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Time-dependent models of grain-forming stellar atmospheres Woodrow, Janice Emily Jean 1981

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TIME-DEPENDENT MODELS OF GRAIN-FORMING STELLAR ATMOSPHERES by JANICE EMILY JEAN WOODROW B . S c , The U n i v e r s i t y of B r i t i s h Columbia M.Sc, The U n i v e r s i t y of B r i t i s h Columbia A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES Department of Geophysics and Astronomy We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA March, 1981 ( c ) J a n i c e Woodrow, 1981 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study. I further agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i cat ion of th is thes is fo r f inanc ia l gain sha l l not be allowed without my written permission. Department of G e o n h v s i r a and Rat-T-onniwy The Univers i ty of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date M a r c h 2 6 , 1881 i i ABSTRACT The atmospheres of many s t a r s are v a r i a b l e to such a degree that they are not adequately represented by e i t h e r s t a t i c or s t e a d y - s t a t e models. In t h i s study, a computer programme capable of c a l c u l a t i n g completely time-dependent model atmospheres of v a r i a b l e s t a r s was developed. S p e c i f i c a l l y , the programme was developed to model the expanding atmsopheres of c o o l , c a r b o n - r i c h s t a r s . The d r i v i n g f o r c e f o r the expansion was r a d i a t i o n pressure a c t i n g on g r a p h i t e g r a i n s which were assumed to form i n the outer l a y e r s of the s t a r . The d e t a i l s of the condensation and growth of the g r a p h i t e g r a i n s as w e l l as the r e s u l t a n t i n t e r a c t i o n s between (a) the g r a i n s and the s t e l l a r r a d i a t i o n f i e l d and (b) the g r a i n s and the ambient gas were i n c l u d e d i n the c a l c u l a t i o n s . The models were c a l c u l a t e d by c o u p l i n g a model atmosphere programme to the hydrodynamical equations needed to d e s c r i b e the flows of the gas and the g r a i n s , and to the equations which govern the n u c l e a t i o n and growth of the g r a i n s . The atmospheres were assumed to be s p h e r i c a l l y symmetrical and to have a grey o p a c i t y . In a d d i t i o n , the thermal time s c a l e was assumed to be n e g l i g i b l e compared to the dynamical time s c a l e . The g r a i n s were assumed to be spheres of g r a p h i t e and to form i n accordance with the Lothe-Pound n u c l e a t i o n theory. The s t e l l a r parameters adopted f o r the models were M = 1.5 VLQ, L = 1. 94x10 4 L 0 , C/H = 1.22x10" 3 and C/O = 1.76. The s u r f a c e - f r e e - e n e r g y adopted f o r the g r a i n s was 1000 ergs/cm 2. Two time-dependent models were c a l c u l a t e d . Model 1 had T* = 2500 K and was f o l l o w e d f o r an e l a p s e d time of 27.3xl0 7 sec. For Model 2, T* = 2400 K and the e l a p s e d time was 56.3x10 7 sec. The s t r u c t u r e of both models was such that carbon vapour was found to be h i g h l y s u p e r s a t u r a t e d i n the outer l a y e r s of the atmospheres and e x t e n s i v e g r a i n condensation occured. The r a d i a t i o n p r e s s u r e on these g r a i n s was s u f f i c i e n t to generate mass flows i n both models. The c a l c u l a t e d mass l o s s r a t e f o r Model 1 was 6.2x10"' M 0/yr, and f o r Model 2, 7.4x10" 8 Mg/yr. In Model 1, the mass flow approached a s t e a d y - s t a t e but i n Model 2, a small amplitude p u l s a t i o n was superimposed upon the outward flow. I n i t i a l l y , t h i s p u l s a t i o n was very i r r e g u l a r but a f t e r an e l a p s e d time of 27x10 7 sec, the model had r e l a x e d i n t o a steady p u l s a t i o n mode with a p e r i o d of 6.48xl0 7 sec and a v e l o c i t y amplitude at the s u r f a c e of 0.8 km/sec. T h i s mode was followed f o r four p e r i o d s d u r i n g which the amplitude of the p u l s e s remained c o n s t a n t . The d r i v i n g f o r c e f o r t h i s p u l s a t i o n appears to be an o p a c i t y c o n t r o l l e d feedback mechanism which operates between the g r a i n - f o r m i n g region of the model and the hydrogen d i s s o c i a t i o n zone. I t was found that i n Model 1, the o p a c i t y of the g r a i n s was too small f o r t h i s mechanism to produce p u l s a t i o n s . In both models, g r a i n n u c l e a t i o n was n e g l i g i b l e at s u p e r s a t u r a t i o n l e v e l s l e s s than 5. As a r e s u l t , the g r a i n -forming r e g i o n was r e s t r i c t e d to l a y e r s f o r which r > 1.82R* i n Model 1, and r > 1.58R* in Model 2. In both cases, the low atmospheric d e n s i t i e s i n the g r a i n - f o r m i n g regions s e v e r e l y l i m i t e d the growth of the g r a i n s f o l l o w i n g t h e i r formation so that the g r a i n s remained very small (a = 5.5x10" 8 cm) and, even i v at the s u r f a c e of the models, only about 75% of the f r e e carbon vapour was i n the form of g r a i n s . The o p t i c a l depth of the g r a i n s at A. = 0.7^ was 5.0xl0" 3 i n Model 1, and l . O x l O " 1 i n Model 2. The p u l s a t i o n i n Model 2 produced a v a r i a t i o n i n the o p t i c a l depth of the g r a i n s of 1.8x10" 2. V TABLE OF CONTENTS A b s t r a c t i i L i s t of Tables v i L i s t of F i g u r e s v i i Acknowledgement v i i i Chapter I INTRODUCTION 2 Chapter II BACKGROUND TO THE PROBLEM 5 Chapter III THE TIME-DEPENDENT MODEL 20 Chapter IV NUMERICAL PROCEDURES 40 Chapter V RESULTS 67 Chapter VI DISCUSSION AND CONCLUSIONS 87 B i b l i o g r a p h y 109 Appendix A NUMERICAL TESTS 115 Appendix B THE NUCLEATION AND GROWTH OF GRAPHITE GRAINS 135 LIST OF TABLES Table I 65 Table II 68 Table III 88 LIST OF FIGURES , 58 F i g u r e 1 ~ 59 F i g u r e 2 72 F i g u r e 3 . 74 F i g u r e 4 . 80 F i g u r e 5 , 84 F i g u r e 6 n 122 F i g u r e A - l • _. „ . 0 125 F i g u r e A-2 _. ^ . , 126 F i g u r e A-3 , . 129 F i g u r e A-4 , c 133 F i g u r e A-5 v i i i ACKNOWLEDGEMENT S p e c i a l thanks are given to my a d v i s o r , Dr. J . R. Auman, and to the Department of Geophysics and Astronomy at UBC fo r t h e i r i n i t i a l encouragement and continued support of my res e a r c h . Thanks are a l s o extended to Dr. F. D. A. Hartwick of the Department of Astronomy, U n i v e r s i t y of V i c t o r i a , f o r s e r v i n g on my t h e s i s committee, and to Dr. Peter Bodenheimer, U n i v e r s i t y of C a l i f o r n i a , Dr. E r i k a Bohm-Vitense, U n i v e r s i t y of Washington and Dr. E. E. S a l p e t e r , C o r n e l l U n i v e r s i t y f o r s e r v i n g as e x t e r n a l examiners f o r the t h e s i s . As w e l l , the f i n a n c i a l support from Zonta I n t e r n a t i o n a l i n the form of an Amelia Ear h a r t F e l l o w s h i p , and from the NRC i s g r a t e f u l l y acknowledged. 1 Chapter I  INTRODUCTION In 1935, Adams and MacCormack noted that the cores of s e v e r a l s t r o n g resonance l i n e s i n four M s t a r s were b l u e - s h i f t e d by about 5 km/sec. On the b a s i s of t h i s evidence, they i n f e r r e d t h a t each s t a r was g r a d u a l l y l o s i n g mass from an expanding c i r c u m s t e l l a r s h e l l . Since these f i r s t o b s e r v a t i o n s , evidence has accumulated that suggests that a l l red g i a n t s t a r s 1 l a t e r than MO are l o s i n g mass to the i n t e r s t e l l a r medium at a r a t e g r e a t e r than 10" 9 M ^ y r (Reimers, 1975; Hagen, 1978). However, while mass l o s s r a t e s are f r e q u e n t l y quoted f o r red g i a n t s , i t i s important to keep i n mind that these r a t e s are ra t h e r p o o r l y determined from the o b s e r v a t i o n a l data. For example, the mass l o s s r a t e determined f o r Betelgeuse by f i v e d i f f e r e n t i n v e s t i g a t o r s ranges from 3.5X10" 5 M^/yr to 1.7X10" 1 M 0/yr (Bernat, 1977). What i s very c l e a r from the o b s e r v a t i o n a l data i s that the flow r a t e s are q u i t e h i g h , and that the atmospheres of such s t a r s cannot be adequately represented by s t a t i c models. One b a s i c q u e s t i o n u n d e r l y i n g the study of mass l o s s from red g i a n t s concerns the cause of the atmospheric flow. Many mechanisms have been suggested ( C a s s i n e l l i , 1979). Among these, i s the suggestion that mass l o s s i n very l a t e red g i a n t s i s the 1 The term, red g i a n t , u n l e s s otherwise q u a l i f i e d , w i l l be understood to i n c l u d e s u p e r g i a n t s as w e l l as g i a n t s throughout t h i s t h e s i s and to imply T e < 4000K. 2 r e s u l t of the formation of atmospheric g r a i n s . According to t h i s h y p o t h e s i s , r a d i a t i o n pressure w i l l expel the g r a i n s and, i f the ambient gas i s s u f f i c i e n t l y coupled to the g r a i n s , i t w i l l be d r i v e n outward as w e l l . When Weymann analyzed t h i s mechanism i n 1962, he r e j e c t e d i t on the b a s i s that there would be i n s u f f i c i e n t c o u p l i n g between the g r a i n s and the gas to expel the atmosphere. In the same year, Hoyle and Wickramasinghe (1962) demonstrated that g r a i n s of g r a p h i t e c o u l d p r e c i p i t a t e i n the atmospheres of c o o l , c a r b o n - r i c h s t a r s , and that such g r a i n s may be the source of much of the i n t e r s t e l l a r reddening m a t e r i a l . However, Weymann's a n a l y s i s appeared to r u l e out r a d i a t i o n p r e s s u r e on g r a i n s as being e f f e c t i v e i n causing mass l o s s from these s t a r s u n t i l Gilman (1972) noted that Weymann had not c o n s i d e r e d the e f f e c t s of c o l l i s i o n s between gas molecules. I t i s through these c o l l i s i o n s , that the momentum that i s t r a n s f e r r e d from the r a d i a t i o n f i e l d to the g r a i n s i s d i f f u s e d through the gas. Gilman demonstrated that the grain-gas momentum c o u p l i n g i s s u f f i c i e n t l y l a r g e under c o n d i t i o n s r e l e v a n t to red g i a n t s that r a d i a t i o n pressure on the g r a i n s can cause a p p r e c i a b l e mass l o s s r a t e s . Although i t has now been adequately demonstrated that r a d i a t i o n d r i v e n mass flows are k i n e m a t i c a l l y p o s s i b l e (Wickramasinghe, et a l , 1966), the co n t r o v e r s y as to whether or not such flows are a c t u a l l y e s t a b l i s h e d i n red g i a n t s remains unresolved (see, f o r example, F u s i - P e c c i and R e n z i n i , 1976, Hagen, 1978, C a s s i n e l l i , 1979). The st r o n g e s t argument i n support of t h i s d r i v i n g mechanism i s the o b s e r v a t i o n of e x t e n s i v e , expanding dust s h e l l s surrounding a l l c o o l g i a n t s 3 ( M e r r i l l , 1977). What remains u n c e r t a i n , i s (a) can the dust g r a i n s r e a l l y form i n the atmosphere of a red g i a n t ? and (b) assuming the c o n d i t i o n s f o r g r a i n formation do p r e v a i l , do the g r a i n s cause the mass flow or are they merely a consequence of i t ? (a) Statement of the Problem The q u e s t i o n of the v i a b i l i t y of the r a d i a t i o n p r e ssure d r i v i n g f o r c e has remained unresolved l a r g e l y because of the lack of a p p r o p r i a t e , e v a l u a t i v e models. Without such models, i t has been impossible to be sure whether or not t h i s mechanism can generate a mass flow of the type a c t u a l l y observed. In the l a s t decade or so, enormous improvements i n the m o d e l l i n g of s t a t i c atmospheres of red g i a n t s have been r e a l i z e d . Attempts to produce dynamic mode-Is have focussed l a r g e l y on the kinematics of the problem and have assumed e i t h e r steady or r e g u l a r l y p u l s a t i n g flows to s i m p l i f y the computational d i f f i c u l t i e s of the models. Such models are i n h e r e n t l y incapable of r e p r e s e n t i n g the i r r e g u l a r flows which are c h a r a c t e r i s t i c of most red g i a n t s . For a proper r e p r e s e n t a t i o n of the atmospheres of red g i a n t s , a completely time-dependent approach i s r e q u i r e d . Consequently, the e s s e n t i a l problem of t h i s t h e s i s was the development of a computer programme capable of g e n e r a t i n g time-dependent model atmospheres. T h i s programme was then used to to t e s t the hy p o t h e s i s that r a d i a t i o n p r e s s u r e on g r a i n s i s a v i a b l e mass l o s s mechanism i n the atmospheres of red g i a n t s . S p e c i f i c a l l y , the programme was designed to model the expanding atmospheres of 4 c o o l , c a r b o n - r i c h , g r a i n - f o r m i n g red g i a n t s by i n t e g r a t i n g the equations of hydrodynamics and those d e s c r i b i n g the n u c l e a t i o n and growth of g r a i n s with the equations used to c o n s t r u c t a model atmosphere. The dynamical p r o p e r t i e s of g r a i n - f o r m i n g atmospheres are numerous and complex. I t was p r e d i c t e d that g r a p h i t e g r a i n s would n u c l e a t e , grow and/or evaporate; r a d i a t i o n pressure a c t i n g on the g r a i n s would d r i v e them and, perhaps, a p o r t i o n of the atmospheric gas outwards; s t r u c t u r a l changes i n the atmosphere would occur; and the formation and growth of the g r a i n s might be s i g n i f i c a n t l y a l t e r e d as a consequence of these s t r u c t u r a l changes. Whether or not these i n t e r a c t i o n s would produce a steady flow was not obvious. T h e r e f o r e , a second and necessary purpose of the study was to see i f the time-dependent models would r e l a x to s t e a d y - s t a t e models s i m i l a r to those c a l c u l a t e d by Lucy (1976) f o r c o o l carbon s t a r s . The model development r e q u i r e d an e x t e n s i v e a n a l y s i s of e x i s t i n g m o d e l l i n g procedures, of t h e o r i e s p e r t a i n i n g to p o s s i b l e mass l o s s mechanisms, and of e m p i r i c a l data r e l a t e d to mass l o s s from red g i a n t s . T h i s data i s summarized i n Chapter I I . The model, and i t s u n d e r l y i n g assumptions and l i m i t a t i o n s , are d e c r i b e d i n Chapter I I I . The s o l u t i o n a l g o r i t h m i s presented i n Chapter IV, f o l l o w e d by the r e s u l t s of the study i n Chapter V. . The d i s c u s s i o n of the r e s u l t s and the c o n c l u s i o n s of the study are presented i n Chapter VI. In a d d i t i o n , the t h e s i s c o n t a i n s two appendices. The r e s u l t s of s e v e r a l numerical t e s t s of the programme are summarized i n Appendix A, and d e t a i l s of the homogeneous n u c l e a t i o n theory are presented i n Appendix B. 5 Chapter II  BACKGROUND TO THE PROBLEM The b a s i c equations and p h y s i c s necessary to c o n s t r u c t model atmsopheres have been known f o r many years but progress in the f i e l d has been slow because of the l a c k of knowledge of fundamental atomic q u a n t i t i e s , and the mathematical d i f f i c u l t i e s i n herent i n the problem (Auer, 1971). For m o d e l l i n g purposes, s t e l l a r atmospheres are g e n e r a l l y assumed to be s t a t i c , plane-p a r a l l e l and h o r i z o n t a l l y homogeneous, to c o n t a i n only continuum o p a c i t i e s , and to s a t i s f y the c o n s t r a i n t s of r a d i a t i v e and l o c a l thermodynamic e q u i l i b r i u m . The r e l a x a t i o n of any of these c o n s t r a i n t s g r e a t l y i n c r e a s e s the time r e q u i r e d to compute a model atmosphere. I d e a l l y , none of these c o n s t r a i n t s should be used to c o n s t r u c t models of M- and N-type s t a r s f o r none of these assumptions i s t r u l y v a l i d f o r red g i a n t s with T e < 3000 K ( T s u j i , 1978). Although some progress has been achieved in t r e a t i n g departures from l o c a l thermodynamic e q u i l i b r i u m , plane-p a r a l l e l geometry and h y d r o s t a t i c e q u i l i b r i u m , r e s e a r c h r e l a t e d to model atmospheres of very c o o l s t a r s has been p r i m a r i l y focussed upon de v e l o p i n g techniques to i n c l u d e the s p e c t r a l l i n e - b l a n k e t i n g e f f e c t s of atomic and molecular l i n e o p a c i t i e s . T h i s e f f o r t has y i e l d e d reasonably a c c u r a t e l i n e - b l a n k e t e d models of a l l but the c o o l e s t l a t e - t y p e s t a r s (Carbon, 1979). The r e s u l t s of t h i s r e s e a r c h w i l l be b r i e f l y reviewed at the b eginning of t h i s c h a p ter. In the second s e c t i o n of t h i s c h apter, the o b s e r v a t i o n a l data on mass l o s s from red g i a n t s 6 w i l l be summarized. T h i s chapter w i l l conclude with a short d i s c u s s i o n of t h e o r i e s and models of suggested mass l o s s mechanisms. (a) Model Atmospheres The o p a c i t y of many o v e r l a p i n g atomic and molecular a b s o r p t i o n l i n e s i s known to a f f e c t both the emergent spectrum of the s t a r and the s t r u c t u r e of the atmosphere i t s e l f . A c o n c i s e review of the e f f e c t s of v a r i o u s molecules on the s t r u c t u r e of s t e l l a r atmospheres has r e c e n t l y been p u b l i s h e d by Gustafsson and Olander (1979). In a more e x t e n s i v e review, Carbon (1979) d i s c u s s e s the p h y s i c s u n d e r l y i n g these complicated l i n e - b l a n k e t i n g e f f e c t s as w e l l as t h e i r s t r u c t u r a l consequences. In that review, Carbon s t a t e s that the p r i n c i p a l s t r u c t u r a l e f f e c t s * o f l i n e - b l a n k e t i n g are a) a h e a t i n g of the continuum forming l e v e l s (T>0.1) of the atmosphere, a phenomenon r e f e r r e d to as "backwarming", and b) a c o o l i n g or h e a t i n g of the s u r f a c e l a y e r s (t<10~ 3), as compared to the temperatures computed f o r these same l e v e l s i n an unblanketed atmosphere with the same e f f e c t i v e temperature. In both of these reviews, i t i s s t r e s s e d that the s t r u c t u r a l e f f e c t s of molecular l i n e -b l a n k e t i n g are s t r o n g l y c o r r e l a t e d with atmospheric composition, m i c r o - t u r b u l e n c e and atmospheric v e l o c i t y g r a d i e n t s , and can a f f e c t the temperatures of the models by s e v e r a l hundred degrees. Among the important c o n c l u s i o n s that have been drawn from the l i n e - b l a n k e t e d models of red g i a n t s are the f o l l o w i n g : 1) The l i n e b l a n k e t i n g of CO c o o l s the s u r f a c e temperature 7 of a l l red g i a n t s (Johnson, 1973; Gustafsson e_t a l , 1975). T h i s c o o l i n g e f f e c t i s very temperature dependent. I t i s small i n models with Te <4500 K, and n e g l i g i b l e i n models with T e <2400 K and Te >55.00 K (Johnson, 1973). 2) The CO c o o l i n g i n models with T e<4500 K i s i n s e n s i t i v e to l a r g e changes i n the o v e r a l l metal abundance of the models suggesting that the c o o l i n g was " s a t u r a t e d " (Gustafsson, et a l , 1975) . 3) The l i n e - b l a n k e t i n g of CO produces only a small backwarming i n the models (Querci and Q u e r c i , 1975). 4) The l i n e - b l a n k e t i n g of a l l but the reddest bands of H^O heats the s u r f a c e l a y e r s of oxygen-rich s t a r s (Carbon, 1979). 5) C^ c o n t r i b u t e s l i t t l e to the s u r f a c e c o o l i n g of models (Querci and Q u e r c i , 1975). 6) The o p a c i t y of TiO leads to a g l o b a l h e a t i n g of the models but t h i s e f f e c t i s s t r o n g l y moderated by the presence of other molecules e s p e c i a l l y CO, H A0 and CN (Krupp et a l , 1978). 7) The s u r f a c e warming of TiO reaches a minimum value i n models with T e =3000 K whereas i t s backwarming e f f e c t decreases m o n o t o n i c a l l y with d e c r e a s i n g e f f e c t i v e temperature (Krupp et a l , 1978). 8) TiO and H^O are l e s s e f f e c t i v e i n backwarming the continuum l a y e r s of the models than are CN and C^ ( T s u j i , 1978, Querci and Q u e r c i , 1975). 9) The backwarming e f f e c t of CN i n c r e a s e s s t e a d i l y with the metal abundance of the models (Gustafsson e_t a l , 1975). The s t r u c t u r e of red g i a n t s i s known to be dependent upon the C 1 2 / C 1 3 r a t i o . Carbon (1974) found that i n c r e a s i n g the 8 G 1 3 abundance from C 1 2 / C 1 3 = 90 to C 1 2 / C 1 3 = 10 decreased the temperature of both the continuum l a y e r s and the s u r f a c e l a y e r s . I t has a l s o been demonstrated that the s t r u c t u r e of carbon s t a r atmospheres i s moderately dependent upon the value of the micro-t u r b u l e n c e used to c a l c u l a t e the models (Gustafsson, et a l . , 1975). In c o n t r a s t , the l i n e b l a n k e t i n g e f f e c t s i n the atmospheres of other l a t e - t y p e s t a r s are only s l i g h t l y a f f e c t e d by v a r i a t i o n s i n the m i c r o - t u r b u l e n c e parameter ( T s u j i , 1978; Carbon, 1974). The p r i n c i p a l e f f e c t of i n c l u d i n g the e x t e n s i o n of the atmosphere i n the models i s to lower the temperature and gas p r e s s u r e near the s u r f a c e (Watanabe and Kodaira, 1978). These i n v e s t i g a t o r s a l s o found t h a t the e x t e n s i o n of the atmosphere enhanced the s p e c t r a of molecules such as TiO and H^O which are formed p r e f e r e n t i a l l y near the s u r f a c e of the models, while i t suppressed the s p e c t r a of molecules such as CO which are formed deeper i n the atmosphere. Such e f f e c t s can c l e a r l y a f f e c t the backwarming and s u r f a c e temperature m o d i f i c a t i o n s of the v a r i o u s molecular s p e c i e s formed i n the s p e c t r a of l a t e - t y p e s t a r s . Although not i n c l u d e d i n t h e i r study, CN i s a l s o formed deeper i n the atmosphere than TiO and H^O so i t seems l i k e l y that the s t r u c t u r a l e f f e c t s of t h i s molecule are l i k e w i s e overestimated by plane p a r a l l e l models. (b) O b s e r v a t i o n s of Mass Loss from Red Giants As Mullan (1978) notes, "the d e t e c t i o n of a c i r c u m s t e l l a r s h e l l i n d i c a t e s unambiguously that the red g i a n t i s l o s i n g mass". The e x i s t e n c e of c i r c u m s t e l l a r s h e l l s i s 9 i n d i c a t e d by the a n a l y s i s of o p t i c a l , i n f r a r e d and r a d i o s p e c t r a of red g i a n t s . The r e s u l t s of such analyses are summarized in t h i s s e c t i o n ; more complete reviews of these t o p i c s are presented by Reimers (1975, 1978), M e r r i l l (1978) and Moran (1976) . Most of the e a r l y o b s e r v a t i o n s of mass l o s s i n red g i a n t s were r e s t r i c t e d to the H and K l i n e s of Ca II which are always the s t r o n g e s t c i r c u m s t e l l a r l i n e s i n M g i a n t s (Deutsch, 1960). However, mass l o s s i s a l s o manifested in the asymmetries or weak P-Cygni-type p r o f i l e s of many st r o n g , low e x c i t a t i o n l i n e s of n e u t r a l metals such as those of Na I at TLX 5890 and 5896, and A l I at K.K 3944 and 3961; i n the l i n e s of s i n g l y i o n i z e d metals such as that of Sr II at A. 4077; and, o c c a s i o n a l l y , i n H ^ (Boesgaard and Hagen, 1979). Such p r o f i l e s are i n t e r p r e t e d as a r i s i n g from the s u p e r p o s i t i o n of the blue-s h i f t e d , c i r c u m s t e l l a r l i n e on the (presumed) u n s h i f t e d p hotospheric l i n e . Numerous i n f r a r e d molecular a b s o r p t i o n bands as w e l l as a g e n e r a l i n f r a r e d excess a r i s i n g from c i r c u m s t e l l a r s h e l l s are d e t e c t e d i n the s p e c t r a of both M and N g i a n t s . The i n f r a r e d excess i s seen in a l l M g i a n t s more luminous than a l i m i t d e f i n e d on the HR diagram by M6 I I I , M5 II and Ml l a b (Reimers, 1975), and i s w e l l c o r r e l a t e d with h i g h l u m i n o s i t i e s and low s u r f a c e temperatures (Cohen and Gaustad, 1973). The emission f e a t u r e s at 9.7^ and 18^u i n these s t a r s are a t t r i b u t e d to the o p t i c a l l y t h i n emission from small s i l i c a t e p a r t i c l e s (Woolf and Ney, 1969). I n f r a r e d emission i s d e t e c t e d i n N s t a r s at 11.5^u and i s a t t r i b u t e d to g r a i n s of SiC ( G i l r a , 1973; T r e f f e r s and 10 Cohen, 1974). However, the primary c o n s t i t u e n t of the dust i n c a r b o n - r i c h c i r c u m s t e l l a r envelopes i s r e l a t i v e l y f e a t u r e l e s s , and i s d e t e c t e d only as a general v e i l i n g of the photospheric l i n e s ( F o r r e s t et a l . , 1975), and a pronounced d e f i c i e n c y i n the blue and u l t r a v i o l e t s p e c t r a l r e gion (Mendoza and Johnson, 1965). T h i s c o n s t i t u e n t i s g e n e r a l l y i d e n t i f i e d as g r a p h i t e g r a i n s , the o p a c i t y of which v a r i e s as A " 2 (Gilman, 1974). The i n f r a r e d data on c i r c u m s t e l l a r dust suggests that the dust i s c o n c e n t r a t e d beyond the main a c c e l e r a t i o n zone of the flow (Reimers, 1975) but the d i s t r i b u t i o n i s d i f f i c u l t to determine a c c u r a t e l y because of the l a c k of sharp s p e c t r a l f e a t u r e s a s s o c i a t e d with the dust ( C a s s i n e l l i , 1979). The evidence f o r mass flows i n red g i a n t s has been enhanced i n the l a s t few years by the o b s e r v a t i o n of both maser and non-maser emission from the c i r c u m s t e l l a r s h e l l s (Wilson and B a r r e t t , 1972; M o r r i s and Alcock, 1972). The dusty c i r c u m s t e l l a r envelopes are i d e a l environments f o r the g e n e r a t i o n of the i n f r a r e d photons necessary to e x c i t e the upper v i b r a t i o n a l l e v e l s of v a r i o u s molecular s p e c i e s . The most thoroughly s t u d i e d l i n e s are the maser l i n e s of OH and H^O (Wilson and B a r r e t t , 1972), and of SiO (Morris et a.1, 1979; Buhl et a l . , 1975). Non-maser emission from SiO has been d e t e c t e d i n the expanding envelopes of 15 red g i a n t s , one of which i s carbon r i c h (Morris et a l . , 1979), and from CO i n a sample of c o o l , oxygen r i c h (Lo and Bechis, 1977) and carbon r i c h (Zukerman et a_l., 1977) s t a r s . One suspected case of weak maser CO emission has been r e p o r t e d ( M o r r i s , 1980). Mass l o s s r a t e s have been d e r i v e d from o b s e r v a t i o n s 11 made i n a l l three s p e c t r a l r e g i o n s . Hagen (1978), Sanner (1976), Boesgaard and Hagen (1979), and Bernat (1977) have c a l c u l a t e d the mass l o s s r a t e s f o r a t o t a l of seventy d i f f e r e n t M s t a r s using o p t i c a l data. They c a l c u l a t e mass l o s s r a t e s ranging from a maximum of 3.8x10" 6 M Q/yr f o r the M1-M2 l a b s t a r , 6 BU Gem (Sanner, 1976) to a minimum of 2x10"' M^/yr f o r the M2-5 I I - I I I s t a r , Peg (Hagen, 1978). Gehrz and Woolf (1971, see, a l s o , Reimers, 1975) d e r i v e d mass l o s s r a t e s from broadband photometric measurements at e f f e c t i v e wavelengths of 3.5, 4.9, 8.4 and 11^ f o r a wide v a r i e t y of M s t a r s . R e p r e s e n t a t i v e r a t e s are 9 x l 0 " 8 M ^ y r for' «c Her, (M5 I b - I l ) , 7X10" 7 M Q/yr f o r ocOri (M2 lab) and 7X10" 6 Mo/yr f o r BC Cyg (M3.5 l a ) . The mean r a t e of mass l o s s obtained f o r Mira v a r i a b l e s was 2x10"' Mg/yr. A trend towards h i g h e r r a t e s f o r the longer p e r i o d , c o o l e r Miras was d e t e c t e d , " i n the r a d i o region', mass i o s s r a t e s are d e r i v e d from the l i n e i n t e n s i t i e s of the non-maser l i n e s . The i n t e n s i t y of such l i n e s i s p r o p o r t i o n a l to fM where, for example, for CO emission, f = • [CO] / [ H A ] , and M i s the r a t e of mass l o s s . An independent measure of e i t h e r f or M i s r e q u i r e d to determine the others For example, assuming s o l a r abundances f o r the envelopes of 15 c o o l , oxygen r i c h s t a r s , M o r r i s e t a l . , (1979) c a l c u l a t e d mass l o s s r a t e s i n the range (1.7x10"' Mg/yr to 2x10" 7 M 0/yr) from the SiO l i n e s . As these r a t e s were a l l one to two orders of magnitude lower than the r a t e s d e r i v e d by Gehrz and Woolf (1971) from i n f r a r e d data, and two to three orders of magnitude lower than the average r a t e s E l i t z e r . et a l . (1976) r e q u i r e to account f o r OH maser emi s s i o n , i t was concluded that up to 99% of the S i i n these envelopes has condensed i n t o 12 g