c o n c e r n i n g the r a d i a t i o n of energy by the m o l e c u l e s i n the gas. ( i i i ) Frames of R e f e r e n c e : I t i s assumed t h a t the f l o w i s stea d y i n the frame of r e f e r e n c e i n which the o b j e c t i s a t r e s t , so i t i s o f t e n c o n v e n i e n t t o . work e i t h e r i n a s p h e r i c a l p o l a r or a c y l i n d r i c a l c o o r d i n a t e system whose o r i g i n i s f i x e d t o the moving o b j e c t (see f i g u r e a - 1 ) . F i g u r e a-1. 35 The ( a , 6 ) c o o r d i n a t e s a r e r e l a t e d t o t h e ( x , r ) c o o r d i n a t e s a c c o r d i n g t o 36 (b) L i n e a r i z e d F l u i d Mechanics The n a t u r a l s t a r t i n g p l a c e f o r a f l u i d m e c h a n i c a l s t u d y of d y n a m i c a l f r i c t i o n , i s w i t h a l i n e a r i z e d t r e a t m e n t . In the frame of r e f e r e n c e i n which the medium i s a t r e s t and the o b j e c t i s moving w i t h speed V, E u l e r ' s e q u a t i o n ( a - 1 ) , the e q u a t i o n of c o n s e r v a t i o n of mass ( a - 2 ) , and the p o l y t r o p i c e q u a t i o n of s t a t e ( a - 7 ) , may be w r i t t e n CL+ P bp + yo C V • - v = O b -a . 7k These a r e combined by t a k i n g ^7-(b-1), | ^ ( b - 2 ) , and u s i n g (b-3) t o e l i m i n a t e p r e s s u r e . The q u a n t i t i e s p, p , and v a r e p e r t u r b e d : p = p. * *p b-M 37 p 0 , v„ , and /° 0 a r e c o n s t a n t s . I t i s assumed t h a t &P<<P0 and S,f> «f0 , and t h a t t h e r e i s a frame o f r e f e r e n c e i n w h i c h t h e u n p e r t u r b e d medium i s a t r e s t : v. =0. When t h e p o l y t r o p i c r e l a t i o n i s p e r t u r b e d t h i s way, i t becomes w h i c h i s i n d e p e n d e n t of t h e p o l y t r o p i c i n d e x . H e r e t h e e x p r e s s i o n f o r t h e sp e e d o f sound i n a f l u i d , h as been u s e d . The l i n e a r i z e d model w i l l t h e r e f o r e y i e l d t h e same r e s u l t , i n d e p e n d e n t o f whether t h e medium i s assumed t o u n d e r g o an a d i a b a t i c or an i s o t h e r m a l change i n t h e p r e s e n c e of t h e o b j e c t . U s i n g e x p r e s s i o n ( a - 1 0 ) , t h e l i n e a r i z e d e q u a t i o n becomes F i g u r e b - 1 d e f i n e s t h e n o t a t i o n . 38 2 = 0 V=0 Figure b-1 In t h i s coordinate system, Poisson's equation i s S}X = - &Cr"> Sc-x^ ) r b -8 where x=z-Vt. Inserting t h i s and the c y l i n d r i c a l Laplacian into (b-7), and defining b-=) yi e l d s ^ • r = SL&tl^ y Q „ ^CrO ^ ( K ) b - K Symmetry considerations dictate that there i s no angular dependence. Equation (b-10) can be solved using Fourier-Hankel transforms (cf. Dokuchaev (1963)) (see appendix A), and the results are as follows. 39 ( i ) S u b s o n i c C a s e : In t h e c a s e where t h e o b j e c t i s moving s u b s o n i c a l l y , The i s o d e n s i t y s u r f a c e s ( p i c t u r e d i n f i g u r e b-2) a r e e l l i p s o i d s c e n t r e d on t h e o b j e c t , w i t h m a j o r a x i s p e r p e n d i c u l a r t o t h e o b j e c t ' s d i r e c t i o n o f m o t i o n . The e c c e n t r i c i t y of t h e e l l i p s o i d s i s fc=V/c, so a s V a p p r o a c h e s t h e s p e e d of sound i n t h e medium, t h e i s o d e n s i t y c o n t o u r s become i n c r e a s i n g l y e c c e n t r i c . F o r V=0 t h e p e r t u r b a t i o n becomes s p h e r i c a l l y s y m m e t r i c a b o u t t h e o b j e c t . I t i s i n t e r e s t i n g t o compare t h i s w i t h t h e r e s u l t s o f Danby and Camm ( 1 9 5 7 ) , and Danby and B r a y (1967) i n s e c t i o n ( c ) of C h a p t e r I . E x p r e s s i o n ( c - 1 ) , d e r i v e d by Danby and Camm f o r t h e d e n s i t y enhancement i n a medium w i t h no t h e r m a l m o t i o n , was n o t sym m e t r i c a b o u t t h e o b j e c t . I n c l u d i n g p r e s s u r e r e s u l t s i n an e x p r e s s i o n w h i c h , i n t h e s u b s o n i c c a s e , i s s y m m e t r i c a b o u t t h e o b j e c t . F u r t h e r m o r e , Danby and B r a y ' s n u m e r i c a l r e s u l t s i n d i c a t e t h a t when t h e r m a l m o t i o n i s i n c l u d e d , t h e s u b s o n i c c o n t o u r s ( i e . V -1 m/kT 1 <1) a r e not c e n t r e d on t h e o b j e c t , whereas i n t h e f l u i d m e c h a n i c a l t r e a t m e n t , t h e y a r e . T h e s e d i f f e r e n c e s a r i s e b e c a u s e i n t h e p r e s e n t c a s e , t h e t r a j e c t o r i e s o f t h e f l u i d e l e m e n t s , and hence t h e d e n s i t y , a r e d e t e r m i n e d by V p as w e l l a s ^ " ^ , w h i l e i n j£>0 so 40 t h e o r b i t t h e o r y b a s e d m o del, even when t h e t h e r m a l m o t i o n was i n c l u d e d , t h e t r a j e c t o r i e s o f t h e p a r t i c l e s were d e t e r m i n e d o n l y by \ 1 1 M F i g u r e b-2. ( i i ) S u p e r s o n i c C a s e : In t h e c a s e where t h e o b j e c t i s moving s u p e r s o n i c a l l y (-^<0), t h e d e n s i t y p e r t u r b a t i o n i s • X 1 s i x r x t-13 41 The i s o d e n s i t y s u r f a c e s a r e h y p e r b o l o i d s w i t h t h e o b j e c t a t t h e c e n t r e ( s e e f i g u r e b - 3 ) . The e c c e n t r i c i t y i s a g a i n e q u a l F i g u r e b-3. Danby and B r a y ' s r e s u l t s a r e s u g g e s t i v e o f t h i s i n t h a t t h e i r c o n t o u r s e x t e n d o n l y downstream o f t h e o b j e c t , and become i n c r e a s i n g l y narrow w i t h l a r g e V. F i g u r e b-4 shows t h e change i n t h e i s o d e n s i t y c o n t o u r s of t h e d e n s i t y p e r t u r b a t i o n as V goes f r o m z e r o t o i n f i n i t y . vj £ = ° F i g u r e b-4. 42 One f e a t u r e w h i c h was n o t p r e s e n t i n t h e s u b s o n i c c a s e i s t h e s i n g u l a r i t y a t xx=-s?rx . I t i n d i c a t e s t h e p r e s e n c e of a s h o c k , w h i c h i s a l i n e w i t h s l o p e ? 2 £ = ^ - i J x , p a s s i n g t h r o u g h t h e o r i g i n , as p i c t u r e d i n f i g u r e b-3. The l i n e a r i z e d t r e a t m e n t c o n t a i n s i n s u f f i c i e n t i n f o r m a t i o n t o d e t e r m i n e whether i t i s a t a i l o r a bow s h o c k . ( i i i ) D r a g F o r c e : I n b o t h t h e s u b s o n i c and s u p e r s o n i c c a s e , t h e r a d i a l d e p e n d e n c e of t h e d e n s i t y p e r t u r b a t i o n i s r o u g h l y 1/a ( s e e f i g u r e a - 1 ) , and i s i n d e p e n d e n t o f t h e p o l y t r o p i c e x p o n e n t . In t h e s u b s o n i c c a s e , t h e p e r t u r b a t i o n i s s y m m e t r i c f o r e and a f t , and t h e r e i s no n e t d r a g f o r c e e x e r t e d on t h e o b j e c t . I n t h e s u p e r s o n i c c a s e , an e n h a n c e d d e n s i t y wake, forms b e h i n d t h e o b j e c t . I t d e c e l e r a t e s t h e o b j e c t a t a r a t e g i v e n by I n s e r t i n g (b-13) f o r t h e d e n s i t y , and i n t e g r a t i n g o v e r a l l volume, y i e l d s ( s e e a p p e n d i x B f o r c a l c u l a t i o n ) : dV = - u n G,"1 M /p. di V 1 ( i - AlhL ) In r The l i n e a r i z e d model t h u s i n d i c a t e s t h a t i n t h e c a s e o f s u b s o n i c m o t i o n , a t e s t o b j e c t e x p e r i e n c e s no d y n a m i c a l f r i c t i o n , w h i l e i n t h e c a s e of s u p e r s o n i c m o t i o n , t h e e x p r e s s i o n f o r i t s d e c e l e r a t i o n d i v e r g e s l o g a r i t h m i c a l l y 43 w i t h d i s t a n c e , a s i n t h e o r b i t b a s e d m o d e l s . T h e r e i s no r e a s o n t o e x p e c t n o n - l i n e a r e f f e c t s t o be i m p o r t a n t f a r from t h e o b j e c t : t h e l i n e a r i z e d model p r o v i d e s an a c c u r a t e d e s c r i p t i o n of t h e o u t e r r e g i o n s of t h e d e n s i t y enhancement. A n o n - l i n e a r i z e d model w i l l s t i l l be e x a m i n e d , however s i n c e t h e r e s u l t s of C h a p t e r I i n d i c a t e t h a t i t i s t h e c u m u l a t i v e e f f e c t s o f d i s t a n t p a r t i c l e s t h a t p l a y an i m p o r t a n t r o l e i n d e c e l e r a t i n g t h e o b j e c t , i t i s n o t e x p e c t e d t h a t any n o n - l i n e a r e f f e c t s w i l l s i g n i f i c a n t l y i n f l u e n c e t h e e x p r e s s i o n f o r t h e d y n a m i c a l f r i c t i o n a l r e a d y o b t a i n e d . 44 (c ) N o n - L i n e a r i z e d F l u i d M e c h a n i c s The r e s u l t s o f s e c t i o n ( b ) , where a l i n e a r i z e d t r e a t m e n t o f d y n a m i c a l f r i c t i o n was p r e s e n t e d , showed t h a t t h e d e n s i t y p e r t u r b a t i o n c r e a t e d by an o b j e c t moving s u b s o n i c a l l y i s s y m m e t r i c , w h i l e i n t h e s u p e r s o n i c c a s e , an e n h a n c e d d e n s i t y wake forms downstream o f t h e o b j e c t . The r a d i a l d e p e n d e n c e o f t h e p e r t u r b a t i o n i s i n d e p e n d e n t of t h e p o l y t r o p i c e x p o n e n t . A l t h o u g h t h e l i n e a r i z e d t r e a t m e n t p r o v i d e s an a c c u r a t e d e s c r i p t i o n o f t h e p e r t u r b a t i o n f a r from t h e o b j e c t , i t i s i n t e r e s t i n g t o see how t h e r e s u l t s o b t a i n e d f r o m a n o n - l i n e a r i z e d a n a l y s i s d i f f e r from t h o s e of t h e l i n e a r i z e d c a s e . I n v i e w o f t h e d i f f i c u l t i e s i n v o l v e d i n a f u l l , n o n - l i n e a r i z e d t r e a t m e n t of t h e d y n a m i c a l i n t e r a c t i o n between a moving o b j e c t and t h e medium t h r o u g h w h i c h i t moves, i t i s • i n s t r u c t i v e t o i n v e s t i g a t e t h e s i m p l i f i e d c a s e i n w h i c h t h e o b j e c t i s a t r e s t w i t h r e s p e c t t o t h e medium. A l t h o u g h t h e r e i s o b v i o u s l y no d y n a m i c a l f r i c t i o n i n t h i s c a s e , t h e r e s u l t s m i g h t be s u g g e s t i v e o f what t o e x p e c t i n t e r m s o f t h e r a d i a l d e p e n d e n c e o f t h e d e n s i t y p e r t u r b a t i o n c a u s e d by a moving o b j e c t . T.o . b e g i n w i t h , t h e i m p l i c a t i o n s o f t h e a s s u m p t i o n t h a t t h e medium u n d e r g o e s a p o l y t r o p i c c hange i n t h e p r e s e n c e o f t h e o b j e c t ' s g r a v i t a t i o n a l f i e l d , a r e e x a m i n e d . A p o l y t r o p i c change i s one t h a t i s c a r r i e d o u t 45 i n f i n i t e s i m a l l y close to thermodynamic equilibrium, and one of the conditions of thermodynamic equilibrium i s hydrostatic equilibrium: outward directed pressure gradients must balance inward directed g r a v i t a t i o n a l forces. In the l i m i t i n g case where the object i s at rest with respect to the medium, and everything i s spherically symmetric, th i s condition determines the r a d i a l dependence of the density enhancement as follows (cf. Cox and G i u l i (1968)). Consider a f l u i d element a distance a from the object, as pictured in figure c-1 . M Figure c-1. The pressure gradient gives r i s e to a force on the surface A of the element which i s ^6 The g r a v i t a t i o n a l f o r c e on t h e e l e m e n t due t o t h e o b j e c t i s The c o n d i t i o n f o r h y d r o s t a t i c e q u i l i b r i u m i s t h a t t h e s e two f o r c e s be e q u a l and o p p o s i t e : E l i m i n a t i n g p r e s s u r e u s i n g t h e p o l y t r o p i c r e l a t i o n (a-9) g i v e s r - x P c-M H e r e t h e e x p r e s s i o n f o r t h e s p e e d o f sound i n a f l u i d , c1- = r^. c- s P° has been u s e d . In t h e a d i a b a t i c c a s e , P=5/3, and s o l v i n g f o r y i e l d s jo - po X_ j _ + c o ^ W t I n t h e i s o t h e r m a l c a s e , T = 1 and s o l v i n g f o r p> g i v e s 47 C 1 a 6/> - />« It i s inter e s t i n g to contrast these results with the results obtained assuming n e g l i g i b l e pressure and temperature. In t h i s case, the p a r t i c l e s in the medium f a l l f r e e l y in the g r a v i t a t i o n a l f i e l d of the object, so the i r motion i s one of r a d i a l streaming towards the object. The ra d i a l acceleration i s <^3= n!~ , and i s due to the gra v i t a t i o n a l force: so the r a d i a l v e l o c i t y of the p a r t i c l e s i s Conservation of mass requires that so a1 and a" c - ° l d a 48 The r a d i a l d e p endence i s t h e same as i n t h e c a s e where p r e s s u r e g r a d i e n t s and g r a v i t a t i o n a l f o r c e s a r e e x a c t l y b a l a n c e d , so t h a t t h e r e i s no i n f l o w o f m a t e r i a l t o w a r d s t h e o b j e c t . T h i s i n d i c a t e s t h a t , a s t h e r e s u l t s of C h a p t e r I s u g g e s t e d , t h e a c c r e t i o n p r o c e s s has no s i g n i f i c a n t e f f e c t on t h e shape of t h e d e n s i t y enhancement c r e a t e d by t h e o b j e c t . O b v i o u s l y t h e r e i s no d r a g f o r c e on t h e o b j e c t i f i t i s a t r e s t r e l a t i v e t o t h e medium, so i n o r d e r t o s t u d y d y n a m i c a l f r i c t i o n , t h e c a s e where t h e o b j e c t i s moving must be e x a m i n e d . Hunt ( 1 9 7 1 ) , S p i e g e l ( 1 9 7 0 ) , and Ruderman and S p i e g e l (1971) p r e s e n t n o n - l i n e a r i z e d a n a l y s e s o f t h e b e h a v i o u r o f a f l u i d i n t h e p r e s e n c e of a g r a v i t a t i n g o b j e c t moving w i t h v e l o c i t y V w i t h r e s p e c t t o t h e medium. Hunt (1971) does a n u m e r i c a l s t u d y o f t h e above s i t u a t i o n , f o r b o t h s u b s o n i c and s u p e r s o n i c m o t i o n , i n t h e c a s e where t h e medium u n d e r g o e s an a d i a b a t i c c h a n g e . He u s e s c o n s e r v a t i o n e q u a t i o n s ( a - 2 ) , ( a - 5 ) , and (a-6) ( w i t h no t h e r m a l c o n d u c t i v i t y term) , and e q u a t i o n o f s t a t e (a-8) w i t h c P - c v = v-(E i s t h e sum o f i n t e r n a l and k i n e t i c e n e r g i e s . ) He c o n s i d e r s t h e o b j e c t t o be a p o i n t s o u r c e , and i n t e g r a t e s t h e f u l l t i m e - d e p e n d e n t f l u i d e q u a t i o n s from a g i v e n i n i t i a l 49 s t a t e u n t i l a s t e a d y s t a t e i s a c h e i v e d ( i n t h e frame of r e f e r e n c e i n w h i c h t h e o b j e c t i s a t r e s t . ) As i n t h e l i n e a r i z e d t r e a t m e n t , he f i n d s t h a t i n t h e s u b s o n i c c a s e , t h e i s o d e n s i t y c o n t o u r s a r e s y m m e t r i c a b o u t t h e o b j e c t , w h i l e i n t h e s u p e r s o n i c c a s e , a d e n s i t y enhancement forms downstream. H i s r e s u l t s do not y i e l d an e x a c t a n a l y t i c e x p r e s s i o n f o r t h i s d e n s i t y enhancement. An a p p r o x i m a t e e x p r e s s i o n d e s c r i b i n g t h e i n n e r r e g i o n may be d e r i v e d , and i t a g r e e s w i t h t h e s p h e r i c a l l y s y mmetric c a s e a l r e a d y d i s c u s s e d : t h e p e r t u r b a t i o n f a l l s o f f as p — _ L c->3 I f t h e d e n s i t y c o n t i n u e d t o f a l l o f f a t t h i s r a t e i n t h e o u t e r r e g i o n s o f t h e p e r t u r b a t i o n , t h e model would y i e l d an e x p r e s s i o n f o r t h e d r a g f o r c e w i t h no d i v e r g e n c e . However, t h e r e i s no r e a s o n t o e x p e c t t h a t i t d o e s t h i s : i t has a l r e a d y been m e n t i o n e d t h a t n o n - l i n e a r e f f e c t s a r e not e x p e c t e d t o p e r s i s t a t l a r g e d i s t a n c e s f r o m t h e o b j e c t ( o r f r o m t h e s h o c k ) . In c o n n e c t i o n w i t h t h e l a t t e r , H u n t ' s r e s u l t s i n d i c a t e t h a t t h e s hock i s a bow s h o c k : t h e l i n e a r i z e d t r e a t m e n t d i d n o t y i e l d enough i n f o r m a t i o n t o d i s t i n g u i s h w h e t h e r i t p r e c e e d e d o r f o l l o w e d t h e o b j e c t . 50 . . S p i e g e l .( 1 970) and Ruderman and S p i e g e l (1971) a l s o p r e s e n t a n o n - l i n e a r i z e d e x a m i n a t i o n o f t h e b e h a v i o u r o f a gas s t r e a m i n g p a s t a g r a v i t a t i n g o b j e c t . They assume t h a t t h e gas u p s t r e a m i s c o l d , and t h e y a d o p t an o r b i t t h e o r y d e s c r i p t i o n . I n p a r t i c u l a r t h e y use t h e e x p r e s s i o n g i v e n by Danby and Camm (1957) ( s e e C h a p t e r I , e q u a t i o n ( c - 1 ) ) t o d e s c r i b e t h e d e n s i t y of t h e i n c i d e n t f l o w . Downstream o f the' o b j e c t t h e y use t h e e q u a t i o n s o f f l u i d m e c h a n i c s t o d e s c r i b e t h e f l o w . In p a r t i c u l a r t h e y use t h e s t e a d y s t a t e c o n s e r v a t i o n e q u a t i o n s (a-2) and ( a - 5 ) . In t h e l a t t e r t h e y i n c l u d e C o r i o l i s and c e n t r i f u g a l f o r c e s , b ut i g n o r e t h e g r a v i t a t i o n a l f i e l d of t h e o b j e c t , a s t h e y assume t h a t t h e f l u i d e l e m e n t s a r r i v e on t h e a c c r e t i o n a x i s s u f f i c i e n t l y f a r downstream t h a t t h e g r a v i t a t i o n a l f o r c e s due t o t h e o b j e c t a r e n e g l i g i b l e . ( T h e i r model i s c o n c e r n e d s p e i f i c a l l y w i t h an e x t e n d e d o b j e c t s u c h as a g a l a x y , r a t h e r t h a n w i t h an o b j e c t w h i c h c a n be t r e a t e d as a p o i n t s o u r c e . ) They a l s o use a p o l y t r o p i c e q u a t i o n of s t a t e , d e r i v e d from ( a - 9 ) . The downstream and u p s t r e a m f l o w s a r e matched a c r o s s a c o n i c a l s h o c k , a s s u g g e s t e d by t h e l i n e a r i z e d t r e a t m e n t . F o r p h y s i c a l r e a s o n s , p e r t a i n i n g m a i n l y t o t h e n a t u r e o f t h e o b j e c t , t h e y assume t h e shock i s a t a i l s h o c k , r a t h e r t h a n a bow s h o c k as H u n t ' s a n a l y s i s i n d i c a t e s . The e x p r e s s i o n they f i n d i o r the d e n s i t y p e r t u r b a t i o n i s /° *- i A Pee) c-ii+ 51 R i s the radius of the object. F(9) depends on the polytropic index , but the r a d i a l dependence does not. The density f a l l s off as 1/a as in the l i n e a r i z e d case, no matter whether the f l u i d i s assumed to be undergoing an adiabatic or an isothermal change. It i s easy to see that, as in the l i n e a r i z e d case, the 1/a behaviour of the density perturbation w i l l lead to an expression for the dynamical f r i c t i o n which diverges logarithmically with distance. The results presented in th i s section indicate that, although a non-linearized treatment provides a more accurate description of the density enhancement in the neighbourhood of the object, and of the shock that forms in the case of supersonic motion, i t does not y i e l d any new information as far as the dynamical f r i c t i o n experienced by the object i s concerned. In the next section, the l i n e a r i z e d model of section (b) i s extended to include the s e l f - g r a v i t y of the medium. 52 (d) A F l u i d M e c h a n i c a l Model of Dynamical F r i c t i o n I n c o r p o r a t i n g the S e l f - G r a v i t y of t h e Medium The problem of i n t e g r a l s which d i v e r g e l o g a r i t h m i c a l l y w i t h d i s t a n c e , e n c o u n t e r e d here i n the c o n t e x t of a massive o b j e c t i n t e r a c t i n g g r a v i t a t i o n a l l y w i t h i t s s u r r o u n d i n g s , a l s o a r i s e s i n c o n n e c t i o n w i t h the i n t e r a c t i o n between a charged p a r t i c l e and the e l e c t r o m a g n e t i c plasma t h r o u g h which i t moves. In the l a t t e r c a s e , the presence of o p p o s i t e l y charged p a r t i c l e s produces a s c r e e n i n g e f f e c t (Debye s c r e e n i n g ) which l i m i t s the s i z e of the r e g i o n which a t e s t p a r t i c l e can i n f l u e n c e , or be i n f l u e n c e d by. There i s t h e r e f o r e a n a t u r a l c u t o f f f o r the d i v e r g i n g i n t e g r a l s a t the Debye l e n g t h . The s i m i l a r i t y between systems of charged p a r t i c l e s i n t e r a c t i n g t h r o u g h Coulomb f o r c e s , and systems of massive o b j e c t s i n t e r a c t i n g t h r o u g h g r a v i t a t i o n a l f o r c e s , has prompted some p e o p l e t o t r y t o f i n d a g r a v i t a t i o n a l a n a l o g of Debye s c r e e n i n g . In e l e c t r o m a g n e t i c plasmas, i t i s the p r e s e n c e of o p p o s i t e l y c h a r g e d p a r t i c l e s t h a t i s r e s p o n s i b l e f o r Debye s c r e e n i n g . In g r a v i t a t i o n a l 'plasmas' however, a l l the p a r t i c l e s have the same 'charge'. A l t h o u g h some a t t e m p t s have been made t o t r e a t o b j e c t s as ' p o s i t i v e ' masses, and empty spaces as ' n e g a t i v e ' masses, such n e g a t i v e masses 53 c o u l d not e f f e c t i v e l y s h i e l d o b j e c t s from the g r a v i t a t i o n a l f i e l d s of o t h e r o b j e c t s . No m a t t e r how r a r e f i e d ( i e . how ' n e g a t i v e ' ) a volume of space between two massive o b j e c t s i s , i t cannot p r e v e n t the o b j e c t s from f e e l i n g each o t h e r ' s p r e s e n c e . I t i s t h e r e f o r e q u e s t i o n a b l e whether a g r a v i t a t i o n a l e q u i v a l e n t of Debye s c r e e n i n g , based on the i d e a of p o s i t i v e and n e g a t i v e masses, i s p o s s i b l e . The c l a i m i s sometimes made ( c f . Ruderman and S p i e g e l ) t h a t i n c l u d i n g the s e l f - g r a v i t y of the medium produces a s h i e l d i n g e f f e c t s i m i l a r t o Debye s c r e e n i n g . In what f o l l o w s , i t i s shown t h a t , a l t h o u g h t h e r e i s no e v i d e n c e f o r such a s c r e e n i n g e f f e c t , i n c l u d i n g the s e l f - g r a v i t a t i o n of the medium does i m p l y a c u t o f f f o r the d i v e r g i n g i n t e g r a l s . However, as w i l l be p o i n t e d o u t , t h i s i s a m a t h e m a t i c a l c u t o f f , and does not have the same p h y s i c a l s i g n i f i c a n c e as the Debye l e n g t h . .. C o n s i d e r a s e l f - g r a v i t a t i n g medium w i t h d e n s i t y p , v e l o c i t y or , and p r e s s u r e p. The e q u a t i o n s of f l u i d mechanics d e s c r i b i n g t h i s medium ar e 54 These are the same as those in section (a), .except that the gr a v i t a t i o n a l potential term on the right hand side of Euler's equation i s due here to the s e l f - g r a v i t y of the medium, rather than to the presence of a massive object. Now suppose something perturbs the medium s l i g h t l y , p, o , and v are changed as in the l i n e a r i z e d case in section (b): (d - 1 ) through (d - 3 ) are combined and l i n e a r i z e d as in section (b), and Poisson's equation, i s used. The result i s 6^ V = o el- h This equation (the Jeans wave equation), i s e s s e n t i a l l y a Klein-Gordon equation with a negative mass term. The general solution i s ft + K e d-1 55 Inserting t h i s in (d-6) y i e l d s the dispersion r e l a t i o n 00 r - c l V •+ u rr Gy© = e> a- 8 which determines the frequency w and the wave number k of the perturbation. There are several p o s s i b i l i t i e s for the temporal and spa t i a l behaviour of the perturbation. Saslaw (1968) for instance, considers a perturbation which grows exponentially with time, and claims that there is a c o l l e c t i v e shielding effect that causes the perturbation to die off quickly at distances greater than the Jeans length. He derives an expression describing the s p a t i a l dependence of the density perturbation which i s a damped sinusoid, and i t i s not clear that t h i s substantiates his claim. The physical explanation he of f e r s for the shielding i s that choosing a s p e c i f i c time dependence forces the perturbation to be s p a t i a l l y coherent, and t h i s coherenence can only be enforced in a region that i s small enough so that information can propagate through i t in a time less than the c h a r a c t e r i s t i c time scale on which the perturbation grows. For further discussion of Jeans theory of 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 , see for example Layzer (1964). For the present purposes, the type of behaviour that i s of interest, i s that in which the perturbation i s marginally 56 stable (time independent). Any perturbation which i s not time-independent leads to behaviour which i s not stationary in the rest frame of the object, and thus invalidates the assumption of s t a t i o n a r i t y , upon which the present discussion i s based. For a time-independent perturbation, LOT = O In t h i s case (d-8) becomes JLl = ur, 6. f>D = ~ J . d kj i s the Jeans number, and i s the Jeans length. The l a t t e r corresponds to the c h a r a c t e r i s t i c size of the largest possible volume of s e l f - g r a v i t a t i n g f l u i d which would be stable under some perturbing influence . The Jeans length thus furnishes an upper cutoff for the diverging i n t e g r a l s . It does not, however, have the same physical si g n i f i c a n c e as the Debye length. Whereas the Debye length delimits the region which a test p a r t i c l e i s capable of e f f e c t i n g or being effected by, the Jeans length delimits the largest region in which an assumption underlying the model, namely that the flow i s stationary in the object's rest frame, remains v a l i d . It does not represent a region outside of which the object's influence cannot be f e l t . In t h i s sense, i t i s a mathematical cutoff rather than a physical one. 57 I n s t e a d o f c o n s i d e r i n g s e p a r a t e l y t h e c a s e o f a m a s s i v e o b j e c t w i t h a n o n - s e l f - g r a v i t a t i n g medium, and t h a t o f a s e l f - g r a v i t a t i n g medium w i t h no m a s s i v e o b j e c t , i t i s i n t e r e s t i n g t o i n c l u d e b o t h i n t h e same m o d e l . R a t h e r t h a n s i m p l y p r o v i d i n g a m a t h e m a t i c a l u p p e r c u t o f f f o r a d i v e r g i n g i n t e g r a l , s e l f - g r a v i t y m i g h t have t h e e f f e c t o f s t o p p i n g t h e i n t e g r a l f r o m d i v e r g i n g i n t h e f i r s t p l a c e . I n c l u d i n g b o t h e f f e c t s i n t h e same l i n e a r i z e d model i n v o l v e s s o l v i n g t h e e q u a t i o n ( s e e a p p e n d i x C) 1 ^ y _ V^Sf - fc3 fy> - in &Nfe W f J UTO A g a i n t h e a s s u m p t i o n has been made t h a t t h e p e r t u r b a t i o n i s t i m e i n d e p e n d e n t . E q u a t i o n ( d - 1 0 ) i s s o l v e d f o r t h e c a s e o f an o b j e c t t r a v e l l i n g s u b s o n i c a l l y , i n a p p e n d i x C. The r e s u l t i s %yo = (^fyo, cos fca (•xl + A l r ^ C 1 (•*>- - ^ ^ V * US? csi- II T h i s d i f f e r s f r o m t h e n o n - s e l f - g r a v i t a t i n g c a s e by a f a c t o r 1 COi The s u p e r s o n i c enhancement i s n o t as e a s i l y f o u n d . However, i n t h e c a s e i n w h i c h s e l f g r a v i t y i s n e g l e c t e d , t h e a n g u l a r d e p e n d e n c e of ' t h e s u b s o n i c and s u p e r s o n i c p e r t u r b a t i o n s i s t h e same. The o n l y d i f f e r e n c e between them i s t h a t i n t h e s u p e r s o n i c c a s e , t h e r e i s a s h o c k a t 58 XL= Si?r"1 , upstream of which, the density i s unenhanced. Since there i s no reason to expect that including s e l f - g r a v i t y w i l l a l t e r t h i s aspect of the non-self-gravitating r e s u l t s , expression (d-11) may be used to examine the drag force exerted on the object by the medium. For t h i s purpose, i t i s convenient to switch to spherical polar coordinates (see section ( a ) ( i i i ) ) . Expression (d-11) then becomes c o s ol-which i s of the form _|_ c o s ^ O -The deceleration i s given as before by Supressing the constants and the angular dependence, t h i s gives a 1 f i C P s k . a \ a ' 1 d c ^ 1 V o. a 59 This integral diverges at a=0, but i f an appropriate inner cutoff ( c . f . Chapter I) i s introduced, then (d-14) y i e l d s a f i n i t e expression for the deceleration of the object. However, as has been noted already, including s e l f - g r a v i t y means that only systems smaller than the Jeans length may be considered: any larger system i s unstable against the object's perturbing influence, and may not be described using a model which assumes stationary behaviour in the object's rest frame. The conclusion i s therefore that including the s e l f - g r a v i t y of an i n f i n i t e medium does not lead to a f i n i t e expression for the dynamical f r i c t i o n suffered by a test object. At best i t furnishes an outer cutoff at the Jeans length, but t h i s cutoff i s mathematical rather than physical. 