r a i n s . The l a r g e d i f f e r e n c e s i n the mass l o s s r a t e s that are given f o r the four s t a r s that are common to a l l the o p t i c a l s t u d i e s (Hagen, 1978) r e f l e c t the s e r i o u s n e s s of the e r r o r s inherent i n a l l of these c a l c u l a t i o n s . Probably, the r a t e s quoted by Hagen, and Boesgaard and Hagen, which are based upon both o p t i c a l and i n f r a r e d data are the most a c c u r a t e l y determined. They are a l s o , c o n s i s t e n t l y the lowest. Most mass l o s s r a t e s quoted in these and other s t u d i e s give values of about 10" 7 Mgj/yr f o r M s u p e r g i a n t s , and 10" 8 M 0/yr f o r M g i a n t s . (c) Mass Loss Mechanisms The o b s e r v a t i o n s of mass l o s s from red g i a n t s suggest a d r i v i n g mechanism that v a r i e s slowly with the s ' t e l l a r parameters but which i s s i m i l a r i n a l l s p e c t r a l and composition s u b - c l a s s e s . There are, b a s i c a l l y , two candidate d r i v i n g mechanisms f o r red g i a n t s ( F u s i - P e c c i and R e n z i n i , 1976): (1) a s o l a r - l i k e wind from a non-thermally heated s t e l l a r corona, and (2) r a d i a t i o n pressure on dust g r a i n s . In a d d i t i o n , s e v e r a l h y b r i d models have been suggested (see, f o r example, Kwok, 1975) as has the a c t i o n of r a d i a t i o n pressure on the molecular bands ( M a c i e l , 1976). S t e l l a r winds have been suggested as a mass l o s s mechanism i n red g i a n t s f o r many years (see, f o r example, .Weymann, 1962, 1963; Parker 1961; C a s s i n e l l i , 1979 and r e f e r e n c e s t h e r e i n ) . T h i s model i s based upon the p o s t u l a t e that energy i s d e p o s i t e d i n a t h i n s h e l l at the base of the corona. The outer atmosphere expands as a r e s u l t of a b u i l d - u p of the 13 pressure g r a d i e n t that ensues. The source of the d e p o s i t e d energy i s not s p e c i f i e d e x p l i c i t l y but i s supposed to a r i s e e i t h e r from turbulence i n h i g h - l y i n g c o n v e c t i v e zones, or from a c o u s t i c or mechanical waves generated i n some manner i n the subphotospheric l a y e r s of the s t a r . I t i s t h i s l a c k of s p e c i f i c a t i o n of the c o r o n a l h e a t i n g mechanism that i s the major shortcoming of the theory (Mullan, 1978). However, Hearn (1975), on the b a s i s of h i s "minimum f l u x c o r o n a l model" argues that the c o r o n a l s t r u c t u r e i s determined by the magnitude of the ( u n i d e n t i f i e d ) mechanical f l u x i n p u t . As an a i d to d e v e l o p i n g s t e l l a r wind models, Reimers (1975) searched fo r a c o r r e l a t i o n between mass l o s s r a t e s and fundamental s t e l l a r parameters. The r e l a t i o n s h i p which appeared to best f i t the e m p i r i c a l data was (Reimers, 1977) dM/dt = 1.4x10" 1 3L/gR ( I I - l ) where L, g and R are a l l i n s o l a r u n i t s . A l t e r n a t i v e l y , t h i s e x p r e s s i o n can be w r i t t e n as 0.5Mv|st = 1 0 _ 1 3 L which, to quote C a s s i n e l l i (1979) " i m p l i e s that a constant f r a c t i o n of the s t e l l a r l u m i n o s i t y i s being used to p r o v i d e the necessary energy f o r escape of the expanding m a t e r i a l " . Reimers (1977) a l s o d e r i v e d the e m p i r i c a l r e l a t i o n s h i p , ^ e r ^ ^ s t ' from the a v a i l a b l e data on mass l o s s r a t e s . Mullan (1976; see a l s o , C a s s i n e l l i , 1979) has m o d i f i e d Hearn's (1975) minimum f l u x theory to model mass l o s s from red g i a n t s . On the b a s i s of t h i s theory, Mullan p r e d i c t s that an e v o l v i n g s t a r should s h i f t r a t h e r suddenly from a c o r o n a l 14 s t r u c t u r e c h a r a c t e r i z e d by a low mass l o s s r a t e l i k e t h a t of the sun, to one dominated by hig h mass l o s s . The t r a n s i t i o n should occur when the sonic p o i n t i n the flow reaches f a r enough below the base of the corona that s p i c u l e s from the r e l a t i v e l y dense, t u r b u l e n t , chromospheric gas enter the supersonic regime of the expanding gas. Observations suggest that such a t r a n s i t i o n e x i s t s (Reimers, 1977). A l s o , Mullan d e r i v e s the mass l o s s r e l a t i o n s h i p dM/dt = 1.6x10"'M R 0 , 5 (11-2) which i s f u n c t i o n a l l y q u i t e d i f f e r e n t from Reimers' r e l a t i o n s h i p , but which a l s o f i t s the o b s e r v a t i o n a l data q u i t e w e l l . Arguments f o r (Mullan, 1978) and a g a i n s t (Vaiana and Rosner, 1978) the Hearn-Mullan model have appeared i n the l i t e r a t u r e . The arguments a g a i n s t t h i s model are d i r e c t e d at some of the assumptions on which i t i s based. Arguments i n support of the model s t r e s s that hot coronae have been d e t e c t e d in some l a t e type s t a r s , and that the s t e l l a r wind mechanism i s the only v i a b l e mass l o s s mechanism i n atmospheres that appear to be too hot to permit s i g n i f i c a n t g r a i n formation. F u r t h e r , Hagen (1978), and Boesgaard and Hagen (1979) f e e l that mass flows o r i g i n a t e i n the chromosphere and, f o r e a r l y M g i a n t s at l e a s t , are best i n t e r p r e t e d as a r i s i n g from a s t e l l a r wind mechanism. Reimers (1975) reached a s i m i l a r c o n c l u s i o n . However, Weymann (1960) and De Jager and De Loore.(1971) s t a t e that while s t a r s with g < 100 cm/sec 2 may have chromospheres, they do not 15 have coronae so i t i s d i f f i c u l t to understand how the Mullan model c o u l d operate i n l a t e type g i a n t s and s u p e r g i a n t s . The idea that r a d i a t i o n p r e s s u r e on dust g r a i n s might l e a d to s i g n i f i c a n t r a t e s of mass l o s s i n red g i a n t s was r e v i t a l i z e d by the suggestion of Hoyle and Wickramasinghe (1962) that i n t e r s t e l l a r g r a p h i t e g r a i n s may o r i g i n a t e i n the atmospheres of carbon s t a r s . Gilman (1974) demonstrated that the g r a i n s most l i k e l y to form i n M g i a n t s are s i l i c a t e s of v a r i o u s compositions (eg. MgSiO^) and m e t a l l i c i r o n , and i n N g i a n t s , g r a p h i t e and s i l i c o n c a r b i d e . The abundant evidence f o r the presence of dust i n such envelopes ( M e r r i l l , 1978) makes a p l a u s i b l e case f o r the o p e r a t i o n of t h i s mechanism. The formation of atmospheric g r a p h i t e g r a i n s has been s t u d i e d by v a r i o u s groups (Donn et a_l. , 1968; Wickramasinghe et. a l . , 1966; F i x , 1969; D o r f e l d and Hudson, 1973; Tabak et a l . , 1975; and' S a l p e t e r , 1974a,b). The d e t a i l s of the process of small g r a i n n u c l e a t i o n have been c l o s e l y examined by Draine (1978), Draine and S a l p e t e r (1977) and Yamamoto and Hasegawa (1977), e s p e c i a l l y f o r the case of s i l i c a t e g r a i n s . The whole t o p i c of the formation and d e s t r u c t i o n of s t e l l a r and i n t e r s t e l l a r dust g r a i n s i s the s u b j e c t of recent reviews by S a l p e t e r (1977) and Greenberg (1978). In a study i n which the p u l s a t i o n a l p e r i o d of an N Mira v a r i a b l e was simulated, F i x (1969) developed a set of growth r a t e equations f o r dust g r a i n s . H i s model p r e d i c t e d copious g r a p h i t e formation d u r i n g p a r t of the c y c l e of a 2050 K s t a r but the study d i d not account f o r the f a c t that the g r a i n s were forming i n a hydrogen r i c h environment. In 1974, S a l p e t e r 16 demonstated that the formation of C^H^ and CH H i n c o o l , carbon r i c h atmospheres reduces the abundance of carbon vapour to the p o i n t that no condensation of g r a p h i t e g r a i n s can occur above a temperature of about 1700 K. He a l s o showed that s i l i c a t e s cannot p r e c e p i t a t e i n oxygen-rich atmospheres above temperatures in the order of 1200 K. These r e s t r i c t i o n s s e v e r e l y l i m i t the c o n d i t i o n s under which g r a i n formation can be expected to occur in s t e l l a r atmospheres. In a second paper, S a l p e t e r (1974b) developed a set of q u a l i t a t i v e r e l a t i o n s h i p s by which the c h a r a c t e r i s t i c s of the mass flow can be p r e d i c t e d , given the b a s i c parameters of the model such as M, L, g r a i n s i z e and composition, and the r e l a t i v e mass f r a c t i o n of condensed m a t e r i a l . These r e l a t i o n s h i p s are very u s e f u l f o r g a i n i n g i n s i g h t i n t o a process which r e s u l t s from .the complex i n t e r a c t i o n of many v a r i a b l e s . Models of r a d i a t i o n d r i v e n mass flows have been c o n s t r u c t e d by Kwok (1975), f o r M g i a n t s , and by Lucy (1976) f o r N s t a r s , f o r a v a r i e t y of r e p r e s e n t a t i v e s t e l l a r parameters. Both s t u d i e s p r e d i c t e d that r a d i a t i o n pressure on g r a i n s produces s i g n i f i c a n t mass l o s s r a t e s . However, both of these s t u d i e s were based upon assumed c h a r a c t e r i s t i c s of the dust g r a i n s . The n u c l e a t i o n and growth of the g r a i n s were not e x p l i c i t l y i n c l u d e d . Kwok's models were r e s t r i c t e d to the o p t i c a l l y t h i n region of the envelope. G r a i n s of a f i x e d s i z e were assumed to form at the base of the s h e l l and grow as they moved outward. Mass l o s s r a t e s of the order of 1x10" 6 M^/yr were p r e d i c t e d p r o v i d e d that the g r a i n s condense at a p o i n t i n the atmosphere where the gas d e n s i t y i s r e l a t i v e l y l a r g e . Since t h i s 17 c o n d i t i o n i s not g e n e r a l l y f u l f i l l e d i n g i a n t s l a t e r than about M3, Kwok suggested that c o n v e c t i v e t u r b u l e n c e c o u l d reduce the pressure s c a l e h e i g h t s i n the l a t e g i a n t s to a degree s u f f i c i e n t f o r the g r a i n mechanism to be e f f e c t i v e . P a r t i a l d e s t r u c t i o n of g r a i n s v i a s p u t t e r i n g when the d r i f t v e l o c i t i e s exceeded 20 km/sec kept the gas flow v e l o c i t i e s of these models w i t h i n the o b s e r v a t i o n a l l i m i t s . Lucy's (1976) models of expanding, c o o l (2000 K< T e<2750 K) carbon s t a r s i n c o r p o r a t e d a b e t t e r treatment of the atmospheric gas than d i d Kwok's, but a g a i n , g r a i n s of a constant r a d i u s (O.Olyu, in t h i s case) were assumed. Sudden g r a i n formation and complete p r e c i p i t a t i o n of the condensable m a t e r i a l at the sonic p o i n t were a l s o assumed. Lucy found that the expanding atmospheres were s t r u c t u r a l l y q u i t e d i f f e r e n t from the comparable s t a t i c ones - the p r e s s u r e at a given temperature was lower, and the e x t e n s i o n , g r e a t e r . The mass l o s s r a t e s p r e d i c t e d by these models ranged from about l x l 0 ~ 7 M^/yr at T e = 2750 K to about 1x10" 5 M Q/yr at T e = 2000 K. A comparision of the emergent s p e c t r a of these models with o b s e r v a t i o n a l data l e d Lucy to conclude that the r e d i s t r i b u t i o n of the r a d i a t i o n i n these models was e x c e s s i v e , probably as a r e s u l t of an o v e r p r o d u c t i o n of g r a i n s , or an underestimate of the g r a i n r a d i u s . The formation and s t r u c t u r e of dusty c i r c u m s t e l l a r s h e l l s by r a d i a t i o n d r i v e n mass flows have been modelled by G o l d r e i c h and S c o v i l l e (1976) and Jones and M e r r i l l (1976). These models demonstrate that by a d j u s t i n g the s i z e of the g r a i n s and the mass l o s s r a t e s (both input parameters f o r these 18 models), the i n f r a r e d s p e c t r a l c h a r a c t e r i s t i c s of red g i a n t s can be a c c u r a t e l y reproduced. M e n i e t t i and F i x (1978) have produced models i n which the r a d i a t i v e and dynamic i n t e r a c t i o n s of the gas and g r a i n s were t r e a t e d e x p l i c i t l y . They c a l c u l a t e d mass l o s s r a t e s ranging from 3.8X10" 7 M^/yr to 4.9x10" 7 l ^ / y r f o r M g i a n t s . A r a d i u s of O.l^u was assumed f o r the s i l i c a t e g r a i n s . In a l l of these i n v e s t i g a t i o n s , the p r o p e r t i e s of the mass flow depended upon the p r o p e r t i e s of the gas at the sonic p o i n t . With d e c r e a s i n g e f f e c t i v e temperature, the sonic p o i n t (which was a l s o assumed to be e s s e n t i a l l y the condensation p o i n t ) moves inward to regions of higher gas d e n s i t y r e s u l t i n g i n g r e a t e r mass l o s s r a t e s i n accordance with the e m p i r i c a l data ( M e r r i l l , 1978). Deguchi (1980) has modelled the n u c l e a t i o n and growth of s i l i c a t e g r a i n s i n c o o l s t e l l a r envelopes. By using a s e r i e s of f i t t i n g equations and i t e r a t i v e techniques, he was a b l e to p r e d i c t the t e r m i n a l flow v e l o c i t y , the g r a i n d e n s i t y , the f i n a l g r a i n s i z e and the amount of condensable Mg remaining i n the envelope f o r s p e c i f i e d values of M, T, L and dM/dt. These models p r e d i c t an upper mass l i m i t of 1.5MQ f o r which the r a d i a t i o n p r e s s u r e a c t i n g on s i l i c a t e g r a i n s can d r i v e mass flows. They a l s o i n d i c a t e that the t e r m i n a l v e l o c i t y of the flow and the f i n a l g r a i n s i z e i n c r e a s e with the mass l o s s r a t e and decrease with l u m i n o s i t y . The models of Kwok (1975), Lucy (1976), G o l d r e i c h and S c o v i l l e (1976), Jones and M e r r i l l (1976), M e n i e t t i and F i x (1976), and Deguchi (1980) a l l assumed s t e a d y - s t a t e flows. However s t e a d y - s t a t e models cannot adequately represent the mass 19 flows from Mira and semi-regular v a r i a b l e s ; they cannot mimic the i r r e g u l a r f l u c t u a t i o n s observed i n many N and l a t e M g i a n t s ; they cannot reproduce the spora d i c mass e j e c t i o n s of R Cor Bor s t a r s ; and they cannot e x p l a i n apparent long term v a r i a t i o n s i n mass l o s s r a t e s . Moreover, even f o r s t a r s such as ©cOri, which have very s t a b l e c i r c u m s t e l l a r s h e l l s , the a c c e l e r a t i o n region of the flow i s o b v i o u s l y v a r i a b l e , and the s h e l l s appear to c o n s i s t of d i s t i n c t l a y e r s (Sanner, 1976). Furthermore, M o r r i s et a l . , (1979) have found that the c i r c u m s t e l l a r expansion v e l o c i t y and the p e r i o d of the u n d e r l y i n g s t a r are c o r r e l a t e d i n red g i a n t s . T h i s c o r r e l a t i o n suggests that the mass l o s s process i t s e l f i s l i n k e d to p u l s a t i o n . No c o r r e l a t i o n has been found between the mass l o s s r a t e and the s t e l l a r type, C/0 r a t i o , or the c o n s t i t u t i o n of the g r a i n s . The a n a l y s i s of such v a r i a b l e atmospheric phenomena r e q u i r e s time-dependent models. Lucy and Solomon (1967) p r e d i c t e d that red g i a n t s become dynam i c a l l y unstable when the mass i n the i o n i z a t i o n zone i s s u f f i c i e n t to reduce the average r a t i o of the s p e c i f i c heats i n the envelope below 4/3. Subsequently, the p o s s i b i l i t y of mass l o s s from dynamically and p u l s a t i o n a l l y u n s t a b l e s t e l l a r envelopes has been e x t e n s i v e l y s t u d i e d (see Tuchman et a l . , 1978 and r e f e r e n c e s c i t e d t h e r e i n ) . Contrary to what was i n i t i a l l y expected, dynamic i n s t a b i l i t y alone produces only s t e a d i l y p u l s a t i n g models accompanied by l i t t l e or no mass l o s s r a t h e r than the r a p i d e j e c t i o n of the whole envelop. R a d i a t i v e l o s s e s at the s u r f a c e e f f e c t i v e l y damp the i n s t a b i l i t y . At e l e v a t e d l u m i n o s i t i e s , however, the i n s t a b i l i t y leads to v i o l e n t o s c i l l a t i o n s and some mass l o s s . 20 Chapter I I I  THE TIME-DEPENDENT MODEL Comparisons among d i f f e r e n t models of s i m i l a r atmospheres are d i f f i c u l t to make a c c u r a t e l y unless the approaches encompass the same g e n e r a l assumptions and s e l e c t i o n of p h y s i c a l parameters. T h i s problem i s p a r t i c u l a r l y pronounced fo r red g i a n t s because of t h e i r d i v e r s e and complex p r o p e r t i e s . Since the time-dependent models to be c a l c u l a t e d i n t h i s study were to be compared to the s t e a d y - s t a t e models c a l c u l a t e d by Lucy (1976), i t was decided to base the models on Lucy's approach i n s o f a r as i t was p o s s i b l e to do so. In p a r t i c u l a r , many of the p h y s i c a l parameters and model assumptions were i d e n t i c a l i n the two approaches. The assumptions and l i m i t a t i o n s of the time-dependent models w i l l be d e s c r i b e d i n t h i s c h a p ter. The most s i g n i f i c a n t d i f f e r e n c e between the two approaches was i n the dynamical treatment of the problem. Where Lucy assumed a c o n d i t i o n of steady flow and sudden g r a i n formation at the sonic p o i n t , the present models i n c o r p o r a t e d a complete time-dependent treatment of the atmospheric flow, of the n u c l e a t i o n and growth of the carbon g r a i n s , and of the r e s u l t a n t i n t e r a c t i o n between the g r a i n s and the gas. The mathematical d e s c r i p t i o n of t h i s approach concludes t h i s c h a p t er. 21 (a) Assumptions and L i m i t a t i o n s of the Models * For the purposes of t h i s study, the f o l l o w i n g assumptions were u t i l i z e d : Assumption 1: Atmospheric Composition The atmosphere was assumed to c o n s i s t of carbon vapour, an i n e r t gas (mainly hydrogen and helium), and, p o s s i b l y , carbon g r a i n s . Normal s o l a r abundances ( A l l e n , 1973) were assumed for a l l elements except that the r e l a t i v e CNO abundances were set at C:N:0:H = 1.22x10" 3:8.71x10" 5:6.94x10" 4:1 to g i v e a C/O r a t i o of 1.76. Assumption 2: S p h e r i c a l Symmetry The atmospheres were assumed to be s p h e r i c a l l y symmetrical and mass flows were assumed to be r a d i a l o nly. In c a l c u l a t i n g the g r a v i t y of the models, the r a d i a l e x tension of the atmosphere was allowed f o r but i t was assumed that the mass of the atmosphere ( = O.1M0) was n e g l i g i b l e compared to that of the s t a r (M* = 1.5M 0), i . e . , g ( r ) = GM*/r 2. Assumption 3: Grey O p a c i t y The atmospheric o p a c i t y of l a t e type g i a n t s i s s t r o n g l y frequency dependent. However, the assumption that the s t e l l a r o p a c i t y was grey was adopted f o r t h i s study to reduce the computation c o s t s of d e v e l o p i n g the computer programme and 22 of producing the models. The o p a c i t i e s of both the gas and the g r a i n s were represented by t h e i r Rosseland means. Assumption 4: L o c a l Thermodynamic E q u i l i b r i u m I t was assumed that the s t e l l a r atmospheres s a t i s f i e d the c o n d i t i o n of l o c a l thermodynamic e q u i l i b r i u m (LTE). The assumption of LTE i s g e n e r a l l y v a l i d when the occupation numbers of the atomic and molecular l e v e l s are determined by c o l l i s i o n a l p r o c e s s e s . However, i n the presence of a strong r a d i a t i o n f i e l d and low atmospheric d e n s i t i e s , the atomic and molecular t r a n s i t i o n s w i l l be dominated by r a d i a t i v e , not c o l l i s i o n a l p r o c e s s e s . I f LTE were not assumed, i t would have been necessary to o b t a i n a s e l f - c o n s i s t e n t , simultaneous s o l u t i o n of both the r a d i a t i v e t r a n s f e r and the s t a t i s t i c a l e q u i l i b r i u m e q u a t i o n s . Assumption 5: Convective S t a b i l i t y The atmospheres were assumed to be c o n v e c t i v e l y s t a b l e . T h i s assumption was j u s t i f i e d on the b a s i s of the w e l l known i n e f f i c i e n c y of c o n v e c t i v e t r a n s p o r t i n the upper atmospheres of red g i a n t s (Auman, 1969). The c o n d i t i o n s necessary f o r g r a i n n u c l e a t i o n are t y p i c a l l y found at small o p t i c a l depths ( i . e . w e l l above the l a y e r s where c o n v e c t i v e mixing i s l i k e l y to occur u n l e s s o v e r s h o o t i n g of the c o n v e c t i v e zone i s s t r o n g ) . Consequently, the models were extended only to the bottom of the H^ d i s s o c i a t i o n zone where the atmosphere was assumed to be only m a r g i n a l l y unstable a g a i n s t c o n v e c t i v e motions. 23 Assumption 6: R a d i a t i v e E q u i l i b r i u m The assumption of r a d i a t i v e e q u i l i b r i u m i m p l i e s that the atmosphere i s s t a t i c ( M i h a l i s , 1978) and that the t o t a l energy t r a n s p o r t i s r a d i a t i v e ( i . e . , the l u m i n o s i t y i s constant throughout the atmosphere). In a n o n - s t a t i c s i t u a t i o n , the assumption of r a d i a t i v e e q u i l i b r i u m can be j u s t i f i e d only i f the thermal time s c a l e i s l e s s than the dynamical time s c a l e and can be set equal to zero. The thermal time s c a l e i s d e f i n e d as where E i s the t o t a l energy i n the atmosphere, and L i s the l u m i n o s i t y . The dynamical time s c a l e i s where AR i s the depth of the atmosphere, and <c> i s an average value f o r the speed of sound f o r the atmosphere. For M- and N-type s t a r s , Tth n a s a value of about 5x10 s sec, and , about 5X10 7 sec. Thus T t h ^ c l a n o ^ r a d i a t i v e e q u i l i b r i u m can be assumed as a f i r s t approximation i n the development of time-dependent models. P h y s i c a l l y , the c o n d i t i o n TtK<t*d means that thermal changes occur more r a p i d l y i n these s t a r s than do dynamical changes. M a t h e m a t i c a l l y , the assumption of r a d i a t i v e e q u i l i b r i u m means that changes i n one p a r t of the atmosphere are i n s t a n t a n e o u s l y communicated to the other p a r t s . As a s p e c i f i c example, changes i n the s u r f a c e o p a c i t y as the r e s u l t of g r a i n formation there are immediately r e f l e c t e d i n a change of E/L ( I I I - D T d = AR/<c> ( I I I - 2 ) ' 24 temperature throughout a l l the lower l a y e r s . Assumption 7: Graphite G r a i n s The carbon g r a i n s were assumed to have the same s t r u c t u r e and p r o p e r t i e s as o r d i n a r y g r a p h i t e . S p e c i f i c a l l y , the vapour p r e s s u r e , the e x t i n c t i o n c o e f f i c i e n t , and the Lothe-Pound f a c t o r of g r a p h i t e were used. In a d d i t i o n , the s u r f a c e - f r e e -energy of the g r a i n s was put equal to 1000 ergs/cm 2, a value Tabak et aJL, (1975) have suggested f o r g r a p h i t e g r a i n s . Assumption 8: S p h e r i c a l G r a i n s The g r a p h i t e g r a i n s were assumed to be s m a l l , non-deformable, s p h e r i c a l p a r t i c l e s u n i f o r m l y d i s p e r s e d throughout the gas. Wickramasinghe (1972) has suggested that g r a p h i t e w i l l form as p l a t e l e t s r a t h e r than spheres at low p r e s s u r e s . However, because of the high degree of s u p e r s a t u r a t i o n r e q u i r e d f o r s i g n i f i c a n t g r a i n n u c l e a t i o n to occur, Donn et a l , (1968) concluded that the g r a i n s w i l l be n e a r l y s p h e r i c a l , non-p o l a r i z i n g p o l y c r y s t a l l i n e g r a i n s ( i . e . , s o o t - l i k e ) . The r e l a t i v e l y f l a t wavelength dependence of the i n t r i n s i c p o l a r i z a t i o n i n N s t a r s supports t h i s c o n c l u s i o n . Assumption 9: C l a s s i c a l , Homogeneous Gr a i n N u c l e a t i o n I t was assumed that the g r a i n s formed e s s e n t i a l l y a c c o r d i n g to the c l a s s i c a l n u c l e a t i o n theory f i r s t developed by Volmer and Webber (1926). T h i s theory i s summarized i n Appendix B. The m o d i f i c a t i o n s that were made to the c l a s s i c a l theory were 25 the i n c l u s i o n of the Lothe-Pound f a c t o r , P , (Lothe and Pound, 1962), the Z e l d o v i c h non-equlibrium f a c t o r , Z*, ( Z e l d o v i c h , 1942), and the allowance f o r the n u c l e a t i o n from a s s o c i a t e d vapours (Katz, et a l . , 1966). S a l p e t e r (1974a) and Draine (1978) have argued that i o n i c n u c l e a t i o n should be r e l a t i v e l y unimportant f o r the formation of g r a p h i t e g r a i n s so only homogeneous n u c l e a t i o n was c o n s i d e r e d . Assumption 10: K i n e t i c a l l y Governed G r a i n Growth The mass exchange r a t e between a vapour and i t s condensed s t a t e i s governed by the slower of the two proce s s e s : (1) the k i n e t i c r a t e at which the vapour molecules c o l l i d e with, and adhere to the g r a i n s u r f a c e , and (2) the r a t e at which the vapour molecules d i f f u s e through the gas to the v i c i n i t y of the g r a i n . G e n e r a l l y , i f an i n e r t gas i s pre s e n t , d i f f u s i v e t r a n s p o r t governs the mass exchange r a t e , whereas i n a pure vapour, d i f f u s i o n i s n o n - e x i s t e n t , and the mass exchange r a t e i s c o n t r o l l e d only by the k i n e t i c f l u x . Although carbon vapour w i l l form only a small f r a c t i o n by mass of the atmosphere, (about 1 0 " 3 ) , d i f f u s i o n w i l l be r a p i d at the low gas p r e s s u r e s and high temperatures of a s t e l l a r atmosphere. Thus, f o r the purposes of t h i s study, i t was assumed that the r a t e of g r a i n growth i s k i n e t i c a l l y c o n t r o l l e d . A s i m i l a r assumption was i m p l i c i t l y made in the d e r i v a t i o n of the homogenous r a t e of n u c l e a t i o n . 26 Assumption 11: Uniform G r a i n V e l o c i t y and S i z e In d e r i v i n g the c o n s e r v a t i o n equations f o r the models, i t was assumed that the g r a i n s were n o n - i n t e r a c t i n g and, consequently, would not c o n t r i b u t e to the atmospheric p r e s s u r e . I t was a l s o assumed that the r a d i i of the g r a i n s c o u l d be represented by a l o c a l e f f e c t i v e v a l u e . These two assumptions p l u s the assumption that the g r a i n p a r t i c l e s were l a r g e enough that Brownian motion c o u l d be ignored, made i t p o s s i b l e to as s i g n a common v e l o c i t y and temperature as w e l l as a common ra d i u s to a l l g r a i n s i n each computational c e l l . Under these assumptions, the growth of the g r a i n s w i l l be u n a f f e c t e d by the processes of coalescence and s h a t t e r i n g . While i t . i s acknowledged that v a r i a t i o n s among p a r t i c l e s i z e s w i l l probably e x i s t in. g r a i n - f o r m i n g r e g i o n s , the a c t u a l spread i n g r a i n s s i z e ' should be r e s t r i c t e d ' by g r a i n - g r a i n c o l l i s i o n s ( S a l p e t e r , 1974b) . Assumption 12: Continuous F l u i d Approximation The g r a i n s were assumed to be momentum coupled to the gas, and the gas to be i n t e r n a l l y momentum coupled. These two c o n d i t i o n s ensure that the momentum absorbed by the g r a i n s from the r a d i a t i o n f i e l d i s un i f o r m l y d i f f u s e d throughout the gas. The assumption of momentum c o u p l i n g i s e q u i v a l e n t to the assumption that the gas and the g r a i n s can be t r e a t e d as continuous f l u i d s with the usual macroscopic p r o p e r t i e s of flow v e l o c i t y , temperature, and d e n s i t y , and that t h e i r flow can be d e s c r i b e d by the o r d i n a r y equations of hydrodynamics. 27 I m p l i c i t l y , the use of the continuous f l u i d approximation fo r the g r a i n s makes the assumption t h a t the r a d i i of the g r a i n s are very s m a l l , and t h e i r number i n each computational c e l l i s r e l a t i v e l y l a r g e . The continuum assumption f o r the g r a i n s might be q u e s t i o n e d when g r a i n n u c l e i are j u s t beginning to form. However, n u c l e a t i o n i s n e g l i g i b l e u n t i l the c r i t i c a l s u p e r s a t u r a t i o n r a t i o i s reached. Then, n u c l e a t i o n proceeds very r a p i d l y . Thus, the number d e n s i t y of the g r a i n s i s e i t h e r so low that they may be n e g l e c t e d or l a r g e enough that the continuum assumption i s v a l i d . I t was p r e d i c t e d that the intermediate s t a t e would be passed through so q u i c k l y , that i t s e f f e c t s would be q u i c k l y damped out and c o u l d be ignored. T h i s p r e d i c t i o n was subsequently confirmed. The use of continuum v a r i a b l e s f o r the gas i m p l i e s that a kind of "smoothing" process i s i n e f f e c t s i n c e the gas v e l o c i t y and temperature may vary s t r o n g l y i n the neighbourhood of a g r a i n that i s moving through the gas and exchanging energy with i t . However, a l l v a r i a b l e s w i l l be c a l c u l a t e d f o r volumes l a r g e enough to c o n t a i n many g r a i n s to ensure the v a l i d i t y of the continuum assumption f o r the g r a i n s , so t h i s smoothing process f o r the gas should a l s o be v a l i d . Assumption 13: Grain Temperature I t was assumed that the g r a i n s and the gas had the same temperature as the l o c a l r a d i a t i o n f i e l d . Lucy (1976) used the same assumption in h i s study. The adoption of Assumptions 3, 4, 6 and 13 g r e a t l y s i m p l i f i e d the procedures r e q u i r e d to c a l c u l a t e the temperature d i s t r i b u t i o n i n the models. 28 (b) Mathematical D e s c r i p t i o n of the Model The flow of a g a s - g r a i n mixture i n a s t e l l a r atmosphere i s d e s c r i b e d by two se t s of coupled equations: one set p e r t a i n i n g to the gas,'and the other, to the g r a i n s . The c o u p l i n g i s pro v i d e d by the t r a n s f e r of mass and momentum between the g r a i n s and gas. Since the atmosphere at some p o i n t i s f r e e of g r a i n s , the homogeneous n u c l e a t i o n of g r a i n n u c l e i as we l l as t h e i r growth and decay w i l l be i n c l u d e d i n the process of mass exchange between the two phases. In t h i s r e s p e c t , the treatment of the problem d i f f e r s from that of c o n v e n t i o n a l treatments of condensing vapours (see, f o r example, Marble, 1969). ( i ) Continuum Conservation Equations condensing to form g r a i n s , seven c o n s e r v a t i o n equations are r e q u i r e d : three d e s c r i b e the c o n s e r v a t i o n of mass, and two each, the c o n s e r v a t i o n of momentum and energy. In E u l e r i a n form, the equations of mass c o n s e r v a t i o n f o r a s p h e r i c a l l y symmetric geometry a r e : To d e s c r i b e the flow of a gas, pa r t of which i s (111 - 3) bp + 1 _£_(r*fu) = Vt r2br 1 ( I I I - 4 ) and ¥ T r ^ r J K r ( I I I - 5 ) 29 where j>vt ^> and f>p are the d e n s i t i e s of the condensing vapour, the t o t a l gas component of the atmosphere, and the g r a i n s , u and Up are the v e l o c i t i e s of the gas and g r a i n s , and Mp i s the l o c a l mass r a t e of g r a i n formation. M p i n c l u d e s both the process of homogeneous n u c l e a t i o n and the processes of g r a i n growth and decay as a r e s u l t of condensation and s u b l i m a t i o n . The f i r s t of the above equations i s the c o n s e r v a t i o n of mass equation f o r the carbon vapour alone, the second, f o r the t o t a l gaseous component of the atmosphere, and the t h i r d , the mass c o n s e r v a t i o n f o r the g r a i n s . I m p l i c i t i n t h i s form of the mass equations i s the assumption that the volume occupied by the g r a i n s i s n e g l i g i b l e . Except p o s s i b l y f o r h y d r o g e n - d e f i c i e n t s t a r s , t h i s c o n d i t i o n should be v a l i d f o r a l l l a t e - t y p e s t a r s . In the absence of g r a i n s , the atmosphere i s assumed to be i n h y d r o s t a t i c e q u i l i b r i u m . As the g r a i n s form and grow, r a d i a t i o n pressure tends to push the g r a i n s outward. The movement of the g r a i n s through the gas, i n t u r n , tends to drag the gas outward as w e l l . Both of these e f f e c t s p l u s the f o r c e of g r a v i t y must be i n c l u d e d i n the equations of motion f o r the gas and g r a i n s . Thus, the a p p r o p r i a t e c o n s e r v a t i o n of momonetum equations a r e : d(pu) + 1 Mr 2pu 2) + bP Jt 7 7'5T y Tr = Fp + R -y*g(r) - M pu (III-6) and = - F p + Rp - f r q ( r ) + M pu (III-7) 30 where Fp re p r e s e n t s the drag f o r c e exerted on the gas by the p a r t i c l e s , R and R Q the r a d i a t i o n pressure e x e r t e d on the gas r and g r a i n s r e s p e c t i v e l y , g ( r ) = GM*/r 2, and P i s the gas pr e s s u r e . The term -Mpu appearing i n the f i r s t of these equations i s the r a t e at which momentum i s removed from the vapour as a r e s u l t of the mass exchange between the carbon vapour and g r a p h i t e g r a i n s . S i m i l a r l y , the r a t e at which the g r a i n s gain momentum as a r e s u l t of t h i s condensation i s MpU. In w r i t i n g these equations, the v i s c o u s s t r e s s tensor f o r the gas has been omitted. T h i s q u a n t i t y should be very s m a l l f o r the hot t h i n gas c h a r a c t e r i s t i c of s t e l l a r atmospheres. Note a l s o that the equation of motion f o r the g r a i n s does not c o n t a i n a term of the form &P/br s i n c e the drag f o r c e a s s o c i a t e d with the gross p r e s s u r e g r a d i e n t i s n e g l i g i b l e f o r very small g r a i n s . Moreover, the g r a i n s were assumed to have no random motion and, t h e r e f o r e , cannot have any pressure a s s o c i a t e d with them. The l a s t two equations needed to complete the system of c o n s e r v a t i o n laws are the c o n s e r v a t i o n of energy equations. The energy equation f o r the gas can be w r i t t e n as _b_{P (e+l/2u 2)} + 1 ^ _ { r 2 p u ( e + l / 2 u 2 + P/p } (111-8) bt r 2 5 r J J = uFp + uR - ujcg(r) + ^ >E(y?,r) + Qp + q where E(j>,r) i s the net he a t i n g r a t e per gram due to r a d i a t i o n , Qp i s the t o t a l heat per u n i t volume t r a n s f e r r e d from the g r a i n s to the gas phase by thermal c o n d u c t i o n , q i s the r a t e that heat i s t r a n s f e r r e d by an exchange of mass, and e i s the i n t e r n a l 31 energy of the gas per gram. S i m i l a r l y , the energy equation f o r the g r a i n s i s hifAe + l / 2 u P 2 ) } + 1 M r 2 f p U p ( e + l / 2 u p 2 ) } (III-9) = -u pF p + u pR p - Up^pg(r) + f p E p ( f p f r ) - Qp + q However, the c o n s t r a i n t s imposed by the assumptions of r a d i a t i v e e q u i l i b r i u m , l o c a l thermodynamic e q u i l i b r i u m and a grey atmosphere, s i g n i f i c a n t l y change the procedures needed to c a l c u l a t e the temperature of the gas. Instead of s o l v i n g the equation of c o n s e r v a t i o n of energy f o r e, the i n t e r n a l energy, and u s i n g e to c a l c u l a t e the temperature, the temperature can be found d i r e c t l y from the equation (Lucy, 1976) T * ( r ) = 0.5T* 4{2W(r) + 3^(r)/2} (111-10) where W(r) i s the r a d i a t i o n d i l u t i o n f a c t o r , T^ , the Rosseland mean o p t i c a l depth a d j u s t e d f o r the s t e l l a r c u r v a t u r e (see Lucy, 1976), and T*, the s t e l l a r t emperature 1. The usual form f o r W(r) i s W(r) = 1/2{1 - ( l - ( R * / r ) 2 } °-5 r>R* ( I I I - l l ) xIn an extended atmosphere, the " r a d i u s " of the s t a r f o r any given o p t i c a l depth v a r i e s g r e a t l y with frequency. Thus, the terms "photosphere" and " e f f e c t i v e temperature" become ambiguous. A c c o r d i n g l y , the photospheric r a d i u s , R*, i s d e f i n e d to be the r a d i u s at which f R e q u a l s 2/3. T* = (L/4 R* 2CT)y 4 i s the temperature at the photosphere and a c t s l i k e an e f f e c t i v e temperature f o r the s t a r . 32 = 1/2 r<R* The d i s c o n t i n u o u s nature of t h i s f u n c t i o n caused numerical problems when used to compute the time-dependent models. A form f o r the d i l u t i o n f a c t o r which i s continuous and which f o r c e s T ( r ) to equal T* when r=R* i s W(r) = 1/2{1 - (1 - ( R b a s e / r ) 2 ) °-5} / { l - (1 - R b a s e / R * ) 2 ) 0 - 5 } (111-12) where Rbase ^ s t * i e r a d i u s of the lower boundary, or base of the atmosphere. T h i s form g i v e s s l i g h t l y higher temperatures throughout the atmosphere than does the standard form f o r W(r), but t h i s e f f e c t i s minimized as long as the photosphere i s c l o s e to the base. For example, f o r r = 1 . 0 6 2 x l 0 1 4 cm, R* = 5 . 5 9 x l 0 1 3 cm and R b < x s e = 5 . 5 1 9 x l 0 1 3 cm, Equation (111-11) g i v e s W = 0.075 whereas Equation (111-12) g i v e s W = 0.086. The p r i n c i p a l advantage of the m o d i f i e d form of W(r) (Equation ( I I I -12)) i s that i t completely removes the d i s c o n t i n u i t y at the photosphere while r e t a i n i n g the e s s e n t i a l p r o p e r t i e s of the standard form. The m o d i f i e d form of W(r) was used to c a l c u l a t e the temperature d i s t r i b u t i o n of a l l the extended models c a l c u l a t e d i n t h i s study. Both the energy equation and the temperature equation rep r e s e n t the c o n s e r v a t i o n of energy. The energy equation, Equation ( I I I - 8 ) , c o n t a i n s the term, (1/r-) 2 d(r 2uP)/dr, which r e p r e s e n t s the work done by r i s i n g volumes of gas on t h e i r s urroundings. In cases where t h i s term i s l a r g e , the gas 33 » temperature may d i f f e r from the temperature of the l o c a l r a d i a t i o n f i e l d . The temperature equation, Equation (111-10), i m p l i e s t h a t the gas flow i s isothermal i n the sense that r i s i n g volumes of gas i n s t a n t a n e o u s l y a d j u s t to the l o c a l temperature. Wood (1979), i n h i s study of p u l s a t i o n and mass l o s s i n Miras, used a temperature equation s i m i l a r to Equation (111-10). For a f i x e d T*, the temperature i n isothermal models i s a f u n c t i o n of W and TflOnly. W, a g e o m e t r i c a l f a c t o r i s t i m e - i n v a r i a n t . For Xf. < 10" 3, the temperature i s a f u n c t i o n p r i m a r i l y of W and i s , t h e r e f o r e , almost time i n v a r i a n t as w e l l . At the very s u r f a c e of the atmosphere, (T R=10 -') even o p a c i t y i n c r e a s e s by f a c t o r s of 10 3 caused by the formation of g r a i n s w i l l not produce s i g n i f i c a n t temperature changes. At depths below the photosphere, the temperature i s determined p r i m a r i l y by T R and becomes very s e n s i t i v e to changes i n o p a c i t y . I t i s at these depths that the temperature s t r u c t u r e of the atmosphere r e f l e c t s the changes i n c o n d i t i o n s caused by mass flows and g r a i n f o rmation. In a d d i t i o n , l a r g e mass flows at the base can change T* r e s u l t i n g i n a change of the temperature s t r u c t u r e throughout the atmosphere. ( i i ) Mass Exchange The t o t a l r a t e at which mass i s exchanged between the s o l i d and vapour phases was w r i t t e n as ( F i x , 1969) 3 Mp = JWa*3f5/3 + Np4ira^ 5-^ZT joCj(yQj - pj) ' (111-13). where 3 J = 2 HO/kT) l / 2 N ( i H ^ Z ^ j y 3 j j 2 ' ) e x p ( - A F * / k T ) (111-14) 34 i s the n u c l e a t i o n r a t e f o r the g r a i n s (see Appendix B). In t h i s equation, J>2 = 2.2gm/cm3 i s the d e n s i t y of g r a p h i t e , N p, the number d e n s i t y of the g r a i n s , a, the g r a i n r a d i u s , -°- = 9x10" 2 4 cm 3, the volume per carbon atom i n g r a p h i t e , °Cjt the s t i c k i n g c o e f f i c i e n t s f o r C, C z, and C 3, N ( l ) the number d e n s i t y of C, and AF* i s the f r e e energy of forming a g r a i n nucleus having a r a d i u s a*. A l s o , JBj = Bu-j/(2irmc j k T ) ^ 2 and JB- = B?y/(2Timc jkT) 1/* where P^j and P£j are the vapour p r e s s u r e s and the s a t u r a t i o n vapour p r e s s u r e s of C, and C 3 . The f i r s t term i n Equation (111-13) a r i s e s from the n u c l e a t i o n of the g r a i n s , and the second, from the growth of e x i s t i n g g r a i n s . The summation i s necessary because carbon forms an a s s o c i a t e d vapour c o n s i s t i n g mainly of C, Cz and C 3. The number d e n s i t y equation of the g r a i n s i s obtained by combining Equation (I I I - 3 ) with Equation (111-13). The r e s u l t i s dNp/dt + r- 2c-(r 2u pNp )/dr = J (111-15) ( i i i ) Momentum Exchange If r a d i a t i o n pressure i s to cause s i g n i f i c a n t mass l o s s , the gas and the g r a i n s must be s t r o n g l y momentum coupled, that i s , the momentum that i s t r a n s f e r r e d to the g r a i n s from the 35 r a d i a t i o n f i e l d must be e f f i c i e n t l y t r a n s f e r r e d from the g r a i n s to the gas molecules. Under these c o n d i t i o n s , the r a d i a t i o n pressure f o r c e w i l l a c c e l e r a t e the g r a i n s outward and the v i s c o u s drag f o r c e w i l l a c c e l e r a t e the gas. Strong momentum c o u p l i n g w i l l p r e v a i l when the g r a i n s are t r a v e l l i n g at c l o s e to t h e i r t e r m i n a l d r i f t v e l o c i t y and when that d r i f t v e l o c i t y i s not too l a r g e compared to the gas v e l o c i t y . I t i s a l s o necessary that the gas be i n t e r n a l l y coupled f o r t h i s mass l o s s to occur so that those gas molecules which d i r e c t l y gain momentum through c o l l i s i o n s with the g r a i n s q u i c k l y d i s t r i b u t e i t to the r e s t of the gas. I t has a l r e a d y been e s t a b l i s h e d that the gas can be t r e a t e d as a continuum. T h i s i m p l i e s that i t i s i n t e r n a l l y coupled. Once the g r a i n s form, momentum c o u p l i n g i s e s t a b l i s h e d very q u i c k l y (Gilman, 1969). The a p p r o p r i a t e form f o r the p a r t i c l e drag f o r c e , Fp , between the gas and the g r a i n s depends upon the Reynolds number and the Mach number f o r the flow. In the present context, the Reynolds number, Re, can be d e f i n e d as Re = |u p-u|ajVTY (111-16) = 2|u p-u|/v t h where \ = ^ a v t h / 2 i - s the v i s c o s i t y of the gas, and v t h = (kT/^u'mH ) V* (£5xl0 5 cm/sec at 1500 K) i s the thermal v e l o c i t y of the gas. If |up-u| i s s m a l l , Re <1, the flow f a l l s i n t o the l i n e a r Stokes regime and the drag f o r c e per u n i t volume 36 F p = 6iraT\|u p-u| (111-17) Under these c o n d i t i o n s , the g r a i n s are s a i d to be p o s i t i o n coupled. For l a r g e d i f f e r e n t i a l v e l o c i t i e s , the drag f o r c e no longer v a r i e s l i n e a r l y with the d r i f t v e l o c i t y , w = (up - u), but becomes F p = N p7Tay(Up - u ) 2 C D / 2 (111-18) where Cp i s an e m p i r i c a l l y determined drag c o e f f i c i e n t . The v e l o c i t y at which the t r a n s i t i o n from the l i n e a r to the q u a d r a t i c form of the drag f o r c e occurs i s i l l - d e f i n e d . Wickramasinghe (1972) suggests that the q u a d r a t i c form be used when w >v t h , and the l i n e a r form, when w < v t h . Lucy (1976) used the q u a d r a t i c form f o r a l l h i s steady flow models even when the d r i f t v e l o c i t y was somewhat lower than the l o c a l thermal v e l o c i t y . In h i s study, he used C]> = 2, which corresponds roughly to a Reynolds number of 3. The i n t e n t of the present study i s to f o l l o w the g r a i n s from the time they are formed u n t i l they are e x p e l l e d from the atmosphere or u n t i l they are evaporated. Thus, the d r i f t v e l o c i t i e s w i l l cover a wide range of v a l u e s . A c c o r d i n g l y , Wickramasinghe's s w i t c h i n g procedure was adopted. . ( i v ) Gas Opacity The o p a c i t y sources i n c l u d e d i n the c a l c u l a t i o n of the Rosseland mean f o r the gas were those of (a) f r e e - f r e e a b s o r p t i o n by H", H" and He", (b) bound-free a b s o r p t i o n by H", 37 (c) R a y l e i g h s c a t t e r i n g by H and H^, (d) Thompson s c a t t e r i n g by f r e e e l e c t r o n s , and (e) bound-bound a b s o r p t i o n by CN. The data compiled by Johnson e t . a l . (1972) were used to c a l c u l a t e the CN o p a c i t y . The c o n t r i b u t i o n s from each o p a c i t y source were computed f o r a set of t h i r t y f r e q u e n c i e s that covered the range from 3.8 m i c r o n s - 1 to 0.035 m i c r o n s - 1 . The frequency i n t e r v a l s were chosen p a r t i a l l y on the b a s i s of g i v i n g a good f i t to the CN bands. The Rosseland mean was then computed from these t h i r t y v a l u e s . The c a l c u l a t i o n of the a b s o r p t i o n c o e f f i c i e n t s r e q u i r e d the knowledge of the e l e c t r o n p r e s s u r e and of the abundances of CN, H, H +, H z and He. These q u a n t i t i e s were computed by s o l v i n g a complicated equation of s t a t e f o r the gas. The f i r s t i o n i z a t i o n l e v e l s of S i , Fe, Mg, A l , Ca, Na, K, Cr and S i n a d d i t i o n to those of H, C, N and 0 were used to c a l c u l a t e d the e l e c t r o n p r e s s u r e . C o n d i t i o n s i n N-type s t a r s are such that many molecular s p e c i e s are p r e s e n t . The abundances of these molecules were obtained by s o l v i n g a l a r g e set of coupled d i s s o c i a t i o n equations. The molecules that were i n c l u d e d i n these c a l c u l a t i o n s were H z, H£, C z, C 3, CO, CN, HCN, C^H, C^H^, C j H , N £ , 0^ , CH, CH^ , CHj , Cj H, Cj H^ , Ci+ , C^ , H , Cij H^> , C^  N^ , C^ N , HC 3N, C ZH 3 , C^H ¥, C 3H 3 and NH. T h i s l i s t i n c l u d e d only two molecular s p e c i e s which c o n t a i n oxygen: 0^ and CO. U n l i k e M s t a r s where many oxygen-bearing s p e c i e s are present, v i r t u a l l y a l l of the oxygen present i n the atmospheres of c a r b o n - r i c h s t a r s i s locked up i n the form of CO. The s e l e c t i o n of the molecular s p e c i e s to be i n c l u d e d i n the equation of s t a t e c a l c u l a t i o n s was based upon the c a l c u l a t i o n s done by T s u j i 38 (1964) f o r carbon s t a r s . The d i s s o c i a t i o n c o n s t a n t s were c a l c u l a t e d from the data i n the JANAF t a b l e s wherever p o s s i b l e . For the remaining molecules the data given by T s u j i was used. (v) G r a i n Opacity The o p a c i t y of the g r a i n s was represented by the Rosseland mean. Wherever p o s s i b l e , the e x t i n c t i o n c o e f f i c i e n t f o r the g r a i n s was found by i n t e r p o l a t i n g i n the t a b l e of c o e f f i c i e n t s c a l c u l a t e d by Wickramasinghe (1967). For the small r a d i i not covered by t h i s t a b l e , the approximate Mie expre s s i o n given by Qei,t (a,A) = -4(2iTaA)Im{ (m 2-l)/(m 2 + 2)} (111-19) was used to c a l c u l a t e the e x t i n c t i o n , Q e^t • In t h i s equation, a i s the g r a i n nucleus, 7i, the wavelength and m = K - 2io"A/c, the complex r e f r a c t i v e index of the g r a i n . T h i s approximate e x p r e s s i o n f o r Q ext (a,\) g i v e s good r e s u l t s p r o v i d e d that 2TTa/A. << 1 where a and A. are expressed i n the same u n i t s . The values of K and d" were taken from the data c o m p l i l e d by Wickramasinghe (1965). For A."1 < 0.47 m i c r o n s " 1 , the approximate e x t i n c t i o n c o e f f i c i e n t , with K = 4 and 2C/c = 10, was used f o r a l l g r a i n r a d i i . The Rosseland mean a b s o r p t i o n c o e f f i c i e n t was c a l c u l a t e d by summing the c o n t r i b u t i o n s at the same t h i r t y f r e q u e n c i e s used to c a l c u l a t e the Rosseland mean f o r the gas but usi n g a s p e c t r a l d i s t r i b u t i o n a p p r o p r i a t e f o r a d i l u t e d T = T* r a d i a t i o n f i e l d . T h i s procedure was adopted i n view of the f a c t that the r a d i a t i o n a c t u a l l y i n t e r c e p t e d by the g r a i n s i n the 39 o p t i c a l l y t h i n upper atmsophere should be c h a r a c t e r i z e d by the photospheric temperature r a t h e r than the l o c a l gas temperature. The o p a c i t y of the g r a i n s i s t y p i c a l l y s e v e r a l orders of magnitude g r e a t e r than that of the gas i n the g r a i n - f o r m i n g regions of l a t e type g i a n t s . T h e r e f o r e , i n such r e g i o n s , the ab s o r p t i o n c o e f f i c i e n t was put equal t o the sum of the Rosseland means f o r the gas and the g r a i n s . The accuracy of t h i s procedure should be at l e a s t commensurate with that of the grey assumption. The r a d i a t i o n pressure f o r c e terms were w r i t t e n as R = AL/47Tr 2c f o r the gas, and (111-20) Rp = K pL/4TTr 2c (111-21) f o r the g r a i n s , where K and Kp are the a p p r o p r i a t e Rosseland mean o p a c i t i e s . 40 Chapter IV  NUMERICAL PROCEDURES The system of equations developed i n the preceding chapters was s o l v e d n u m e r i c a l l y on UBC's Amdahl 470 V6-II computer. The s o l u t i o n of the dynamical p o r t i o n of the programme was based on the a l g o r i t h m s used by E i l e k (1975), and those suggested by Harlow and Amsden (1975) f o r multiphase flow problems. A d e s c r i p t i o n of these s o l u t i o n procedures i s c o n t a i n e d i n t h i s c h a p ter. The numerical techniques were sub j e c t e d to a number of t e s t s p r i o r to the c a l c u l a t i o n of the g r a i n - f o r m i n g models. These t e s t s are summarized in Appendix A. (a) The S o l u t i o n A l g o rithm P h y s i c a l l y , the time-dependent equations d e s c r i b e a non-viscous, compressible, two-phase condensable f l u i d flow i n one space dimension. M a t h e m a t i c a l l y , the equations comprise a system of n o n - l i n e a r , f i r s t order, p a r t i a l d i f f e r e n t i a l e q u a t i o n s . The I m p l i c i t C o n t i n u o u s - F l u i d E u l e r i a n (ICE) numerical technique was used to s o l v e t h i s system of equations. The developers of t h i s method, Harlow and Amsden (1968), a s s e r t t hat i t i s a n u m e r i c a l l y s t a b l e and e f f i c i e n t means of c a l c u l a t i n g t r a n s i e n t f l u i d flows i n a l l v e l o c i t y ranges. Of p a r t i c u l a r importance to t h i s study i s the f a c t that the ICE method can be used to c a l c u l a t e t r a n s o n i c flows. No s p e c i a l techniques are r e q u i r e d to handle shock waves. The ICE method i s 41 based upon an i m p l i c i t , f i n i t e - d i f f e r e n c e approximation to the no n - l i n e a r equations of hydrodynamics. To s t a r t the c a l c u l a t i o n s , an i n i t i a l model i s r e q u i r e d . Subsequent models are c a l c u l a t e d through a sequence of small but f i n i t e time s t e p s . Each model r e p r e s e n t s the c o n f i g u r a t i o n of the atmosphere at some d e f i n i t e time a f t e r the s t a r t of the c a l c u l a t i o n s . To c a l c u l a t e the f i n i t e d i f f e r e n c e approximations to the space d e r i v a t i v e s appearing i n the equations, an E u l e r i a n mesh of computational c e l l s was i n t r o d u c e d . A l l f i e l d v a r i a b l e s were ce n t e r e d i n these c e l l s except the v e l o c i t i e s of the g r a i n s and the gas. These two v a r i a b l e s were centered on the c e l l boundaries. Throughout the c a l c u l a t i o n s , the values of the c e l l -c e n t ered v a r i a b l e s were approximated at the c e l l boundaries by t h e i r l o g a r i t h m i c averages, but the v e l o c i t i e s were approximated at the c e l l c e n t e r s by t h e i r s t r a i g h t means. The t y p i c a l l y l a r g e range of magnitudes of the c e l l - c e n t e r e d v a r i a b l e s made the f i r s t procedure mandatory, whereas the t y p i c a l l y l i m i t e d range of v e l o c i t i e s p ermitted the use of the second. A l s o , Wood (1979) notes that the use of l o g a r i t h m i c averages f o r c e l l c e n tered v a r i a b l e s i s necessary to ensure that s t a b l e s o l u t i o n s are obtained f o r the hydrodynamic equations i n the presence of the e x t e r n a l body f o r c e s of g r a v i t y and r a d i a t i o n p r e s s u r e . ( i ) V a r i a b l e Depth Steps The depth v a r i a b l e , r , was r e p l a c e d by the E u l e r i a n c o - o r d i n a t e , x, which was d e f i n e d as x = oC + r (IV-1) 42 where x=l,2,3,.....s. oC , j$ , and V are a d j u s t a b l e parameters, and s i s the number of c e l l s used to compute the model. In terms of x, the r a d i a l depth step was dr = ( r t )dx = h Ldx (IV-2) and thus, with % < 1, i n c r e a s e d with i n c r e a s i n g r a d i u s . T h i s v a r i a t i o n i n c e l l s i z e was u s e f u l i n reducing the t o t a l number of c e l l s needed f o r the c a l c u l a t i o n s . Small c e l l s were r e q u i r e d near the photosphere where the f i e l d v a r i a b l e s changed r a p i d l y with depth, whereas l a r g e r c e l l s were s u i t a b l e near the s u r f a c e . Wood (1979) used a s i m i l a r v a r i a b l e zone spacing t o compute h i s models. The parameters, oC , y3 > a n <3 ^ were a d j u s t e d f o r each e f f e c t i v e temperature to g i v e a s u i t a b l e g r i d . oC and ^ were read i n t o the model. Ji was c a l c u l a t e d as J3 = (x* -°C)/(R**) (IV-3) where x* i s the value x i s to have at the photosphere. G e n e r a l l y , the g r i d was set up such that x* = 6 or 7, and tf = —3. oC was a d j u s t e d u n t i l x = s at r =RS, the r a d i u s s e l e c t e d as the upper boundary, or s u r f a c e , of the model. As an i l l u s t r a t i o n , f o r T* = 2400K and s =99, the parameters used were oc= 120, ft = - 1 . 9 1 6 x l 0 4 3 , and = -3. 43 ( i i ) O p a c i t y Tables The Rosseland mean gas o p a c i t y , and the p a r t i a l p r e s s u r e s of C, C^, and C 3 were c a l c u l a t e d as f u n c t i o n s of T and f> at the beginning of the study, and s t o r e d i n t a b l e s . The t a b l e s were c a l c u l a t e d f o r three v a l u e s of the carbon abundance: C/H = 1.22xl0" 3, 0.957xl0" 3, and 0.694xl0" 3 to allow f o r the e f f e c t s of the d e p l e t i o n of gaseous carbon i n the presence of the g r a p h i t e g r a i n s . A three-way i n t e r p o l a t i o n scheme was designed to compute the values of the a b s o r p t i o n c o e f f i c i e n t of the gas and the p a r t i a l p r e s s u r e s of gaseous carbon from the values s t o r e d i n the t a b l e s . ( i i i ) The I n i t i a l Model The i n i t i a l models r e q u i r e d to begin the c a l c u l a t i o n s of the time dependent models, were c a l c u l a t e d assuming h y d r o s t a t i c e q u i l i b r i u m . The pressure d i s t r i b u t i o n was given by the equation dP/dr = ^ {GM*/r 2 - KL/( 471r 2 c )} = f{q(r) - K L/(4TTr 2c)} (IV-4) The o p t i c a l depth of the atmosphere was given by the d i f f e r e n t i a l equation dlogT/dr = - (K/(R*/r) 2}/2.302585093T (IV-5) and Equation (111-10) was used to c a l c u l a t e the temperature. 