60 (e) Summary In t h i s c h a p t e r , f l u i d m e c h a n i c a l models o f d y n a m i c a l f r i c t i o n were s t u d i e d . I t was assumed t h a t t h e medium c o u l d be m o d e l e d a s a n o n - v i s c o u s , c o m p r e s s i b l e f l u i d . I t s u n p e r t u r b e d d e n s i t y was assumed t o be u n i f o r m , and i n t h e frame o f r e f e r e n c e i n w h i c h t h e o b j e c t t r a v e l s w i t h v e l o c i t y V, t h e medium was assumed t o be a t r e s t . In s e c t i o n (b) t h e l i n e a r i z e d e q u a t i o n s o f f l u i d m e c h a n i c s , d e s c r i b i n g a n o n - s e l f - g r a v i t a t i n g f l u i d , were a n a l y s e d . The s u b s o n i c d e n s i t y enhancement t u r n s o u t t o be s y m m e t r i c , w h i l e i n t h e s u p e r s o n i c c a s e , an e n h a n c e d wake forms a l o n g t h e downstream a x i s . I n b o t h c a s e s t h e d e n s i t y f a l l s o f f r o u g h l y as 1/a, i n d e p e n d e n t o f whether t h e medium i s assumed t o u n d e r g o an a d i a b a t i c o r an i s o t h e r m a l c h a n ge, i n t h e p r e s e n c e of t h e o b j e c t ' s g r a v i t a t i o n a l f i e l d . The s u p e r s o n i c p e r t u r b a t i o n e x e r t s a d r a g f o r c e on t h e o b j e c t w h i c h d e c e l e r a t e s i t a t t h e r a t e e- i a In s e c t i o n ( c ) a n o n - l i n e a r i z e d t r e a t m e n t was r e v i e w e d b r i e f l y . To b e g i n w i t h , t h e s p h e r i c a l l y s y m m e t r i c c a s e was e x a m i n e d . I t was f o u n d t h a t , when p r e s s u r e g r a d i e n t s a r e c o n s i d e r e d t o be n e g l i g i b l e compared t o g r a v i t a t i o n a l 61 f o r c e s , t h e d e n s i t y f a l l s o f f a s /°~V Q>>'i . When p r e s s u r e g r a d i e n t s and g r a v i t a t i o n a l f o r c e s e x a c t l y b a l a n c e , t h e d e n s i t y enhancement f a l l s o f f a s / > ~ ( i - * l j i n t n e a d i a b a t i c c a s e , and f>*- «- V c" i n t h e i s o t h e r m a l c a s e . The f a c t t h a t i n t h e a d i a b a t i c c a s e , t h e r a d i a l d e p e n d e n c e o f t h e d e n s i t y enhancement i s t h e same whether o r n o t t h e r e i s an i n f l o w o f m a t e r i a l t o w a r d s t h e o b j e c t s u g g e s t s t h a t , a t l e a s t i n t h e a d i a b a t i c c a s e , t h e a c c r e t i o n p r o c e s s d o e s n o t i n f l u e n c e t h e d y n a m i c a l f r i c t i o n t h a t t h e o b j e c t e x p e r i e n c e s . The c o n c l u s i o n o f t h e o r b i t b a s e d m o dels o f C h a p t e r I , namely t h a t i t i s t h e c u m u l a t i v e e f f e c t o f d i s t a n t m a t e r i a l t h a t p l a y s t h e d o m i n a n t r o l e i n d e c e l e r a t i n g t h e o b j e c t , i s t h u s r e i n f o r c e d by t h e a d i a b a t i c r e s u l t s . A r e v i e w o f two n o n - l i n e a r i z e d a n a l y s e s of t h e f l u i d e q u a t i o n s i n t h e c a s e of a moving o b j e c t , i n d i c a t e d t h a t , i n t h e r e g i o n n e a r t h e o b j e c t , t h e r a d i a l d e p endence of t h e enhancement depends on t h e p o l y t r o p i c e x p o n e n t , but t h a t i n t h e r e g i o n f a r f r o m t h e o b j e c t , i t does n o t . N o n - l i n e a r i z e d e f f e c t s s h o u l d not be i m p o r t a n t f a r f r o m t h e o b j e c t , and as i n d i c a t e d i n C h a p t e r I , t h e d o m i n a n t i n f l u e n c e i n d e c e l e r a t i n g t h e o b j e c t i s due t o t h e c u m u l a t i v e e f f e c t o f m a t e r i a l f a r f r o m t h e o b j e c t . The n o n - l i n e a r i z e d a n a l y s i s t h e r e f o r e y i e l d s no new i n f o r m a t i o n as f a r as d y n a m i c a l f r i c t i o n i s c o n c e r n e d . In s e c t i o n ( d ) , t h e l i n e a r i z e d model of s e c t i o n (b) was e x t e n d e d t o i n c l u d e s e l f - g r a v i t y . I t was d e m o n s t r a t e d t h a t i n c l u d i n g s e l f - g r a v i t y d o e s n o t p r o v i d e s h i e l d i n g e f f e c t a n a l o g o u s t o Debye s c r e e n i n g , and d o e s n o t s t o p t h e 62 i n t e g r a l s f r o m d i v e r g i n g . I t d o e s however, p r o v i d e a c u t o f f a t t h e J e a n s l e n g t h , w h i c h d e l i m i t s t h e maximum s i z e o f a volume of s e l f - g r a v i t a t i n g gas t h a t would be s t a b l e a g a i n s t p e r t u r b a t i o n s . T h i s c u t o f f i s r e a l l y more m a t h e m a t i c a l t h a n p h y s i c a l , i n t h a t any l a r g e r r e g i o n would be u n s t a b l e under t h e p e r t u r b i n g i n f l u e n c e o f t h e o b j e c t , and t h e a s s u m p t i o n o f s t a t i o n a r y b e h a v i o u r u n d e r l y i n g t h e model, c o u l d n o t be m a i n t a i n e d . The p r i n c i p l e c o n c l u s i o n s t o be drawn from t h i s a r e t h a t an o b j e c t moving s u b s o n i c a l l y t h r o u g h a n o n - v i s c o u s medium e x p e r i e n c e s no d y n a m i c a l f r i c t i o n , w h i l e t h e e x p r e s s i o n f o r t h e d e c e l e r a t i o n o f an o b j e c t moving s u p e r s o n i c a l l y d i v e r g e s l o g a r i t h m i c a l l y w i t h d i s t a n c e , i n d e p e n d e n t o f whether o r n o t t h e p r e s s u r e o r s e l f - g r a v i t y o f t h e medium i s i n c l u d e d , and i n d e p e n d e n t o f whether t h e medium i s assumed t o u n d e r g o an a d i a b a t i c o r an i s o t h e r m a l c hange i n t h e p r e s e n c e o f t h e o b j e c t . 63 CHAPTER III MODELS OF DYNAMICAL FRICTION BASED ON BINARY ENCOUNTERS Introduction In Chapters I and II, models of dynamical f r i c t i o n were presented, in which a test object was assumed to be t r a v e l l i n g through a medium composed of dust or gas. This chapter and the next examine models of dynamical f r i c t i o n where the medium through which the test object tra v e l s i s composed of ' f i e l d ' objects of roughly the same mass as the test object. It i s convenient to separate these models into two types: those which employ the two-body approximation, and those which do not. The present chapter i s devoted to the former type. The basic assumption underlying these models i s that a test object t r a v e l l i n g through a system of f i e l d objects, may be viewed as undergoing binary encounters with f i e l d objects. In.section (a), a model i s presented in which i t i s assumed that the c o l l i s i o n s are a l l complete, and a l l the momentum transfer occurs at the point of closest approach. The result of t h i s model i s the same as those of the or b i t 64 b a s e d m o dels o f C h a p t e r I . T h i s i s t o be e x p e c t e d s i n c e t h e e x p r e s s i o n s f o r t h e d y n a m i c a l f r i c t i o n o f t h e t e s t o b j e c t , f o u n d i n C h a p t e r I were i n d e p e n d e n t o f t h e mass o f t h e f i e l d p a r t i c l e s : a l l f i e l d p a r t i c l e s f o l l o w t h e same t r a j e c t o r i e s , r e g a r d l e s s o f t h e i r mass. E s p e c i a l l y i n t h e l i m i t o f d i s t a n t m a t e r i a l , t h e t e s t o b j e c t s h o u l d have no way o f d i s t i n g u i s h i n g whether t h e mass i t f e e l s i s smoothed o u t , a s d u s t o r g a s , o r c l u m p e d i n t o o t h e r f i e l d o b j e c t s . I t i s a r g u e d however, a t t h e end o f s e c t i o n ( a ) , t h a t t h e o r b i t m o d e l s , t h e f l u i d m o d e l s , and t h i s s i m p l e b i n a r y e n c o u n t e r model a l l p o r t r a y a s t a t i o n a r y s i t u a t i o n w h i c h , i f c o l l i s o n s w i t h i n f i n i t e i m p a c t p a r a m e t e r s a r e i n c l u d e d , t a k e s an i n f i n i t e t i m e t o e s t a b l i s h . To a v o i d t h i s , t h e t i m e d e p e n d e n c e , p a r t i c u l a r l y o f d i s t a n t e n c o u n t e r s , must be i n c l u d e d . M o d e l s i n c o r p o r a t i n g t h i s t i m e d e p e n d e n c e a r e p r e s e n t e d i n s e c t i o n ( b ) . They r e s u l t i n e x p r e s s i o n s f o r t h e d e c e l e r a t i o n o f t h e t e s t o b j e c t , i n w h i c h t h e d i v e r g e n c e w i t h d i s t a n c e i s r e p l a c e d by a d i v e r g e n c e w i t h t i m e . The r e s u l t s a r e summarized i n s e c t i o n ( c ) . 65 (a) A Two-Body C a l c u l a t i o n o f D y n a m i c a l F r i c t i o n i n w h i c h C o m p l e t e C o l l i s i o n s a r e Assumed In t h i s s e c t i o n , a method due t o C h a n d r a s e k h a r , f o r c a l c u l a t i n g t h e d y n a m i c a l f r i c t i o n e x p e r i e n c e d by a t e s t o b j e c t u n d e r g o i n g s u c c e s s i v e b i n a r y e n c o u n t e r s w i t h f i e l d o b j e c t s , i s p r e s e n t e d . I t i n v o l v e s a c a l c u l a t i o n of t h e n e t change i n t h e v e l o c i t y of a t e s t o b j e c t (mass M, v e l o c i t y V) p a s s i n g t h r o u g h a s p h e r i c a l l y s y m m e t r i c d i s t r i b u t i o n o f n o n - i n t e r a c t i n g f i e l d o b j e c t s (mass m, v e l o c i t y v ) , whose p e c u l i a r v e l o c i t i e s obey a M a x w e l l i a n d i s t r i b u t i o n . T h i s i s e q u i v a l e n t t o t h e s i t u a t i o n d e s c r i b e d i n C h a p t e r I , s e c t i o n ( c ) , i n w h i c h t h e t e s t o b j e c t was assumed t o t r a v e l t h r o u g h a s y s t e m o f p a r t i c l e s whose t h e r m a l v e l o c i t i e s obey a M a x w e l l i a n d i s t r i b u t i o n . The r e s u l t i s , as i n C h a p t e r I , an e x p r e s s i o n f o r t h e d e c e l e r a t i o n of t h e t e s t o b j e c t w h i c h d i v e r g e s l o g a r i t h m i c a l l y w i t h i m p a c t p a r a m e t e r . The change i n v e l o c i t y t h a t t h e t e s t o b j e c t s u f f e r s d u r i n g a s i n g l e b i n a r y e n c o u n t e r w i t h a f i e l d o b j e c t i s d e t e r m i n e d u s i n g o r b i t t h e o r y , and t h e e f f e c t o f t h e e n t i r e s y s t e m i s c a l c u l a t e d by summing o v e r t h e i n d i v i d u a l e n c o u n t e r s i n a s u i t a b l e way. The d e t a i l s o f t h e c a l c u l a t i o n a r e l e f t o u t o f t h e p r e s e n t d i s c u s s i o n : t h e y may be f o u n d i n C h a n d r a s e k h a r ( 1 9 4 3 a ) , and i n h i s book P r i n c i p l e s of S t e l l a r  D y n a m i c s . D u r i n g a s i n g l e b i n a r y e n c o u n t e r , a t e s t o b j e c t 66 e x p e r i e n c e s v e l o c i t y c h a n g e s v s S and v x , p a r a l l e l t o and p e r p e n d i c u l a r t o i t s d i r e c t i o n o f m o t i o n r e s p e c t i v e l y . T h e s e i n c r e m e n t s i n v e l o c i t y may be c a l c u l a t e d u s i n g s t a n d a r d o r b i t t h e o r y ( s e e S t e l l a r D y n a m i c s ) . When summed o v e r a l a r g e number o f s u c h e n c o u n t e r s , v x v a n i s h e s a s e x p e c t e d from symmetry c o n s i d e r a t i o n s , b u t v X N d o e s n o t . The n e t change i n t h e v e l o c i t y o f t h e t e s t o b j e c t , AV, due t o a s u c c e s s i o n o f b i n a r y e n c o u n t e r s , i s c a l c u l a t e d f o r a t i m e i n t e r v a l A t w h i c h i s l o n g compared w i t h t h e d u r a t i o n o f a s i n g l e c o l l i s i o n ( i e . l o n g enough f o r t h e most d i s t a n t e n c o u n t e r s t o be c o m p l e t e d ) , b u t s h o r t compared w i t h t h e t i m e r e q u i r e d t o s i g n i f i c a n t l y a l t e r t h e t e s t o b j e c t ' s v e l o c i t y . T h i s c a l c u l a t i o n i n v l o v e s i n t e g r a t i o n s o v e r t h r e e a n g l e s p a r a m e t r i z i n g t h e e n c o u n t e r , o v e r t h e im p a c t p a r a m e t e r , and o v e r t h e i n i t i a l v e l o c i t i e s o f t h e f i e l d o b j e c t s . To f a c i l i t a t e t h e i n t e g r a t i o n s o v e r t h e a n g l e s , s e v e r a l s i m p l i f i c a t i o n s a r e made. To b e g i n w i t h , i t i s assumed t h a t t h e d i s t r i b u t i o n of t h e v e l o c i t i e s o f t h e f i e l d o b j e c t s i s s p h e r i c a l l y s y m m e t r i c . ( C h a n d r a s e k h a r (1943a) d i s c u s s e s b r i e f l y t h e t a s k of g e n e r a l i z i n g t h i s t o a random d i s t r i b u t i o n . ) I n t h e i n t e g r a l o v e r i m p a c t p a r a m e t e r s , i t i s assumed t h a t P h y s i c a l l y , a s i n C h a p t e r I , t h i s amounts t o c o n s i d e r i n g 67 o n l y e n c o u n t e r s w i t h i m p a c t p a r a m e t e r much g r e a t e r t h a n t h e d i s t a n c e a t w h i c h a f i e l d o b j e c t w i t h mass M, t r a v e l l i n g w i t h v e l o c i t y V ( ( v - V ) , (v+V) ), j u s t r e m a i n s g r a v i t a t i o n a l l y bound t o t h e t e s t o b j e c t . T h i s e x c l u d e s c l o s e e n c o u n t e r s , b u t i t i s a r g u e d t h a t s u c h e n c o u n t e r s a r e s u f f i c i e n t l y i n f r e q u e n t , t h a t t h e a p p r o x i m a t i o n i s j u s t i f i e d . A more d e t a i l e d argument t o t h i s e f f e c t i s i n c l u d e d i n s e c t i o n ( b ) . O n l y t h e 'dominant' t e r m i n t h e i n t e g r a t i o n i s r e t a i n e d ( s e e S t e l l a r D y n a m i c s , p p . 