44 The equations were i n t e g r a t e d using the m o d i f i e d E u l e r method with r (or x) as the independent v a r i a b l e . T h i s i n t e g r a t i o n method uses g r a d i e n t s c a l c u l a t e d at the mid-points of the i n t e g r a t i o n g r i d as does the dynamical programme. As i n the dynamic programme, l o g a r i t h m i c averages were used to c a l c u l a t e the v a r i a b l e s at these p o i n t s . T h i s procedure gave b e t t e r v a l u e s f o r the mid-point q u a n t i t i e s i n s i t u a t i o n s where the g r a d i e n t s were steep than d i d the procedure of using s t r a i g h t means. To s t a r t the c a l c u l a t i o n of these i n i t i a l models, i t was necessary to s p e c i f y a r a d i u s , R, an e l e c t r o n p r e s s u r e , P e, and an o p t i c a l depth, Z0, f o r the s t e l l a r s u r f a c e . The s u r f a c e value of the o p t i c a l depth was set at 1 0 " 1 0 . The s u r f a c e r a d i u s was a d j u s t e d to give a s u r f a c e temperature of about 1300 K. The s u r f a c e e l e c t r o n pressure was s p e c i f i e d as a f u n c t i o n of g r a v i t y and temperature by the r e l a t i o n s h i p P e = C + 0.51og(g) - 2.2 0 (IV-6) where C i s a constant whose value was g e n e r a l l y found to l i e between -3 and -4. The value of C was a d j u s t e d f o r each model u n t i l the inward i n t e g r a t i o n gave T = 2/3 at r = R*. A s i m i l a r method of c a l c u l a t i n g the s u r f a c e e l e c t r o n p r e s s u r e was used by Mihalas (1967). The s o l u t i o n s of the i n i t i a l , h y d r o s t a t i c models were i t e r a t e d u n t i l AP/P<10-6 and 4t/r<10"' at each p o i n t . Six to ten i t e r a t i o n s per c e l l were g e n e r a l l y r e q u i r e d . Great care had to be taken to ensure that the models were a c c u r a t e l y i n 45 h y d r o s t a t i c e q u i l i b r i u m before they were t r a n s f e r r e d to the dynamic programme. S l i g h t d e v i a t i o n s from e q u i l i b r i u m , e s p e c i a l l y at the base of the atmosphere, generated u n d e s i r a b l e r e l a x a t i o n p u l s e s and/or shock waves i n the dynamic models. (See Appendix A f o r examples.) Moreover, the r e l a x a t i o n p u l s e s d i s s i p a t e d very slowly e s p e c i a l l y i n the top h a l f of the atmosphere where the damping f o r c e s were very s m a l l . In i t s present s t a t e , the h y d r o s t a t i c programme can generate high temperature (20 000K), constant o p a c i t y models which, when run on the dynamic programme i n the absence of a c c e l e r a t i n g f o r c e s , develop v e l o c i t i e s of l e s s than 10" 7 cm/sec over i n t e g r a t i o n p e r i o d s of 10 7 sec. The convergence of the low temperature models was good to one p a r t i n 10 4. T h i s l i m i t was imposed by the i n t e r p o l a t i o n r o u t i n e used to o b t a i n the gas o p a c i t y . . ( i v ) The Hydrodynamic Equations The procedure used to - c a l c u l a t e the time-dependent models i s summarized i n Table I. In the f i r s t p a r t of the c y c l e , the g r a i n v a r i a b l e s were c a l c u l a t e d e x p l i c i t l y . In the second p a r t , the t r a n s p o r t of the gas and carbon vapour, and the momentum of both the gas and the g r a i n s were c a l c u l a t e d i n an i m p l i c i t i t e r a t i o n loop. In the f i n a l p a r t of the c y c l e , the temperature d i s t r i b u t i o n and the d e n s i t y of the carbon vapour were c a l c u l a t e d . The f i r s t s tep i n the e x p l i c i t p o r t i o n of the c y c l e was to s o l v e the equation (IV-7) 46 f o r N £ + 1 , where N c = C/H i s the r e l a t i v e number d e n s i t y of carbon atoms in the gaseous s t a t e . In t h i s equation, mH i s the atomic mass of carbon, QQS i s the t o t a l number of gas p a r t i c l e s per hydrogen atom e x c l u d i n g carbon,ju', i s the mean mass per p a r t i c l e of the gas, and i s the d e n s i t y of carbon vapour remaining in the gas. Knowing N" + 1 and using the temperature c a l c u l a t e d i n the p r e v i o u s model, i t was then p o s s i b l e to o b t a i n the p a r t i a l p r e s s u r e s , P^j , of C, C x, and C 3 from the t a b l e c a l c u l a t e d f o r t h i s purpose. Next, the s a t u r a t e d vapour p r e s s u r e s , Pjfj, of C, C z, and C 3, (see, F i x , 1969), and the s u p e r s a t u r a t i o n r a t i o , S = P^r,/P%-, were c a l c u l a t e d . Then Equations (B-4) and (111-14) were s o l v e d to o b t a i n the s i z e of the g r a i n n u c l e i , a * * + 1 , and the g r a i n n u c l e a t i o n r a t e , J " + l . The next step was to s o l v e Equations (111-15) and (III-5) to o b t a i n the advanced number d e n s i t y , N £ + 1 , and mass d e n s i t y , fp+y of the g r a i n s . G r a i n p r o d u c t i o n was c o n s i d e r e d to be s i g n i f i c a n t only i n the r e g i o n where N p>10" 1 6. T h i s c o n d i t i o n set the lower boundary of the g r a i n forming r e g i o n i n the models. Next, the equation a n + 1 = {3j? n + V ( N £ + M i f f s ) } 1 / 3 (IV-8) was used to c a l c u l a t e the advanced e f f e c t i v e r a d i u s of the g r a i n s . The f i n a l step i n t h i s p o r t i o n of the c y c l e was to c a l c u l a t e the advanced mass r a t e of g r a i n formation, M £ + 1 , by s o l v i n g Equation (111-13). In the second p a r t of the computational c y c l e , the v a r i o u s f i e l d v a r i a b l e s approached t h e i r f i n a l , updated values 47 through a s e r i e s of sweeps through an i t e r a t i o n loop. The f i r s t step i n t h i s procedure was to f i n d an i n i t i a l value f o r the advanced p r e s s u r e . T h i s was accomplished by s o l v i n g the second order, i m p l i c i t , f i n i t e d i f f e r e n c e equation obtained by combining the mass and momentum d i f f e r e n c e equations f o r the gas ( E i l e k , 1975). The ZIP d i f f e r e n c i n g method (Harlow and Amsden, 1968) was used to d i f f e r e n c e the c o n v e c t i v e terms i n the momentum equations, and the i n t e r p o l a t e d donor c e l l method (Harlow and Amsden, 1975) was used to d i f f e r e n c e the c o n v e c t i v e terms appearing i n the c o n t i n u i t y equations. The advanced d e n s i t y , i . e . the d e n s i t y corresponding to the model i n the process of being c a l c u l a t e d , was e l i m i n a t e d from the pressure equation by i n t r o d u c i n g an approximate e x p r e s s i o n f o r the advanced p r e s s u r e , ' p n + 1 , 1 given by In t h i s e x p r e s s i o n , the s u b s c r i p t s r e f e r to the g r i d p o i n t s , and the s u p e r s c r i p t s , to the time s t e p s . T h i s d e f i n i t i o n of p " + 1 i s v a l i d when changes i n the equation of s t a t e are determined p r i n c i p a l l y by changes i n d e n s i t y . In cases where the temperature or the i n t e r n a l energy changes r a p i d l y , the e x p r e s s i o n d e f i n i n g P n + 1 can s t i l l be used, but the convergence of the s o l u t i o n w i l l be slow. *The t i l d e q u a n t i t i e s are used to represent the f i e l d v a r i a b l e s of the advanced model d u r i n g the i t e r a t i o n process s i n c e these q u a n t i t i e s do not r e a l l y r e p r e s e n t the advanced model u n t i l the i t e r a t i o n loop has converged. (IV-9) 48 The p r e c i s e value of C" i n Equation (IV-9) i s not s p e c i f i e d t h e o r e t i c a l l y . Rather, i t i s found by making short t e s t runs with d i f f e r e n t values f o r C*. In some problems, the value chosen f o r C" s t r o n g l y a f f e c t s the convergence of the s o l u t i o n . In the case of the present problem, however, convergence seemed to be i n s e n s i t i v e to the p r e c i s e value of C? as long as i t was c l o s e t o Pi/j>i . Hence, i t was de c i d e d t o set C" equal to 1.125xPi/^" f o r a l l models. Test runs at T* = 2400 K i n d i c a t e d that t h i s value f o r Cf gave the f a s t e s t r a t e of convergence. E i l e k (1975) used c [ = 1 . 5 P i / f ? . The pressure equation along with the a p p r o p r i a t e boundary c o n d i t i o n s was s o l v e d f o r the pressure throughout the atmosphere. The next step i n the s o l u t i o n was to c a l c u l a t e the advanced d e n s i t y from the equation which d e f i n e s P n + 1 . F i n a l l y , the momentum equations were so l v e d simultaneously f o r the g r a i n and gas v e l o c i t i e s . (v) I t e r a t i v e Procedures Having obtained i n i t i a l v a l u e s f o r P" + 1, j>n*1, t f n t l and Up* 1, the next step i n the c a l c u l a t i o n c y c l e was to check the accuracy of the s o l u t i o n and, i f necessary, determine the increment to be added to each f i e l d v a r i a b l e f o r the next sweep through the i t e r a t i o n loop. The c o n t i n u i t y equation f o r the gas component of the atmosphere was used as the b a s i s f o r determining these increments s i n c e i t was not e x p l i c i t y used i n the s o l u t i o n a l g o r i t h m . The procedure was to d e f i n e a f u n c t i o n , qL, as 49 g. = - J>V) _ (IV-10) where (fc*1 ~ ?i) i s t n e c a l c u l a t e d change in d e n s i t y , and (fi*1 ~ ft )' i s the change i n d e n s i t y that a c t u a l l y s a t i s f i e s the c o n t i n u i t y equation. Defined i n t h i s way, the f u n c t i o n , g i , i s the measure of the accuracy of the d e n s i t y c a l c u l a t i o n . I f \g-L |/^>1.5xl0" 3 , new v a l u e s of f i * 1 were c a l c u l a t e d at a l l depths by s o l v i n g the system of equations represented by 1 9L + E ( * g ; = o i , j = 2 , 3 , . . . , s - i ( I V - I D J The c o e f f i c i e n t s of t h i s Newton-Raphson s t y l e r e l a x a t i o n matrix were ob t a i n e d by n u m e r i c a l l y d i f f e r e n t i a t i n g the c o n t i n u i t y equation with respect to a l l f?L. was set equal to zero at the boundaries. During the passes through the i t e r a t i o n loop, the temperature was h e l d c o n s t a n t , so changes i n pressure and d e n s i t y were r e l a t e d by the r e l a t i o n s h i p ( p n + i + A p 0 + 1 ) = f ( ^ * + 1 + d/>n + 1 , T ° , N £ + 1 ) (IV-12) where the f u n c t i o n , f, r e p r e s e n t s the equation of s t a t e . Instead of f o r m a l l y c a l c u l a t i n g the equation of s t a t e each time the p r e s s u r e was to be incremented, the updated pressure was xThe over-under r e l a x a t i o n parameter that appears i n the usual f o r m u l a t i o n of the Newton-Raphson method f o r f i n d i n g the zeros of a set of equations was put equal to one i n t h i s study. T h e r e f o r e , i t has not been w r i t t e n i n t h i s e q uation. 50 i n t e r p o l a t e d from a t a b l e g i v i n g the pressure as a f u n c t i o n of j>, T, and N c . O c c a s i o n a l l y , the pressure had to be a d j u s t e d even when S^p"*1 was n e g l i g i b l e because of the method used to i n i t i a l i z e P n + 1 . In such cases, however, the s o l u t i o n converged a f t e r j u s t one adjustment of P n + 1 . &P n + 1 was always put equal to zero at the boundaries. The l a s t v a r i a b l e s to be incremented were the v e l o c i t i e s . The increments to the v e l o c i t i e s were found by s u b s t i t u t i n g the incremented d e n s i t i e s and p r e s s u r e s i n t o the momentum equations and s o l v i n g again f o r the v e l o c i t i e s . I t e r a t i v e accumulations of changes to the f i e l d v a r i a b l e s continued u n t i l | g i | / f p t and |APj/Pi +1 were both l e s s than 1.5x10" 3 i n each computational c e l l . Once these c o n d i t i o n s had been met, the changes i n the f i e l d v a r i a b l e s were c o n s i d e r e d to be s u f f i c i e n t l y small that the t i l d e ' q u a n t i t i e s had converged to t h e i r f i n a l v alues f o r the model. ( v i ) O p t i c a l Depth A f t e r the i t e r a t i o n loop had converged, the next step i n the c y c l e was to c a l c u l a t e the advanced o p t i c a l depth and temperature d i s t r i b u t i o n s f o r the atmosphere. The o p t i c a l depth equation was d i f f e r e n c e d i n e x a c t l y the same way as the mass and momentum equations except that an e x p l i c i t f o r m u l a t i o n was used. The i n t e g r a t i o n of the o p a c i t y equation i n both the h y d r o s t a t i c and the dynamic programme was complicated by the f a c t t h a t a d e n s i t y i n v e r s i o n e x i s t s near the photosphere i n these models. T h i s i n v e r s i o n o c c u r r e d i n e x a c t l y the same region that the g r a d i e n t s of the f i e l d v a r i a b l e s such as P, f and T 51 s t a r t e d to i n c r e a s e r a p i d l y with depth. T h i s d e n s i t y i n v e r s i o n was r e l a t e d to v a r i a t i o n s i n the molecular abundances of CN and H^. For the chosen model parameters, the molecular abundance of CN peaks i n the re g i o n near the photosphere. Above t h i s r e g i o n , the d e n s i t y of the models decreases r a p i d l y with r a d i u s , while the temperature changes very s l o w l y . With the formation of the CN molecule, the o p a c i t y , and hence the temperature of the atmosphere begins to inc r e a s e r a p i d l y inward. The r i s i n g temperature t r i g g e r s a r a p i d d i s s o c i a t i o n of the atmospheric molecules, i n c l u d i n g . The mean molecular weight of the atmosphere decreases as t h i s d i s s o c i a t i o n occurs and t h i s i n i t i a t e s the d e n s i t y i n v e r s i o n . Deeper i n the atmosphere as other molecules i n c l u d i n g CN d i s s o c i a t e , the steep temperature g r a d i e n t i s maintained by the. o p a c i t y of H~ and the region of low d e n s i t y i s extended inward to the base of the models. With the standard g r i d s p a c i n g , the c a l c u l a t e d values of the o p t i c a l depth i n the r e g i o n of the d e n s i t y i n v e r s i o n o s c i l l a t e d between two extremes without showing any sign of converging to a s i n g l e value even when as many as one hundred i t e r a t i o n s were performed. I t was found necessary to introduce i n t ermediate g r i d p o i n t s to reduce the s i z e of the depth step i n t h i s r e g i o n of the models to a t t a i n convergence of the o p t i c a l depth equation. G e n e r a l l y , eighteen intermediate p o i n t s per c e l l were used. The s o l u t i o n was i t e r a t e d at each i n t e r m e d i a t e g r i d p o i n t u n t i l i t converged before the c a l c u l a t i o n a t the next, p o i n t was begun. I f the i n t e g r a t i o n had not converged, a new value f o r the o p t i c a l depth was d e f i n e d as 52 t t + ( = 0.75t:;( + 0 . 2 5 ^ , (IV-13) the temperature was r e c a l c u l a t e d , and the i n t e g r a t i o n procedure repeated u n t i l &XL/Ti <10~ 6. G e n e r a l l y , four to s i x i t e r a t i o n s were r e q u i r e d to o b t a i n t h i s degree of accuracy even with the use of the in t e r m e d i a t e p o i n t s . To ensure that the o p t i c a l depths c a l c u l a t e d by both programmes were i d e n t i c a l , i t was necessary to r e t u r n and r e c a l c u l a t e the o p t i c a l depths i n the h y d r o s t a t i c programme a f t e r the equations had been s o l v e d at a l l depth p o i n t s . In the h y d r o s t a t i c programme, the o p t i c a l depth equation was s o l v e d s i m u l t a n e o u s l y with the d e n s i t y equation, whereas i n the dynamic programme, the d e n s i t i e s were known at a l l depth p o i n t s before the o p t i c a l depth equation was i n t e g r a t e d . .Before t h i s c y c l i n g was i n t r o d u c e d , the temperature at the lower boundary c a l c u l a t e d by the two programmes c o u l d d i f f e r by as much as 400 K. A f t e r the advanced temperatures were c a l c u l a t e d , the temperature at T = 2/3 was checked to see i f i t e q u a l l e d T*. If i t d i d not, a new e f f e c t i v e temperature was d e f i n e d as T* = 0.5(T* + T(r=2/3)) (IV-14) and a new photospheric r a d i u s , R*, was d e f i n e d as R* = L/(4flVTT* 4) °-5 1 (IV-15) and the c a l c u l a t i o n s of T and T were repeated u n t i l £ T * / T * < 10"*. 53 (b) Boundary C o n d i t i o n s A t o t a l of three boundary c o n d i t i o n s were r e q u i r e d to s o l v e the dynamical gas e q u a t i o n s : the p r e s s u r e and o p t i c a l depth at the s u r f a c e , and the p r e s s u r e at the base. 1 However, because of the f i n i t e d i f f e r e n c e scheme adopted, i t was a l s o necessary to s p e c i f y the value of the v e l o c i t y at the base and the s u r f a c e of the models. 2 A d d i t i o n a l boundary c o n d i t i o n s were r e q u i r e d to s o l v e the d i f f e r e n t i a l equations governing the formation and growth of the carbon g r a i n s . The boundary c o n d i t i o n s had to be d e f i n e d i n such away that the s o l u t i o n s to the equations were unique as w e l l as c l o s e approximations to the s o l u t i o n s that would e x i s t i n the absence of the a r t i f i c i a l boundaries. In p a r t i c u l a r , the boundary c o n d i t i o n s had to be chosen such t h a t they minimized the amplitude of waves r e f l e c t e d from the a r t i f i c i a l boundaries. U n f o r t u n a t e l y , such tr a n s p a r e n t boundary c o n d i t i o n s f o r general c l a s s e s of wave equations are n o n - l o c a l i n both space and time, and are not u s e f u l f o r p r a c t i c a l c a l c u l a t i o n s , (Engquist, e t . a l . , 1977). Consequently, i t was necessary to determine approximate boundary c o n d i t i o n s that would minimize r e f l e c t i o n s . A great d e a l of research went i n t o d e v e l o p i n g a set of boundary JA d i s c u s s i o n of boundary c o n d i t i o n s s u i t a b l e f o r the ICE method i s given i n Harlow and Amsden,(1968). 2 F o r an i l l u s t r a t i o n of a s i m i l a r s i t u a t i o n i n which e x t r a boundary c o n d i t i o n s were r e q u i r e d because of the d i f f e r e n c e scheme used, see Engquist, e t . a l . , (1974). T h i s a r t i c l e a l s o c o n t a i n s a d i s c u s s i o n of how to s e l e c t boundary c o n d i t i o n s and of t h e i r i n f l u e n c e upon the s o l u t i o n of the equations of hydrodynamics. 54 c o n d i t i o n s that s a t i s f i e d t h i s requirement. ( i ) O p t i c a l Depth As noted e a r l i e r , the s u r f a c e o p t i c a l depth, Xa, f o r the i n i t i a l models was set equal to 1 0 " 1 0 . The inward i n t e g r a t i o n of the o p t i c a l depth equation converges very r a p i d l y , even with a poor i n i t i a l value f o r Ta . Since the value of To cannot be determined a n a l y t i c a l l y , the top two p o i n t s of the i n i t i a l model were always d i s c a r d e d before the model was t r a n s f e r r e d to the dynamical programme to e l i m i n a t e any p o s s i b l e e r r o r s i n the o p t i c a l depth s c a l e . The o p t i c a l depth of the uppermost p o i n t that was r e t a i n e d became the f i x e d boundary value f o r the dynamical c a l c u l a t i o n s . A constant s u r f a c e o p t i c a l depth worked very w e l l f o r g r a i n - f r e e t e s t models but was o b v i o u s l y i n a p p r o p r i a t e when g r a i n s were present. Because of t h e i r very l a r g e o p a c i t y , the formation of g r a p h i t e g r a i n s can i n c r e a s e the o p t i c a l depth by three to four orders of magnitude. Moreover, the s u r f a c e o p t i c a l depth was s e n s i t i v e to the d e n s i t y of the g r a i n s both at and above the s u r f a c e . Allowance f o r t h i s s e n s i t i v i t y was made by d e f i n i n g T 0 i n these models as I* = 1 0 " 2 K ( s ) R * 2 / r ( s ) (IV-16) In t h i s equation, K ( s ) i s the sum of the Rosseland mean a b s o r p t i o n c o e f f i c i e n t s of the g r a i n s and the gas at the s u r f a c e . T h i s d e f i n i t i o n of T0 i m p l i c i t l y assumes that the g r a i n d e n s i t y above the s u r f a c e boundary f a l l s o f f as r - 2 . Such a 55 d e n s i t y f a l l o f f i s p h y s i c a l l y reasonable and has been assumed by Gilman (1972) and by S a l p e t e r (1974b) as w e l l as other authors when a n a l y z i n g g r a i n forming atmospheres. In the absence of g r a i n s , the d e f i n i t i o n of T„ g i v e s a value f o r the s u r f a c e o p t i c a l depth that i s very c l o s e to the value t r a n s f e r r e d from the i n i t i a l models. In the presence of g r a i n s , Xa has a value that appears to be a smooth extension of the v a l u e s one and two steps below the s u r f a c e . ( i i ) Pressure The boundary c o n d i t i o n s e s t a b l i s h e d f o r the second order p r e s s u r e equation were designed to minimize the generation of r e f l e c t e d p u l s e s at the upper and lower boundaries. The procedure used to e s t a b l i s h these boundary c o n d i t i o n s was based upon the f a c t that under c e r t a i n c o n d i t i o n s , the equations of gas dynamics (that i s , the equation of c o n t i n u i t y , the momentum equation, and the equation of s t a t e ) can be grouped together and w r i t t e n i n the form of the a c o u s t i c or wave e q u a t i o n . 1 (See, f o r example, P o l o z h i y , 1967.) Assume f o r the moment that the wave equation i s used to determine the displacement of a s t r e t c h e d s t r i n g from i t s e q u i l i b r i u m p o s i t i o n . Under these circumstances, 1 T h i s t r a n s f o r m a t i o n i s r i g o r o u s only at low flow v e l o c i t i e s i n the absence of body f o r c e s . A l s o , the t r a n s f o r m a t i o n i s based upon the assumption t h a t changes occur i n the gas a d i a b a t i c a l l y and that the gas obeys the p e r f e c t gas law. Since a l l of these assumptions are v i o l a t e d i n the s t e l l a r atmospheres that are being modelled here, the pressure equation w i l l have only the g e n e r a l form and c h a r a c t e r i s t i c s of the wave equ a t i o n . T h i s was one of the c o m p l i c a t i n g f a c t o r s i n determining s u i t a b l e boundary c o n d i t i o n s . 56 i t i s w e l l known that i f one end of the s t r i n g i s h e l d r i g i d , a pulse f l o w i n g to that boundary w i l l be p e r f e c t l y r e f l e c t e d , 180° out of phase with the i n c i d e n t p u l s e (see, f o r example, Z e l ' d o v i c h and R a i z e r , 1966). At a transparent boundary, in c o n t r a s t , a p u l s e t r a v e l l i n g along a s t r e t c h e d s t r i n g w i l l produce no r e f l e c t e d p u l s e at a l l but w i l l pass through the boundary completely unchanged. At a l l other boundaries, p a r t of the p u l s e w i l l be t r a n s m i t t e d or absorbed, and p a r t r e f l e c t e d . Numerical t e s t s i n d i c a t e d that the dynamic models r e a c t e d to imposed boundary c o n d i t i o n s i n a manner very l i k e the s t r e t c h e d s t r i n g . The data p l o t t e d i n F i g u r e 1 show the behaviour of a wave generated by a small d e n s i t y p e r t u r b a t i o n i n an otherwise h y d r o s t a t i c model. With the base pressure h e l d c o n s t a n t , the pulse was almost p e r f e c t l y r e f l e c t e d . Waves were s i m i l a r l y r e f l e c t e d at the s u r f a c e when the pressure at that boundary was kept c o n s t a n t . T e s t s a l s o i n d i c a t e d t h at h o l d i n g the v e l o c i t y equal to zero at the boundaries a l s o generated r e f l e c t e d waves. An example of such a t e s t i s p l o t t e d i n F i g u r e 2. In order to simulate t r a n s p a r e n t boundaries f o r the dynamic models, t h e r e f o r e , i t was necessary to p r e d i c t the change, A P, i n the boundary v a l u e s of the p r e s s u r e as waves approached the edge of the model. To i l l u s t r a t e how t h i s was done, c o n s i d e r the procedure used at the s u r f a c e boundary. The f i r s t s t e p was to d e f i n e two standard or r e f e r e n c e pressures c a l l e d PSUR and PSURl. These two q u a n t i t i e s were the p r e s s u r e s that would e x i s t at the s u r f a c e and at one g r i d p o i n t below the s u r f a c e i f the atmosphere were i n h y d r o s t a t i c e q u i l i b r i u m . These 57 FIGURES 1 and 2 Boundary C o n d i t i o n T e s t s These f i g u r e s show how the dynamic models r e a c t e d to two d i f f e r e n t boundary c o n d i t i o n s . In F i g u r e 1, the gas pressure at the base was h e l d c o n s t a n t . In F i g u r e 2, the gas v e l o c i t y at the base was h e l d equal to zero. The curves i n these f i g u r e s r e c o r d the behaviour of the model f o l l o w i n g a 0.001% i n c r e a s e , i n the d e n s i t y i n the 11 +* c e l l . In the c o n f i g u r a t i o n shown by the d o t t e d curve, the p u l s e s c r e a t e d by the d e n s i t y p e r t u r b a t i o n are f l o w i n g away from the l l * * c e l l . In the c o n f i g u r a t i o n shown by the s o l i d curve, the pu l s e that was i n i t i a l l y f l o w i n g towards the base boundary has been r e f l e c t e d with l i t t l e change i n amplitude and i s propagating towards the s u r f a c e . 58 oo UD cr CM o CM CD CD CD CD CD CD I VELOCITY KM/SEC FIGURE 1 BOUNDARY CONDITION TEST 59 VELOCITY KM/SEC FIGURE 2 BOUNDARY CONDITION TEST 60 two reference' p r e s s u r e s were set equal to P(s) and P ( s - l ) i n the i n i t i a l model. The second step i n the procedure, was to c a l c u l a t e the advanced s u r f a c e pressure by p u t t i n g Psn + 1 = PSUR {1 + (PX2-PX1 )/PXl} (IV-17) = PSUR + A P where PX1 = PSUR + (c°(s) + u n ( s ) ) (IV-18) x(PSURl - PSUR)At/Ar PX2 = P n ( s ) + ( c n ( s ) + u°(s)) (IV-19) x ( P n ( s - l ) - P n ( s ) ) & t / a r and A P = PSUR{(PX2 - PXD/PXl} (IV-20) De f i n e d i n t h i s way, PX2 i s the pressure at that downstream p o i n t on the wave in the n t h model which w i l l be propagated to the s u r f a c e by the n + l s l " model. The d i f f e r e n c e , PX2-PX1, re p r e s e n t s the s i z e of the pulse that i s superimposed upon the e q u i l i b r i u m p ressure s t r u c t u r e at t h i s same point.. A pulse of t h i s same magnitude, ( i . e . , A P) but s c a l e d by the f a c t o r , PSUR/PX1, to allow f o r the decrease i n d e n s i t y i n the atmosphere, i s then added to PSUR, the e q u i l i b r i u m s u r f a c e 61 pressure i n the n+l"*model, to get the r e q u i r e d value f o r P J + 1 . Thus the e f f e c t of the approaching wave i s taken i n t o account by a d j u s t i n g the s u r f a c e pressure by an amount equal to Ap. The above procedure al l o w s f o r the pressure v a r i a t i o n s at the s u r f a c e boundary a r i s i n g from approaching waves and thereby prevents the generation of r e f l e c t e d p u l s e s . However, i t does not allow f o r the s t r u c t u r a l changes i n the atmosphere that accompany mass flows, nor f o r the s t r u c t u r a l changes that accompany the formation of g r a p h i t e g r a i n s . These changes were i n c o r p o r a t e d i n t o the d e f i n i t i o n s of both PSUR and PSUR1 by p u t t i n g PSUR n + 1 = PSURn + &t[(l-$){u£ (uj "Fp s )} + uj + 1Mp j + 1-Fp s + 1 ) } ] (IV-21) - d t a r [ (l-4>)Mr (GM* - K 3 L ) r s 2 4tr c + 4>M" + 1(GM* - K S + 1 L ) ] and PSURl n + 1 = PSURl n + At[ (l-4 >){uS.,(uJ. lMp s. ( - Fps.,')} + ^ {u^MuS.VMp^^Z-FpV.*)}] (IV-22) - AtAr{(l-^)M p^., (GM*- Kfi/L) + *Mj5V |(GM* - K-tVLl} 41Tc where K i s the Rosseland mean a b s o r p t i o n c o e f f i c i e n t f o r the gas 62 only, Fp i s the drag f o r c e , and (p i s the r e l a t i v e time c e n t e r i n g c o e f f i c i e n t used to d i f f e r e n c e the momentum equ a t i o n . The base r e f e r e n c e p r e s s u r e s , PBASE and PBASE1, were a d j u s t e d i n a s i m i l a r manner. These adjustments ensure that the r e f e r e n c e p r e s s u r e s r e t a i n t h e i r o r i g i n a l r e l a t i o n s h i p to one another. S p e c i f i c a l l y , the d i f f e r e n c e s (PSUR1 - PSUR)/£r 5 and (PBASE -PBASE|)/^r, s a t i s f y the equation of h y d r o s t a t i c e q u i l i b r i u m . P h y s i c a l l y , these adjustments allow f o r changes i n the p r o j e c t e d d e n s i t y above the boundaries. The term, M pu, i n these equations, r e p r e s e n t s the momentum that i s t r a n s f e r r e d to that p o r t i o n of the carbon vapour that condenses to g r a p h i t e . The term, Fp, re p r e s e n t s the momentum that i s t r a n s f e r r e d to the gas as the g r a i n s move through i t . The remaining terms allow f o r the e f f e c t s of the change i n the s u r f a c e gas d e n s i t y caused by g r a i n formation. I m p l i c i t w i t h i n these adjustments to PSUR and PSUR1 i s the assumption that the weight of the atmosphere above the s u r f a c e remains c o n s t a n t . However, some v a r i a t i o n i n the weight above the s u r f a c e can be expected to occur, e s p e c i a l l y i n the e a r l y stages of g r a i n formation. Some allowance f o r such changes was made by using M p(s) i n s t e a d of M p(s+l/2) i n the d e n s i t y adjustment term f o r PSUR. Since the r a t i o of g r a i n s to gas was always small f o r the models, the g r a i n r e l a t e d adjustments made to PSUR and PSUR1 were always s e v e r a l orders of magnitude sm a l l e r than those r e l a t e d to the gas dynamics. ( i i i ) V e l o c i t y Boundary C o n d i t i o n s Only two boundary c o n d i t i o n s were s y s t e m a t i c a l l y 63 t e s t e d f o r the v e l o c i t y boundary c o n d i t i o n s . One method was to put u ( l ) = u ( 2 ) , and u ( s + l ) = u ( s ) , that i s , to assume that A u / ^ r = 0 at the boundaries. The other, was to e x t r a p o l a t e the inner v e l o c i t i e s to get a s u r f a c e v e l o c i t y . Both methods were found to work q u i t e w e l l . T e s t s i n d i c a t e d that the e x t r a p o l a t i o n method s l i g h t l y reduced the amplitude of r e f l e c t e d waves when they were present i n some of the e a r l i e r t e s t s , but c o n s i s t e n t l y produced s u r f a c e v e l o c i t i e s which appeared to be too l a r g e i n comparison with the v e l o c i t i e s at neighbouring p o i n t s . N e i t h e r method i n t e r f e r e d with the s t a b i l i z i n g of perturbed models. Since the two methods d i f f e r e d so l i t t l e , i t was decided to use the s i m p l i e r method of p u t t i n g u ( l ) = u(2) and u(s+l) = u ( s ) . A s i m i l a r procedure was suggested by Harlow and Amsden (1968) f o r c o n t i n u a t i o n boundaries f o r both the MAC and ICE methods. ( i v ) Boundary C o n d i t i o n s f o r the G r a i n Equations Four p a r t i a l d i f f e r e n t i a l equations, Equation (IV-7), (111-15), (I I I - 5 ) and (II I - 3 ) were s o l v e d to o b t a i n the g r a i n f i e l d v a r i a b l e s , N t, Np, f?p, and f>v. For each equation a boundary c o n d i t i o n was r e q u i r e d . These c o n d i t i o n s were s p e c i f i e d at the base of the g r a i n r e g i o n as N c = C/H (=1.22x10" 3) 64 and A = Ntmc/(yu' (QQS + N c ) ) Because of the nature of the d i f f e r e n c e equations that were used, a d d i t i o n a l boundary c o n d i t i o n s were r e q u i r e d to c a l c u l a t e the v a l u e s of these v a r i a b l e s i n the s u r f a c e c e l l . These a d d i t i o n a l c o n d i t i o n s a l l had the general form where A rep r e s e n t s the v a r i o u s g r a i n v a r i a b l e s . In a d d i t i o n , N c was c o n s t r a i n e d such that Nt>0/H. When N c became equal to 0/H, i t was assumed that a l l gaseous carbon remaining i n the atmosphere at that p o i n t was i n the form of CO and was not a v a i l a b l e f o r g r a i n formation or growth. AlogA/ar = constant (IV-23) 65 Table I FLOW CHART FOR THE TIME-DEPENDENT PROGRAMME  Se c t i o n I E x p l i c i t C a l c u l a t i o n of the G r a i n V a r i a b l e s a) C a l c u l a t e N c, the r e l a t i v e number d e n s i t y of the gaseous carbon (Equation ( I V - 7 ) ) . b) Look up the p a r t i a l p r e s s u r e s and c a l c u l a t e the vapour pr e s s u r e s of C, C z, and C s . c) C a l c u l a t e a*, the r a d i u s of the g r a i n n u c l e i (Equation (B-4)). d) C a l c u l a t e J , the g r a i n n u c l e a t i o n r a t e (Equation ( I I I -14)). e) C a l c u l a t e the d e n s i t y , , and number d e n s i t y , Np , of the g r a i n s (Equations ( I I I - 5 ) and (111-15)). f) C a l c u l a t e the average g r a i n r a d i u s , a (Equation ( I V - 8 ) ) . g) C a l c u l a t e the mass r a t e of g r a i n formation, Mp (Equation (111-13)). S e c t i o n II I m p l i c i t S o l u t i o n of the Dynamic Equations a) I n i t i a l i z e the advanced p r e s s u r e , P (Equation (A-4)). b) C a l c u l a t e an i n i t i a l value f o r the advanced d e n s i t y (Equation ( I V - 9 ) ) . c) Solve the two momentum equations (Equations ( I I I - 6 ) and ( I I I - 7 ) ) simultaneously to f i n d i n i t i a l v alues f o r gas and g r a i n v e l o c i t i e s , u and u p . 