6 2 - 6 4 ) . T h i s l e a d s t o t h e s i g n i f i c a n t r e s u l t t h a t , t o t h i s a c c u r a c y , t h e o n l y c o n t r i b u t i o n s t o t h e d e c e l e r a t i o n of t h e t e s t o b j e c t , a r e f r o m f i e l d o b j e c t s whose i n i t i a l v e l o c i t y i s l e s s t h a n t h a t o f t h e t e s t o b j e c t . In o t h e r words, i n o r d e r t o e x p e r i e n c e a d e c e l e r a t i o n , t h e t e s t o b j e c t must be t r a v e l l i n g f a s t e r t h a n t h e a v e r a g e v e l o c i t y v of o b j e c t s i n t h e s y s t e m : A c c o r d i n g t o k i n e t i c t h e o r y , t h e a v e r a g e s p e e d o f t h e members of a s y s t e m i s g i v e n by T h e r e f o r e , i n o r d e r t o be d e c e l e r a t e d , t h e t e s t o b j e c t ' s v e l o c i t y must s a t i s f y t h e i n e q u a l i t y 68 But s i n c e ( c f . C h a p t e r I , s e c t i o n ( c ) ) r t h i s i n e q u a l i t y i s s i m p l y V ^ c Thus, t o a good a p p r o x i m a t i o n , t h e o b j e c t o n l y e x p e r i e n c e s a d e c e l e r a t i o n i f i t i s t r a v e l l i n g s u p e r s o n i c a l l y . T h i s i s t h e same r e s u l t a s t h a t y i e l d e d by t h e f l u i d m e c h a n i c a l model of C h a p t e r I I . F i n a l l y , i t i s assumed t h a t t h e f i e l d o b j e c t s ' v e l o c i t i e s obey a M a x w e l l i a n d i s t r i b u t i o n , and t h a t t h e u n p e r t u r b e d number of f i e l d o b j e c t s p e r u n i t volume, N, i s c o n s t a n t . The r e s u l t i n g e x p r e s s i o n f o r t h e n e t change i n v e l o c i t y of t h e t e s t o b j e c t i n t h e i n t e r v a l o f t i m e A t , i s A M = - Urr Nm NM Q»x \C tA- £ A s i v r a - * 4 S= CO Y*? i s t h e mean s q u a r e v e l o c i t y o f t h e f i e l d p a r t i c l e s , and H e r e , <x i s t h e same q u a n t i t y e n c o u n t e r e d i n C h a p t e r I , 69 s e c t i o n ( c ) : <X - V and Cll(<x) i s t h e e r r o r f u n c t i o n : 4rT 0 T h i s may be compared w i t h t h e e q u i v a l e n t e x p r e s s i o n d e r i v e d i n C h a p t e r I , by w r i t i n g mN=/P„/ and a s s u m i n g m « M . Then (a-7) becomes w h i c h i s t h e same as C h a p t e r I , e x p r e s s i o n ( a - 6 ) , f o r s»GM/Vx , e x c e p t t h a t t h e mean v e l o c i t y of t h e f i e l d o b j e c t s i s l ^ i * , whereas i n ( a - 6 ) i t was V, and t h e f a c t o r K has been i n t r o d u c e d . T h i s f a c t o r K depends on t h e same q u a n t i t y t h a t c r o p p e d up i n t h e o r b i t b a s e d m o del, when t h e t h e r m a l m o t i o n o f t h e p a r t i c l e s was i n c l u d e d ( s e e C h a p t e r 1 ( c ) ) . An a n a l y t i c e x p r e s s i o n f o r t h e d e c e l e r a t i o n of t h e t e s t o b j e c t was n o t o b t a i n e d i n t h a t c a s e , b u t i t was a r g u e d t h a t i n c l u d i n g t h e t h e r m a l m o t i o n o f t h e f i e l d p a r t i c l e s was n o t e x p e c t e d t o a l t e r .the r e s u l t s s i g n i f i c a n t l y . In t h e above a n a l y s i s , t h e p e c u l i a r m o t i o n s of f i e l d o b j e c t s were i n c l u d e d , and t h e r e s u l t s a r e , e x c e p t f o r K, t h e same a s t h e r e s u l t s f o u n d by i g n o r i n g t h e p e c u l i a r 70 m o t i o n s . T h u s , t h e c l a i m o f C h a p t e r I , s e c t i o n ( c ) , t h a t t h e r m a l m o t i o n among t h e f i e l d p a r t i c l e s does n o t i n f l u e n c e s i g n i f i c a n t l y t h e d y n a m i c a l f r i c t i o n s u f f e r e d by t h e t e s t o b j e c t , i s s u b s t a n t i a t e d . L ee (1968) a l s o c a l c u l a t e s t h e d y n a m i c a l f r i c t i o n s u f f e r e d by a t e s t o b j e c t u s i n g a method b a s e d on summing o v e r i n d e p e n d e n t b i n a r y e n c o u n t e r s , and l i k e C h a n d r a s e k h a r , he d e r i v e s an e x p r e s s i o n w h i c h d i v e r g e s l o g a r i t h m i c a l l y w i t h d i s t a n c e . The o r b i t m odels o f C h a p t e r I , t h e f l u i d m o d e l s o f C h a p t e r I I , and t h e b i n a r y e n c o u n t e r model j u s t d i s c u s s e d , a r e a l l b a s e d on t h e a s s u m p t i o n t h a t t h e f l o w o f t h e medium i n t h e r e s t frame o f t h e t e s t o b j e c t i s s t a t i o n a r y . The i m p l i c a t i o n s of t h i s a s s u m p t i o n a r e most e a s i l y s e e n i n t h e c o n t e x t of t h e o r b i t m o d e l s . ( R e f e r t o t h e d i s c u s s i o n a t t h e b e g i n n i n g of C h a p t e r I , s e c t i o n ( a ) . ) I f one i m a g i n e s t h e f l o w o f p a r t i c l e s p a s t t h e o b j e c t t o be ' t u r n e d on' a t some p a r t i c u l a r • t i m e , t h e n i n o r d e r f o r t h e s t a t i o n a r y s i t u a t i o n d e s c r i b e d i n t h e model t o be e s t a b l i s h e d , t h e f i r s t wedge o f p a r t i c l e s must have c o m p l e t e d t h e i r t r a j e c t o r i e s f r o m u p s t r e a m t o downstream i n f i n i t y ( i e . between t h e a s y m p t o t e s o f t h e i r t r a j e c t o r i e s ) . P a r t i c l e s w i t h s m a l l i m p a c t p a r a m e t e r s w i l l r e a c h t h e i r d o wnstream a s y m p t o t e r e l a t i v e l y q u i c k l y . P a r t i c l e s w i t h l a r g e i m p a c t p a r a m e t e r s w i l l t a k e l o n g e r . I t f o l l o w s t h a t , i f i m p a c t p a r a m e t e r s out t o i n f i n i t y a r e c o n s i d e r e d , t h e n i t w i l l t a k e an i n f i n i t e amount o f t i m e f o r t h e s e p a r t i c l e s t o 71 r e a c h t h e i r downstream a s y m p t o t e s , and t h u s , . an i n f i n i t e amount of t i m e must p a s s b e f o r e a s t a t i o n a r y s i t u a t i o n i s e s t a b l i s h e d . In t h e c o n t e x t of t h e b i n a r y e n c o u n t e r m o d e l s , t h e i n t e r v a l o f t i m e r e q u i r e d f o r i n f i n i t e l y d i s t a n t e n c o u n t e r s t o be c o m p l e t e d , i s i n f i n i t e . Hunt (1971) n o t e s t h e same phenomenon i n c o n n e c t i o n w i t h h i s n o n - l i n e a r i z e d f l u i d m e c h a n i c a l a n a l y s i s . I n i n t e g r a t i n g t h e t i m e d e p e n d e n t f l u i d e q u a t i o n s f r o m a g i v e n i n i t i a l s t a t e , u n t i l a s t e a d y s t a t e i s r e a c h e d , he f i n d s t h a t f o r s m a l l r , t h e s o l u t i o n s become s t e a d y q u i t e q u i c k l y , w h i l e f o r l a r g e r , t h e y t a k e a p r o p o r t i o n a l l y l o n g t i m e . The n e x t s e c t i o n d e s c r i b e s m o d i f i e d b i n a r y e n c o u n t e r m o d e ls w h i c h c i r c u m v e n t t h i s p r o b l e m by t a k i n g i n t o a c c o u n t t h e t i m e d e p e n d e n c e o f d i s t a n t c o l l i s i o n s . 72 (b) Time Dependent C o l l i s i o n s As p o i n t e d o u t i n t h e p r e v i o u s s e c t i o n , t o e s t a b l i s h a s t a t i o n a r y r e g i m e i n w h i c h a l l e n c o u n t e r s may be c o n s i d e r e d t o be c o m p l e t e , r e q u i r e s an i n f i n i t e amount o f t i m e . In t h i s s e n s e , t h e s t a t i o n a r y m o d e ls c a n n e v e r s t r i c t l y a p p l y t o any r e a l s i t u a t i o n . To g e t a r o u n d t h i s p r o b l e m , t h e t i m e d e p e n d e n c e , p a r t i c u l a r l y o f d i s t a n t e n c o u n t e r s , must be i n c o r p o r a t e d i n t h e m o d e l . O s t r i k e r and D a v i d s e n ( 1 9 6 8 ) , and Henon (1958) p r e s e n t s u c h m o d e l s . A t t h i s s t a g e , a new f e a t u r e , r e p r e s e n t i n g an e v o l u t i o n t o w a r d s a f u l l y s t a t i s t i c a l m odel, i s i n t r o d u c e d : r a t h e r t h a n d e t e r m i n i n g t h e change i n t h e v e l o c i t y o f a t e s t o b j e c t , i t i s t h e e x p e c t a t i o n v a l u e o f t h i s q u a n t i t y t h a t i s c a l c u l a t e d . - The b a s i c s t r u c t u r e of t h i s c a l c u l a t i o n i s as f o l l o w s ( c . f . Henon ( 1 9 5 8 ) ) . The i n i t i a l s t a t e o f t h e s y s t e m o f o b j e c t s i s d e s c r i b e d by t h e f u n c t i o n p = txcr> dr b-1 w h i c h g i v e s t h e p r o b a b i l i t y t h a t a t t i m e t=0, t h e r e i s an o b j e c t whose c o o r d i n a t e s i n phase s p a c e l i e between f and The change i n t h e v e l o c i t y o f t h e t e s t o b j e c t a f t e r a 73 time T r due to i t s interaction with a single f i e l d object, i s given by where T) i s some function of the f i e l d object's position and v e l o c i t y . In r e a l i t y , the test object i s acted upon simultaneously by a large number of f i e l d objects. However, to avoid the complexities of a f u l l N-body problem, two approximations are made: the interaction of the f i e l d objects with one another i s ignored, and i t i s assumed that the t o t a l change, AV in the test object's v e l o c i t y , may be calculated by adding together the changes due to the individual encounters: AV = Z S CO The former approximation, concerning the s e l f - i n t e r a c t i o n of the medium, was examined in Chapter II, section (d) in the context of a f l u i d mechanical model, and w i l l be discussed further in Chapter V. The l a t t e r approximation i s j u s t i f i e d in the case of distant encounters, where the individual changes are small, but not in the case of close encounters. (The terms 'close' and 'distant' are defined further on.) However, given that the p r o b a b i l i t y of an object undergoing more than one close encounter i s remote, the approximation may s t i l l be used. 74 To cal c u l a t e the expectation value of A V , a function g(t-) i s defined. It represents the perturbation in the test object's v e l o c i t y due to the element of phase space d r . Thus, g ( f ) i s equal to 1} ( T ) i f d r contains a f i e l d object (probability p), and zero i f i t does not (probability 1-p). Equation (b-3) may be rewritten: where the integration extends over a l l phase space. The expectation value of AV i s therefore Since (b-5) becomes (with (b-1)), Thus, < A ^ > = [ \cn * (^  d tr b-8 Similar expressions may be obtained for A V ^ and A V T . The second moments, <*V C A.V->, may also be calculated. 75 For example, from (b-4), 3 * ^ So b- 10 It was assumed that the f i e l d objects are non-interacting, so their v e l o c i t i e s and positions must be uncorrelated. g„(f,) and g*( r i ) are therefore independent, and It turns out ( c f Henon) that, for times small compared with the relaxation time of the system, < A V> i s n e g l i g i b l e compared with < ( A V ) x >. From the d e f i n i t i o n of g ( - r ) , Expression (b-11) therefore becomes < ( A \ I O X > = ( V cn <x(ri dr 76 Similar expressions hold for the other second moments. It should be noted that <( AV„) > represents the dispersion of A V around i t s expectation value, and i s not the same as < A ( V x l )>, the expectation value of the change in the object's squared v e l o c i t y ( i e . i t s kinetic energy). The dispersion < ( 4 V ) 1 > i s related to the d i f f u s i o n process, or the randomization of the test object's peculiar v e l o c i t y . The quantity of interest here i s <LV >, the change in the test object's v e l o c i t y in i t s d i r e c t i o n of motion. To fi n d < A V* >, an expression for T ) , the change in the v e l o c i t y due to a single encounter, must be calculated, and then the integration over a l l phase space must be performed. Two d i f f e r e n t methods are employed by Henon to determine ^ ( T - ) , depending on whether the encounters in question are close or distant. For close encounters, the impact parameter, s, i s small compared with the distance t r a v e l l e d by the f i e l d object in time T: s < < u^T t>-14. (w=lv-vl). In this case, the v e l o c i t i e s at the beginning and end of the interaction are approximately equal to . their asymptotic v e l o c i t i e s at i n f i n i t y , and orbit theory i s used ( c f . Chapter I ) . For distant encounters, a perturbation technique i s employed, which i s accurate as long as the deflections procuced by individual encounters are very small. Since 77 S> = G O-*^ b- is i s the impact distance at which a maximum defl e c t i o n occurs, the perturbation technique may be used for impact parameters such that C0>?