66 d) Check the s o l u t i o n accuracy of the c o n t i n u i t y equation and i n c r e m e n t ^ i f necessary (Equation (IV-11)). e) Check the s o l u t i o n accuracy of the equation of s t a t e and increment P i f necessary (Equation (IV-12)). f) If e i t h e r j> or 1? were incremented, r e t u r n to c to increment u and Up. g) Repeat c, d, e, and f u n t i l convergence i s obtained at a l l l e v e l s . S e c t i o n III E x p l i c i t C a l c u l a t i o n of the Temperature a) C a l c u l a t e the o p t i c a l depth d i s t r i b u t i o n (Equation (IV-5 ) ) . b) C a l c u l a t e the temperature of the gas and g r a i n s . c) C a l c u l a t e the advanced carbon vapour d e n s i t y , (Equation (III-3)),. I t e r a t e t h i s s o l u t i o n u n t i l convergence i s o b t a i n e d . d) Store the model and prepare f o r the next c y c l e . 67 Chapter V  RESULTS Two time-dependent models were c a l c u l a t e d , one with T* = 2500 K, and the other with T* = 2400 K. These models w i l l be r e f e r r e d to as model 1 and model 2 r e s p e c t i v e l y . For both of these models, M = 1.5M@ and Mt>0| = -6 (which corresponds to L = 1 . 9 4 X 1 0 4 L Q ) . A t o t a l of 3600 time steps with A t = 8x10* sec were r e q u i r e d to generate model 1, and 4800 time steps with A t = 1.25x10 s sec, to generate model 2. Each time s t e p r e q u i r e d approximately 3 seconds of computer time. Table II g i v e s some of the p h y s i c a l parameters of these models at the photosphere (R*,P*, e t c . ) , at the sonic p o i n t ( S c , e t c . ) , and a t the s u r f a c e of the models ( T 5 , e t c . ) . The s u b s c r i p t , c, denoting the sonic p o i n t , i s used as a reminder that the condensation of the g r a i n s becomes s i g n i f i c a n t at t h i s l e v e l i n the atmosphere. The sonic p o i n t marks the base of what w i l l be c a l l e d the g r a i n - f o r m i n g region of the atmosphere i n the remainder of t h i s study. The symbols appearing i n Table II t h a t have not yet been d e f i n e d are : "r\_= (r (r = 10~ 3 )-R*)/R*, the atmospheric e x t e n s i o n expressed as a percentage, D, the column d e n s i t y of the g r a i n s , &m, the o p t i c a l t h i c k n e s s of the g r a i n s , and Pj , the p u l s a t i o n p e r i o d . (a) I n i t i a l Time-Dependent Behaviour of the Models The time-dependent behaviour of the two models i s summarized i n F i g u r e s 3 and 4. I n i t i a l l y , the two models TABLE II P r o p e r t i e s of the Time-Deper ident Models Model 1 Model 2 Photosphere 1 T* 2500 2400 l o g P* 2. 67 2 .90 R*/R© 740 803 l o g g* -1. 12 -1 .20 vest 27. 8 26 .7 i \ ( r = 10- 3) 7. 6% 7 .7% Sonic P o i n t T c 1400 1436 l o g P c -6. 14 -5 .05 Rc/R* 1. 83 1 .58 T 5. 42(--5) 1 .15(--3) Sc 5. 3 5 .1 v c 2. 8 2 .9 w c 14. 65 3 .2 a* c 5. 57 (--8) 5 .57(--8) a c 5. 65(--8) 5 .64(--8) continued TABLE II (Continued) P r o p e r t i e s of the Time-Dependent Models Model 1 Model 2 Surface l o g Ps Rs/R* s s w« °5 c/o s Am (TV. = 0.7^) SAm(K = 0.7//) dM/dt 1257 -6.95 2.24 4.69(-6) 7.51 12.94 0.0 28.49 5.15(-8) 5.53(-8) 1.17 1.6(-7) 1.6(-7) 5.0(-3) 6.2(-9) 1316 -5.85 1.86 1.79(-4) 6.66 14.25 • 0.8 8.43 5.22(-8) 5.5K-8) 1.24 1.6(-6) 3.4(-6) 1.0(-1) 1.8(-2) 6.48(7) 7.4(-8) ^ g s u n i t s are used throughout except that v e l o c i t i e s are km/sec and dM/dt i s i n s o l a r masses/year. 70 e x h i b i t e d s i m i l a r c h a r a c t e r i s t i c s . Both were i n i t i a t e d as g r a i n -f r e e , h y d r o s t a t i c models. In both, the carbon vapour i n the s u r f a c e l a y e r s was h i g h l y s u p e r s a t u r a t e d . Ss = 16.0 i n model 1 and S s = 11.9 i n model 2. Test runs i n d i c a t e d t hat the dynamical response of the atmosphere to the sudden formation of g r a i n s i n the upper l a y e r s would be the g e n e r a t i o n of a massive outward flow of the whole atmosphere. (See Lucy, 1976). However, the c o n d i t i o n s necessary f o r sudden g r a i n formation throughout the s u r f a c e l a y e r s are probably not p h y s i c a l l y r e a l i z a b l e . In a d d i t i o n , the time r e q u i r e d f o r the model to r e l a x from t h i s a r t i f i c a l impulse i s long, and the computer c o s t s r e q u i r e d to f o l l o w the model while i t i s r e l a x i n g are very h i g h . For these reasons, the r a t e of g r a i n n u c l e a t i o n was a r t i f i c a l l y reduced i n the i n i t i a l stages of both models by s e t t i n g J ( t , r ) =°c( t)j(r) ( V - l ) where J ( r ) i s obtained from Equation (111-14), and ^ ^ ( t ) = ssc"'1 (t) + 4 . 0 x l 0 - 8 A t n i f oe°-»(t) < 1, otherwise, c>c n(t) = 1. By c o n t r o l l i n g the value of J ( t , r ) i n t h i s manner, the adjustment of the atmosphere to the presence of the g r a i n s was spread over a time i n t e r v a l of 10 7 sec and the e f f e c t on the dynamical p r o p e r t i e s of the models was much l e s s c a t a s t r o p h i c than would have been the case i f u n c o n t r o l l e d g r a i n n u c l e a t i o n i n the h i g h l y s u p e r s a t u r a t e d atmosphere had been p e r m i t t e d to occur. The dynamical e f f e c t s of the i n i t i a l b urst of g r a i n formation are best i l l u s t r a t e d i n F i g u r e 3 where the v e l o c i t y at the base of model 1 i s p l o t t e d as a f u n c t i o n of time. As Lucy 71 FIGURE 3 Model 1 The data p l o t t e d i n t h i s f i g u r e are the gas v e l o c i t y at the s u r f a c e (upper c u r v e ) , the o p t i c a l depth at the sonic p o i n t (middle c u r v e ) , and the gas v e l o c i t y at the base of the model (lower c u r v e ) . The p u l s a t i o n amplitude of each of these v a r i a b l e s d i m i n i s h e d as the model approached a s t e a d y - s t a t e . The mass l o s s r a t e f o r t h i s model was 6.2x10'' M^/yr. 72 te (Xl0"5) CD LP* CD LO " i 1 i r 73 FIGURE 4 Model 2 The data p l o t t e d i n t h i s f i g u r e are the gas v e l o c i t y at the s u r f a c e (upper c u r v e ) , the o p t i c a l depth at the sonic p o i n t (middle c u r v e ) , and the gas v e l o c i t y at the base of the model (lower c u r v e ) . I n i t i a l l y , the p u l s a t i o n of the base and the s o n i c p o i n t were out of phase l e a d i n g to the d i s t o r t i o n s evident in the three curves at the p o i n t s l a b e l l e d (b) and ( c ) . A f t e r an e l a p s e d time of 3 0 x l 0 7 sec, a steady p u l s a t i o n with a p e r i o d of 6.48xl0 7 sec was e s t a b l i s h e d . The average mass l o s s r a t e f o r t h i s model was 7.4x10" 8 M^/yr. 74 -=3" 1—I i—I i — | SURFACE VELOCITY KM/SEC BASE VELOCITY CM/SEC FIGURE 4A MODEL 2 75 csj rc(xio-3) SURFACE VELOCITY KM/SEC BASE VELOCITY CM/sEC FIGURE 4B MODEL 2 76 (1976) noted i n the d i s c u s s i o n of h i s s t a t i c models, the a d d i t i o n of the g r a i n a b s o r p t i o n to the o p t i c a l depth of the atmosphere g i v e s a higher temperature i n the g r a i n - f r e e regions at a given pressure than that obtained f o r the g r a i n - f r e e , h y d r o s t a t i c i n i t i a l model. Or, c o n v e r s e l y , one may regard the e f f e c t of the g r a i n s as that of lowering the pressure r e q u i r e d to maintain h y d r o s t a t i c e q u i l i b r i u m at a given temperature. The atmosphere a d j u s t s to t h i s lower p r e s s u r e requirement by moving mass upwards. T h i s process produced the f i r s t p u l s e i n the s u r f a c e v e l o c i t y shown i n F i g u r e 3. The remaining p u l s e s are r e l a x a t i o n p u l s e s that were generated as the atmosphere a d j u s t e d to a new e q u i l i b r i u m s t a t e . The i n i t i a l outward p u l s e generated by the formation of g r a i n s i n the atmosphere i s a l s o evident i n F i g u r e 4 f o r model 2 but the subsequent behaviour of t h i s model was much d i f f e r e n t from that of model 1. T h i s behaviour w i l l be d i s c u s s e d i n s e c t i o n ( b ) . In both models, the s u r f a c e l a y e r s a c c e l e r a t e d r a p i d l y to very high v e l o c i t i e s reaching a peak v e l o c i t y of 28.0 km/sec at t = 4.5x10 6 sec i n model 1, and 25.5 km/sec at t = 7.4x10' sec, i n model 2. These v e l o c i t i e s decayed r e l a t i v e l y s l owly to 12.0 km/sec and 13.8 km/sec r e s p e c t i v e l y . The a c c e l e r a t i o n of the g r a i n s was much g r e a t e r than that of the gas i n both models. The g r a i n s reached a peak v e l o c i t y of 113.2 km/sec at t = 1.6x10' sec i n model 1, and 39.3 km.sec at t = 6.0x10' sec i n model 2. Even with the r a t e of n u c l e a t i o n a r t i f i c a l l y reduced, g r a i n formation was very r a p i d i n the i n i t i a l stages of the models. G r a i n formation reduced the carbon vapour 77 s u p e r s a t u r a t i o n to an " e q u i l i b r i u m " value c l o s e to 5.0 w i t h i n 10 4 sec of the s t a r t of the model c a l c u l a t i o n s . The s u p e r s a t u r a t i o n remained c l o s e to t h i s e q u i l i b r i u m value throughout the remainder of the model c a l c u l a t i o n s . T h i s i n i t i a l b u rst of g r a i n formation i n the c a r b o n - r i c h atmosphere r e s u l t e d in a p e r i o d of c l o s e to 10* sec i n which the g r a i n d e n s i t y and, consequently, the d r i v i n g f o r c e on the upper atmosphere, were r e l a t i v e l y h i g h . However, the r a p i d outward flow of the g r a i n s r e l a t i v e to the gas, p l u s the r a p i d d e p l e t i o n of the f r e e carbon vapour caused by t h e i r formation q u i c k l y r e s u l t e d i n a r e d u c t i o n of the d e n s i t y of the g r a i n s formed in the i n i t i a l b u r s t . Formation of new g r a i n s was c o n t r o l l e d by the r a t e at which c a r b o n - r i c h m a t e r i a l flowed i n t o the g r a i n - f o r m i n g r e g i o n . T h i s process r e s u l t e d i n a much reduced g r a i n d e n s i t y ( f ? / f = 8.7x10" 4 as compared with fP/f = 2.1x10" 3) and, consequently, a weaker c o u p l i n g between the g r a i n s and the gas, lower gas flow v e l o c i t i e s and higher d r i f t v e l o c i t i e s i n both models. (See Table I I ) . These p r o p e r t i e s are r e f l e c t e d i n the d e c e l e r a t i o n of the s u r f a c e flow v e l o c i t i e s at the p o i n t s l a b e l l e d (a) i n F i g u r e s 3 and 4. (b) Subsequent Time-Dependent Behaviour of the Models Both models entered t h e i r second dynamic phase near t = 3x10 7 sec. T h i s phase was c h a r a c t e r i z e d by a s m a l l amplitude p u l s a t i o n which was superimposed upon the outward flow of mass from the atmosphere. In model 1, the pulses were very steady and had an i n i t i a l amplitude of 0.6 km/sec but they decayed r a p i d l y , and by t = 27x10 7 sec, the model had r e l a x e d to almost a steady-78 s t a t e flow (see F i g u r e 3). In c o n t r a s t , the p u l s e s i n model 2 were i n i t a l l y very i r r e g u l a r , but by t = 27x10 7 sec the model had r e l a x e d i n t o a steady p u l s a t i o n mode with a p u l s e amplitude of 0.8 km/sec (see F i g u r e 4 ) . In both models, the p u l s a t i o n had a p e r i o d of about 750 days (6.48x10 7 s e c ) . Each pulse took approximately 750 days to reach the sonic p o i n t of the atmosphere and a f u r t h e r 170 days to reach the s u r f a c e . The d i f f e r e n c e between the dynamical c h a r a c t e r i s t i c s of the two time-dependent models was evident below the g r a i n -forming region as w e l l as at the s u r f a c e of the models. An examination of F i g u r e 4 shows that the base of model 2 i n i t i a l l y appeared to be r e l a x i n g i n a manner s i m i l a r to the r e l a x a t i o n of the base of model 1. However, the s m a l l peak at t = 10x10 7 sec, the l a r g e amplitude pulse c e n t e r e d at t = 15x10 7 sec, and the d i s t o r t e d peak centered at t = 2 7 x l 0 7 sec, were a l l i n d i c a t i o n s that the base p u l s a t i o n s were being d r i v e n , and t h a t the d r i v i n g f o r c e was not i n i t i a l l y i n phase with the p u l s a t i o n s . A f t e r t = 3 0 x l 0 7 sec, the pulses at the base became very uniform and showed no evidence of decaying. The constancy of the p u l s e s was a l s o an i n d i c a t i o n that they were being d r i v e n . Each pulse o r i g i n a t i n g at the base of the model c o u l d be f o l l o w e d as i t propagated with i n c r e a s i n g amplitude towards the s u r f a c e . For example, F i g u r e 5 shows the v e l o c i t y s t r u c t u r e of model 2 at t = 36.6xl0 7 sec and again at t = 40.0xl0 7 sec e x a c t l y h a l f a p e r i o d l a t e r . The time-dependence of Tc , the o p t i c a l depth at the sonic p o i n t , of models 1 and 2 i s p l o t t e d i n F i g u r e s 3 and 4. These f i g u r e s demonstrate the degree of v a r i a b i l i t y of the model 79 FIGURE 5 V e l o c i t y D i s t r i b u t i o n The two curves show v ( r ) at times separated by e x a c t l y h a l f of the p u l s a t i o n p e r i o d f o r model 2. The bump at the photosphere i n both curves i s produced by the presence of the d e n s i t y i n v e r s i o n j u s t below the photosphere. Note that the v e r t i c a l a x i s i n t h i s f i g u r e i s | v | l o g ( | v | ) / v . 80 IVI LOG (I VI CM/SEC)/V ( FIGURE 5 VELOCITY DISTRIBUTION 81 parameters at the base of the g r a i n - f o r m i n g r e g i o n s . The uniform p u l s a t i o n s observed at the s u r f a c e and the base of model 2 as i l l u s t r a t e d by the v e l o c i t y i n F i g u r e 4 was found to be c h a r a c t e r i s t i c of a l l the v a r i a b l e s throughout the atmosphere i n t h i s model. (c) Mass Loss Rates The mass l o s s r a t e f o r model 1 a f t e r i t had r e l a x e d to a steady flow was 6.2x10"' Mg/yr. The average r a t e of mass l o s s from model 2 was 7.4x10"" MQ/yr. Both of these r a t e s are almost two orders of magnitude lower than those c a l c u l a t e d by Lucy (1976) on the b a s i s of s t e a d y - s t a t e models but l i e w i t h i n the o b s e r v a t i o n a l l i m i t s f o r M s t a r s , 2x10"' Mg/yr to 3.8x10" 5 M^/yr, (Sanner, 1976). The o n l y c a r b o n - r i c h s t a r s f o r which mass l o s s r a t e s have been determined from o b s e r v a t i o n a l data are extreme i n f r a r e d o b j e c t s such as CRL 2688 (Lo and Bechis, 1976) and IRC +10216 (Kwan and H i l l , 1977) which have very l a r g e mass l o s s r a t e s (~1.0xl0~ 5 Mg/yr). However, M o r r i s et a_l, (1979) have suggested that the mass l o s s mechanism i n l o n g - p e r i o d v a r i a b l e s i s independent of s t e l l a r type or C/O r a t i o and independent of the composition of the g r a i n s so i t i s p o s s i b l e that the range of mass l o s s r a t e s f o r N-type s t a r s encompasses the same range as that observed f o r M-type s t a r s . (d) Grain S i z e The e f f e c t i v e g r a i n r a d i u s , a, remained c l o s e to the c r i t i c a l r a d i u s a*, at a l l l e v e l s i n the g r a i n - f o r m i n g r e g i o n . 82 T h i s r e s u l t was a d i r e c t consequence of the very low gas pressure i n the g r a i n - f o r m i n g r e g i o n of both models. (See Table I I ) . Under such c o n d i t i o n s , g r a i n growth i s exceedingly slow (see Equation (B-27)). Furthermore, i t was found that a* v a r i e d very l i t t l e throughout the g r a i n - f o r m i n g r e g i o n i n both models. For example, in model 2, a* = 5.57x10*' cm at the sonic p o i n t and 5.19x10-" cm at the s u r f a c e , while the e f f e c t i v e g r a i n r a d i u s at these two p o i n t s was 5.63x10"" cm and 5.51x10"" cm r e s p e c t i v e l y . G r a p h i t e g r a i n s of t h i s s i z e c o n t a i n about 75 carbon atoms. (N = 4 a3/3-^, = 9 x10" 2 4 cm 3). While i t was not p o s s i b l e to f o l l o w each group of g r a i n s that formed at each time step as an i n d i v i d u a l group, i t was p o s s i b l e to i s o l a t e those p a r t i c l e s which formed throughout the g r a i n - f o r m i n g region at an a r b i t r a r y time step and to f o l l o w the d e t a i l s of t h e i r movement and growth u n t i l they escaped from the s u r f a c e of the atmosphere. The r e s u l t of such a procedure i s shown i n F i g u r e 6. The d i s t r i b u t i o n of g r a i n s i z e s was s t r o n g l y skewed towards the small end of the spectrum both when the g r a i n s were formed, and when they l e f t the atmosphere. However, the d i s t r i b u t i o n s show that those g r a i n s that form f a i r l y deep i n the atmosphere with a* = 7.8x10"" cm, can succeed in growing to r e l a t i v e l y l a r g e r a d i i (designated by a W f t^ in Table II) and would, t h e r e f o r e , c o n t a i n a s i g n i f i c a n t p r o p o r t i o n of the mass of the g r a i n s were they not formed i n such small numbers. The number d e n s i t y of g r a i n s was g r e a t e s t at the s o n i c p o i n t of the atmospheres. The r e l a t i v e number d e n s i t y of the l a r g e s t g r a i n s p l o t t e d i n F i g u r e 6 to the number d e n s i t y at the s o n i c p o i n t i s 10 5 s m a l l e r than the r e l a t i v e masses of the g r a i n s at these two FIGURE 6 Gr a i n S i z e D i s t r i b u t i o n T h i s curve shows the s i z e d i s t r i b u t i o n of the g r a i n s l e a v i n g s u r f a c e of the model. For a > 5.5x10" 8 cm the curves can approximately f i t t e d by the f u n c t i o n N p ( a )<*> a " 1 7 . 84 85 p o i n t s . Thus, the use of an e f f e c t i v e r a d i u s to represent the s i z e of a l l the g r a i n s at a given l e v e l i n the atmosphere appeared to be an ac c e p t a b l e approximation of the a c t u a l s i z e d i s t r i b u t i o n . (e) S u p e r s a t u r a t i o n The formation of the g r a p h i t e g r a i n s only p a r t l y reduced the s u p e r s a t u r a t i o n r a t i o of the carbon vapour i n the gr a i n - f o r m i n g regions of the models. In model 1, the carbon vapour s u p e r s a t u r a t i o n was reduced from i t s i n i t i a l value of 16.0 to 7.5 at the s u r f a c e , and from 6.4 to 5.4 at the sonic p o i n t . In model 2, the corresp o n d i n g r e d u c t i o n s were from 11.9 to 6.6, and from 5.5 to 5.1. No s i g n i f i c a n t g r a p h i t e formation o c c u r r e d at s u p e r s a t u r a t i o n r a t i o s l e s s than 5.0. For example, in model 2, at t = 53.5x10 7 sec, the number d e n s i t y of the g r a i n s was 1.6x10"4/cm3 at the l e v e l where S = 4.0, whereas the number d e n s i t y was 27.0/cm 3 two depth steps higher i n the atmosphere where S = 5.1 (f) D r i f t V e l o c i t y The d r i f t v e l o c i t y of the g r a i n s was l a r g e i n both models (29.0 km/sec and 8.5 km/sec at the s u r f a c e ) implying that the g r a i n s were momentum coupled to the gas but not p o s i t i o n coupled. One important consequence of the high d r i f t v e l o c i t i e s was that the t o t a l carbon (gas p l u s g r a i n s ) present i n any l a y e r of the g r a i n - f o r m i n g region was l e s s than that i m p l i e d by a C/O r a t i o of 1.76, i . e . , s i n c e carbon was l e a v i n g the atmosphere 86 f a s t e r than the r e s t of the s t e l l a r m a t e r i a l , the s u r f a c e l a y e r s were being d i f f e r e n t i a l l y d e p l e t e d of carbon. The d r i f t v e l o c i t i e s were not so l a r g e , however, as to imply s e r i o u s g r a i n damage by s p u t t e r i n g ( S a l p e t e r , 1977). (g) O p t i c a l Thickness of the Gr a i n s Assuming that f>p v a r i e s as r " 2 , D, the column d e n s i t y of the g r a i n s , was 1.6x10" 7 gm/cm2 f o r model 1, and 3.4x10" 6 gm/ cm 2 f o r model 2. These d e n s i t i e s correspond to an o p t i c a l t h i c k n e s s i n the g r a i n s at A = 0.7/j. of 5.0xl0" 3 and l . O x l O " 1 r e s p e c t i v e l y . In model 2, the p u l s a t i o n s produced a v a r i a t i o n , SAm, i n the o p t i c a l t h i c k n e s s of the g r a i n s of 1.8x10" 2 at K = 0.7/jl. 87 Chapter VI  DISCUSSION AND CONCLUSIONS The r e s u l t s presented i n the p r e v i o u s chapter c o n f i r m the h y p o t h e s i s that r a d i a t i o n p r essure a c t i n g on g r a p h i t e g r a i n s can generate mass flows i n c a r b o n - r i c h red g i a n t s and that the r e s u l t a n t mass l o s s r a t e s l i e i n the range obtained from o b s e r v a t i o n a l data. In model 1, where the g r a i n d e n s i t y was low, the flow that was generated e v e n t u a l l y r e l a x e d to a steady-s t a t e . In model 2, the g r a i n d e n s i t y was a f a c t o r of 20 higher and a s t e a d i l y p u l s a t i n g flow was generated. T h i s flow took on some of the c h a r a c t e r i s t i c s a s s o c i a t e d with s m a l l amplitude, l o n g - p e r i o d , v a r i a b l e carbon s t a r s . The behaviour of the two models w i l l be di.scussed i n t h i s c h a p ter. F o l l o w i n g t h i s d i s c u s s i o n , some of the assumptions l i s t e d i n Chapter III w i l l be examined to estimate t h e i r probable e f f e c t s on the models. T h i s chapter w i l l conclude with a p r e s e n t a t i o n of the major c o n c l u s i o n s of the study. (a) Time-dependent Models Versus Steady State Models The parameters used to c a l c u l a t e model 1 (T* = 2500 K) were chosen to make a d i r e c t comparison between i t and two of Lucy's (1976) s t e a d y - s t a t e models p o s s i b l e . The p r o p e r t i e s of these three models are summarized i n Table I I I . The e s s e n t i a l d i f f e r e n c e s between the two m o d e l l i n g approaches were that Lucy assumed (a) that the outflow of the gas and the g r a i n s was time-TABLE I I I P r o p e r t i e s of the Time-Dependent and Time-Independent Models Time-Dependent Time-Independent Photosphere 1 T* R*/R© l o g P* L * / L G C/0 M*/M© 2500 740 2.67 1.94(4) 1.76 1.5 2500 740 2.29 1.94(4) 1.76 1.5 2500 740 2.44 1.94(4) 1.76 1.5 . Sonic P o i n t 1400 l o g Pt -6.14 Rc/R* 1.83 S t 5.3 t c 5.42( -5) v t 2.8 5.65( -8) 1660 •3.37 1.43 2.0 6.0(-2) 3.2 1.0(-6) 1585 -3.90 1.45 3.5 2.0(-2) 3.2 1.0(-6) .. .continued TABLE III (Continued) P r o p e r t i e s of the Time-Dependent and Time-Independent Models Time-Dependent Time-Independent " S u r f a c e " 2 l o g Ps Rs/R* am<v* Am(/\ = 0.7/0 dM/dt 1257 -6.95 2.24 5.53(-8) 1.6(-7) 12.94 28.49 5.0 1.6(-7) 5.0(-3) 6.2(-9) 1349 -4.93 1.90 1.0(-6) 1.0(-6) 39.1 7.8 8.6 2.6(-2) 3.62 1.4(-6) 1318 -5.44 1.94 1.0(-6) 1.0(-6) 37.8 13.2 8.2 8.4(-3) 1.18 4.4(-7) xCgs u n i t s are used throughout except that the v e l o c i t i e s are km/sec and dM/dt i s i n s o l a r masses/year. 2The p o i n t at which v 2 = l / 2 v 2 t e r has been d e s i g n a t e d as su r f a c e of Lucy's models f o r the purpose of t h i s comparison. 3 6 = A.L/41TGMC. 90 independent, (b) that g r a i n formation occurred only at the sonic p o i n t , (c) that a l l carbon not c o n t a i n e d i n CO e x i s t e d i n the form of g r a p h i t e g r a i n s above the sonic p o i n t , (d) that the g r a i n s a l l had a r a d i u s of 1x10' 6 cm, and ( f ) that the c r i t i c a l s u p e r s a t u r a t i o n r a t i o , S*, necessary fo r the formation of g r a p h i t e g r a i n s i n the atmosphere was 2.0 i n h i s model #2, and 3.5 i n h i s model #5. None of these assumption were used in the c a l c u l a t i o n of the time-dependent model. From an examination of the data given i n Table I I I , i t i s c l e a r that Lucy's assumptions have a profound i n f l u e n c e upon the c a l c u l a t e d p r o p e r t i e s of the expanding atmosphere. However, Lucy's key assumption, that the mass flow i s t i m e - i n v a r i a n t , was, i n f a c t , v e r i f i e d at 2500 K. T h i s r e s u l t was somewhat f o r t u i t o u s . I f the comparison between the two m o d e l l i n g approaches had been undertaken at a lower e f f e c t i v e temperature, the assumption of a steady flow would not have been v a l i d a t e d . The most s i g n i f i c a n t d i f f e r e n c e between the time-dependent and the s t e a d y - s t a t e models i s the c a l c u l a t e d r a t e of mass l o s s . S a l p e t e r (1974b) has argued that i n the case of a r a d i a t i o n - p r e s s u r e - d r i v e n flow, dM/dt oC Xc. T h i s r e l a t i o n s h i p i s approximately s a t i s f i e d by both the s t e a d y - s t a t e and the time-dependent models. Thus, the q u e s t i o n of why the mass l o s s r a t e s d i f f e r i s b a s i c a l l y one of determining why T t i s so much smaller i n the time-dependent model than i t was i n the s t e a d y - s t a t e models. Lucy's assumption that S* i s e i t h e r 2.0 or 3.5 determined the l o c a t i o n of the sonic p o i n t i n h i s models. In the time-dependent model, i n c o n t r a s t , no assumptions were made as to the value of S*. Instead, the equations governing the 91 n u c l e a t i o n and growth of the g r a i n s were sol v e d e x p l i c i t l y at each time s t e p and the sonic p o i n t was determined by the response of the atmosphere to the presence of the g r a p h i t e g r a i n s . 1 With t h i s procedure, i t was found that the sonic p o i n t always o c c u r r e d j u s t above the l e v e l where S was 5.0. At t h i s l e v e l , both f t and the mass d e n s i t y of carbon vapour a v a i l a b l e to form g r a p h i t e g r a i n s were about two orders of magnitude smal l e r than the corresponding q u a n t i t i e s at the sonic p o i n t i n both of Lucy's models. F u r t h e r , only a small f r a c t i o n of the carbon vapour that was a v a i l a b l e at the sonic p o i n t i n model 1 a c t u a l l y was i n c o r p o r a t e d i n t o the g r a p h i t e g r a i n s . Both of these f a c t o r s .lowered Xt and, consequently, dM/dt i n the time-dependent model. (b) Uniform G r a i n Size. * * I t was observed that i n both time-dependent models, a and a* v a r i e d very l i t t l e with depth i n the atmosphere. T h i s r e s u l t can be best understood by c o n s i d e r i n g the f o l l o w i n g sequence of s t e p s . Assume that at a given r a d i u s , r , i n the g r a i n - f o r m i n g r e g i o n , the atmosphere c o n s i s t s of carbon vapour, g r a p h i t e g r a i n s and an i n e r t gas. Assume a l s o , that S ( r ) > l but that S ( r ) < S * ( r ) . Under these c o n d i t i o n s , the only process by which the carbon vapour can be reduced i s by the growth of g r a p h i t e g r a i n s . At the very low d e n s i t i e s which p r e v a i l e d i n xAs i s demonstrated i n the next s e c t i o n , the form of the n u c l e a t i o n equation as w e l l as the value adopted f o r <S both i n f l u e n c e the l o c a t i o n of the sonic p o i n t . 92 the g r a i n - f o r m i n g r e g i o n , t h i s growth i s almost n e g l i g i b l e (da/dt ^ 1 0 " 1 5 cm/sec). Thus, S(r) w i l l remain c l o s e t o , but j u s t s l i g h t l y l e s s than, S*(r) u n t i l the l a y e r under c o n s i d e r a t i o n moves upward to a region of lower temperature. At t h i s new r a d i u s , r ' , S ( r ' ) > S * ( r ' ) and the g r a i n n u c l e a t i o n w i l l proceed to reduce the carbon vapour u n t i l S ( r ' ) < S * ( r 1 ) . T h i s process of short " b u r s t s " of g r a i n formation w i l l be repeated c o n t i n u o u s l y as the l a y e r moves outward. The r a t e at which the n u c l e i are formed i s c o n t r o l l e d by the outward flow r a t e of the atmosphere, but a l l n u c l e i are formed in a vapour whose s u p e r s a t u r a t i o n r a t i o j u s t exceeds the l o c a l value of S*. Since both S* and T were slowly v a r y i n g f u n c t i o n s of depth i n the upper l a y e r s of the atmosphere a f t e r the f i r s t b u r s t of g r a i n formation, the r a d i i of the g r a i n n u c l e i were almost independent of the l e v e l at which they were- formed. The low r a t e of g r a i n growth accounted f o r the f a c t t h a t , except f o r the few g r a i n s formed w e l l below the sonic p o i n t , the g r a i n s were only s l i g h t l y l a r g e r than the n u c l e i . (c) P u l s a t i o n Mechanism The f o r c e d r i v i n g the p u l s e s i n model 2 was t r a c e d to an o p a c i t y c o n t r o l l e d feedback mechanism o p e r a t i n g between the hydrogen d i s s o c i a t i o n zone and the g r a i n - f o r m i n g r e g i o n . Under the assumptions used i n t h i s study, T = T ( r , T ) and = 0. Consequently, v a r i a t i o n s i n the o p a c i t y of the model at a p a r t i c u l a r l e v e l were manifested as. immediate changes i n the temperature s t r u c t u r e at a l l lower l e v e l s . However, s i n c e the atmospheric o p a c i t y was a f u n c t i o n of temperature, an i n c r e a s e 93 i n temperature produced an i n c r e a s e i n the l o c a l gas o p a c i t y which, i n t u r n , i n c r e a s e d the temperature of the gas s t i l l more. For example, at the s o n i c p o i n t i n model 2, Atc = 2x10' 4 (At/X = 17%). T h i s v a r i a t i o n had i n c r e a s e d to A T = 5x10" 1 (AT/T = 3%) at the base of the model. Above the photosphere, the small v a r i a t i o n s i n T had l i t t l e e f f e c t on the s t r u c t u r e of the atmosphere, . but i n the hydrogen d i s s o c i a t i o n zone, the temperature s t r u c t u r e was very s e n s i t i v e to changes i n X . For example, with AT t = 2 x l 0 - 4 , A T was 11 K at the photosphere and 20 K near the base of the hydrogen d i s s o c i a t i o n zone. The r e s u l t a n t s h i f t i n the d i s s o c i a t i o n e q u i l i b r i u m of i n c r e a s e d P by 0.1% and generated an outgoing compression p u l s e . S i m i l a r l y , a decrease i n X generated an outgoing r a r e f a c t i o n p u l s e . As noted i n Chapter V, Np was almost n e g l i g i b l e below the s o n i c p o i n t whereas, above i t , N p was almost constant with depth. I t was a l s o at t h i s p o i n t that the a b s o r p t i o n due to the g r a i n s and the f r a c t i o n a l changes i n the o p t i c a l depth due to v a r i a t i o n s i n Np were the l a r g e s t . With the a r r i v a l of a compression wave from the base of the atmosphere, the gas d e n s i t y at the sonic p o i n t i n c r e a s e s and generates an i n c r e a s e i n the g r a i n n u c l e a t i o n and growth r a t e s at that l e v e l . The i n c r e a s e i n g r a i n p r o d u c t i o n has two consequences. The f i r s t , as can be i n f e r r e d from the d i s c u s s i o n i n s e c t i o n ( a ) , i s to i n c r e a s e the r a t e of mass l o s s from the s u r f a c e . The second, i s to i n c r e a s e the o p t i c a l depth throughout the r e g i o n below the sonic p o i n t and to generate a new compression p u l s e i n the hydrogen d i s s o c i a t i o n zone. The r a r e f a c t i o n f o l l o w i n g the 94 compression has j u s t the opposite e f f e c t at the sonic p o i n t , d e p r e s s i n g r a t h e r than s t i m u l a t i n g g r a i n p r o d u c t i o n by d e c r e a s i n g the l o c a l gas d e n s i t y . Thus, the p u l s e s generated at the base of the atmosphere modulate the g r a i n p r o d u c t i o n r a t e throughout the g r a i n - f o r m i n g region of the atmosphere. These changes, i n t u r n , generate new p u l s e s at the base and complete the feedback loop. The feedback mechanism appears capable of m a i n t a i n i n g r a d i a l p u l s a t i o n s p r o v i d e d that /ATt2 1x10" 4. In the case of model 1, T C was small ( =5.5xl0 - 5) and there was l i t t l e feedback between the sonic p o i n t and the base. T h i s f a c t i s seen most c l e a r l y by examining the behaviour of the base of the atmosphere. A f t e r the i n i t i a l l a r g e p u l s e , the base v e l o c i t y showed no s i g n of f o r c e d o s c i l l a t i o n s but d i s p l a y e d a s e r i e s of c h a r a c t e r i s t i c r e l a x a t i o n o s c i l l a t i o n s of d e c r e a s i n g amplitude. Each pulse had l e s s e f f e c t at the s u r f a c e than the p r e v i o u s one as can be seen by examining the behaviour of the s u r f a c e v e l o c i t y as given i n F i g u r e 3. E v e n t u a l l y the flow approached a s t e a d y - s t a t e i n t h i s model. The e f f e c t of the feedback mechanism was much d i f f e r e n t i n model 2. In t h i s case, the a r r i v a l of the f i r s t compression p u l s e at the s o n i c p o i n t i n c r e a s e d the o p t i c a l depth there by Af t= 6x10" 4 as i s shown i n F i g u r e 4. The f l u c t u a t i o n of T C was not i n i t i a l l y i n phase with the r e l a x a t i o n o s c i l l a t i o n s that were commencing at the base, and, consequently c o u n t e r a c t e d the base p u l s a t i o n s . (See F i g u r e 4 ) . However, a f t e r 3 p e r i o d s , the d r i v i n g mechanism had f o r c e d the base to s h i f t i n t o phase with the f l u c t u a t i o n s i n the g r a i n - f o r m i n g r e g i o n whereafter a 95 n e a r l y constant amplitude, atmospheric p u l s a t i o n was maintained fo r 4 p e r i o d s . The i n t e r a c t i o n between the base and the sonic p o i n t i s seen most c l e a r l y at p o i n t s (a) (as d i s c u s s e d i n Chapter V), and (b) and (c) i n F i g u r e 4. At p o i n t ( b ) , the p u l s e t h a t was generated at the base of the atmosphere d u r i n g the i n i t i a l stages of the model had reached the sonic p o i n t and s t i m u l a t e d an i n c r e a s e i n g r a i n formation. The i n c r e a s e i n Xc, i n t u r n , c o u n t e r a c t e d the r e l a x a t i o n pulse at the base of the atmosphere. The i r r e g u l a r p a t t e r n of the p u l s e o r i g i n a t i n g at the base at p o i n t (b) generated the i r r e g u l a r p a t t e r n i n the v a r i a t i o n of T T at p o i n t ( c ) , which, i n t u r n , a f f e c t e d the c h a r a c t e r i s t i c s of the base pulse being d r i v e n then. E v e n t u a l l y , these i r r e g u l a r i t i e s d i e d out and the p u l s e s assumed a r e g u l a r p a t t e r n . " (d) D i s c u s s i o n of the Assumptions ( i ) R a d i a t i v e E q u i l i b r i u m The assumption that the atmosphere was everywhere i n r a d i a t i v e e q u i l i b r i u m i m p l i e d that the thermal time s c a l e was zero and that T = T(r,T). The important consequence of adopting t h i s assumption was that i t i m p l i e s that the gas was completely coupled to the r a d i a t i o n f i e l d and t h a t changes i n the model s t r u c t u r e c o u l d be i n s t a n t a n e o u s l y r e f l e c t e d i n temperature changes elsewhere i n the atmosphere. I m p l i c i t i n t h i s assumption, i s the assumption that energy sources other than the r a d i a t i o n f i e l d were n e g l i g i b l e . S p e c i f i c a l l y , the e f f e c t s of 96 a d i a b a t i c c o o l i n g of the expanding atmosphere, the k i n e t i c energy of the flow and the d i s s o c i a t i o n energy were a l l n e g l e c t e d on the b a s i s that a l l of these energy sources were small r e l a t i v e to the l u m i n o s i t y . For example, the energy change accompanying the d i s s o c i a t i o n of i n model 2 was estimated to be approximately 1 % of the t o t a l thermal energy of the gas. However, the r e l a x a t i o n of the r a d i a t i v e e q u i l i b r i u m assumption may have s e v e r a l important consequences. F i r s t , a thermal time l a g between the changes i n f t and the d i s s o c i a t i o n of H^ may be e s t a b l i s h e d . Second, the amplitude of the waves generated i n the H^ zone may be d i m i n i s h e d s i n c e changes i n the i n t e r n a l energy of the l a y e r w i l l tend to c o u n t e r a c t any thermal v a r i a t i o n i n the zone. T h i r d , a d i a b a t i c c o o l i n g i n the s u r f a c e l a y e r s may tend t o magnify the amplitude of the p u l s e s somewhat. T e n t a t i v e l y , i t 'seems l i k e l y t h a t ' the r e l a x a t i o n of t h i s assumption w i l l supress but not e l i m i n a t e the p u l s e d r i v i n g mechanism. ( i i ) The Grey Assumption The predominant e f f e c t of assuming a grey o p a c i t y was that the s u r f a c e temperatures of both models were probably s e v e r a l hundred degrees too h i g h f o r the chosen v a l u e s of T*. T h i s estimate was d e r i v e d as f o l l o w s . The s u r f a c e temperature of a grey, plane p a r a l l e l model i s determined by the f u n c t i o n T/T* = 0 . 8 1 . For blanketed, plane p a r a l l e l models of a s o l a r composition, T s u j i ( 1 9 7 6 ) f i n d s that T 0/T* = 0.76 f o r T* < 4 0 0 0 K where T e i s the temperature at T = 1 0 " 3 . For a b l a n k e t e d s p h e r i c a l model of s i m i l a r composition and 97 T* = 2800 K, Watanabe and Kodaira (1979) o b t a i n T D = 1800 K, or about 300 K lower than the temperature obtained from T s u j i ' s r e l a t i o n s h i p ( i . e . , T 0/T* = 0.61), and 600 K lower than that which would be given by a grey, plane p a r a l l e l model. For both of the time-dependent models, W "= 3.15xl0' 1 at t s 10" 3 which g i v e s a r a t i o of T„/T* = 0.75 or about the same s u r f a c e temperature as T s u j i ' s plane p a r a l l e l , blanketed models. Thus, i f the s u r f a c e c o o l i n g e f f e c t s i n c a r b o n - r i c h and oxygen-rich s t a r s are s i m i l a r , blanketed s p h e r i c a l models of c a r b o n - r i c h s t a r s should be about 200 K to 300 K c o o l e r than the s p h e r i c a l grey models. The f a c t that the T* = 3400 K, blanketed, carbon s t a r models of Querci and Querci (1975) and the e q u i v a l e n t s o l a r abundance models of T s u j i (1978) have n e a r l y i d e n t i c a l s u r f a c e temperatures,, g i v e s some c r e d i b i l i t y to the above argument. However, s i n c e i t was not p o s s i b l e to make a d i r e c t comparison between grey and blanketed s p h e r i c a l models of c a r b o n - r i c h s t a r s , only the d i r e c t i o n of the temperature s h i f t should be taken as r e l i a b l e . On the other hand, Watanabe and Kodaira (1979) f i n d that the s u r f a c e temperature i n t h e i r T* = 2800 K model i s r a t h e r i n s e n s i t i v e to l a r g e v a r i a t i o n s (3% to 17%) i n the atmospheric e x t e n s i o n . I f such a t r e n d p e r s i s t e d to even lower temperatures, i t i s p o s s i b l e that the e r r o r i n the s u r f a c e temperature of the grey models i s lower than the estimated 200 K to 300 K. An i n c r e a s e i n the e f f e c t i v e temperature of the two models would b r i n g them c l o s e r to the center of R i c h e r ' s (1980) l u m i n o s i t y d i s t r i b u t i o n f o r N s t a r s i n the LMC. Mass l o s s r a t e s o b t a i n e d from blanketed models at 2500 K and 2400 K would l i k e l y 98 be one to two orders of magnitude l a r g e r than the ones obtained with the grey models pr o v i d e d that the temperature of the g r a i n n u c l e i can be approximated by the ambient gas temperature. (See s e c t i o n ( v i ) ) . ( i i i ) Convective S t a b i l i t y Although c o n v e c t i o n has been demonstrated to c a r r y a very small f r a c t i o n of the energy f l u x i n the atmospheres of red g i a n t s (Auman, 1969), c o n v e c t i v e motions c o u l d have s e v e r a l important consequences on the behaviour of the g r a i n - f o r m i n g atmospheres. (1) The formation of l a r g e c o n v e c t i o n c e l l s suggested by Harm and S c h w a r z c h i l d (1975) c o u l d l e a d to a "lumpy" g r a i n d i s t r i b u t i o n ( S a l p e t e r , 1977). G r a i n s should form p r e f e r e n t i a l l y above the u p w e l l i n g regions at r a d i i l a r g e r than the r a d i a t i v e models p r e d i c t . I f i t i s assumed t h a t the c e l l s have a higher temperature and a lower d e n s i t y than the surrounding gas, then the g r a i n s formed in the c e l l would have a l a r g e r r a d i u s but a smaller growth r a t e than the c o n v e c t i v e l y s t a b l e models p r e d i c t . Since the e f f e c t of n e i t h e r of these f a c t o r s would be very l a r g e , the s i z e d i s t r i b u t i o n of the g r a i n s would be v i r t u a l l y u n a f f e c t e d by the presence of c o n v e c t i o n p r o v i d e d that the r a d i a t i o n p r e s s u r e f o r c e a c t i n g on the g r a i n s was s u f f i c i e n t to prevent them from being dragged back i n t o the p h o t o s p h e r i c l a y e r s of the s t a r by the c o n v e c t i v e c i r c u l a t i o n . If the g r a i n s do c i r c u l a t e back to the inner atmospheric r e g i o n , one of two p o s s i b l e processes can be expected to occur. E i t h e r the g r a i n w i l l be p a r t i a l l y or t o t a l l y d e stroyed by e v a p o r a t i o n , or, the growth r a t e of the g r a i n w i l l be g r e a t l y enhanced i f i t 99 e n t e r s a region i n which the ambient gas d e n s i t y i s h i g h and the carbon vapour abundance has not been s e r i o u s l y d e p l e t e d by g r a i n formation. Under both of these c o n d i t i o n s , the g r a i n - s i z e d i s t r i b u t i o n would be a f f e c t e d to a s i g n i f i c a n t degree. In the f i r s t case, the l o c a l C/0 r a t i o would be r a i s e d and t h i s , i n t u r n , would enhance the formation of g r a i n s when c o n v e c t i o n once again l i f t e d the element to the s u r f a c e . In the second case, the number d e n s i t y of l a r g e g r a i n s (10" 5 cm to 10" 6 cm) would be g r e a t l y enhanced and that of small g r a i n s , reduced, b r i n g i n g the s i z e d i s t r i b u t i o n of the g r a i n s i n t o c l o s e r agreement with that which has been suggested as g i v i n g the best f i t to the e x t i n c t i o n by i n t e r s t e l l a r g r a i n s of mixed composition (Mathis et a l , 1977). (2) In regions where the c o n v e c t i v e energy f l u x i s not n e g l i g i b l e , the temperature d i s t r i b u t i o n becomes p a r t i a l l y decoupled from the o p t i c a l depth of the r e g i o n . Under these c o n d i t i o n s , the feedback mechanism o p e r a t i n g i n model 2 would become l e s s e f f i c i e n t or, perhaps, be damped completely even i f the c o n v e c t i o n d i d not extend higher than the d i s s o c i a t i o n zone. Conversely, i f the gas remains s t r o n g l y coupled to the r a d i a t i o n f i e l d i n the hydrogen i o n i z a t i o n zone, the p o s s i b i l i t y of g e n e r a t i n g p u l s e s i n t h i s zone a r i s e s . (3) A t h i r d e f f e c t that c o n v e c t i o n c o u l d have on the models i s the g e n e r a t i o n of a c o u s t i c a l waves i n the atmosphere l e a d i n g to: the formation of a hot s t e l l a r chromosphere. A hot chromosphere would tend to evaporate the g r a i n s . I t i s p o s s i b l e t h a t the i n f r a r e d e m i s s i v i t y of the s m a l l g r a p h i t e g r a i n s i s s u f f i c i e n t to prevent the formation of a chromosphere by c o o l i n g 100 the expanding s h e l l as r a p i d l y as a c o u s t i c a l energy i s d e p o s i t e d w i t h i n i t . ( i v ) G r a i n Composition The morphology of s t e l l a r and i n t e r s t e l l a r carbon g r a i n s has been the subject of much debate. Czyzak and Santiago (1973) have argued that such g r a i n s cannot have a s t r u c t u r e l i k e that of g r a p h i t e on the grounds that (1) the temperature and pressure c o n d i t i o n s r e q u i r e d f o r g r a i n n u c l e a t i o n to be p o s s i b l e w i l l y i e l d only h i g h l y d i s o r d e r e d carbon g r a i n s , and (2) the c r y s t a l l i z a t i o n of these d i s o r d e r e d g r a i n s r e q u i r e s higher temperatures and/or pressures than the g r a i n s are l i k e l y to encounter. However, Stephens (1980) has shown that d i s o r d e r e d amorphous or g l a s s y carbon g r a i n s with r a d i i £ 5x10" 7 cm have a UV e x t i n c t i o n peak that i s r e d - s h i f t e d r e l a t i v e to the 2200 A peak i n the i n t e r s t e l l a r reddening curve. He a l s o has shown that the UV e x t i n c t i o n peak f o r carbon g r a i n s s h i f t s towards the v i o l e t as the degree of c r y s t a l l i n i t y i n c r e a s e s and/or the g r a i n s i z e i s decreased. In h i s study, Stephens i n t e r p r e t e d t h i s data as i n d i c a t i n g that i n t e r s t e l l a r carbon g r a i n s are more l i k e l y to have a s t r u c t u r e resembling that of g r a p h i t e than of amorphous carbon and i t i s tempting to use t h i s same argument as a j u s t i f i c a t i o n of the assumption of the formation of g r a p h i t e g r a i n s i n the models. However, Stephen's c o n c l u s i o n i s not n e c e s s a r i l y v a l i d i f the g r a i n s are smaller than 5x10" 7 cm as was the case in both time-dependent models. Indeed, the e x i s t e n c e of the e x t i n c t i o n peak at 2200 A may be i n t e r p r e t e d as suggesting that i n t e r s t e l l a r carbon g r a i n s have r a d i i ^ 10" 7 cm. 101 The values of the s u r f a c e - f r e e - e n e r g y , <f, the Lothe-Pound f a c t o r , and the o p t i c a l c o n s t a n t s used to c a l c u l a t e the g r a i n o p a c i t y were a l l d e r i v e d from the assumption that the carbon g r a i n s had the s t r u c t u r e of g r a p h i t e . The adopted value of o~ (1000 ergs/cm 2) has been suggested as a r e p r e s e n t a t i v e value f o r g r a p h i t e (Tabak et. a_l, 1975). However, the u n c e r t a i n t y a s s o c i a t e d with choosing t h i s value f o r <5 i s l a r g e enough that i t should be as r e p r e s e n t a t i v e of n u c l e i of amorphous carbon s o l i d s as i t i s of g r a p h i t e p a r t i c l e s . The a p p r o p r i a t e form of the Lothe-Pound replacement f a c t o r f o r carbon s t r u c t u r e s other than g r a p h i t e has not been determined (Tabak, et a_l, 1975). I f the g r a i n s a r e , i n f a c t , amorphous i n s t r u c t u r e , P c o u l d have been i n e r r o r by s e v e r a l orders of magnitude without s e r i o u s l y changing the dynamic behaviour of the models. (v) Lothe-Pound N u c l e a t i o n Theory Tabak et a_l, (1975) have argued that the Lothe-Pound form of the n u c l e a t i o n equation should be a p p l i c a b l e to the formation of g r a p h i t e g r a i n s . They s t a t e that the u n c e r t a i n t i e s r e g a r d i n g the s p e c i f i c s t r u c t u r e of the g r a i n s and the c h o i c e of an a p p r o p r i a t e value f o r Cf should outweigh the u n c e r t a i n t i e s inherent i n the use of the Lothe-Pound theory. They note a l s o , that n u c l e a t i o n experiments on non-polar m a t e r i a l s g i v e r e s u l t s that are i n good agreement with the p r e d i c t i o n s of the Lothe-Pound theory. More r e c e n t l y , Draine (1979) has s e r i o u s l y questioned the v a l i d i t y of the Lothe-Pound theory e s p e c i a l l y i n 102 the case of the n u c l e a t i o n of very small c l u s t e r s (N*^20), where N* i s the number of carbon atoms per c r i t i c a l nucleus. While h i s o b j e c t i o n s do not at f i r s t appear to apply to the present study s i n c e i t was found that N* = 70, i t was s t i l l p o s s i b l e that the use of the Lothe-Pound theory may have r e s u l t e d i n an erroneous n u c l e a t i o n r a t e . As an a l t e r n a t i v e to the Lothe-Pound theory, Draine suggests that f o r the n u c l e a t i o n of atomic s o l i d s , the s u r f a c e - f r e e - e n e r g y should be r e p l a c e d by 6/2 where 6 i s the s u r f a c e - f r e e - e n e r g y of the bulk s o l i d , and t h a t f be set equal to 1. To compare these two d i f f e r e n t d e s c r i p t i o n s of the n u c l e a t i o n of small g r a i n s , c o n s i d e r the n u c l e a t i o n r a t e given by each when T •= 1446 K, S = 5.0, and P^, = 1.4x10" 1 1 dynes/cm 2. Under these c o n d i t i o n s , the Lothe-Pound theory g i v e s a* =5.6x10-" cm, and P = 5 x l 0 2 ' . P _ 1 i s comparable to the e x p o n e n t i a l term i n the r a t e equation, exp(-AF*/kT) = 2 . 3 x l 0 - 2 ' . The Lothe-Pound. n u c l e a t i o n r a t e , J(LP,C=1000), i s equal to 8 . 6 x l 0 - ' sec and i s e s s e n t i a l l y given by the remaining pre-e x p o n e n t i a l terms i n Equation (111-14). When Draine's suggested n u c l e a t i o n r a t e equation i s used, a* i s reduced to 2 . 8 x l 0 _ " cm, N* = 10, exp(-AF*/kT) = 2x10" 4, and the n u c l e a t i o n r a t e , J(D,cr/2=500) , i s reduced to 4 x l 0 _ 1 ° s e c " 1 . On the b a s i s of t h i s comparison, i t would appear that the Lothe-Pound theory overestimates the n u c l e a t i o n r a t e . However, i n h i s a n a l y s i s , Draine assumed that C = 1400 ergs/cm 2 f o r s p h e r i c a l carbon g r a i n s . Using t h i s value f o r cT and the same valu e s f o r T, S and P/xri as above, P becomes 6xl0 3 0 , a* = 7 . 8 x l 0 - 8 cm, exp(-AF*/kT) = 3x10"", N* = 220, and J(LP,ff=1400) i s reduced to ~ 3 x l 0 _ 5 6 s e c - 1 . In t h i s case, reducing O*to.cr/2 f and assuming 103 p = 1, reduces the e x p o n e n t i a l term to 1 0 " 1 0 and i n c r e a s e s the n u c l e a t i o n r a t e by a f a c t o r of 10* 1 to J(D, CJ/2=700) = 1 0 " 1 5 s e c " 1 . Thus, the n u c l e a t i o n r a t e i s much more s e n s i t i v e to the chosen value of C5 than i t i s to the use of the Lothe-Pound theory. T h i s r e s u l t i s s i m i l a r to that reached by Deguchi (1980) who found that the number d e n s i t y of s i l i c a t e g r a i n s formed in steady flows was i n s e n s i t i v e to the form of the n u c l e a t i o n e q u a t i o n . When <3 i s l a r g e , J i s a l s o extremely s e n s i t i v e to the value of l n ( S ) appearing i n the e x p o n e n t i a l term of the n u c l e a t i o n equation. For example, an i n c r e a s e i n S to 9.5 (a f a c t o r of 1.9) i s a l l that i s r e q u i r e d to i n c r e a s e J (LP,C =1400) to J (D, Cf/2 = 700). A comparison between the two r a t e equations cannot be r e s t r i c t e d to comparing only the values of the term P exp(- AF*/kT) . I f , f o r example, p' = 10" S P had been used to " c o r r e c t " J(LP) such that J (LP ,o"=1000) = J (D,o72=500) , the base of the g r a i n forming r e g i o n would have been s h i f t e d from S c = 5.0 to =5.9 i n the models and the mass l o s s r a t e would have been decreased i n model 2 by about 15%. A l s o , the e f f e c t i v e g r a i n r a d i u s would have been equal to a = 5x10" 8 cm. I f , i n s t e a d , J(D,0/2=500) had been used to o b t a i n J , the e f f e c t i v e g r a i n r a d i i i n both models would have been reduced to a = 2.8x10" 8 cm and the s e n s i t i v i t y of J to l n ( S ) would have been c o n s i d e r a b l y reduced s i n c e i n t h i s case exp(-AF*/kT)—* 1. For example, f o r the case under c o n s i d e r a t i o n , i n c r e a s i n g S by a f a c t o r of 2 would in c r e a s e the e x p o n e n t i a l term only by a f a c t o r of ~ 1 0 2 . However, the p r e - e x p o n e n t i a l term c o n t a i n s the f a c t o r (P/u-, / T ) 2 and t h i s term v a r i e s s t r o n g l y with depth i n the 104 atmosphere with the r e s u l t t h a t J(D,CT/2 = 500) at T = 1513 K, S = 3.5 i s comparable to J(D,cT/2 = 500) at T =1446, S = 5.0. Thus, while J(D,tf/2=500) < J (LP ,o"=1000) , the use of J(D) would have moved the base of the g r a i n - f o r m i n g region deeper i n t o the atmosphere so that the o v e r a l l e f f e c t of using the Lothe-Pound theory may have been to underestimate the mass l o s s r a t e of the two models and to overestimate the r a d i i of the g r a i n n u c l e i . ( v i ) G r a i n Temperature The temperature of both g r a p h i t e and amorphous carbon smoke i s e l e v a t e d above the e q u i v a l e n t black body temperature by the greenhouse e f f e c t . T h i s f a c t was ignored when i t was assumed that the gas and g r a i n s had the same l o c a l temperature. Using the Mie theory with 'the o p t i c a l c o n s t a n t s of g r a p h i t e , the v i b r a t i o n a l temperature of the g r a i n s can be obtained by s o l v i n g the energy balance equation 4ffa 2 {Q p(a,T*)W0T* 4 + n(2kT/7r/*'mH ) °-soCk(T - T p )} = 4ua 2Qp(a,T p) ( V - l ) In t h i s e quation, Q p i s the Plank mean e x t i n c t i o n c o e f f i c i e n t f o r the g r a i n s , n, the number d e n s i t y of the gas p a r t i c l e s , and oc, the accomodation c o e f f i c i e n t . Under c o n d i t i o n s r e p r e s e n t a t i v e of the g r a i n - f o r m i n g r e g i o n s of the two models, i t was found that the c o l l i s i o n a l term was n e g l i g i b l e compared with the r a d i a t i o n a l h e a t i n g term, i . e . , the g r a i n s were not t h e r m a l l y coupled to the gas. As a consequence of t h i s non-LTE e f f e c t , i t was estimated that the v i b r a t i o n a l temperature of the 105 g r a i n s c o u l d be e l e v a t e d by as much as 150 K above the temperature of the ambient gas. T h i s temperature d i f f e r e n c e i s l i k e l y an upper l i m i t f o r the f o l l o w i n g reasons. (1) The Mie theory may not g i v e v a l i d r e s u l t s f o r g r a i n s with r a d i i = 5x10" 8 cm (Draine, 1981a). (2) The g r a i n s probably have l a t t i c e d e f e c t s which c o u l d i n c r e a s e t h e i r i n f r a r e d e m i s s i v i t y and decrease t h e i r temperature. (3) M o r p h o l o g i c a l l y , the g r a i n s may resemble amorphous carbon smoke p a r t i c l e s more c l o s e l y than they do g r a p h i t e p a r t i c l e s and e m p i r i c a l l y , carbon smoke p a r t i c l e s have been found to have a much smal l e r greenhouse e f f e c t that g r a p h i t e p a r t i c l e s (Draine, 1981b). (4) In a non-grey atmosphere, J^, the mean i n t e n s i t y , and B-o, the Plank f u n c t i o n , may d i f f e r by a l a r g e amount so that the g r a i n a b s o r p t i o n term appearing i n the above equation may be overestimated. If the greenhouse e f f e c t a l s o e l e v a t e s the temperature of the unstable c l u s t e r s by 150 K, then g r a i n n u c l e a t i o n i n both models would have been prevented. In t h i s case, the f a c t that carbon g r a i n s appear to form in c a r b o n - r i c h red g i a n t s suggests two a l t e r n a t i v e hypotheses: e i t h e r (1) A T < 150 K f o r atmospheric carbon p a r t i c l e s , or (2) the carbon condenses on a nucleus of some other m a t e r i a l . SiC has been suggested as a l i k e l y candidate f o r the seed n u c l e i (see, f o r example, Draine, 1981a) s i n c e i t has an i n v e r s e greenhouse e f f e c t and, i n i t i a l l y , the g r a i n s would be c o o l e r than the ambient gas. The hypothesis of SiC seeding i s c o n s i s t e n t with the f a c t t h a t those carbon s t a r s having an i n f r a r e d excess appear a l s o to have the 11.2/U a b s o r p t i o n f e a t u r e that has been a t t r i b u t e d to a b s o r p t i o n by SiC 106 g r a i n s . However, i f the temperature d i f f e r e n c e between the g r a i n s and the gas i s e s t a b l i s h e d a f t e r the g r a i n s are formed, then, although e v a p o r a t i o n would dominate g r a i n growth, the g r a i n s with a = 5x10" 8 cm would be able to escape from the atmosphere be f o r e being evaporated. The p o s s i b i l i t y t h a t the gas was t h e r m a l l y coupled t o the g r a i n s was i n v e s t i g a t e d by examining the energy balance equation a p p r o p r i a t e f o r the gas. The h e a t i n g term of t h i s equation w i l l be dominated by the r a d i a t i o n f i e l d p r o v i d e d that n'n < ( n ' n ) c r ; t = K (T* ) WCT* */{ 4TTa2 ( 2kT/if/t' mH ) 0-so<?kT} (VI-2) where n' i s the number d e n s i t y of the g r a i n s per gram of s t e l l a r m a t e r i a l . Using the Plank mean f o r K. and W = 0.1, T = 1500 K, a = 5.5xl0" 8 cm and T* = 2400 K, (n'n) M;+ = 8.97xl0 2 8 gm^cm" 3. whereas i n the g r a i n - f o r m i n g region of model 2, n'n = 1 . 