-In general however, O i 1 so the regions in which the two methods are v a l i d , overlap. A distance 1 i s therefore chosen such that Gn(/* + N O ^ c I <L<L l a - 1 * and t h i s distance i s used as the change over point between the two methods. It turns out that 1 does not appear in the f i n a l r e s u l t s , so i t s value i s unimportant. The perturbation technique i s as follows. X and x are the coordinates of the test object (mass M) and a f i e l d object (mass m) respectively, at some time t. They are expanded in a series with respect to the g r a v i t a t i o n a l constant G: 78 b - \°i x = Xo + l , G •» -xiG,1 +• • E q u i v a l e n t e x p r e s s i o n s h o l d f o r t h e y - d i r e c t i o n . I f G were z e r o ( i e . i f M and m d i d n o t i n t e r a c t ) , t h e n X, x, Y, and y would be X - X. - o X = X . + c o + b - i o v = o Here M i s assumed t o r e m a i n a t t h e o r i g i n of t h e c o o r d i n a t e s y s t e m , w i s t h e r e l a t i v e v e l o c i t y o f M and m, w h i c h i s i n t h e x d i r e c t i o n , and x i s m's i n i t i a l p o s i t i o n . The e q u a t i o n d e s c r i b i n g M's m o t i o n i s X = GA m (x - X S ) b - * i E q u i v a l e n t e x p r e s s i o n s h o l d f o r x, Y, and y . E x p r e s s i o n s (b-19) a r e i n s e r t e d i n t o ( b - 2 1 ) : t h e h i g h e r o r d e r t e r m s d e c r e a s e v e r y r a p i d l y when e x p r e s s i o n (b-16) i s s a t i s f i e d . The f u n c t i o n ^ ( T ) i s t h e change a f t e r t i m e T, i n t h e t e s t o b j e c t ' s v e l o c i t y due t o a s i n g l e e n c o u n t e r . F o r a 79 distant encounter, i t i s therefore given by (from (b-19) and (b-20)), where X,, X t, ... are found by integrating the expressions for X,, X», which are the c o e f f i c i e n t s of successive powers of G, found from (b-21). Having found an expression for ^J* ( T ), Henon proceeds to perform the integrations over a l l phase space. One set of integrals extends over impact parameters 0<s<l, and the other over l<s< «>. The former set contains logarithmic terms in which 1 comes into the argument. The l a t t e r set also contains logarithmic terms, and 1 and wT are present in the argument. When the expressions for s<l and s>l are added, 1 cancels out, and the results are, after retaining only the dominant terms, v Vs-a. v T 3 v e1 C C N U r ^ do Q (-vr^  T J - 1 * ciXr + I a C \ r ) v d"U" b- J-4 Here, v i s the mean value of the f i e l d object's v e l o c i t i e s . The divergence with impact parameter has been replaced by a divergence with time, and the rate of deceleration of 80 the test object i s proportional to TlogT, rather than just to T. Ostriker and Davidsen a r r i v e at similar r e s u l t s , but rather than using a perturbation technique, they take into account e x p l i c i t l y the time dependence of distant encounters. They assume that the momentum transfer in a single distant encounter, in an in t e r v a l of time T, i s given by i Ap IIs) - j ^Ct) eW fa-IS o F i s the force exerted on the f i e l d object by the test object, but here i t i s taken to be a function of time: C\P\X -t l - S l M ^ ' * Close encounters are treated as in section (a), where a l l the momentum transfer i s assumed to occur at the point of closest approach. The major approximation underlying t h i s method, i s that the f i e l d objects follow l i n e a r t r a j e c t o r i e s . This allows considerable mathematical s i m p l i f i c a t i o n , and although i t leads to an inaccurate treatment of close encounters, i t i s argued that t h i s doesn't matter since, as demonstrated in previous chapters using a stationary approach, i t i s the cumulative e f f e c t of distant material that i s important. The 81 l i n e a r a p p r o x i m a t i o n i s more a c c u r a t e i n t h e c a s e o f d i s t a n t e n c o u n t e r s , and O s t r i k e r and D a v i d s e n c l a i m t h a t i t c a n be shown t h a t t h e d i f f e r e n c e between t h e r e s u l t s u s i n g l i n e a r o r b i t s , and t h o s e u s i n g h y p e r b o l i c o r b i t s , i s n e g l i g i b l e . I t w i l l be s e e n t h a t t h i s c l a i m i s s u b s t a n t i a t e d i n t h e c a s e o f <( & V ) 1 >, by t h e agreement between O s t r i k e r and D a v i d s e n ' s r e s u l t s , and t h o s e of Henon, w h i c h were f o u n d by t a k i n g i n t o a c c o u n t t h e d e v i a t i o n s o f f i e l d o b j e c t s f r o m l i n e a r t r a j e c t o r i e s . I t i s n o t s u b s t a n t i a t e d i n t h e c a s e of < AV>. The q u a n t i t y < ( A p ) 1 > i s c a l c u l a t e d by summing o v e r t i m e d e p e n d e n t e n c o u n t e r s d e s c r i b e d by (b-25) and ( b - 2 6 ) , as f o l l o w s . E x p r e s s i o n (b-26) i s i n s e r t e d i n t o ( b - 2 5 ) , t h e i n t e g r a t i o n i s p e r f o r m e d , and t h e r e s u l t i s s q u a r e d . The f i e l d o b j e c t s a r e assumed t o be r a n d o m l y d i s t r i b u t e d , w i t h c o n s t a n t number d e n s i t y N. T h e i r v e l o c i t i e s a r e assumed t o s a t i s f y t h e s p h e r i c a l l y s y m m e t r i c d i s t r i b u t i o n where v i s t h e a v e r a g e v e l o c i t y o f t h e f i e l d o b j e c t s . The i n t e g r a t i o n ? (i^ O) = u r n ; 1 i s t h e n p e r f o r m e d , u s i n g a s y m p t o t i c e x p a n s i o n s i n t h e p a r a m e t e r 82 ( A p ) 1 i s g i v e n by (b-25) and ( b - 2 6 ) . r e p r e s e n t s t h e c o o r d i n a t e s o f a f i e l d o b j e c t i n p h a s e s p a c e ( t h e t e s t o b j e c t i s i n i t i a l l y a t t h e o r i g i n ) , and t(7r) i s t h e p r o b a b i l i t y d i s t r i b u t i o n i n p h a s e s p a c e , d e s c r i b e d a b o v e . The r e s u l t , k e e p i n g o n l y t h e d o m i n a n t t e r m , i s The c o n t r i b u t i o n f o r c l o s e e n c o u n t e r s i s d e m o n s t r a t e d by O s t r i k e r and D a v i d s e n t o be n e g l i g i b l e compared t o t h i s . E x p r e s s i o n (b-30) c o n t a i n s no f a c t o r K, s i n c e t h e v e l o c i t i e s o f t h e f i e l d o b j e c t s were n o t assumed t o be M a x w e l l i a n . I f t h e mean v e l o c i t y o f t h e o b j e c t s , v , i s t a k e n t o be V, ( a s i t was i n s e c t i o n ( a ) ) , t h e n (b-30) a g r e e s w i t h (a-12) w i t h one n o t a b l e d i f f e r e n c e : as i n Henon's r e s u l t s , t h e d i v e r g e n c e w i t h d i s t a n c e h a s been r e p l a c e d by a d i v e r g e n c e w i t h t i m e . I t s h o u l d be r e m a r k e d t h a t t h e r e i s a m i s l e a d i n g a s p e c t t o O s t r i k e r and D a v i d s e n ' s work. I t was p o i n t e d out p r e v i o u s l y t h a t <( ^V) > i s t h e d i s p e r i s o n o f AV a r o u n d i t s e x p e c t a t i o n v a l u e , < AV>, w h i l e < A ( V X ) > , an e n t i r e l y d i f f e r e n t q u a n t i t y , r e p r e s e n t s t h e e x p e c t e d v a l u e o f t h e n e t c h ange i n t h e o b j e c t ' s k i n e t i c e n e r g y . O s t r i k e r and D a v i d s e n 83 however, take • ( Ap) x to be the change in the energy of the test object due to a single encounter, but then erroneously associate < ( A p ) 1 > with the t o t a l change in i t s energy due to a l l encounters. Ostriker and Davidsen's method could be used to calculate an expression for <AV> simply by integrating the quantity AP over a l l phase space, rather than (Lpf • ! t i s e a s i l y seen however, that such a c a l c u l a t i o n would lead to the result < A v - o The density of the system was assumed to be i n i t i a l l y uniform, and the approximation that the f i e l d objects follow li n e a r t r a j e c t o r i e s means that i t must remain uniform. The net force acting of the test object therefore cancels out by symmetry. 84 ( c ) Summary C a l c u l a t i n g t h e d y n a m i c a l f r i c t i o n e x p e r i e n c e d by a t e s t o b j e c t , u s i n g t h e b i n a r y e n c o u n t e r method and a s s u m i n g s t a t i o n a r y b e h a v i o u r ( i e . c o m p l e t e e n c o u n t e r s ) , l e a d s a s e x p e c t e d , t o t h e same r e s u l t s a s C h a p t e r s I and I I : t h e r a t e a t w h i c h a t e s t o b j e c t i s d e c e l e r a t e d d i v e r g e s l o g a r i t h m i c a l l y w i t h d i s t a n c e . T h e s e r e s u l t s a r e i n d e p e n d e n t o f t h e mass of t h e f i e l d o b j e c t s ( p a r t i c l e s ) . However, t h e f a c t t h a t i t t a k e s an i n f i n i t e amount o f t i m e f o r a s t a t i o n a r y s i t u a t i o n t o be e s t a b l i s h e d , i s n o t e n t i r e l y s a t i s f a c t o r y , so models t a k i n g i n t o a c c o u n t t h e t i m e d e p e n d e n c e , p a r t i c u l a r l y of d i s t a n t e n c o u n t e r s , were e x a m i n e d . T h e s e l e d t o e x p r e s s i o n s f o r - t h e d y n a m i c a l " f r i c t i o n e x p e r i e n c e d by t h e t e s t o b j e c t i n w h i c h t h e d i v e r g e n c e w i t h d i s t a n c e i s r e p l a c e d by a d i v e r g e n c e w i t h t i m e , so t h e t e s t o b j e c t i s d e c e l e r a t e d a t a r a t e p r o p o r t i o n a l t o T l o g T , r a t h e r t h a n j u s t t o T. I t was d e m o n s t r a t e d t h a t t h e e x p r e s s i o n f o r <( A V ) 1 >, t h e d i s p e r s i o n o f AV a b o u t i t s mean, i s i n s e n s i t i v e t o whether o r n o t t h e t r a j e c t o r i e s o f t h e f i e l d o b j e c t s a r e assumed t o be l i n e a r . A s s u m i n g l i n e a r t r a j e c t o r i e s l e a d s t o t h e r e s u l t < A v > = o as e x p e c t e d f r om symmetry c o n s i d e r a t i o n s . 85 T h e r e i s a n o t h e r s c h o o l w h i c h q u e s t i o n s t h e use of t h e a s s u m p t i o n o f b i n a r y e n c o u n t e r s a l t o g e t h e r . I n s t e a d , i t i s c l a i m e d t h a t t h e t e s t o b j e c t s h o u l d be v i e w e d a s a c t e d upon by a s t o c h a s t i c a l l y f l u c t u a t i n g f o r c e a r i s i n g f r o m t h e v a r y i n g c o m p l e x i o n o f f i e l d o b j e c t s i n t h e t e s t o b j e c t ' s i m m e d i a t e n e i g h b o u r h o o d . Such a ' s t o c h a s t i c ' model i s d e s c r i b e d i n t h e n e x t c h a p t e r . 86 CHAPTER IV STOCHASTIC MODELS OF DYNAMICAL FRICTION I n t r o d u c t i o n I n C h a p t e r I I I , t h e d y n a m i c a l f r i c t i o n s u f f e r e d by a t e s t o b j e c t t r a v e l l i n g t h r o u g h a s y s t e m o f f i e l d o b j e c t s was c a l c u l a t e d , a s s u m i n g t h a t t h e t e s t o b j e c t c o u l d be v i e w e d a s u n d e r g o i n g b i n a r y e n c o u n t e r s w i t h t h e f i e l d o b j e c t s . An a l t e r n a t e a p p r o a c h , d e v e l o p e d p r i m a r i l y by C h a n d r a s e k h a r , i s t o view t h e t e s t o b j e c t a s a c t e d upon by a s t o c h a s t i c a l l y f l u c t u a t i n g f o r c e F, a r i s i n g from t h e v a r y i n g c o m p l e x i o n o f f i e l d o b j e c t s s u r r o u n d i n g i t . A l t h o u g h i t i s i m p o s s i b l e t o p r e d i c t e x a c t l y what F w i l l be a t any p a r t i c u l a r i n s t a n t , i t l e n d s i t s e l f w e l l t o a s t a t i s t i c a l d e s c r i p t i o n . I t i s t h e r e f o r e t h e a i m o f t h i s a p p r o a c h t o d i s c o v e r t h e s t a t i s t i c a l p r o p e r t i e s o f F, and t o use them t o a n a l y s e t h e d y n a m i c a l i n t e r a c t i o n between t h e t e s t o b j e c t and t h e r e s t o f t h e s y s t e m . The t e s t o b j e c t i n t h i s a p p r o a c h i s a ran d o m l y c h o s e n member of a s y s t e m o f s i m i l a r o b j e c t s , whereas i n t h e models of C h a p t e r s I and I I , i t was an ' i n t e r l o p e r ' . The q u e s t i o n 87 therefore arises whether the si t u a t i o n described by thi s approach i s equivalent to that of the previous chapters. To answer th i s question, i t i s i n s t r u c t i v e to consider the binary encounter models. The equivalence between the binary encounter models and the or b i t based and f l u i d mechanical models has already been demonstrated ( c . f . Chapter I I I ) : the test object i s incapable of distinguishing between a medium consisting of dust or gas, and one consisting of massive f i e l d objects. On the other hand, the binary encounter models and the stochastic models are equivalent: in both cases the medium consists of objects of roughly the same mass as the test object, and the arguments concerning the deceleration of the l a t t e r apply to any member of the system. The test object has no special q u a l i t i e s which set i t apart from the f i e l d objects. Since the binary encounter models describe the same sit u a t i o n as that described by the orbit based and f l u i d mechanical models, and also by the stochastic model, i t follows that the stochastic model i s indeed equivalent to the models of Chapters I and II, in which the test object i s an interloper rather than a member of the system. There i s an extensive l i t e r a t u r e describing the s t a t i s t i c a l approach (cf. Chandrasekhar (1941-44), Chandrasekhar and von Neumann (1942), Kandrup (1980), Lee (1968)): none of the d e t a i l s w i l l be included in t h i s discussion. Since i t draws heavily on ideas from the theory of Brownian motion, a d i s t i l l a t i o n of the fundamentals of t h i s theory i s presented in section (a). 88 S e c t i o n (b) d e m o n s t r a t e s how t h e y may be a p p l i e d t o a g r a v i t a t i o n a l s y s t e m , and i n p a r t i c u l a r , u s e d t o d e r i v e an e x p r e s s i o n f o r t h e d y n a m i c a l f r i c t i o n e x p e r i e n c e d by a t e s t o b j e c t . The r e s u l t s a r e summarized i n s e c t i o n ( c ) . 89 (a) F u n d a m e n t a l N o t i o n s o f t h e T h e o r y o f B r o w n i a n M o t i o n Many of the basic ideas of the s t a t i s t i c a l approach are borrowed from the theory of Brownian motion, so before proceeding to discuss the properties of the s t o c h a s t i c a l l y fluctuating force in the context of a g r a v i t a t i o n a l system, the fundamentals of Brownian motion w i l l be reviewed b r i e f l y . The theory of Brownian motion was o r i g i n a l l y developed to describe the motion of a 'Brownian' p a r t i c l e (large on a molecular scale) moving through a f l u i d . The Brownian p a r t i c l e c o l l i d e s with the molecules in the f l u i d , and thus suffers a succession of random changes in v e l o c i t y . The sequence of c o l l i s i o n s that the Brownian p a r t i c l e undergoes constitutes a Markoff process: the future 'state' of the p a r t i c l e s is dependent only on i t s present state, and not on i t s past h i s t o r y . If F(T) i s the force acting on i t at time t, then the force i t experiences at time t + A,t, F(t + At), i s to a high degree, independent of F ( t ) . In p a r t i c u l a r , the c o r r e l a t i o n between F(t) and F(t + &t) may be shown to be proportional to e- , where T i s the mean l i f e of the force ( c f . Doob,l942). A standard s t a r t i n g place for a study of Brownian motion i s with the assumption that the motion of the Brownian p a r t i c l e may be described as a d i f f u s i o n process: 90 t h e r e i s a f u n c t i o n W ( v , t ; v 0 ,t„ ) , w h i c h d e s c r i b e s t h e p r o b a b i l i t y t h a t a p a r t i c l e w i l l have v e l o c i t y v a t t i m e t , g i v e n t h a t i t had v e l o c i t y v 0 a t t i m e t D , a n d w h i c h s a t i s f i e s t h e d i f f u s i o n e q u a t i o n : where q i s t h e d i f f u s i o n c o e f f i c i e n t : a = j_ I l FI ^ T C F ^ > u . k  U b I t i s e a s i l y v e r i f i e d t h a t t h e G a u s s i a n d i s t r i b u t i o n , i s a s o l u t i o n o f e q u a t i o n ( a - 1 ) . The e x p e c t a t i o n v a l u e < t^v") x>, may be c a l c u l a t e d : I n s e r t i n g ( a -3) i n t o ( a - 4 ) , p e r f o r m i n g t h e i n t e g r a t i o n , and u s i n g d e f i n i t i o n ( a - 2 ) , t h e r e s u l t i s However t h i s has an u n p l e a s a n t f e a t u r e : t h e e x p e c t a t i o n v a l u e o f t h e s q u a r e d change i n t h e p a r t i c l e ' s v e l o c i t y becomes a r b i t r a r i l y l a r g e f o r l o n g i n t e r v a l s o f t i m e . 