2 1 x l 0 2 S gm"1cm"3. I f the Rosseland mean i s used f o r K, (n'n)c<.^ i s reduced to 2.8xl0 2 6 gm~lcm"s. In a non-grey model, the value of (n'n)Cr,-f would probably l i e somewhere between these two extremes. I t i s apparent, t h e r e f o r e , that g a s - g r a i n c o l l i s i o n s would have been i n e f f e c t i v e i n h e a t i n g the gas i n the models. At lower temperatures, however, n'n w i l l be l a r g e r and the gas may become t h e r m a l l y coupled to the g r a i n s . I t i s a l s o noted that i f J(D) i s s u b s t i t u t e d f o r J ( L P ) , then the g r a i n s may form deeper i n the atmosphere, n'n w i l l be much l a r g e r than that o b t a i n e d i n the present models, and c o l l i s i o n a l h e a t i n g of the gas may tend to e q u i l i b r a t e the gas and g r a i n temperatures. 107 (e) C o n c l u s i o n s The major c o n c l u s i o n s a r i s i n g from t h i s study a r e : 1) R a d i a t i o n pressure a c t i n g on g r a i n s i s a v i a b l e mass l o s s mechanism in c o o l , c a r b o n - r i c h red g i a n t s . At T* = 2500 K, t h i s mass l o s s mechanism generated a mass l o s s r a t e of 6.2x10" M©/yr, and at T* = 2400 K, a mass l o s s r a t e of 7.4xl0" 8 M Q/yr. 2) The i n t r i n s i c v a r i a b i l i t y of c o o l , c a r b o n - r i c h s t a r s may be d i r e c t l y l i n k e d to the formation of atmospheric g r a i n s . The p u l s a t i o n of model 2 appeared to be s t a b l e with a p e r i o d of 750 days which i s c o n s i s t e n t with observed p e r i o d s of v a r i a b l e carbon s t a r s . The source of the p u l s e s appears to be an o p a c i t y c o n t r o l l e d feedback loop a c t i n g between the g r a i n - f o r m i n g region of the atmosphere, and the hydrogen d i s s o c i a t i o n zone. 3) G r a i n s formed i n the atmospheres of c a r b o n - r i c h s t a r s have r a d i i t h a t are <10" 7 cm. Such g r a i n s are much smal l e r than the s i l i c a t e g r a i n s that appear to be the cause of the i n t e r s t e l l a r p o l a r i z a t i o n of s t a r l i g h t . However, the computed s i z e of the carbon g r a i n s i s c o n s i s t e n t with (a) the l o c a t i o n of the 2200 A peak i n the i n t e r s t e l l a r e x t i n c t i o n curve, (b) the sharp i n c r e a s e i n t h i s curve i n the UV r e g i o n , and (c) the f a c t t h a t the UV e x t i n c t i o n i s g r e a t e r i n the a p p a r e n t l y carbon s t a r r i c h areas of the LMC than i t i s i n the M i l k y Way Galaxy (Nandy, et a l , 1980) . 4) The mass l o s s r a t e s d e r i v e d from both s t e a d y - s t a t e and time-dependent models are very s e n s i t i v e to the assumptions used i n t h e i r c o n s t r u c t i o n . By assuming that S t = 2.0, Lucy (1976) c a l c u l a t e d a mass l o s s r a t e of 1.4x10" 6 Mg/yr, whereas, the 108 time-dependent model p r e d i c t s that at T* = 2500 K, S c = 5.3 and dM/dt = 6 . 2 x l 0 - ' M©/yr. However, the degree of g r a i n formation i n the time-dependent models was very dependent upon the form of the n u c l e a t i o n r a t e equation that was used and the value chosen f o r C5 . T h i s dependency i s the major u n c e r t a i n t y a s s o c i a t e d with the d e t e r m i n a t i o n of these mass l o s s r a t e s . 109 BIBLIOGRAPHY Abraham, F. F., Homogeneous N u c l e a t i o n Theory, (New York: Academic P r e s s ) , 1974. Adams, W. S., and MacCormack, E., Ap. J . , 81, 119, 1935. A l l e n , C. W., A s t r o p h y s i c a l Q u a n t i t i e s , (London: Athlone P r e s s ) , 1973. Auer, L. H., J . Quant. S p e c t r o s c . Rad. T r a n s f . , 11, 573, 1971. Auman, J . R., Ap. J . , 157, 799, 1969. Becker, R. , and Doring, W., Ann. Physik, 24, 719, 1935. Bernat, A. P., Ap. J . , 213, 756, 1977. Boesgaard, A. M., and Hagen, W., Ap. J . , 231, 128, 1979. Buhl, D., Snyder, L. E., Lovas, F. J . , and Johnson, D. R., Ap. J . ( L e t t e r s ) , 201, L29, 1975. Carbon, D. F., Ap. J . , 187, 135, 1974. , Ann. Rev. A s t r o n . Astrophys., 17, 513, 1979. C a s s i n e l l i , J . P., Ann. Rev. A s t r o . Astrophys., 17, 275, 1979. Cohen, M., and Gaustad, J . E., Ap. J . ( L e t t e r s ) , 186, L131, 1973. Czyzak, S. J . And Santiago, J . J . , Ap. and Space Sc i . , 23, 443, 1973. Deguchi, S., Ap. J . , 236, 567, 1980. De Jager, C , and De Loore, C , Ap. and Space Sc i . , 11, 284, 1971. Deutsch, A. J . , In S t a r s and S t e l l a r Systems, V o l . 6, ed., G r e e n s t e i n , J . L., (Chicago: U n i v e r s i t y of Chicago P r e s s ) , 1960. Donn, B., Wickramasinghe, N. C , Hudson, J . P., and Stecher, T. P., Ap. J . , 153, 451, 1968. D o r f e l d , W. G., and Hudson> J . B., Ap. J . , 186, 725, 1973. Draine, B. T., Ap. and Space S c i . , 65, 313, 1979. ______ 1981a, In p r e s s . •• 1981b, P r e p r i n t . 110 Draine, B. T., and S a l p e t e r , E. E., J . Chem. Phys., 67, 2230, 1977. E i l e k , J . , Ph. D. T h e s i s , (U. B. C : Vancouver), 1975. E l i t z e r , M., G o l d r e i c h , P., and S c o v i l l e , N., Ap. J . , 205, 384, 1976. E n q u i s t , B., Gustafsson, B., and Vreeburg, J . , "A D i f f e r e n c e Method f o r a Non-Linear Mixed H y p e r b o l i c - P a r a b o l i c Problem  I", ( U p s a l l a U n i v e r s i t y : Department of Comp. Science) 1974. , "Numerical S o l u t i o n of a PDE-System D e s c r i b i n g a C a t a l y t i c Converter"" ( U p s a l l a U n i v e r s i t y : Department of Comp. Science ) 1977. F i x , J . D., M. N. R. A. S., 146, 37, 1969. F o r r e s t , W., J . , G i l l e t t , F. C , and S t e i n , W. A., Ap. J . , 195, 423, 1975. F u s i - P e c c i , F., and R e n z i n i , A., A s t r o . Astrophys., 46, 447, 1976. Gehrz, R. D., and Woolf, N. J . , Ap. J . , 165, 285, 1971. Gilman, R. C , Ap. J . ( L e t t e r s ) , 155, L185, 1969. , Ap. J . , 178, 423, 1972. , Ap. J . Suppl., 28, 397, 1974. G i l r a , D. P., i n Proceedings of the IAU Colloquium No. 52, " I n t e r s t e l l a r Dust and Rel a t e d T o p i c s " , ed., Greenberg, J . M. , and Van de H u l s t , H. C , (Dordrecht-Hoiland: D. R e i d e l ) , 1973. G o l d r e i c h , P., and S c o v i l l e , N., Ap. J . , 205, 144, 1976. Good, R. J . , J . Phys. Chem., 61, 810, 1957. Greenberg, J . M., In Cosmic Dust, ed., McDonnell, J . A. M., ( C h i c h e s t e r : John Wiley and Sons), 1978. Gustafsson, B., and Nissen, P. E., A s t r o n . Astrophys., 19, 261, 1972. Gustafsson, B., B e l l , R. A., E r i k s s o n , K., and Nordlund, A., As t r o n . Astrophys., 42, 407, 1975. Gustafsson, B., and Olander, N., Phys. S c i . , 20, 570, 1979. Hagen, W., Ap. J . Suppl., 38, 1, 1978. Harlow, F. H., and Amsden, A. A., J . Comp. Phys., 3, 80, 1968. I l l , J . Comp. Phys., 18, 440, 1975. Hearn, A. G., A s t r o n . Astrophys., 40, 355, 1975. Heasley, J . N., Ridgway, S. T., Carbon, D. F., Mil k e y , R. W., and-Hall> D. N. B., Ap. J . , 219, 970, 1978. H i l l , S. J . , and W i l l s o n , L. A., Ap. J . , 229, 1029, 1979. H i r t , C. W., J . Comp. Phys., 2, 339, 1968. Hoyle, F., and Wickramasinghe, N. C , M. N. R. A. S., 124, 417, 1962. Johnson, H. R., Ap. J . , 180, 81, 1973. Johnson, H. R., Marenin, I. R., and P r i c e , S., J . Quant.  S p e c t r o s c . Rad. T r a n s f . , 12, 189, 1972. Jones, T. W., and M e r r i l l , K. M., Ap. J . , 209, 509, 1976. Krupp, B. M., C o l l i n s , J . G., and Johnson, H. R., Ap. J . , 219, 963, 1978. Katz, J . , S a l t s b u r g , H., and R e i s s , H., J . C o l l o i d . S c i . , 21, 560, 1966. Kwan, J . , and H i l l , F., Ap. J . , 216,781, 1977. Kwok, S., Ap. J . , 198, 583, 19?5. Lo, K. Y., and Bechis, K. P., Ap. J . ( L e t t e r s ) , 218, L27, 1977. Lothe, J . , and Pound, G. M., J . Chem. Phys., 36, 2080, 1962. Lucy, L. B., Ap. J . , 205, 482, 1976. Lucy, L. B., and Solomon, P. M., A. J . , 72, 310, 1967. M a c i e l , W. J . , As t r o n . Astrophys., 48, 27, 1976. Marble, F. M., As t r o n . A c t a . , 14, 585, 1969. Mathis, J . S., Rumpl, W., and Nordsieck, K. H., Ap. J . , 217, 425, 1977. M e n i e t t i , J . D., and F i x , J . D., Ap. J . , 224, 961, 1978. Mendoza, E. E., and Johnson, H. L., Ap. J . , 141, 161, 1965. M e r r i l l , K. M., In Proceedings of the IAU Colloquium No. 42, "The I n t e r a c t i o n of V a r i a b l e S t a r s with t h e i r Environment", ed., Kippenhahn, R. K., Rahe, J . , and Strohmeier, W., (Erlangen-Nurnberg: Astronomisches I n s t i t u t der U n i v e r s i t a t ) , 1978. 112 M i h a l i s , D., Ap. J . , 141, 564, 1965. , Meth. Comp. Phys., 7, 1, 1967. , S t e l l a r Atmospheres, (San F r a n s i s c o : W. H. Freeman and ,. Co.), 1978. Moran, J . M., In F r o n t i e r s of A s t r o p h y s i c s , ed., A v r e t t , E. H., (Cambridge: Harvard U n i v e r s i t y P r e s s ) , 1976. M o r r i s , M., Ap. J . , 197, 603, 1975. , Ap. J . , 236, 823, 1980. M o r r i s , M. , and Alcock, C , Ap. J . , 218, 687, 1977. M o r r i s , M., Redman, R., Reid, M. J . , and D i c k i n s o n , D. F., Ap. J_:_, 229, 257, 1979. Mullan, D. J . , Ap. J . , 209, 171, 1976. , Ap. J . , 226, 151, 1978. Nandy, K., Morgan, D. H., W i l l i s , A.,J., Wilson, R., Gondhalekar, P. M., and Houziaux, L., Nature, 283, 725, 1980 Parker,' E. N. , Ap. J . , 134, 20, 1961. P o l o z h i y , G. N., Equations of Mathematical P h y s i c s , (New York: Hayden Book Co.), 1967 Q u e r c i , F., and Q u e r c i , M., A s t r o n . Astrophys., 39, 113, 1975. Ramsey, L. W., Ap. J . , 215, 827, 1977. Reimers, D., In Problems i n S t e l l a r Atmospheres and Envelopes, ed., Baschek, B., Kegel, W. H., and T r a v i n g , G., ( B e r l i n : S p r i n g e r - V e r l o g ) , 1975. , A s t r o n . Astrophys., 54, 485, 1977. , In Proceedings of the IAU Colloquium No. 42, "The I n t e r a c t i o n of V a r i a b l e S t a r s with t h e i r Environment", ed., Kippenhahn, R., Rathe, J . , and Strohmeier, W., (Erlangen-Nurnberg: Astronomisches I n s t i t u t der U n i v e r s i t a t ) , 1978. R i c h e r , H. B., 1981, In p r e s s . Richtmeyer, R. 0., and Morton, K. W.,'. D i f f e r e n c e Methods f o r  I n i t i a l - V a l u e Problems, (New York! I n t e r s c i e n c e ) , 1967. S a l p e t e r , E. E., Ap. J . , 193, 579, 1974a. , Ap. J . , 193, 585, 1974b. 113 , Ann. Rev. A s t r o n . Astrophys., 15, 297, 1977. Sanner, F., Ap. J . Suppl., 32, 115, 1976. , Ap. J . ( L e t t e r s ) , 211, L35, 1977. , Ap. J . , 219, 538, 1978. Schwarzchild, M., Ap. J . , 195, 137, 1975. Stephens, J . R., Ap. J . , 237, 450, 1980. Tabak, R. G., H i r t h , J . P., Meyrick, G., and Roark, T. P., Ap. Jk, 196, 457, 1975. Thorn, R., and Winslow, G., J . Chem. Phys., 26, 186, 1957. T r e f f e r s , R., and Cohen, M., Ap. J . , 188, 545, 1974. T s u j i , T., Ann. Tokyo A s t r o n . Obs., Ser. 2, 9, 1, 1964. , Pub. A s t r o n . Soc. Jpn., 28, 543, 1976. , A s t r o n . Astrophys., 62, 29, 1978. Tuchman, Y., Sack, N., and Barkat, Z., Ap. J . , 219, 183, 1978. Vaiana, G. S., and Rosen,. R. , Ann. Rev. A s t r o n . Astrophys., 16, 393, 1978. Volmer, M. , and Webber, A.', Z. Phys. Chem., 119, 277, 1926. Watanabe, T., and Kodaira, K., Pub. A s t r o n . Soc. Jpn., 30, 21, 1978. , Pub. A s t r o n . Soc. Jpn., 31, 61, 1979. Weymann, R. J . , Ap. J . , 132, 452, 1960. , Ap. J . , 136, 844, 1962. , Ann. Rev. As t r o n . Astrophys., 1, 97, 1963. Wickramasinghe, N. C , Nature, 207, 366, 1965. , I n t e r s t e l l a r G r a i n s , (London: Chapman H a l l ) , 1967. , M. N. R. A. S., 159, 269, 1972. Wickramasinghe, N. C , Donn, B. D., and Stecher, T. P., Ap. J . , 146, 590, 1966. Wilson, A. H., Thermodynamics and S t a t i s t i c a l Mechanics, (Cambridge: Cambridge U n i v e r s i t y P r e s s ) , 1957. Wilson, W. J . , and B a r r e t t , A. H., A s t r o n . Astrophys., 17, 385, 114 1972. Wood, P. R., Ap. J . , 227, 220, 1979. Woolf, N. J . , and Ney, E. P., Ap. J . ( L e t t e r s ) , 155, L181, 1969. Yamamoto, T., and Hasegawa, H., Prog. Theoret. Phys., 58, 816, 1977. Z e l d o v i c h , J . , J . Exp. Theor. Phys., 12, 525, 1942. Z e l ' d o v i c h , Ya. B., and R a i z e r , Ya. P., Ph y s i c s of Shock Waves  and High Temperature Phenomena, (London: Academic P r e s s ) , 1966. Zuckerman, B., G i l r a , D. P., Turner, B. E., M o r r i s , M., and Palmer, P., Ap. J . ( L e t t e r s ) , 205, L15, 1976. Zuckerman, B., Palmer, P., G i l r a , D. P., Turner, B. E., and M o r r i s , M., Ap. J . ( L e t t e r s ) , 220, L53, 1978. Zuckerman, B., Palmer, P., M o r r i s , M., Turner, B. E., G i l r a , D. P., Bowers, P. F., and Gilmore, W., Ap. J .  ( L e t t e r s ) 211, L397, 1977. 115 Appendix A NUMERICAL TESTS In the course of i t s development, the computer programme was subjected to a number of t e s t s to determine i t s behaviour under s p e c i f i e d c o n d i t i o n s and to t e s t the s t a b i l i t y and accuracy of the s o l u t i o n a l g o r i t h m s . Numerical t e s t s were a l s o made t o s e l e c t optimum v a l u e s f o r some of the computation parameters. For a l l of these t e s t s , the programme was m o d i f i e d to compute model atmospheres of hot, pure hydrogen s t a r s . The only source of o p a c i t y i n c l u d e d i n these models was e l e c t r o n s c a t t e r i n g . (a) S t a b i l i t y t e s t s ( i ) A r t i f i c i a l Mass D i f f u s i o n and V i s c o s i t y Although the ICE method can handle the propagation of shock waves without the a d d i t i o n of a r t i f i c i a l v i s c o s i t y terms, i t i s necessary to add some form of a r t i f i c i a l d i s s i p a t i o n to the system of equations to prevent the development of the numerical i n s t a b i l i t i e s which can occur at the sonic p o i n t s and s t a g n a t i o n p o i n t s , (Richtmeyer and Morton, 1967). Such i n s t a b i l i t i e s are produced by the n e a r - d i s c o n t i n u i t i e s that r e s u l t from the very l a r g e v a l u e s of |du/<br| that accompany shock t r a n s i t i o n s . V a r i o u s forms have been suggested f o r t h i s term. The form used i n t h i s study, was 116 (I?) = - ^ { f ^ ^ ^ ] where "Kj = (1/r- + 8x)hjix 1 + (3u? + ( l - 2 0 ) c 2 ) S t / 2 (A-2) A s i m i l a r term was added to the momentum equation f o r the g r a i n s . T h i s form of the a r t i f i c i a l v i s c o s i t y term was f i r s t suggested by H i r t (1968) f o r q u a s i l i n e a r equations and was used by E i l e k (1975) i n the dynamical study of S e y f e r t g a l a x i e s . Although t h i s form of bq/6t does not n e c e s s a r i l y equal zero i n steady flows, t e s t runs i n d i c a t e d that i t does become n e g l i g i b l e under steady flow c o n d i t i o n s i n comparison with the other terms in the momentum equation. Moreover, near shock t r a n s i t i o n s , H i r t ' s form of bq/br i s very s e n s i t i v e to the v e l o c i t y g r a d i e n t , bu/br, and, consequently, i s very e f f e c t i v e i n damping numerical shock p e r t u r b a t i o n s . In some a p p l i c a t i o n s of the ICE method, an a r t i f i c i a l mass d i f f u s i o n term,X , i s added to the c o n t i n u i t y equation to f u r t h e r s t a b i l i z e the equations. E i l e k (1975) and Wood (1979), fo r example, used such a term. However, t e s t runs i n d i c a t e d that the use of such a term a c t u a l l y d i s t o r t e d the r e s u l t s of these c a l c u l a t i o n s r a t h e r than s t a b i l i z i n g them. For t h i s reason, a mass d i f f u s i o n was not i n c l u d e d i n the c a l c u l a t i o n s . The models proved to be q u i t e s e n s i t i v e to the value of the a r t i f i c i a l v i s c o s i t y c o e f f i c i e n t . Test runs using a 117 perturbed model were made with ^-i= ^ -max, 0.1 A. and 10/1 max , where Am** repr e s e n t s the maximum value of the a r t i f i c i a l v i s c o s i t y c o e f f i c i e n t c a l c u l a t e d f o r the model (Eqn. (A-2)). These t e s t s c l e a r l y showed that v a l u e s of A i that are too l a r g e can i n t r o d u c e i n s t a b i l i t i e s i n t o the s o l u t i o n r a t h e r than removing those generated by other means, while p u t t i n g Ai = 0.1 Xn&t r e s u l t e d i n too l i t t l e s t a b i l i z a t i o n when shock waves were present i n the flow. Thus, i t was decided to c a l c u l a t e a value f o r Ai f o r each computational c e l l r a t h e r than t r y to f i n d a constant value of A i that c o u l d be used to c a l c u l a t e the a r t i f i c i a l v i s c o s i t y term at a l l p o i n t s . ( i i ) Time C e n t e r i n g Some of the d i f f e r e n c e equations used i n the programme were w r i t t e n such that they c o n t a i n e d both forward and backward time d i f f e r e n c e s . For example, the c o n t i n u i t y equation f o r the gas was w r i t t e n as f ' n 7 - f ? = 0 _ { r t l , ( f u ) ^ V - r ^ f u ^ y ) (A-3) + ( l ^ r j M ^ u t f . t - r ^ f u C i ) 5xh t r^2 - 0M"pr - (l-d)Mji where 9 i s the r e l a t i v e time c e n t e r i n g c o n s t a n t . A s i m i l a r c o n s t a n t , <f>, was used to d i f f e r e n c e the momentum equations. Thus, the gas momentum equation, (Eqn. ( I I I - 6 ) ) , was w r i t t e n as 118 A t + (l-lO){F^ 4 i-(Mpu)r*0 + where R&4 = _ 1 {u?4i(rL»/ffuJ - r^.f-.u?,,)} - /5:4GM*/r?4l - (*q/»r) £ 4 t c o n t a i n s only terms known from the p r e v i o u s c y c l e . W r i t t e n i n the above forms, the d i f f e r e n c e equations are s t a b l e p r o v i d e d that 9 and 0 l i e between 0.5 and 1.0. Te s t s were used to choose optimum valu e s f o r d and 0 . Tuchman e t . a_l. (1978) s t a t e that the c o n s e r v a t i o n equations are so l v e d most a c c u r a t e l y when d =0= 0.5 but because of s t a b i l i t y c o n s i d e r a t i o n s , they a c t u a l l y used 0 =<P = 0.6 i n t h e i r c a l c u l a t i o n s . In the present study, the choice of & =0= 0.6 a l s o proved to gi v e the best compromise between s t a b i l i t y and accuracy. Test runs i n which d and <p were set equal to 0.5 developed l a r g e i n s t a b i l i t i e s i n the v e l o c i t y p r o f i l e when the h y d r o s t a t i c e q u i l i b r i u m of the model was perturbed s l i g h t l y . The t e s t s i n which $ and <p were put equal to 0.7 appeared to be " o v e r - s t a b i l i z e d " i n the sense that the p e r t u r b a t i o n pulse that was generated by a s l i g h t adjustment to the d e n s i t y d i s t r i b u t i o n was supressed by the equations. = , <fr (P;-P;+,) + (i-d>)(p?-pr,,) 119 ( i i i ) The Courant C o n d i t i o n T e s t s were used to see whether or not time steps l a r g e r than those given by the Courant c o n d i t i o n , At<Ar/|u+c|, c o u l d be used. Two types of t e s t s were performed. The f i r s t type was done on unperturbed models, and the second, on perturbed models. For the f i r s t t e s t , a model that departed only very s l i g h t l y from h y d r o s t a t i c e q u i l i b r i u m was used. T h i s model was f o l l o w e d with the dynamic programme u n t i l i t reached h y d r o s t a t i c e q u i l i b r i u m f i r s t with A t equal to 2/3 A t t , where A t t i s the time step given by the Courant c o n d i t i o n , and second, with A t equal to 4/3 A t L . A f t e r the same ela p s e d time, the two runs produced models which were i n d i s t i n g u i s h a b l e . The t e s t on the perturbed model gave q u i t e d i f f e r e n t r e s u l t s . For t h i s t e s t At was put equal to 2/3At c and 2At c . In the f i r s t of these runs, the pulse t r a v e l l e d through the s u r f a c e without any s i g n of i n s t a b i l i t y throughout the atmosphere. But i n the second run, when the p e r t u r b a t i o n reached the base of the atmosphere, a s e r i e s of r a p i d o s c i l l a t i o n s were generated. These o s c i l l a t i o n s grew from -2X10 2 cm/sec to -2X10 6 cm/sec a f t e r which the base v e l o c i t y decreased slowly reaching 1.65x10* cm/sec at the end of the run. I t was apparent, t h e r e f o r e , that the Courant c o n d i t i o n had to be used to determined the maximum s i z e of the time s t e p that c o u l d be used fo r each model. Because A t t depends both upon the speed of sound and the flow v e l o c i t y , i t s value can vary a p p r e c i a b l y throughout the atmosphere. A c c o r d i n g l y , A t t was c a l c u l a t e d at each g r i d p o i n t fo r both the gas and the g r a i n flow v e l o c i t i e s , and At was set 120 equal to 0.99 times the minimum value of Atc. When shock waves were i n v o l v e d , At was put equal to 0.5 times the minimum value of A t c as suggested by Richtmeyer and Morton (1967) to ensure that the equations remained s t a b l e . (b) Boundary C o n d i t i o n T e s t s The boundary c o n d i t i o n s were e x t e n s i v e l y t e s t e d with models whose l u m i n o s i t y was time-dependent. The r e s u l t s of one t e s t of these boundary c o n d i t i o n s i s shown i n F i g u r e ( A - l ) . For t h i s t e s t , the l u m i n o s i t y was i n i t i a l l y set a t L=9. 0x10'L^,, which was j u s t 9% below the s t a b i l i t y l i m i t f o r the s t a r being modelled. T h i s model was perturbed by i n c r e a s i n g L by 0.003%. As a r e s u l t of t h i s p e r t u r b a t i o n , a pulse was generated at the base of the atmosphere. The data p l o t t e d i n the f i g u r e show the r e l a x a t i o n p u l s e s at the base. No s i g n of any r e f l e c t e d waves were observed i n t h i s and s i m i l a r t e s t s . The second curve p l o t t e d i n t h i s f i g u r e serves to demonstrate how s e n s i t i v e the models were to the f o r m u l a t i o n of the boundary c o n d i t i o n s . For t h i s run, the ref e r e n c e p r e s s u r e , PBASEl, was d e f i n e d as PBASEl n + 1 = PBASEl n + &t { (.1-0) u£ />£( qx - K^/^Cv*C ) .+ <pul+ xf1* 1 ( g a - *Y 1L/4fTr 2c) } (A-5) The c a l c u l a t i o n s i n t h i s t e s t were extended to t=6.0xl0'sec d u r i n g which the base v e l o c i t y showed no s i g n of s t a b i l i z i n g . The d e f i n i t i o n of PBASEl"* 1 was only one of many f a c t o r s to which the boundary c o n d i t i o n s were s e n s i t i v e . Other f a c t o r s 121 FIGURE A - l Base V e l o c i t y of Perturbed Model T h i s f i g u r e shows the s e n s i t i v i t y of the dynamic models to the procedure used to c a l c u l a t e the boundary c o n d i t i o n s . The only parameter which was d i f f e r e n t i n the two models p l o t t e d i n t h i s f i g u r e was the r e f e r e n c e p r e s s u r e , PBASE1, used to c a l c u l a t e the pressure boundary c o n d i t i o n at the base. Curve (a) shows the time-dependence of the v e l o c i t y at the base of the model when PBASE1 was c a l c u l a t e d on the b a s i s of the net l o s s of mass from the second c e l l (see Chapter V ) . A f t e r the i n i t i a l p e r t u r b a t i o n of the l u m i n o s i t y , t h i s model was r e l a x i n g to a steady s t a t e . Curve (b) shows the behaviour of the v e l o c i t y at the base when PBASE1 was c a l c u l a t e d on the b a s i s of the average mass l o s s from the second c e l l (Equation A-5). T h i s model showed no si g n s of r e l a x i n g to a steady s t a t e . 122 O CO O ^ H 6 ^ UD _ o P X C M O C D C D ST t—I O C D - - C D O L O O ) O C M M M C_> LU CO CD C D X LO CD CM CD CNI I VELOCITY (102) CM/SEC FIGURE A - l BASE VELOCITY OF PERTURBED MODEL 123 i n c l u d e d the d e f i n i t i o n of Cf used i n the c a l c u l a t i o n of P.p*,' the e r r o r t o l e r a n c e s used to check the s o l u t i o n of the equation of s t a t e and the c o n t i n u i t y c o n d i t i o n , and the method of i n t e r p o l a t i n g i n the p r e s s u r e s to f i n d the value of Z used i n the s p e c i f i c a t i o n of the boundary p r e s s u r e s . For example, the use of l o g v a l u e s f o r the i n t e r p o l a t i o n precedure generated l a r g e , r a p i d , growing o s c i l l a t i o n s i n the base v e l o c i t i e s . The behaviour of the s u r f a c e v e l o c i t i e s of the high l u m i n o s i t y model i s shown in F i g u r e (A-2) and F i g u r e (A-3). The pulse generated at the base of the atmosphere had grown i n t o a shock wave by the time i t reached the s u r f a c e . Again, the f i r s t p u l se was f o l l o w e d by a s e r i e s of r e l a x a t i o n p u l s e s . The second f i g u r e shows the behaviour of the v e l o c i t y near the s u r f a c e d u r i n g the passage of one of these shock waves. Again, no evidence of r e f l e c t e d waves can be seen i n the d a t a . (c) Luminosity T e s t s The a b i l i t y of the dynamic programme to t r e a t time-dependent problems and shock wave phenomena was e x t e n s i v e l y t e s t e d before the c a l c u l a t i o n of g r a i n - f o r m i n g models was i n i t i a t e d . Three types of t e s t s were run. In the f i r s t , the s t e l l a r l u m i n o s i t y was i n c r e a s e d at a constant r a t e . In the second, the l u m i n o s i t y was f i x e d at a value j u s t below the Eddington l i m i t and p e r t u r b e d s l i g h t l y . In the t h i r d t e s t , the s t e l l a r l u m i n o s i t y was i n c r e a s e d u n t i l i t exceeded the Eddington s t a b i l i t y l i m i t . The computer programme appeared t o handle each s i t u a t i o n very w e l l . For the f i r s t t e s t , the model parameters were set at 124 FIGURES A-2 and A-3 Surface V e l o c i t y of a Perturbed Model These f i g u r e s show the passage of shock waves through the s u r f a c e of a T* = 20 000 K model. In F i g u r e A-2, s i x s u c c e s s i v e shocks of slowly d e c r e a s i n g amplitude are shown. The passage of the f o u r t h of these shock, waves through the upper l a y e r s of the model i s shown i n F i g u r e A-3. The curves l a b e l l e d (a) to (e) • show the v e l o c i t y d i s t r i b u t i o n of the outer l a y e r s of the model at s u c c e s s i v e time i n t e r v a l s . There was no evidence of waves r e f l e c t e d from the s u r f a c e boundary f o l l o w i n g the passage of these shocks. 125 -J 1— 1 I L 2 4 6 8 10 TIME (Xl05) SEC FIGURE A-2 SURFACE VELOCITY OF PERTURBED MODEL 126 80 85 90 GRID NUMBER FIGURE A-3 SURFACE VELOCITY 127 M*=30.3Mo, T*=20 OOOK, lo g g=4.0, and R*=8.99R©. The extension of the atmosphere wasAg=4.6% which i s q u i t e modest compared t o that encountered i n red g i a n t s , but l a r g e enough that the model v a r i a b l e s covered many orders of magnitude. The d e n s i t y , f o r example, ranged from 2.2x10"' gm/cm3 at the base to 2.1x10" 2 3 gm/cm3 at the s u r f a c e . The s t a r t i n g l u m i n o s i t y f o r t h i s t e s t was L=l.179x10 4LQ and was in c r e a s e d at a ra t e given by L n + 1 = I/Ml + 8xlO" 6 A t ) (A-6) Th i s r a t e i n c r e a s e d L by about 10% over one thermal time s c a l e . A run d u r i n g which L in c r e a s e d by 4% i s p l o t t e d i n Fi g u r e (A-4). It i s c l e a r from t h i s data that a constant i n c r e a s e i n l u m i n o s i t y produces a mass flow of almost constant v e l o c i t y throughout the e n t i r e atmosphere. I t a l s o produces an in c r e a s e in both the e f f e c t i v e temperature and the ra d i u s of the photosphere. The su r f a c e p u l s e s seen i n t h i s data are an example of the d i f f i c u l t i e s t h a t r e s u l t e d when the i n i t i a l model was not p e r f e c t l y h y d r o s t a t i c . I t was a f t e r t h i s set of runs, that a v a r i e t y of methods were i n c o r p o r a t e d i n t o both programmes to ensure that the h y d r o s t a t i c programme c o u l d produce models i n which the c o n d i t i o n |dP/dr -y?g|<10" 1 0 was s a t i s f i e d throughout the e n t i r e atmosphere, and that the dynamic programme c o u l d r e t a i n such models i n h y d r o s t a t i c e q u i l i b r i u m f o r prolonged p e r i o d s i n the absence of a c c e l e r a t i n g f o r c e s . For the second t e s t , the model parameters that were used were T*=29 926K, M*=3OM0, R*=35.1Ro, l o g g=4.0 and K. = 0.4 cm2/gm. The l u m i n o s i t y was set at equal to 0.9L t r i t where 128 FIGURE A-4 High Luminosity Model Over s h o r t time p e r i o d s , a steady i n c r e a s e i n the l u m i n o s i t y of a T* = 20 000 K model produced a mass flow of almost constant v e l o c i t y throughout the lower h a l f of the model. The pulse at the s u r f a c e of t h i s model was generated.by small d e p a r t u r e s from h y d r o s t a t i c e q u i l i b r i u m i n the i n i t i a l model. The three curves show the v e l o c i t y d i s t r i b u t i o n o f the- model at s u c c e s s i v e time i n t e r v a l s . 