91 To g e t a r o u n d t h i s p r o b l e m , t h e s t a n d a r d p r o c e d u r e i s t o i n t r o d u c e d y n a m i c a l f r i c t i o n . I n p a r t i c u l a r , i t i s assumed t h a t t h e r a t e o f change i n v e l o c i t y of t h e p a r t i c l e i s t h e sum o f two p a r t s : olv- = F L+~) - p - v a ~ k T h i s i s t h e L a n g e v i n e q u a t i o n . The f i r s t t e r m i s due t o t h e s t a t i s t i c a l l y f l u c t u a t i n g f o r c e , and t h e s e c o n d r e p r e s e n t s a d e c e l e r a t i o n due t o d y n a m i c a l f r i c t i o n . ( p> i s t h e c o e f f i c i e n t o f d y n a m i c a l f r i c t i o n . ) I t s h o u l d be n o t e d t h a t t h e r e a r e two e f f e c t s o p e r a t i n g . One i s r e s p o n s i b l e f o r t h e e v o l u t i o n o f t h e q u a n t i t y < A v>, and l e a d s t o a s y s t e m a t i c d e c e l e r a t i o n o f t h e t e s t o b j e c t . T h i s i s d y n a m i c a l f r i c t i o n . The o t h e r i s r e s p o n s i b l e f o r t h e e v o l u t i o n of t h e q u a n t i t y <(AV) >. T h i s i s n o t d y n a m i c a l f r i c t i o n : i t c o r r e s p o n d s t o t h e d i s p e r s i o n o f A v a r o u n d i t s mean v a l u e , and i s r e l a t e d t o t h e d i f f u s i o n p r o c e s s i n v e l o c i t y s p a c e . I t r e p r e s e n t s t h e r a n d o m i z a t i o n o f t h e t e s t o b j e c t ' s p e c u l i a r v e l o c i t y , and n o t a n e t c hange i n t h e o b j e c t ' s k i n e t i c e n e r g y . T h i s l a t t e r i s < ^ ( v 5 - )>, w h i c h i s a d i f f e r e n t q u a n t i t y . I n B r o w n i a n m o t i o n t h e p h y s i c a l e f f e c t r e s p o n s i b l e f o r d y n a m i c a l f r i c t i o n i s v i s c o s i t y . The f l u c t u a t i n g f o r c e i s r e s p o n s i b l e f o r t h e change i n < ( A v ) a >, and i s g o v e r n e d by a p r o b a b i l i t y d i s t r i b u t i o n ^ ^ v . ~ ) , w h i c h g i v e s t h e p r o b a b i l i t y o f t h e p a r t i c l e u n d e r g o i n g a t r a n s i t i o n f r o m one v e l o c i t y t o a n o t h e r , and w h i c h i s assumed t o be G a u s s i a n . 92 N ote t h a t t h i s i s d i f f e r e n t f r o m W ( v , t ) , w h i c h g o v e r n s t h e p r o b a b i l i t y o f t h e o c c u r r e n c e o f a v e l o c i t y v a t a g i v e n t i m e . The l a t t e r may be c a l c u l a t e d u s i n g t h e f o r m e r , from t h e i n t e g r a l e q u a t i o n T h i s i n t e g r a l e q u a t i o n i s u s e d t o d e r i v e a m o d i f i e d form o f d i f f u s i o n e q u a t i o n ( a - 1 ) , namely t h e F o k k e r - P l a n c k e q u a t i o n . F i n a l l y , t h e r e q u i r e m e n t t h a t t h e M a x w e l l i a n d i s t r i b u t i o n be an e x a c t s o l u t i o n of t h e F o k k e r - P l a n c k e q u a t i o n i n t h e l i m i t t —> 0 0 , imposes t h e r e l a t i o n between q and p, : A l t h o u g h o n l y a few of t h e r u d i m e n t a r y n o t i o n s and e q u a t i o n s of B r o w n i a n t h e o r y have been s k e t c h e d , t h e y s h o u l d be s u f f i c i e n t t o p r o v i d e a b a s i s f o r t h e d i s c u s s i o n o f g r a v i t a t i o n a l B r o w n i a n m o t i o n , i n t h e n e x t s e c t i o n . 93 (b) G r a v i t a t i o n a l B r o w n i a n M o t i o n and t h e C a l c u l t a t i o n of D y n a m i c a l F r i c t i o n R e t u r n i n g t o t h e g r a v i t a t i o n a l c a s e , i t i s c l e a r t h a t i t d i f f e r s f r o m B r o w n i a n m o t i o n on a m o l e c u l a r l e v e l i n some f u n d a m e n t a l ways. To b e g i n w i t h , as m e n t i o n e d i n t h e p r e v i o u s s e c t i o n , d y n a m i c a l f r i c t i o n i n m o l e c u l a r B r o w n i a n m o t i o n i s due t o v i s c o s i t y : t h e B r o w n i a n p a r t i c l e i s s l o w e d down by c o l l i d i n g w i t h m o l e c u l e s w h i c h l i e i n i t s p a t h . A c c o r d i n g l y , i t f e e l s a f o r c e F o n l y when i t comes i n t o d i r e c t c o n t a c t w i t h a m o l e c u l e . The f o r c e f e l t by a m a s s i v e t e s t o b j e c t , on t h e o t h e r hand, i s due t o a l o n g r a n g e i n t e r a c t i o n . D y n a m i c a l f r i c t i o n d o e s n o t a r i s e from t h e t e s t o b j e c t c o l l i d i n g w i t h f i e l d o b j e c t s : t h e s y s t e m s g e n e r a l l y of i n t e r e s t a r e s u f f i c i e n t l y d i f f u s e t h a t t h e p r o b a b i l i t y o f t h i s h a p p e n i n g i s v a n i s h i n g l y s m a l l . R a t h e r , t h e o b j e c t i s d e c e l e r a t e d by t h e d r a g f o r c e e x e r t e d on i t by a r e g i o n of e n h a n c e d d e n s i t y w h i c h forms i n i t s wake ( c f . C h a p t e r s I t h r o u g h I I I ) . T h i s i s a l o n g r a n g e i n t e r a c t i o n . One of t h e m a n i f e s t a t i o n s o f i t s l o n g r a n g e n a t u r e i s t h e c o r r e l a t i o n between two f o r c e s a c t i n g a t t i m e s t and t 0 : .F+- and F o . Whereas i n t h e m o l e c u l a r c a s e , t h i s c o r r e l a t i o n d e c r e a s e s as e. , i n t h e g r a v i t a t i o n a l c a s e , t h e c o r r e l a t i o n o n l y d e c r e a s e s as 1/t ( s e e C h a n d r a s e k h a r , 1944). 94 In a f i r s t a t t e m p t a t a s t a t i s t i c a l a n a l y s i s o f a g r a v i t a t i o n a l s y s t e m , C h a n d r a s e k h a r ( 1 9 4 3 ) , a n d C h a n d r a s e k h a r and von Neumann (1942) c a l c u l a t e d e x p r e s s i o n s f o r W(F), t h e p r o b a b i l i t y o f an i n s t a n t a n e o u s f o r c e F a c t i n g on t h e t e s t o b j e c t , and T ( F ) , t h e mean l i f e of s u c h a f o r c e . The l a t t e r i n v o l v e s t h e q u a n t i t y W ( F): t h e p r o b a b i l i t y t h a t t h e f o r c e e x p e r i e n c e d by t h e o b j e c t h as an i n s t a n t a n e o u s * r a t e o f change F = d F / d t . K a n d r u p (1980) c a l c u l a t e s an e x p r e s s i o n f o r t h e d y n a m i c a l f r i c t i o n s u f f e r e d by a t e s t o b j e c t u s i n g t h e s e two q u a n t i t i e s , i n t h e f o l l o w i n g way. From d e f i n i t i o n ( a - 2 ) , cj_ = ( F 1 TCF^> b-i T h i s e x p e c t a t i o n v a l u e i s g i v e n by < P ' L T C F ) > - [ \Alt?) F 1 TCF^> d F b-3i The e x p r e s s i o n s f o r W(F) and T ( F ) d e r i v e d by C h a n d r a s e k h a r and C h a n d r a s e k h a r and von Neumann, a r e i n s e r t e d i n t o ( b - 2 ) , and t h e i n t e g r a t i o n i s p e r f o r m e d . The r e s u l t i n g e x p r e s s i o n f o r t h e d i f f u s i o n c o e f f i c i e n t i s b-3 D c i s t h e mean i n t e r - o b j e c t s e p a r a t i o n . I t was n o t i n t r o d u c e d a s a c u t o f f f o r a d i v e r g i n g i n t e g r a l . 95 R e l a t i o n (a-8) y i e l d s an e x p r e s s i o n f o r the c o e f f i c i e n t of d y n a m i c a l f r i c t i o n : The e q u a t i o n s 4 &-Vc> = p, VC A T f o l l o w from the i n t e g r a t e d form of the L a n g e v i n e q u a t i o n ( c . f . K a n d rup): b-7 I n s e r t i n g (b-3) and (b-4) i n t o (b-5) and (b-6) r e s p e c t i v e l y , y i e l d s U-ui| f ^ 1 n A-r or 2r\ b-s < & V ^ N > - SL •'It fa1^ rx A+ £ a 3..X3 &.^ v*> b-<1 These e x p r e s s i o n s a r e f i n i t e : no d i v e r g e n c e s were 96 e n c o u n t e r e d i n a r r i v i n g a t them. T h e s e r e s u l t s c o n t r a d i c t s a l l t h o s e o b t a i n e d so f a r , w h i c h i n d i c a t e t h a t i t i s t h e c u m u l a t i v e e f f e c t o f d i s t a n t m a t e r i a l w h i c h i s t h e dom i n a n t f a c t o r i n d e c e l e r a t i n g t h e o b j e c t . E x p r e s s i o n s (b-8) and (b-9) i m p l y t h a t o n l y f i e l d o b j e c t s w i t h i n a b o u t t h e mean i n t e r - o b j e c t d i s t a n c e ( i e . o n l y t h e n e a r e s t n e i g h b o u r s ) i n f l u e n c e t h e t e s t o b j e c t . K a n d r u p ' s c a l c u l a t i o n made use o f t h e q u a n t i t i e s W(F) and T ( F ) . The l a t t e r was c a l c u l a t e d on t h e p r e m i s e t h a t a g r a v i t a t i o n a l M a r k o f f p r o c e s s p o s e s s e s t h e u s u a l c h a r a c t e r i s t i c t h a t c o r r e l a t i o n s between s u b s e q u e n t f o r c e s - t / - r d i e of a s e . However, as C h a n d r a s e k h a r p o i n t e d o u t i n 1 944, " W h i l e t h e s p e c i f i c a t i o n of t h e s e moments o f F a r e s u f f i c i e n t f o r t h e p u r p o s e s of d e t e r m i n i n g t h e i n s t a n t a n e o u s r a t e s o f change o f F t h a t a r e t o be e x p e c t e d , t h e y a r e v e r y f a r f r o m p r o v i d i n g a l l t h e i n f o r m a t i o n t h a t i s n e c e s s a r y f o r a c o m p l e t e s t a t i s t i c a l d e s c r i p t i o n o f t h e f l u c t u a t i n g f o r c e a c t i n g on a s t a r . F o r t h e e n t i r e s t o c h a s t i c v a r i a t i o n of F w i t h t i m e c a n be d e s c r i b e d f u l l y o n l y i n t e r m s o f t h e a v e r a g e f o r c e F + a c t i n g a t any l a t e r t i m e t , g i v e n t h a t a f o r c e o f some p r e s c r i b e d i n t e n s i t y a c t e d a t t i m e t=0. In o t h e r words, we need a c o m p l e t e ' i n t e g r a t i o n ' o f t h e s t o c h a s t i c e q u a t i o n s o f F." In t h i s p a p e r , C h a n d r a s e k h a r c a l c u l a t e s t h e r e q u i r e d a u t o c o r r e l a t i o n f u n c t i o n W(F +.,F ( >), t h e p r o b a b i l i t y t h a t a t e s t o b j e c t w i l l f e e l a f o r c e F+ a t t i m e t , g i v e n t h a t i t f e l t a f o r c e F c a t t i m e t=0. I t has t h e c h a r a c t e r i s t i c 97 a l r e a d y m e n t i o n e d , t h a t i n s t e a d o f d e c r e a s i n g a s e. , i t f a l l s o f f as 1/t a t l a r g e t . Lee (1968) p o i n t s o u t t h a t " T h i s f a n t a s t i c 'memory' o f a f o r c e i s i n c o m p a t i b l e w i t h t h e a s s u m p t i o n o f b r i e f e n c o u n t e r s and w i t h t h e use o f t h e D c u t o f f . " The e f f e c t of a g i v e n e n c o u n t e r d i e s o f f so s l o w l y t h a t t h e n e x t e n c o u n t e r may b e g i n b e f o r e i t has e n d e d . T h u s , a s a r g u e d i n s e c t i o n (b) o f C h a p t e r I I I , d i s t a n t e n c o u n t e r s c a n n o t be c o n s i d e r e d t o be c o m p l e t e . Lee u s e s t h e a u t o c o r r e l a t i o n f u n c t i o n t o f i n d an e x p r e s s i o n f o r t h e d y n a m i c a l f r i c t i o n s u f f e r e d by a t e s t o b j e c t , i n t h e f o l l o w i n g way. The f o r c e a c t i n g on a t e s t o b j e c t a t some i n i t i a l t i m e i s A s s u m i n g t h a t d i s t a n t f i e l d o b j e c t s t r a v e l a l o n g l i n e a r t r a j e c t o r i e s , t h e f o r c e a c t i n g a t some l a t e r t i m e t i s The a u t o c o r r e l a t i o n f u n c t i o n and i t s f i r s t and s e c o n d moments a r e c a l c u l a t e d u s i n g C h a n d r a s e k h a r ' s method. The f i r s t moment, <Fi>, t u r n s o u t t o be z e r o . T h i s i s e x p e c t e d : a s a c o n s i d e r a t i o n of t h e o r b i t b a s e d o r f l u i d m e c h a n i c a l m o d e ls i n d i c a t e s , i f t h e m o t i o n o f t h e p a r t i c l e s ( o r f l u i d 98 e l e m e n t s ) were u n p e r t u r b e d , no d e n s i t y enhancement w o u l d form i n t h e wake of t h e o b j e c t , and t h e r e c o u l d be no d y n a m i c a l f r i c t i o n . The s e c o n d moment, <F+ F=>, doe s n o t v a n i s h , and can be u s e d t o c a l c u l a t e t h e q u a n t i t y <(&v)x >: t h e d i s p e r s i o n of /iv a b o u t i t s mean. The change i n t h e o b j e c t ' s v e l o c i t y i n an i n t e r v a l o f t i m e T, i s g i v e n by S q u a r i n g t h i s , o o <5 + The e x p e c t a t i o n v a l u e i s t h e r e f o r e o <F + F+- > depends on t h e i n t e r v a l o f t i m e s between t T h i s a l l o w s (b-14) t o be w r i t t e n ( c f . L e e ) and t . o 99 I n s e r t i n g t h e e x p r e s s i o n f o r <F G F 5 > d e r i v e d a c c o r d i n g t o C h a n d r a s e k h a r 1 s method ( c . f . L e e ) , and i n t e g r a t i n g o v e r s, y i e l d s f (IVI) d e s c r i b e s t h e i n i t i a l d i s t r i b u t i o n o f v e l o c i t i e s i n t h e s y s t e m , and i s l e f t u n s p e c i f i e d . T i s an ' a p p r o p r i a t e ' i n n e r c u t o f f . T h i s e x p r e s s i o n i s o f t h e f o r m T l o g T , i n a greement w i t h t h o s e d e r i v e d i n t h e t i m e d e p e n d e n t b i n a r y e n c o u n t e r m o d e l s . 