129 FIGURE A-4 VELOCITY KM/SEC HIGH LUMINOSITY MODEL 130 L c n > = (4trcGM/KM0)L© (A-7) = 9 . 9 3 x l O 5 L 0 i s the Eddington s t a b i l i t y l i m i t f o r the atmosphere. 1 Under these c o n d i t i o n s , the atmosphere was expected to be very s e n s i t i v e to small changes i n the model parameters. T h i s p r e d i c t i o n was t e s t e d by i n c r e a s i n g the l u m i n o s i t y of the model by 0.003%. I t s behaviour was then followed f o r l . l x l O 7 sec (about 5 dynamical time p e r i o d s ) . The l u m i n o s i t y i n c r e a s e f i r s t produced a s m a l l , outward flow. The g r e a t e s t a c c e l e r a t i o n o c c u r r e d at the base where the pressure p e r t u r b a t i o n was g r e a t e s t . T h i s behaviour was f o l l o w e d by a s e r i e s of decaying o s c i l l a t i o n s as the programme sought to r e - e s t a b l i s h h y d r o s t a t i c e q u i l i b r i u m under dynamically unstable c o n d i t i o n s . (See F i g u r e ( A - l ) . ) Each o s c i l l a t i o n generated a pulse t h a t , had become a shock wave by the time i t had reached the s u r f a c e . Near the end of each post-shock p e r i o d , the i n f a l l v e l o c i t y of the s u r f a c e m a t e r i a l exceeded the speed of sound. Since the s u r f a c e o s c i l l a t i o n s appeared t o be decaying i t was concluded that i f the c a l c u l a t i o n s were continued long enough, the s u r f a c e would e v e n t u a l l y s t a b i l i z e as i t had i n the p r e v i o u s t e s t . However, because of the slowness with which the model appeared to be s t a b i l i z i n g , the c a l c u l a t i o n of f u r t h e r models f o r t h i s t e s t was concluded to be i m p r a c t i c a l at t h i s time. For the t h i r d l u m i n o s i t y t e s t , the model parameters *In the f i r s t lumimosity t e s t , P (=1.179x10 4L©/9.9x10 5LQ) was equal to 0.01 and the atmosphere was r a d i a t i o n a l l y q u i t e s t a b l e . 131 used were M*=30Mo, T*=20 OOOK, L=9X10 5L o, l o g g=2.0 and R*=49RQ. T h i s model was a l s o very c l o s e to being d y n a m i c a l l y unstable (L/L c<-,+ =0.9) . I t was p r e d i c t e d that an i n c r e a s e i n the l u m i n o s i t y l a r g e enough f o r i t to exceed the Eddington l i m i t would l e a d to spontaneous mass l o s s . In a d d i t i o n , i t .was probable that numerical i n s t a b i l i t i e s r e l a t e d to r e a l p h y s i c a l i n s t a b i l i t i e s might a r i s e ( M i h a l i s , 1978). These p r e d i c t i o n s were t e s t e d by i n c r e a s i n g the l u m i n o s i t y i n grad u a l steps u n t i l i t reached 10.16X10 s LQ ( = 1.02Lc»;t ). The i n c r e a s e was c o n t r o l l e d by the equation L n + 1 = L n (1 +oc4t) (A-8) where <*? was in c r e a s e d by 2x10" 7 f o r each of the f i r s t 35 models, and decreased by the same amount f o r each of the second 35 models to g i v e a smooth i n c r e a s e i n l u m i n o s i t y . A f t e r the 7 0 ^ model, oC was put equal to zero, and the r e s u l t a n t behaviour of the model was fol l o w e d . As expected, the model proved to be q u i t e u n s t a b l e t o mass l o s s . The data p l o t t e d i n F i g u r e (A-5) show the gas v e l o c i t y f o r the l a s t model i n which the l u m i n o s i t y was i n c r e a s e d , and for a model 2x10* sec l a t e r . As expected, the atmosphere co n t i n u e s to be a c c e l e r a t e d outward even when AL was equal to zero. For the second of these two models, the v e l o c i t y was ten times the sound v e l o c i t y at the base of the atmosphere, and about one-half the sound v e l o c i t y at the s u r f a c e . Under these c o n d i t i o n s , the data gave no i n d i c a t i o n of any problems at the sonic p o i n t , and the programme seemed p e r f e c t l y capable of 132 FIGURE A-5 Perturbed Dynamically Unstable Model-The l u m i n o s i t y of t h i s model was r a i s e d above the Eddington s t a b i l i t y l i m i t . A f t e r t = t , , the l u m i n o s i t y of the model was kept constant but the atmosphere continued to a c c e l e r a t e outwards. 133 _! —I I I l_ C D oo to -=r CNI VELOCITY KM/SEC FIGURE A-5 PERTURBED DYNAMICALLY UNSTABLE MODEL 134 t r e a t i n g t r a n s s o n i c flows of t h i s type. The f a c t t h at the gas flow d i d not become steady when AL was reduced to zero, was taken to be f u r t h e r evidence that the dynamic programme was working p r o p e r l y . T h e o r e t i c a l l y , i t can be shown that steady t r a n s s o n i c flows cannot be produced or maintained by the c o n d i t i o n that p >1 throughout the atmosphere, (Mihalas, 1978). The data p l o t t e d i n F i g u r e (A-5) show that the whole atmosphere was a c c e l e r a t e d outward d u r i n g the time p e r i o d covered by t h i s data. The behaviour of t h i s model was f o l l o w e d u n t i l t=7.85X10* sec. During t h i s i n t e r v a l , the v e l o c i t y at the base of the atmosphere reached a maximum of 1.6X10' cm/sec and then began to decrease s l o w l y . T h i s behaviour generated an outgoing p u l s e which was propagated at the l o c a l speed of sound. For the model at which the c a l c u l a t i o n s were terminated, the v e l o c i t y of the peak of the p u l s e was 3. 57x10 7 cm/sec". On the b a s i s of the behaviour e x h i b i t e d by models i n other t e s t runs, i t was expected that the peak v e l o c i t y of t h i s p u l s e would be i n c r e a s e d by s e v e r a l more orders of magnitude before i t reached the s u r f a c e . Since no p r o v i s i o n s were made i n the programme to handle r e l a t i v i s t i c v e l o c i t i e s , the c a l c u l a t i o n s were terminated at t h i s p o i n t . With the completion of these t e s t s , the computer programme was judged to be capable of coping with a wide v a r i e t y of dynamical problems. 135 Appendix B THE NUCLEATION AND GROWTH OF GRAPHITE GRAINS The c l a s s i c a l theory of homogeneous n u c l e a t i o n i s based upon the model that the vapour c o n t a i n s c l u s t e r s of molecules bound together by i n t e r m o l e c u l a r f o r c e s . I t i s assumed that these c l u s t e r s , A't, c o n t a i n i n g ( i ) molecules, are formed by a s i n g l e molecule, A,, combining with a c l u s t e r c o n t a i n i n g ( i - 1 ) molecules, Aj.-| , or, through the evaporation of a s i n g l e molecule from a c l u s t e r , A'L + i , c o n t a i n i n g i+1 molecules. Assuming that e q u i l i b r i u m c o n d i t i o n s p r e v a i l , the c o n c e n t r a t i o n s of c l u s t e r s of v a r i o u s s i z e s i s given by a Boltzmann-like d i s t r i b u t i o n f u n c t i o n N ( i ) = N(l)exp(-AF; /kT) (B-l) where N ( i ) i s the number d e n s i t y of the Ai c l u s t e r s , N ( l ) , the number d e n s i t y of the monomers, (carbon atoms i n the case of g r a p h i t e f o r m a t i o n ) , and A F i , the change i n the Helmholtz p o t e n t i a l with respect to the f r e e vapour s t a t e . The above equation i s a l s o known as the law of mass a c t i o n . A F i may be c o n s i d e r e d as the f r e e energy of formation of a; c l u s t e r of i molecules from the vapour s t a t e p r o v i d e d that the vapour i s maintained at a constant temperature and p r e s s u r e . Under these c o n d i t i o n s , t h e f r e e energy may be w r i t t e n as A F- = era 0 iV» - i kTln (S) (B-2) 136 where the s u p e r s a t u r a t i o n r a t i o , S, i s the r a t i o of the pressure of the vapour, P„r, to the e q u i l i b r i u m vapour pressure of the s o l i d , P£, i . e . S = Pv/P%, 6 i s the s p e c i f i c s u r f a c e - f r e e - e n e r g y i n ergs/cm 2, and a 0 i s the s u r f a c e area per s u r f a c e molecule (Abraham, 1974). An a l t e r n a t e and more u s e f u l form f o r the energy of formation i s A F t = 4tra 2cT- (4fra3/3-°-)kTln(S) ) (B-3) where the c l u s t e r c o n t a i n i n g i molecules i s assumed to be s p h e r i c a l with r a d i u s a. The molecular volume i n the s o l i d s t a t e i s denoted by - /M=4fta 3/3i) • The f i r s t term i n t h i s e x p r e s s i o n f o r A F T r e p r e s e n t s the s u r f a c e - f r e e - e n e r g y or " s u r f a c e t e n s i o n " of the c l u s t e r , and the second term, the c o n t r i b u t i o n made by the b u l k - f r e e - e n e r g y change. For very small v a l u e s of i , A F T " i s p o s i t i v e , whereas f o r l a r g e i , i t i s n e g a t i v e . P h y s i c a l l y , t h i s means that small c l u s t e r s are unstable and decay r a p i d l y , while l a r g e c l u s t e r s grow spontaneously. For some c r i t i c a l r a d i u s , a*, A F l has a maximum va l u e . The a d d i t i o n of j u s t one more molecule to a nucleus of t h i s c r i t i c a l s i z e decreases the f r e e energy of the c l u s t e r and leads to i r r e v e r s i b l e growth of the g r a i n at the expense of the s u p e r s a t u r a t e d vapour. For a given temperature and degree of s u p e r s a t u r a t i o n , the c r i t i c a l r a d i u s i s found by d i f f e r e n t i a t i n g AF; with respect to the r a d i u s , a, and s e t t i n g the d e r i v a t i v e equal to z e r o . T h i s g i v e s a* = 2-a«r/.(kTln(S)) (B-4) 137 T h i s e x p r e s s i o n f o r a* r e v e a l s the fundamental r e l a t i o n s h i p between the c r i t i c a l r a d i u s and the s u p e r s a t u r a t i o n r a t i o : namely, the l a r g e r the s u p e r s a t u r a t i o n r a t i o , the smaller the c r i t i c a l r a d i u s . Thus extremely few c r i t i c a l c l u s t e r s are present to act as n u c l e i f o r g r a i n formation u n t i l S i s q u i t e l a r g e . The f r e e energy of forming a c r i t i c a l s i z e nucleus i s A F * = 4lT(a*) 2cT- (4ir(a*) ,/3- r t-)kTln(S) = 4 i r ( a * ) 2cT/3 (B-5) AF* r e p r e s e n t s a b a r r i e r to the spontaneous growth of g r a i n s . P h y s i c a l l y , t h i s b a r r i e r a r i s e s from the formation of an i n t e r f a c e between a c l u s t e r and the vapour. The c r i t i c a l r e a c t i o n i n the formation of g r a i n s , t h e r e f o r e , i s that i n which c l u s t e r s exceed the c r i t i c a l s i z e by the a d d i t i o n of a s i n g l e molecule. The r a t e of t h i s c r i t i c a l r e a c t i o n i s c a l l e d the n u c l e a t i o n r a t e , J , and i s d e f i n e d as J =ocCN* (B-6) where N* i s the e q u i l i b r i u m number d e n s i t y of the c r i t i c a l s i z e d c l u s t e r s , C = 0.25N(1) (8kT/frm) V 2 i s the the c o l l i s i o n r a t e between the vapour molecules and the c l u s t e r s , and oC, the s t i c k i n g c o e f f i c i e n t , i s the p r o b a b i l i t y that a c o l l i d i n g molecule w i l l s t i c k to the nu c l e u s . Making the a p p r o p r i a t e s u b s t i t u t i o n s y i e l d s 138 J =°0 47r(a*) 2PN(l)exp(-AF*/kT) (B-7) (2trmkT) °-5 Since the c o l l i s i o n r a t e depends upon the p r e s s u r e , i n c r e a s i n g the p r e s s u r e and, t h e r e f o r e , the s u p e r s a t u r a t i o n r a t i o , enhances the n u c l e a t i o n r a t e . However, the main i n f l u e n c e of the s u p e r s a t u r a t i o n r a t i o occurs i n the determination of AF*. A small change i n S changes J by many orders of magnitude. The above expr e s s i o n of J i s the " c l a s s i c a l " one and does not i n c l u d e the e f f e c t s of such processes as t u n n e l i n g . I t i s d e r i v e d from the work of Becker and Doring (1935). B e t t e r r e p r e s e n t a t i o n s of the n u c l e a t i o n r a t e i n c o r p o r a t e a number of m o d i f i c a t i o n s to the c l a s s i c a l form. These are d e s c r i b e d i n the f o l l o w i n g s e c t i o n s . (a) N o n - e q u i l i b r i u m The use of a Boltzmann-like d i s t r i b u t i o n f u n c t i o n f o r the s i z e d i s t r i b u t i o n of the c l u s t e r s i s a c c u r a t e only when the c o n d i t i o n s f o r thermodynamic e q u i l i b r i u m p r e v a i l . In t h i s case, dN(i)/e)t = 0. However, once the c l u s t e r s exceed the c r i t i c a l s i z e , they are removed from the e q u i l i b r i u m composition of the " c l u s t e r gas" and become s t a b l e n u c l e i . T h i s r e d u c t i o n of N* i s allowed f o r by the i n t r o d u c t i o n of a n o n - e q u i l i b r i u m f a c t o r , Z*, which i s d e f i n e d as Z* = [a 0cr/(9trkT(i*) */3) ] V2 = [41^(3-^-/4^) V V ( 9 7 r k T ( i * ) 4 / 3 ) ] V 2 (B-8) 139 T h i s f a c t o r a l s o a l l o w s f o r the f a c t that c l u s t e r s having a > a* have a non-zero p r o b a b i l i t y of e v a p o r a t i n g . Normally, Z* i s of the order of 10" 1 to 10" 2 . Donn et a l . (1965) used Z* = 10" 2 as a t y p i c a l v alue f o r g r a p h i t e . S u b s t i t u t i n g the above e x p r e s s i o n f o r Z* and us i n g the i d e n t i t y a 0 = 4T(3- n-/4fr) V 3, the e x p r e s s i o n f o r the n u c l e a t i o n r a t e becomes j = 2<*(<3/kT) °'s [-a-P/(2TTmkT) °-5 ]N(1)exp(-AF*/kT) (B-9) The Z e l d o v i c h n o n - e q u i l i b r i u m f a c t o r i s important i n these c a l c u l a t i o n s because i t produces a r e d u c t i o n i n the n u c l e a t i o n r a t e by one to two orders of magnitude. Thus, a p p r e c i a b l e n u c l e a t i o n r e q u i r e s somewhat high e r s u p e r s a t u r a t i o n r a t i o s than those p r e d i c t e d by the c l a s s i c a l theory. Note, however, t h a t the e x p o n e n t i a l terra c o n t a i n i n g the s u p e r s a t u r a t i o n r a t i o i s the same i n both theories,.(compare equations (B-7) and (B-9)), and s i n c e t h i s f a c t o r dominates the k i n e t i c s and the thermodynamics of the n u c l e a t i o n process, only a s l i g h t i n c r e a s e i n s u p e r s a t u r a t i o n i s r e q u i r e d to o f f s e t the e f f e c t of the n o n - e q u i l i b r i u m f a c t o r . (b) A s s o c i a t e d Vapours Thus f a r , i t has been assumed that the condensing vapour i s non-associated, with the vapour molecules o c c u r i n g only as monomers i n the absence of c l u s t e r i n g . However, i t i s w e l l known that carbon vapour c o n s i s t s of C, C^, C 3 and p o s s i b l y l a r g e r molecules. In t h e i r i n v e s t i g a t i o n of n u c l e a t i o n i n a s s o c i a t e d vapours, Katz et a l . (1966) found that a s s o c i a t i o n 140 r e s u l t s i n two opposing e f f e c t s . The f i r s t e f f e c t i s an i n c r e a s e in the condensation r a t e , and the second (and l a r g e r ) e f f e c t i s an i n c r e a s e i n the a c t i v a t i o n b a r r i e r . The o v e r a l l e f f e c t i s a r e d u c t i o n i n the rate of n u c l e a t i o n and, consequently, an in c r e a s e i n the c r i t i c a l s u p e r s a t u r a t i o n r e q u i r e d f o r a n u c l e a t i o n r a t e equal to that i n a non-associated vapour. Assuming that the vapour c o n s i s t s of monomers, dimers and t r i m e r s , the n u c l e a t i o n r a t e f o r an a s s o c i a t e d vapour as developed by Katz e_t. a l . i s J = 2(tf7kT) V'NUH*- i f ("CJ/SJ j 2 )exp(-AF*/kT) ) (B-10) where J5j = P^/(2ffmjkT)V»_ ( B - l l ) In these e x p r e s s i o n s , PAT) i s the p a r t i a l p r essure of the j-mer and ©ey, the a p p r o p r i a t e s t i c k i n g c o e f f i c i e n t . For carbon vapour, Thorn and Winslow (1957) quote the value s oC, = 0.37,oC^= 0.34, and °C 3= 0.08. The sum over j i n c r e a s e s the n u c l e a t i o n r a t e - an e f f e c t which seems reasonable f o r a s s o c i a t e d vapours s i n c e the a s s o c i a t i o n complexes are i n d i s t i n g u i s h a b l e from the c l u s t e r s but, u n l i k e the c l u s t e r s , the a s s o c i a t i o n complexes are s t a b l e . However, the important e f f e c t of the a s s o c i a t i o n i s the term P^/P^y used to determine the s u p e r s a t u r a t i o n r a t i o . For non-a s s o c i a t e d vapours, S = P^/P*, where P^ i s the t o t a l p a r t i a l p r e ssure of the vapour and P£- i s the e q u i l i b r i u m vapour pr e s s u r e of the s o l i d . I f the a s s o c i a t i o n i s a p p r e c i a b l e , BJ-//P£-I < P/r/P% 141 and the a c t i v a t i o n energy i s c o n s i d e r a b l y i n c r e a s e d . For example, i f the vapour c o n s i s t s almost e n t i r e l y of dimers, the a c t i v a t i o n energy b a r r i e r , AF*, i s the square of the value given by the c l a s s i c a l theory f o r the non-associated vapour. T h i s e f f e c t i s expected to be q u i t e important when c o n s i d e r i n g the condensation of carbon vapours i n s t e l l a r atmospheres s i n c e T s u j i ' s (1964) c a l c u l a t i o n s i n d i c a t e that much of the f r e e carbon i s i n the form of and C 3 molecules. (c) The Lothe-Pound Fa c t o r The c l a s s i c a l theory of homogeneous n u c l e a t i o n i s based upon the standard p o s t u l a t e s of thermodynamics. However, thermodynamic f u n c t i o n s are d e r i v e d f o r macroscopic systems, and t h e i r use ^  i n d e s c r i b i n g the p r o p e r t i e s of small c l u s t e r s of molecules i s debatable. N e v e r t h e l e s s , the c l a s s i c a l Becker-Doring n u c l e a t i o n r a t e i s found to agree w e l l with e m p i r i c a l v a l u e s f o r many l i q u i d s i n c l u d i n g that f o r water. G e n e r a l l y , these l i q u i d s are p o l a r , s t r o n g l y hydrogen-bonded l i q u i d s , and have a low s u r f a c e entropy (Good, 1957). However, Abraham (1974) has suggested that the apparent success of the Becker-Doring theory i n p r e d i c t i n g the n u c l e a t i o n r a t e of p o l a r l i q u i d s i s f o r t u i t o u s . C e r t a i n l y i t g r o s s l y underestimates the n u c l e a t i o n r a t e of non-polar l i q u i d s . Lothe and Pound (1962) suggested t h a t the f a i l u r e of the Becker-Doring theory a r i s e s because i t c o n s i d e r s the c l u s t e r s to be at r e s t when i n f a c t they form a kind of macro-molecule gas, so that t h e i r t r a n s l a t i o n , v i b r a t i o n and r o t a t i o n must be taken i n t o c o n s i d e r a t i o n . T h i s Lothe-Pound c o r r e c t i o n 142 i n c r e a s e s the n u c l e a t i o n r a t e by 10 to 20 orders of magnitude (Deguchi, 1980). Using q u a n t u m - s t a t i s t i c a l arguments to c a l c u l a t e the ab s o l u t e entropy of the c l u s t e r s , Lothe and Pound deduced t h a t the f r e e energy of formation of the c l u s t e r s i s given by the expre s s i o n A Ft = 4lTa2dT - (4fr/3- n-)a 3kTlnS - k T l n Z f m n S - k T l n Z r o t - A F r e p (B-12) i n s t e a d of the expre s s i o n i n Equation (B-3). Z + r a n s and Zro+ , the t r a n s l a t i o n a l and r o t a t i o n a l p a r t i t i o n f u n c t i o n s r e s p e c t i v e l y , are given by Zfrans = (kT/P^,) (2?rmi*kT) 3/ 2/h 3 (B-13) where (kT/Rvi) i s a n o r m a l i z i n g f a c t o r , and Zrol = ( 8 i r 2 l k T ) 3' 2/h 3 (B-14) where I = (8frma*5 )/15-Fi-t i s the moment of i n e r t i a f o r s p h e r i c a l c l u s t e r s . Lothe and Pound i n t r o d u c e d a f a c t o r , 4 F r e p , c a l l e d the replacement f r e e energy, to remove some of the e r r o r inherent i n usi n g thermodynamic f u n c t i o n s to d e s c r i b e the p r o p e r t i e s of the c l u s t e r s . In t h e i r o r i g i n a l paper, they argued t h a t s i x degrees of freedom are needed to e n e r g i z e the c l u s t e r . Consequently, s i x degrees of v i b r a t i o n a l freedom i n the c l u s t e r must be removed from the f r e e energy of the c l u s t e r . T h i s i s rep r e s e n t e d by the 143 term, A F r e p i n Equation (B-12). They approximated t h i s decrease by d e a c t i v a t i n g one monomer molecule g i v i n g A F r e p = Ts (B-15) where s, the molecular entropy, i s of the order of 5k f o r many l i q u i d s . An a d d i t i o n a l term of kTln(27Ti)/2 was suggested to account f o r the s e p a r a t i o n of the c l u s t e r s . Thus, they suggested that AFrep = 5kT + kT l n ( 2 l U ) / 2 (B-16) As noted by Tabak, et a l . (1975), the form Lothe and Pound chose f o r the replacement f r e e energy has generated much debate. Most of i t a p p l i e s to the a p p r o p r i a t e form to use f o r p o l a r molecules f o r which the Lothe-Pound theory over-estimates the r a t e of n u c l e a t i o n . For many non-polar m a t e r i a l s , the Lothe-Pound theory worked very w e l l . Moreover, Abraham (1974) has suggested t h a t the Lothe-Pound theory should be a p p l i c a b l e to c l u s t e r s of simple atoms. The Lothe-Pound theory should, t h e r e f o r e , be a p p l i c a b l e to the n u c l e a t i o n of g r a p h i t e g r a i n s . A c c o r d i n g l y , i t was worthwhile s e a r c h i n g f o r the best p o s s i b l e form f o r the replacement f r e e energy. Tabak et a l . (1975) suggested the form A F r f p = k T l n Z ' V l b (B-17) where Z'v,b i s the v i b r a t i o n a l p a r t i t i o n f u n c t i o n of one molecule. U n f o r t u n a t e l y , the exact a n a l y t i c a l e x p r e s s i o n f o r the 144 p a r t i t i o n f u n c t i o n f o r a simple, t h r e e - d i m e n s i o n a l c r y s t a l i s not known, so Tabak, et a l . , (1975) adopted the s i n g l e molecule p a r t i t i o n f u n c t i o n of a one-dimensional harmonic o s c i l l a t o r of c l a s s i c a l frequency = k# c/h where c9c i s a c h a r a c t e r i s t i c temperature. The e x p r e s s i o n f o r t h i s p a r t i t i o n a l f u n c t i o n i s ZVb = exp(-0 C/2T)/{1 " exp(-6> c/T)} (B-18) At h i g h temperatures, i . e . T>>0 C, Z'^b = kT/lW. Even when T = , Z'^ Jb = kT/hi 1 i s a reasonable approximation. T h i s method of approximating the p a r t i t i o n f u n c t i o n of c r y s t a l s i s used i n the Born-Karman theory of s o l i d s . At low temperatures, t h i s theory i s approximated by the Debye theory, and 9c = &t> , the Debye temperature, whereas at h i g h temperatures, the E i n s t e i n model i s used, and ©c = ®f , the E i n s t e i n temperature. Under these c o n d i t i o n s , the replacement f r e e energy becomes, 1 flFr(p = 6kTln(T/£ t) (B-19) P h y s i c a l l y , the replacement f r e e energy allows f o r the f a c t that the t o t a l number of i n t e r n a l degrees of freedom f o r a small c l u s t e r of N molecules i s (3N-6) not 3N as i s assumed f o r l a r g e systems i n d e r i v i n g the Helmholtz p o t e n t i a l , A F of a m a c r o s c o p i c a l l y s t a t i o n a r y drop. For small systems, the c o r r e c t i o n , AFrep i s very important. 'In t h e i r paper, Tabak, et a l . , (1975) d i d not i n c l u d e the term exp(- 0c/2T) i n t h e i r e x p r e s s i o n f o r Z'„i"b. 145 Recently, normal-mode techniques have been used to approximate the replacement f r e e energy of i d e a l i z e d m i c r o c r y s t a l l i t e s . Since carbon condenses to a c r y s t a l r a t h e r than to a l i q u i d drop as has been assumed i n the d i s c u s s i o n of n u c l e a t i o n up to t h i s p o i n t , i t i s reasonable that these new estimates f o r AF r* p should d e s c r i b e carbon n u c l e a t i o n even more a c c u r a t e l y than the above equation. For s p h e r i c a l m i c r o c r y s t a l l i t e s , the exact, normal-mode techniques are approximated very a c c u r a t e l y by means of a c l a s s i c a l . E i n s t e i n theory of c r y s t a l l i n e s o l i d s g e n e r a l i z e d t o apply to the s u r f a c e s of m i c r o c r y s t a l l i n e c l u s t e r s (Abraham, 1974). In terms of the E i n s t e i n model, the replacement f r e e energy i s w r i t t e n as A F r e p = - k T l n { i V>exp(L) ( T / 0 f )L } (B-20) where L = 7.1 f o r s p h e r i c a l m i c r o - c r y s t a l l i t e s . The E i n s t e i n temperature appearing i n t h i s equation i s r e l a t e d to the Debye temperature by the simple r e l a t i o n (Wilson, 1957) Qt = (3/5)^0i> (B-21) For carbon, &t> = 1850 K and de = 1433 K. In terms of the Lothe-Pound theory, t h e r e f o r e , the n u c l e a t i o n rat e i s given by the e x p r e s s i o n 146 J = V 2(tf/kT) V»N(l)-n-Zf(oCjy8jj»)exp.(-^F*/kT) (B-22) where T = [ Z t r a o J Z r o + / Z v l b ] (B-23) [T- ^ P^, 1 ($e/e)7>1 (1287T 4m 2k a/15-<Mi 4) 3- 2a* 7- 5 ] i s the Lothe-Pound (or s t a t i s t i c a l weight) f a c t o r . The b a s i c d i f f e r e n c e between the Becker-Doring and the Lothe-Pound t h e o r i e s i s t h e i r r e p r e s e n t a t i o n of the t o t a l s u r f a c e - f r e e - e n e r g y . T y p i c a l l y , the e x p r e s s i o n used i n the Becker-Doring theory o v e r - e s t i m a t e s the s u r f a c e - f r e e - e n e r g y by g r e a t l y o v e r - e s t i m a t i n g the number of s u r f a c e molecules i n the d r o p l e t . For non-polar l i q u i d s , t h i s over-estimate i s removed by the Lothe-Pound theory. However, f o r p o l a r l i q u i d s , the Lothe-Pound theory f a i l s to account f o r important d e v i a t i o n s of the s u r f a c e p r o p e r t i e s of the c r i t i c a l nucleus from those of the bulk l i q u i d . For these l i q u i d s , the Lothe-Pound theory g i v e s the c o r r e c t n u c l e a t i o n r a t e only when the s u r f a c e t e n s i o n f o r the c l u s t e r s i s assumed to be about 20% g r e a t e r than that of the bulk l i q u i d . For g r a p h i t e formation, such a c o r r e c t i o n i s not expected to be necessary, and the n u c l e a t i o n of g r a p h i t e g r a i n s should f o l l o w the Lothe-Pound d e s c r i p t i o n . The c h o i c e of an a p p r o p r i a t e value f o r the s u r f a c e energy of g r a p h i t e i s c o m p l i c a t e d by i t s c r y s t a l s t r u c t u r e . For i t s b a s a l s u r f a c e , (5 = 40 to 170 ergs/cm 2, while f o r i t s prism s u r f a c e , CT = 4000 to 4800 ergs/cm 2. F o l l o w i n g Tabak et a l . , (1975) a v a l u e of 0~ = 1000 ergs/cm 2 was used f o r the s p h e r i c a l n u c l e i that were assumed f o r the purposes of t h i s study. T h i s 147 value f o r 6 was a l s o used by F i x (1969) and by Donn et a l . (1968). (d) Condensation Once the s u p e r s a t u r a t i o n r a t i o , S, reaches i t s c r i t i c a l v a l u e , l a r g e numbers of n u c l e i w i l l form and begin to grow by a c c r e t i o n of f r e e atoms impinging upon t h e i r s u r f a c e s . At the same time, the e v a p o r a t i o n of atoms w i l l decrease the s i z e of the g r a i n s . Assuming that the g r a i n s are s p h e r i c a l , the net r a t e of change of the volume of the g r a i n i s dVp/dt = 4flra2-rv'£ j°C;(C: - E; ) (B-24) r j= ( where Vp i s the volume of the g r a i n , and Cj and E j , the condensation and e v a p o r a t i o n f l u x e s of the j-mer. For g r a i n s , the e v a p o r a t i o n f l u x i s Ej = P*j/(2nmjkT) V» (B-25) and the condensation f l u x i s Cj = P4rj/(2trmjkT) V 2 (B-26) Using the f a c t that V p = 4fla 3/3 f o r s p h e r i c a l g r a i n s , the r a d i a l growth of the g r a i n can be w r i t t e n as 3 da/dt = -rt-^E j e ^ C j - E j ) (B-27) 148 which i s independent of the s i z e of the g r a i n . As the gains grow, they d e p l e t e the vapour of the condensing s p e c i e s . T h i s q u i c k l y decreases the degree of s u p e r s a t u r a t i o n , e f f e c t i v e l y stopping the n u c l e a t i o n process. Growth of the g r a i n s a l r e a d y formed, however, co n t i n u e s u n t i l Cj = Ej , i . e . u n t i l Rvj = P^' f o r a l l v a l u e s of j . At the same time, by reducing the p a r t i a l p r e s s u r e s of the j-mers, the growth of the g r a i n s w i l l a f f e c t the d i s s o c i a t i o n e q u i l i b r i u m of the molecules c o n t a i n i n g atoms of the condensing s p e c i e s . T h i s process must be c o n s i d e r e d i n determining the f i n a l s i z e t h at the g r a i n w i l l o b t a i n . E v e n t u a l l y , of course, a new e q u i l i b r i u m s t a t e w i l l be reached i n which the presence of the s o l i d m a t e r i a l i s accounted f o r . 

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