100 (c ) Summary In t h i s c h a p t e r , a s t o c h a s t i c model o f t h e e v o l u t i o n o f a system, of g r a v i t a t i n g o b j e c t s , d e v e l o p e d p r i m a r i l y by C h a n d r a s e k h a r , was p r e s e n t e d . The model b o r r o w s many o f i t s i d e a s from t h e t h e o r y o f B r o w n i a n m o t i o n , but a l s o d i f f e r s f r o m t h e l a t t e r i n some f u n d a m e n t a l ways. In t h i s a p p r o a c h , a t e s t o b j e c t i s v i e w e d as a c t e d upon by a s t o c h a s t i c a l l y f l u c t u a t i n g f o r c e F, a r i s i n g f r o m t h e v a r y i n g c o m p l e x i o n o f f i e l d o b j e c t s s u r r o u n d i n g i t . E a r l y a t t e m p t s a t a n a l y s i n g t h e b e h a v i o u r of t h e t e s t o b j e c t under t h e i n f l u e n c e o f t h e f l u c t u a t i n g f o r c e , i n v o l v e d t h e q u a n t i t i e s W(F) and T ( F ) . The f o r m e r i s t h e p r o b a b i l i t y o f t h e o b j e c t e x p e r i e n c i n g an i n s t a n t a n e o u s f o r c e F. The l a t t e r i s t h e d u r a t i o n o f t h e f o r c e , and depends on t h e moments of t h e b i v a r i a t e d i s t r i b u t i o n w ( F , F ) . F i s t h e i n s t a n t a n e o u s r a t e o f change of F. An a n a l y s i s b a s e d on t h e s e q u a n t i t i e s y i e l d s e x p r e s s i o n s f o r t h e d y n a m i c a l f r i c t i o n , < Av>, and t h e q u a n t i t y < ( A V ) 1 >, w h i c h a r e p r o p o r t i o n a l t o T l o g D 6 , where D 6 i s t h e mean i n t e r - o b j e c t s e p a r a t i o n ( c . f . K a n drup, 1980). T h i s c o n t r a d i c t s t h e r e s u l t s o b t a i n e d up t o t h i s p o i n t , i n t h a t i t i m p l i e s t h a t i t i s t h e m a t e r i a l i n t h e immediate v i c i n i t y of t h e o b j e c t t h a t p l a y s t h e dom i n a n t r o l e i n d e c e l e r a t i n g i t , r a t h e r t h a n t h e c u m u l a t i v e e f f e c t o f d i s t a n t m a t e r i a l . 101 However, C h a n d r a s e k h a r (1944) p o i n t s o u t t h a t a c o m p l e t e a n a l y s i s must be b a s e d n o t on w(F) and T ( F ) , b u t on W ( F + , F 0 ) , t h e p r o b a b i l i t y o f t h e t e s t o b j e c t e x p e r i e n c i n g a f o r c e F + a t t i m e t , g i v e n t h a t i t e x p e r i e n c e d a f o r c e F 0 a t t i m e t „ . C h a n d r a s e k h a r d e r i v e s an e x p r e s s i o n f o r t h i s f u n c t i o n , and Lee (1968) u s e s i t t o f i n d t h e d y n a m i c a l f r i c t i o n e x p e r i e n c e d by t h e o b j e c t . He assumes t h a t f i e l d o b j e c t s f o l l o w l i n e a r t r a j e c t o r i e s , w h i c h l e a d s t o t h e r e s u l t t h a t t h e r e i s no d y n a m i c a l f r i c t i o n . However, t h e q u a n t i t y < ( A v ) > does n o t v a n i s h , and t h e e x p r e s s i o n f o r i t t h a t he a r r i v e s a t i s of t h e f o r m T l o g T , i n agreement w i t h t h e t i m e d e p e n d e n t b i n a r y e n c o u n t e r m o d e l s o f c h a p t e r I I I . The n e x t and f i n a l c h a p t e r c o n t a i n s a summary o f t h e v a r i o u s r e s u l t s , a d i s c u s s i o n of t h e d i s c r e p a n c i e s between them, and s u g g e s t i o n s f o r f u r t h e r i n v e s t i g a t i o n s . 1 0 2 CHAPTER V SUMMARY AND CONCLUSIONS I n t r o d u c t i o n Under c e r t a i n c i r c u m s t a n c e s , a m a s s i v e o b j e c t t r a v e l l i n g t h r o u g h a medium may e x p e r i e n c e a d e c e l e r a t i o n due t o t h e d y n a m i c a l i n t e r a c t i o n between i t and t h e medium. In C h a p t e r s I t h r o u g h IV, a v a r i e t y o f d i f f e r e n t m o d e l s of t h i s e f f e c t were p r e s e n t e d . S e c t i o n (a) o f t h i s c h a p t e r summarizes t h e r e s u l t s y i e l d e d by t h e v a r i o u s m o d e l s . S e c t i o n (b) c o n t a i n s a d i s c u s s i o n of t h e d i s c r e p a n c i e s between t h e s e r e s u l t s , and a s s e s s e s t h e v a l i d i t y of e a c h o f t h e m o d e l s . F i n a l l y , i n s e c t i o n ( c ) , p o i n t s r e q u i r i n g f u r t h e r i n v e s t i g a t i o n a r e s u g g e s t e d . 103 (a) Summary of R e s u l t s C h a p t e r s I and I I d e s c r i b e d m o d e l s o f t h e d y n a m i c a l f r i c t i o n e x p e r i e n c e d by a m a s s i v e o b j e c t t r a v e l l i n g t h r o u g h a medium c o n s i s t i n g o f f i e l d p a r t i c l e s much l e s s m a s s i v e t h a n i t s e l f . I n p a r t i c u l a r , t h e m o d e l s o f C h a p t e r I were b a s e d p r i m a r i l y on o r b i t t h e o r y . In t h e s i m p l e s t o f t h e s e m o d e l s , t h e f i e l d p a r t i c l e s were assumed t o be c o l l i s i o n l e s s , and have no t h e r m a l m o t i o n , and t h e i r m u t u a l g r a v i t a t i o n was i g n o r e d . Some m o d i f i c a t i o n s were made t o i n c l u d e c o l l i s i o n s downstream o f t h e o b j e c t , where t h e p a r t i c l e s , f o l l o w i n g h y p e r b o l i c t r a j e c t o r i e s i n t h e o b j e c t ' s g r a v i t a t i o n a l f i e l d , were f o c u s e d a l o n g t h e downstream a x i s . F u r t h e r m o d i f i c a t i o n s were made t o i n c l u d e t h e r m a l m o t i o n . The m a t h e m a t i c a l c o m p l e x i t i e s i n v o l v e d i n i n c o r p o r a t i n g t h e r m a l m o t i o n s u g g e s t t h a t a f l u i d m e c h a n i c a l a p p r o a c h m i g h t be more a p p r o p r i a t e , so s e v e r a l a n a l y s e s o f t h e i n t e r a c t i o n between a g a s e o u s medium and a m a s s i v e o b j e c t were d i s c u s s e d i n C h a p t e r I I . B o t h t h e l i n e a r i z e d and n o n - l i n e a r i z e d e q u a t i o n s of f l u i d m e c h a n i c s were examined, and t h e l i n e a r i z e d c a s e was e x t e n d e d t o i n c l u d e t h e s e l f - g r a v i t y o f t h e medium. In a l l o f t h e m o d e l s , t h e medium's b e h a v i o u r was assumed t o be s t a t i o n a r y i n t h e o b j e c t ' s r e s t f r a m e . The s i m p l e o r b i t b a s e d m o dels y i e l d e d an e x p r e s s i o n f o r t h e d y n a m i c a l f r i c t i o n s u f f e r e d by a t e s t o b j e c t w h i c h 104 depends on the object's mass and v e l o c i t y , and on the density of the medium. Including thermal motion introduced a dependence on the quantity V/4kT/xn , however the mathematical complexities were such that an analytic expression for the object's deceleration could not be derived. The f l u i d mechanical approach yielded two r e s u l t s : for subsonic motion, the object experiences no dynamical f r i c t i o n , while in the case of supersonic motion, the expression for i t s deceleration i s the same as that of the orbit based models, including a dependence on the r a t i o V/c, equivalent to the dependence on V/ V kT/m' . In a l l cases the expressions diverge logarithmically with distance. Chapters III and IV dealt with the situ a t i o n in which the medium through which a test object travels consists of objects of roughly the same mass as i t s e l f . In Chapter I I I , the interaction between the test object and f i e l d objects was described in terms of binary encounters. The simplest of these models assumed that the encounters are complete, and that the f i e l d objects' peculiar v e l o c i t i e s obey a Maxwellian d i s t r i b u t i o n . The result of t h i s model i s equivalent to the res u l t s of the orbit based models, including the dependence on the quantity W m/kT1 . Furthermore, the model indicates that an object w i l l only be decelerated i f i t s i n i t i a l v e l o c i t y i s greater than the average v e l o c i t y of the objects in the system. It was demonstrated (see Chapter III) that t h i s i s equivalent 105 t o t h e f l u i d m e c h a n i c a l r e s u l t t h a t o n l y an o b j e c t t r a v e l l i n g s u p e r s o n i c a l l y w i l l e x p e r i e n c e d y n a m i c a l f r i c t i o n . In a l l t h e m o d e ls m e n t i o n e d so f a r , a s t a t i o n a r y f l o w i n t h e o b j e c t ' s r e s t frame i s assumed. However, i t was a r g u e d i n C h a p t e r I I I t h a t s u c h a s t a t i o n a r y f l o w t a k e s an i n f i n i t e t i m e t o e s t a b l i s h . B i n a r y e n c o u n t e r m o d e l s t a k i n g i n t o a c c o u n t t h e t i m e d e p e n d e n c e o f d i s t a n t e n c o u n t e r s were t h e r e f o r e e x a m i n e d . T h e s e l e d t o e x p r e s s i o n s f o r t h e d y n a m i c a l f r i c t i o n i n w h i c h t h e d i v e r g e n c e w i t h d i s t a n c e i s r e p l a c e d by a d i v e r g e n c e w i t h t i m e . C h a p t e r IV p r e s e n t e d a s t o c h a s t i c model of d y n a m i c a l f r i c t i o n , w h i c h i s b a s e d l a r g e l y on i d e a s b o r r o w e d from t h e t h e o r y o f B r o w n i a n m o t i o n . The t e s t o b j e c t i s v i e w e d a s a c t e d upon by a s t o c h a s t i c a l l y f l u c t u a t i n g f o r c e a r i s i n g f r o m t h e v a r y i n g d i s t r i b u t i o n o f f i e l d o b j e c t s s u r r o u n d i n g i t . An e a r l y a t t e m p t a t a n a l y s i n g t h e p r o p e r t i e s of t h e f l u c t u a t i n g f o r c e y i e l d e d e x p r e s s i o n s f o r W(F), t h e p r o b a b i l i t y o f t h e t e s t o b j e c t e x p e r i e n c i n g an i n s t a n t a n e o u s f o r c e F, and T ( F ) , t h e d u r a t i o n o f t h a t f o r c e . T h i s l a t t e r q u a n t i t y d e pends on t h e b i v a r i a t e d i s t r i b u t i o n W ( F , F ) : t h e p r o b a b i l i t y t h a t t h e t e s t o b j e c t w i l l f e e l an i n s t a n t a n e o u s f o r c e F, whose i n s t a n t a n e o u s r a t e o f c hange i s F. I n c o n t r a d i c t i o n t o a l l t h e o t h e r r e s u l t s , t h e e x p r e s s i o n f o r t h e d y n a m i c a l f r i c t i o n of t h e t e s t o b j e c t d e r i v e d u s i n g w(F) and T ( F ) i s f i n i t e : t h e d e c e l e r a t i o n i s p r o p o r t i o n a l t o T l o g D o , where D G i s t h e mean i n t e r - o b j e c t s e p a r a t i o n . 106 I t was p o i n t e d o ut by C h a n d r a s e k h a r however, t h a t W(F) and W(F,F) a r e n o t s u f f i c i e n t t o d e s c r i b e t h e e v o l u t i o n o f t h e v e l o c i t y d i s t r i b u t i o n . What i s needed i s t h e a u t o c o r r e l a t i o n f u n c t i o n , W(F t,F C ): t h e p r o b a b i l i t y t h a t t h e t e s t o b j e c t w i l l f e e l a f o r c e F + a t t i m e t , g i v e n t h a t i t f e l t a f o r c e F„ a t t i m e t „ . U s i n g t h i s f u n c t i o n t a k e s i n t o a c c o u n t t h e l o n g r a n g e n a t u r e o f g r a v i t a t i o n a l f o r c e s . A d e r i v a t i o n of t h e d y n a m i c a l f r i c t i o n b a s e d on t h i s q u a n t i t y r e s u l t s i n an e x p r e s s i o n f o r t h e q u a n t i t y < ( A V ) i > w h i c h i s p r o p o r t i o n a l t o T l o g T . The d y n a m i c a l f r i c t i o n , < A.V>, i s z e r o , due t o t h e a s s u m p t i o n t h a t t h e f i e l d o b j e c t s f o l l o w l i n e a r t r a j e c t o r i e s . T h u s , t h e r e a r e two p r e d o m i n a n t r e s u l t s : t h o s e w h i c h a r e o f t h e f o r m T l o g X , and t h o s e w h i c h a r e o f t h e f o r m T l o g T . B o t h o f t h e s e i n d i c a t e t h a t i t i s t h e c u m u l a t i v e e f f e c t o f d i s t a n t m a t t e r t h a t p l a y s t h e dom i n a n t r o l e i n a l t e r i n g t h e t e s t o b j e c t ' s m o t i o n . In a d d i t i o n , t h e r e i s one c o n t r a d i c t o r y r e s u l t , w h i c h i s a f i n i t e e x p r e s s i o n i n d i c a t i n g t h a t o n l y m a t e r i a l n e a r t h e o b j e c t , i n p a r t i c u l a r o n l y i t s n e a r e s t n e i g h b o u r s , have any e f f e c t i n s l o w i n g i t down. Re a s o n s f o r t h e d i f f e r e n c e s between t h e s e r e s u l t s a r e d i s c u s s e d i n t h e n e x t s e c t i o n . 1 0 7 (b) D i s c u s s i o n o f t h e D i f f e r e n c e s between t h e R e s u l t s T h r e e t y p e s of e x p r e s s i o n f o r t h e d e c e l e r a t i o n of an o b j e c t due t o d y n a m i c a l f r i c t i o n have been o b t a i n e d : t h o s e o f t h e f o r m T l o g X , t h o s e of t h e form T l o g T , and one o f t h e form T l o g D „ . The d i f f e r e n c e between t h e TlogDo and T l o g T e x p r e s s i o n has a l r e a d y been d i s c u s s e d ( s e e C h a p t e r I V ) : i t i s due t o b a s i n g t h e d e r i v a t i o n on t h e q u a n t i t i e s W(F) and T ( F ) , r a t h e r t h a n on t h e a u t o c o r r e l a t i o n f u n c t i o n W(F4.,F 0). T ( F ) i s d e r i v e d on t h e a s s u m p t i o n t h a t g r a v i t a t i o n a l B r o w n i a n m o t i o n i s c h a r a c t e r i z e d by t h e same c o r r e l a t i o n between f o r c e s a c t i n g a t s u b s e q u e n t t i m e s , a s m o l e c u l a r B r o w n i a n m o t i o n : namely, t h e c o r r e l a t i o n d i e s o f f a s e As shown by C h a n d r a s e k h a r , W ( F + , F 0 ) f o r a g r a v i t a t i o n a l s y s t e m f a l l s o f f o n l y a s 1 / t , and as p o i n t e d o u t by L e e , t h i s i s n o t c o m e n s u r a t e w i t h a s h o r t d i s t a n c e c u t o f f s u c h as D o . F u r t h e r m o r e , no agreement w o u l d be p o s s i b l e - between a T l o g D 0 r e s u l t and t h e r e s u l t s of a f l u i d m e c h a n i c a l o r o r b i t b a s e d m o d e l . In t h e s e , t h e r e i s o n l y one o b j e c t : D 0, t h e a v e r a g e i n t e r - o b j e c t s e p a r a t i o n , h as no meaning i n t h e c o n t e x t o f a model d e s c r i b i n g a s i n g l e o b j e c t t r a v e l l i n g t h r o u g h a c l o u d o f d u s t o r g a s . The d i s c r e p a n c y between t h e T l o g X and T l o g T r e s u l t s c a n be t r a c e d t o one p a r t i c u l a r a s s u m p t i o n : t h e T l o g X m o d e l s a l l 108 assume t h a t t h e f l o w of t h e medium p a s t t h e t e s t o b j e c t i s s t a t i o n a r y i n t h e t e s t o b j e c t ' s r e s t f r a m e . In t h e T l o g T b i n a r y e n c o u n t e r m o d e l s , no s u c h a s s u m p t i o n i s made: t h e t i m e d e p e n d e n c e o f d i s t a n t e n c o u n t e r s i s e x p l i c i t l y i n c l u d e d . In t h e T l o g T s t o c h a s t i c m o d el, t i m e d e p e n d e n c e i s i n c o r p o r a t e d by a n a l y s i n g t h e a u t o c o r r e l a t i o n f u n c t i o n W(F+,F 0) r a t h e r t h a n t h e q u a n t i t i e s W(F) and T ( F ) . The f o r m e r d e s c r i b e s t h e c o r r e l a t i o n between f o r c e s a c t i n g o v e r w i d e l y s e p a r a t e d t i m e s , w h i l e t h e l a t t e r two depend o n l y on t h e i n s t a n t a n t e o u s f o r c e and i t s i n s t a n t a n e o u s r a t e of c h a n g e . I t was d e m o n s t r a t e d i n C h a p t e r I I I t h a t s t a t i o n a r y b e h a v i o u r t a k e s an i n f i n i t e amount of t i m e t o e s t a b l i s h , so t h e T l o g X m o d e l s c a n n o t s t r i c t l y be a p p l i e d t o any r e a l s i t u a t i o n . I t s h o u l d be n o t e d t h a t a l l t h e s e m o d els assume a .medium w h i c h h a s an i n i t i a l l y u n i f o r m d e n s i t y . I n t h e o r b i t and f l u i d m e c h a n i c a l m o d e l s , t h e d e n s i t y becomes n o n - u n i f o r m as an e n h a n c e d r e g i o n forms downstream o f t h e o b j e c t . The change i n t h e momentum of t h e t e s t o b j e c t may be f o u n d by c a l c u l a t i n g t h e d r a g f o r c e e x e r t e d on i t by t h i s enhancement. In O s t r i k e r and D a v i d s e n ' s t i m e d e p e n d e n t e n c o u n t e r m o d el, and i n t h e s t o c h a s t i c m o d e l s , t h e a s s u m p t i o n i s made t h a t d i s t a n t f i e l d o b j e c t s f o l l o w l i n e a r t r a j e c t o r i e s . W i t h no p e r t u r b a t i o n o f t h e i r o r b i t s , t h e r e c a n be no enhancement downstream of t h e t e s t o b j e c t , no d r a g f o r c e on i t , and t h e r e f o r e no d y n a m i c a l f r i c t i o n . 109 However < ( A v ) l > , the dispersion of Av about i t s mean, does not vanish on assuming li n e a r t r a j e c t o r i e s , and i t has been demonstrated by the authors of the time dependent binary encounter models ( c f . Henon, Ostriker and Davidsen; also Lee) that c a l c u l a t i n g the quantity <(Av)'~ > using the approximation that the distant f i e l d objects t r a v e l along unperturbed t r a j e c t o r i e s , does not a l t e r the resulting expression s i g n i f i c a n t l y . If c o r r elations between f i e l d objects ( i e . s e l f - g r a v i t y ) are included , then the TlogX and TlogT results can be shown to be equivalent in the following way. It has already been demonstrated ( c . f . Chapter II) that including the s e l f - g r a v i t y of the medium in the case of a stationary f l u i d mechanical model, introduces an outer cutoff at the Jeans length. This i s a mathematical cutoff in the sense that i t delimits the largest possible region in which the assumption of stationary behaviour remains v a l i d . However, i t has physical s i g n i f i c a n c e in the sense that, according to Jeans i n s t a b i l i t y theory, no stable s e l f - g r a v i t a t i n g systems with dimensions larger than the Jeans length can e x i s t . Henon and Ostriker and Davidsen, (also see Prigogine and Severne (1966)), give arguments to the ef f e c t that, in the expressions which diverge logarithmically with time, the Jeans time, T-j., which i s the Jeans length divided by the c h a r a c t e r i s t i c v e l o c i t y of p a r t i c l e s in the system, provides an upper cuto f f . The arguments are as follows. Because of the time dependence in the argument of the 1 10 l o g a r i t h m , t h e p e r t u r b a t i o n i n t h e v e l o c i t y of a t e s t o b j e c t i s n o t s i m p l y d i r e c t l y p r o p o r t i o n a l t o t i m e , b u t i n c r e a s e s f a s t e r . I f t h e f i e l d o b j e c t s a r e i n i t i a l l y i n d e p e n d e n t , c o r r e l a t i o n s between them d e v e l o p , and c o l l e c t i v e b e h a v i o u r e n s u e s . However, c o l l e c t i v e b e h a v i o u r on a s c a l e g r e a t e r t h a n t h e J e a n s l e n g t h r e s u l t s i n i n s t a b i l i t i e s . S t a r t i n g w i t h a l a r g e , homogeneous p o p u l a t i o n of o b j e c t s , t h e t i m e s c a l e f o r f o r m i n g a g r a v i t a t i o n a l l y bound s y s t e m (whose d i m e n s i o n s a r e d e t e r m i n e d by t h e J e a n s l e n g t h ) i s t h e J e a n s t i m e . A f t e r t h e s y s t e m has f ormed, t h e o b j e c t s w i l l o r b i t i t on a t i m e s c a l e T 7 . A f t e r s e v e r a l c r o s s i n g t i m e s , t h e c o r r e l a t i o n s e x i s t i n g a t t h e t i m e o f f o r m a t i o n , T j , w i l l have been d e s t r o y e d . T h i s s u g g e s t s t h a t c u m u l a t i v e e f f e c t s l e a d i n g t o t h e T l o g T b e h a v i o u r , happen o n l y f o r a t i m e T < T j , and t h a t f o r T>T T, t h e v e l o c i t y o f t h e t e s t o b j e c t c h a n g e s a t a r a t e d i r e c t l y p r o p o r t i o n a l t o T, and e q u a l t o t h e e x p r e s s i o n g i v e n by t h e f o r m u l a s i f T i n t h e argument o f t h e l o g a r i t h m i s r e p l a c e d w i t h T 3 . The J e a n s t i m e t h e r e f o r e comes i n as an o u t e r c u t o f f , y i e l d i n g an e x p r e s s i o n e q u i v a l e n t t o t h a t i n w h i c h t h e d i v e r g e n c e was w i t h d i s t a n c e , a n d t h e J e a n s l e n g t h was i n t r o d u c e d as an o u t e r c u t o f f . In t h e n e x t s e c t i o n , p o s s i b l e d i r e c t i o n s f o r f u r t h e r i n v e s t i g a t i o n o f d y n a m i c a l f r i c t i o n a r e s u g g e s t e d . 111 ( c ) S u g g e s t i o n s f o r F u r t h e r I n v e s t i g a t i o n s In t h e p r e v i o u s s e c t i o n , a b r i e f argument was g i v e n , i n d i c a t i n g why, i n t h e c a s e o f t h e s t o c h a s t i c m o d e l , t h e T l o g T e x p r e s s i o n f o r d y n a m i c a l f r i c t i o n was t o be c o n s i d e r e d more a c c u r a t e t h a n t h e T l o g D e x p r e s s i o n . A more t h o r o u g h i n v e s t i g a t i o n o f t h e s t o c h a s t i c model i s n e e d e d t o s u b s t a n t i a t e t h i s a rgument. L i k e w i s e , a c l o s e r e x a m i n a t i o n o f t h e r o l e o f s e l f - g r a v i t y and c o r r e l a t i o n s , i s n e e d e d t o t i g h t e n t h e c o n n e c t i o n between t h e T l o g T and T l o g X r e s u l t s . A f u r t h e r s t u d y o f t h e r o l e of a s s u m p t i o n s c o n c e r n i n g t h e M a r k o f f i a n n a t u r e o f g r a v i t a t i o n a l e n c o u n t e r s , c o u l d p r o v e i n t e r e s t i n g . C h a n d r a s e k h a r ' s e a r l y work l e a n s q u i t e h e a v i l y on a M a r k o f f i a n a n a l y s i s o f t h e t y p e u s e d i n B r o w n i a n m o t i o n . Agekyan, whose work was n o t d i s c u s s e d h e r e , has d e v e l o p e d a s t o c h a s t i c model s i m i l a r t o C h a n d r a s e k h a r ' s . T h i s model i s b a s e d on t h e a s s u m p t i o n of a M a r k o f f i a n e n c o u n t e r p r o c e s s , and i t , l i k e C h a n d r a s e k h a r ' s e a r l y work, l e a d s t o r e s u l t s w h i c h i n d i c a t e t h e e x i s t e n c e o f a c a n c e l l a t i o n o f t h e e f f e c t s of d i s t a n t m a t e r i a l , so t h a t o n l y n e a r b y o b j e c t s c o n t r i b u t e t o t h e e v o l u t i o n of t h e q u a n t i t i e s < AV> and <( A V ) 1 >. The work o f P r i g o g i n e and S e v e r n e (1966) was n o t d i s c u s s e d h e r e e i t h e r . However, i t i s b a s e d on t h e p r e m i s e t h a t i n g r a v i t a t i o n a l s y s t e m s , t h e c o l l i s i o n p r o c e s s i s 112 n o n - M a r k o f f i a n . T h i s i s e s s e n t i a l l y t h e same a s s u m p t i o n as i n t h e t i m e d e p e n d e n t b i n a r y e n c o u n t e r m o d e l s , where c o l l i s i o n s a r e no l o n g e r c o n s i d e r e d t o be c o m p l e t e and i n d e p e n d e n t . C h a n d r a s e k h a r ' s l a t e r work seems t o f o l l o w t h e s e l i n e s a s w e l l , c o n c e n t r a t i n g on t h e a u t o c o r r e l a t i o n f u n c t i o n W ( F + , F o ) , w h i c h i n d i c a t e s t h a t t h e c o r r e l a t i o n between two " f o r c e s a c t i n g a t d i f f e r e n t t i m e s d e c r e a s e s o n l y as 1/t, r a t h e r t h a n a s , as i n a u s u a l M a r k o f f p r o c e s s . A p o s s i b i l i t y f o r g e t t i n g r i d o f t h e d i v e r g e n c e a l t o g e t h e r , w h i c h has n o t been c o n s i d e r e d h e r e , i s t o i n c l u d e t h e e x p a n s i o n o f t h e U n i v e r s e . T h i s i s m o t i v a t e d i n p a r t by t h e s i m i l a r i t y between t h e p r e s e n t s i t u a t i o n and O l b e r ' s p a r a d o x . The f o r m e r d e a l s w i t h 1/r f o r c e s , w h i l e t h e l a t t e r d e a l s w i t h r a d i a t i o n whose i n t e n s i t y f a l l s o f f a s 1/r . O l b e r ' s p a r a d o x i s d i s p e l l e d by i n c l u d i n g t h e e x p a n s i o n o f t h e U n i v e r s e . 113 APPENDIX A S o l u t i o n o f t h e L i n e a r i z e d E q u a t i o n D e s c r i b i n g a P e r t u r b a t i o n C r e a t e d by a M a s s i v e O b j e c t T r a v e l l i n g t h r o u g h a N o n - S e l f - G r a v i t a t i n g Gas S t a r t i n g w i t h e q u a t i o n (b - 1 0 ) o f C h a p t e r I I , l e t A - X and A-2 Then (A-1) i s L e t r 114 E x p a n d i n g f> i n F o u r i e r - H a n k e l i n t e g r a l s g i v e s CO ' r / 9 r 2JT CO cO A - t I n s e r t i n g (A-5) and (A-6) i n t o ( A - 4 ) : T a k i n g d/dk and d/dx, and c a n c e l l i n g t h e k's and «- 's y i e l d s br1- r "b<" r A - ? U5 However, 3 CXI r ^77 SO 2-77 (&> + .0-MxN) Using (A-6) to invert (A-9), O — + V A - l o 116 T h e r e a r e two c a s e s : (a) s u b s o n i c (-^">0), and (b) s u p e r s o n i c (s><0) (a) S u b s o n i c c a s e : C o n s i d e r t h e i n t e g r a l o v e r k: (A - 1 0 ) becomes X 1 S i n c e i n t e g r a l s o v e r i n f i n i t e l i m i t s , o f odd f u n c t i o n s a r e z e r o , t h i s becomes f - (. O <1 X <!. t'52-x~ tt-tOX 7: "> L.n-r 11? o C> C K <• SIS' F o r a l l c a s e s , S/s C r , • - J L U H / > C A ' The i s o d e n s i t y s u r f a c e s a r e r e p r e s e n t e d by - Cor\ sV^ V^ T h e s e a r e e l l i p s e s w i t h m i n o r a x i s p a r a l l e l t o t h e x - a x i s . (b) S u p e r s o n i c c a s e : The i n t e g r a l 118 CO & i c d i o h a s a s i n g u l a r i t y a t £ = • - + i n t h e c o m p l e x p l a n e . The i n t e g r a l i s t a k e n i n t h e s e n s e of i t s p r i n c i p a l v a l u e , and t h e n o n - s i n g u l a r p a r t i s added t o t h e F o u r i e r - H a n k e l t r a n s f o r m . 1 (3D (X) = iJT ^ J 0 C-^rs) 3- "2-= o (A - 1 0 ) becomes CD D r o p p i n g odd t e r m s a s b e f o r e , r 0.5 , * » 119 T - c OC Xc SLr-H = o o <C \ < 52. r I g n o r i n g t h e i m a g i n a r y s o l u t i o n l e a v e s 120 APPENDIX B D r a g F o r c e e x e r t e d on a T e s t O b j e c t by a N o n - S e l f - G r a v i t a t i n g F l u i d ( L i n e a r i z e d M o d e l ) In t h e s u p e r s o n i c c a s e , t h e d e n s i t y enhancement c r e a t e d by an o b j e c t i n a n o n - s e l f - g r a v i t a t i n g f l u i d i s ( s e e A p p e n d i x A ) : The d r a g f o r c e e x e r t e d on t h e o b j e c t by t h i s enhancement i s g i v e n by so t h e o b j e c t ' s d e c e l e r a t i o n i s f B-3. y<sur 1 2 1 I n s e r t i n g (B-1 ) , t - V4 P e r f o r m i n g t h e z - i n t e g r a t i o n (remembering t h a t x = z - V t ) , X = d~x \ - ix>- V + I n s e r t i n g t h e l i m i t s g i v e s E x p a n d i n g t h e s e c o n d t e r m i n a T a y l o r s e r i e s a r o u n d Vt ~° y i e l d s Oo X = r ot r P e r f o r m i n g t h e r - i n t e g r a t i o n l e a v e s cAV = - u nCnXV\Pa I l - Ixsi} ' \ r T h u s , t h e r a t e a t w h i c h t h e o b j e c t i s d e c e l e r a t e d d i v e r g e s l o g a r i t h m i c a l l y w i t h d i s t a n c e . 122 APPENDIX C Solution of the Equation Describing the Density Enhancement Created by an Object T r a v e l l i n g Through a Self-Gravitating Medium The equation describing the behaviour of a s e l f - g r a v i t a t i n g medium in the presence of a massive object i s (for a marginally stable perturbation, w =0): i - ^ ~ K ¥ = ^ M P o un kn c - i where vf* * + 1 ^ £ ^ £ V+ (C -1 ) i s Let _ S 2 > = I - \/_* c 2 123 r c 4 So r Tr si? hx ' r Using Fourier-Hankel transforms: CD CJ -co Inserting these in (C-2), taking d/dk and d/dx, and canc e l l i n g J*. e- y i e l d s 124 With 6c bcx (C-4) becomes Now, a, " ^ " + ( x s . I r 3<-r X5 125 So P - & C- <o 5-'T ^ 5 2 > -t £S ^ I n v e r t i n g t h e t r a n s f o r m : (r,7r> -03 00 f •1X5 J T h e r e a r e two c a s e s : t h e s u b s o n i c c a s e ( > 0 ) , and t h e s u p e r s o n i c c a s e ( - s £ < 0 ) . (a) S u b s o n i c C a s e : ( i ) \ - i n t e g r a t i o n : tor a* * V- -C 126 So • i , ( H ^ -( i i ) k - i n t e g r a t i o n : R e w r i t e (C-8): (Ley7" " L t U r ^ X T h e r e a r e two c a s e s : ( I ) k^Oc, , and ( I I ) k*>k^ . ( I ) k v<k^ : 1 2 ? (C-9) becomes = (> I (lr>|X<1 V" Keeping only the real part, = - c - l o a 1 I*1 J C - l I Now, So - 3 _ l COS Cos (II) kNk^ : •isC-129 [ £• M.O.I. (H) p. 45" *t2^>] *>P = i cos k , ( t l + n . V ^ c-r* 7 c ^ " ( - x N ^ V - V ^ The density perturbation i s the same at distances greater than the Jeans length as at distances less than the Jeans length, and i s C - \ H This d i f f e r s from the non-self-gravitating case by the factor BIBLIOGRAPHY Agekyan,T.A. (1959) S o v . 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P a r i s 1955 P r i g o g i n e , I and S e v e r n e , G . (1966) P h y s i c a 32,1376 P r i g o g i n e , I and S e v e r n e , G . (1968) B u i . A s t r . P a r i s 3,273 Ruderman,M.A. a n d S p i e g e l , E . A . (1971) A p J 165,1 Saslaw,W.C. (1968) MNRAS 141,1 S p i e g e l , E . A . i n Symposium on C o s m i c a l Gas Dyna m i c s . 6 t h , Y a l t a , 1969, e d . H . J . H a b i n g , H o l l a n d , D . R e i d e l Pub. 1970 Watson,G.N. A T r e a t i s e on t h e T h e o r y o f B e s s e l F u n c t i o n s , 2nd.ed., New Y o r k , M a c m i l l a n , 1945 

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